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V. Bagro, S. A. Kruglyak In [Kruglyak S. A., Samoĭlenko Yu. S.: On structure theorems for a family of idempotents. Ukrainskii Matematicheskii Zhurnal 50 (4), (1998) (Russian), Kruglyak S. A., Samoĭlenko Yu. S.: On the complexity of description of representations of \(*\)-algebras generated by idempotents. [in:] Proc. of the American Mathematical Society, vol. 128, 1655–1664, AMS, 2000, Kruglyak S. A.: A majorization relation for \(*\)-categories and \(*\)-wild categories. [in:] Proc. of the Fifth Intern. Conference "Symmetry in Nonlinear Math. Physics", 2004], \(*\)-algebras and \(*\)-categories over the field \(\mathbb{C}\) of complex numbers were quasi-ordered with respect to the complexity of the structure of their \(*\)-representations with a majorization relation \(\succ\). A notion of \(*\)-wildness was also introduced there for an algebra (a category) if the algebra majorizes the \(*\)-algebra \(C^{*}(\mathcal{F}_2)\). In this paper, we discuss some methods for proving that an algebra is \(*\)-wild and obtain criteria for certain "standard" \(*\)-categories (ensembles with an orthogonality condition) to be \(*\)-wild. algebras, categories and functors, representations 2004 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol24iss1art2 https://www.opuscula.agh.edu.pl/vol24/1/art/opuscula_math_2402.pdf Discretization of the stationary distribution of heat in the non-homogeneous body Bogusław Bożek We give a short survey on the theory of the mixed boundary-value problem for the stationary Fourier equation in a non-homogeneous medium defined on any Lipschitz domain \(\Omega\subset\mathbb{R}^n\) (\(n\geq 2\)). The compatibility condition for the thermal flux has been established by the standard procedure of integration the divergence. elliptic partial differential equations, stationary distribution of heat, discretization method 2004 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol24iss1art3 https://www.opuscula.agh.edu.pl/vol24/1/art/opuscula_math_2403.pdf Euler's Beta function diagonalized and a related functional equation Bodgan Choczewski, Anna Wach-Michalik Euler's Gamma function is the unique logarithmically convex solution of the functional equation \[\varphi(x+1)=x\varphi(x),\quad x\in\mathbb{R}_{+};\quad \varphi(1)=1,\] cf. the Proposition. In this paper we deal with the function \(\beta :\mathbb{R}_{+}\to\mathbb{R}_{+}\), \(\beta (x):=B(x,x)\), where \(B(x,y)\) is the Euler Beta function. We prove that, whenever a function \(h\) is asymptotically comparable at the origin with the function \(a\log +b\), \(a\gt 0\), if \(\varphi :\mathbb{R}_{+}\to\mathbb{R}_{+}\) satisfies equation \[\varphi(x+1)=\frac{x}{2(2x+1)}\varphi(x),\quad x\in\mathbb{R}_{+}\] and the function \(h\circ \varphi\) is continuous and ultimately convex, then \(\varphi =\beta\). Euler's Beta function, diagonalization, functional equations, convex functions 2004 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol24iss1art4 https://www.opuscula.agh.edu.pl/vol24/1/art/opuscula_math_2404.pdf P_{m}-saturated graphs with minimum size Aneta Dudek, A. Paweł Wojda By \(P_m\) we denote a path of order \(m\). A graph \(G\) is said to be \(P_m\)-saturated if \(G\) has no subgraph isomorphic to \(P_m\) and adding any new edge to \(G\) creates a \(P_m\) in \(G\). In 1986 L. Kászonyi and Zs. Tuza considered the following problem: for given \(m\) and \(n\) find the minimum size \(sat(n;P_m)\) of \(P_m\)-saturated graph and characterize the graphs of \(Sat(n;P_m)\) - the set of \(P_m\)-saturated graphs of minimum size. They have solved this problem for \(n\geq a_m\) where \(a_m=\begin{cases}3\cdot 2^{k-1}-2 &\quad\text{ if }\quad m=2k,\, k\gt 2\\ 2^{k+1}-2 &\quad\text{ if }\quad m=2k+1,\, k\geq 2\end{cases}\). We define \(b_m=\begin{cases}3\cdot 2^{k-2} &\quad\text{ if }\quad m=2k,\, k\geq 3\\ 3\cdot 2^{k-1}-1 &\quad\text{ if }\quad m=2k+1,\, k\geq 3\end{cases}\) and give \(sat(n;P_m)\) and \(Sat(n;P_m)\) for \(m\geq 6\) and \(b_m\leq n\leq a_m\). graph, saturated graph, extremal graph 2004 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol24iss1art5 https://www.opuscula.agh.edu.pl/vol24/1/art/opuscula_math_2405.pdf Pricing of a defaultable coupon bond in an extended Merton's model Ewa Frankiewicz Three alternative approaches to the valuation of a defaultable coupon bond in an extended Merton's model are given. Probabilistic approach yields a closed-form expression for the arbitrage price of this bond. A boundary value problem method is based on the concept of an CD-extended generator for Markov processes. The third approach relies on a recursive procedure method in which at every step a suitable Cauchy problem is solved. arbitrage valuation, Markov processes, contraction semigroup, generators 2004 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol24iss1art6 https://www.opuscula.agh.edu.pl/vol24/1/art/opuscula_math_2406.pdf The general differential-geometric structure of multidimensional Delsarte transmutation operators in parametric functional spaces and their applications in soliton theory. Part 2 J. Golenia, Y. A. Prykarpatsky, A. M. Samoilenko, A. K. Prykarpatsky The structure properties of multidimensional Delsarte transmutation operators in parametric functional spaces are studied by means of differential-geometric tools. It is shown that kernels of the corresponding integral operator expressions depend on the topological structure of related homological cycles in the coordinate space. As a natural realization of the construction presented we build pairs of Lax type commutive differential operator expressions related via a Darboux-Backlund transformation having a lot of applications in soliton theory. Some results are also sketched concerning theory of Delsarte transmutation operators for affine polynomial pencils of multidimensional differential operators. Delsarte transmutation operators, parametric functional spaces, Darboux transformations, inverse spectral transform problem, soliton equations, Zakharov-Shabat equations, polynomial operator pencils 2004 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol24iss1art7 https://www.opuscula.agh.edu.pl/vol24/1/art/opuscula_math_2407.pdf Difference methods for infinite systems of hyperbolic functional differential equations on the Haar pyramid Danuta Jaruszewska-Walczak We consider the Cauchy problem for infinite system of differential functional equations \[\partial_tz_k(t,x)=f_k(t,x,z,\partial_xz_k(t,x)),\;k\in\mathbf{N}.\] In the paper we consider a general class of difference methods for this problem. We prove the convergence of methods under the assumptions that given functions satisfy the nonlinear estimates of the Perron type with respect to functional variables. The proof is based on functional difference inequalities. We constructed the Euler method as an example of difference method. initial problems, infinite systems of differential functional equations, difference functional inequalities, nonlinear estimates of the Perron type 2004 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol24iss1art8 https://www.opuscula.agh.edu.pl/vol24/1/art/opuscula_math_2408.pdf Edge decompositions of multigraphs into multi-2-paths Jan Kratochvil, Zbigniew Lonc, Mariusz Meszka, Zdzisław Skupień We establish the computational time complexity of the existence problem of a decomposition of an instance multigraph into isomorphic 3-vertex paths with multiple edges. If the two edge multiplicities are distinct, the problem is NPC; if mutually equal then polynomial. edge decomposition, multigraph, multipath, path, time complexity 2004 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol24iss1art9 https://www.opuscula.agh.edu.pl/vol24/1/art/opuscula_math_2409.pdf Li's criterion for the Riemann hypothesis - numerical approach Krzysztof Maślanka There has been some interest in a criterion for the Riemann hypothesis proved recently by Xian-Jin Li [Li X.-J.: The Positivity of a Sequence of Numbers and the Riemann Hypothesis. J. Number Theory 65 (1997), 325-333]. The present paper reports on a numerical computation of the first 3300 of Li's coefficients which appear in this criterion. The main empirical observation is that these coefficients can be separated in two parts. One of these grows smoothly while the other is very small and oscillatory. This apparent smallness is quite unexpected. If it persisted till infinity then the Riemann hypothesis would be true. Riemann zeta function, Riemann hypothesis, Li's criterion, numerical methods in analytic number theory 2004 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol24iss1art10 https://www.opuscula.agh.edu.pl/vol24/1/art/opuscula_math_2410.pdf Some theorems of Rabinowitz type for nonlinearizable eigenvalue problems Jolanta Przybycin We discuss the structure of the solution set for nonlinearizable eigenvalue problems in a Hilbert space. nonlinear eigenvalue problem, bifurcation point, bifurcation interval 2004 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol24iss1art11 https://www.opuscula.agh.edu.pl/vol24/1/art/opuscula_math_2411.pdf Corona theorem and isometries Krzysztof Rudol The aim of this note is to discuss a new operator theory approach to Corona Problem. An equivalent operator problem invariant under unitary equivalence is stated. The related condition involves certain joint spectra of commuting subnormal operators. A special case leading to isometries is studied. As a result one obtains a relatively short proof of Corona Theorem for a wide class of domains in the plane, where Marshall's Theorem on the approximation by inner functions holds. Hardy classes, Taylor's joint spectra, cluster sets 2004 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol24iss1art12 https://www.opuscula.agh.edu.pl/vol24/1/art/opuscula_math_2412.pdf Instability and the formation of bubbles and the plugs in fluidized beds Piotr Schulz This is an review paper, particulary concentrate on results not many researches by reason that are explain in the text. We consider stability of disperse, two-phase flow (gas-solid particles or liquid-solid particles) linear and non-linear. In particular we discuss the result of Anderson, Sundareson and Jackson (1995) [Anderson K., Sundareson S., Jackson R.: Instabilities and the formation of bubbles in fluidized beds. J. Fluid Mech. 303 (1995), 327-366] that for vertical dispersion flow one- and two-dimensional, they attack problem growing disturbances directly by numerical integration of equations of motion from given initial conditions (using computer Cray C-90). In principle, this would allow authors to explore all aspects of dynamical behaviour of fluidized beds. It is interesting mechanism of periodic plug describing by Anderson et al. and attest by other researchers. Second part of paper is more general, dedicate the problem of linear stability of uniformly fluidized state ("fluidized bed"). We make the most important stages of calculations (after to Jackson (2000) [Jackson R.: The Dynamics of Fluidized Particles. Cambridge University Press 2000]) and demonstrate that the majority (but not all) of fluidized beds with parameters having technical importance is unstable, or stable in narrow interval of wave numbers \(k\). multiphase flow, bubbling, linear and nonlinear instability, dispersion relation, periodic plugs (or slugs) 2004 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol24iss1art13 https://www.opuscula.agh.edu.pl/vol24/1/art/opuscula_math_2413.pdf A local existence theorem of the solution of the Cauchy problem for BBGKY chain of equations represented in cumulant expansions in the space E_{ξ} Myhaylo O. Stashenko, Halyna M. Hubal It is proved convergence of solution in cumulant expansions of the initial value problem for BBGKY chain of equations of non-symmetrical one-dimensional system of particles which interact via a short-range potential in the space \(E_{\xi}\) of the sequences of continuous bounded functions. non-symmetrical particle systems, space of the sequences of continuous bounded functions, BBGKY chain of equations, cumulant 2004 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol24iss2 https://www.opuscula.agh.edu.pl/om-vol24iss2art1 https://www.opuscula.agh.edu.pl/vol24/2/art/opuscula_math_2414.pdf A note on a list colouring of hypergraphs Ewa Drgas-Burchardt In the note we present two results. The first of them gives a sufficient condition for a colouring of a hypergraph from an assigned list. It generalises the analogous fact for graphs. The second result states that for every \(k\geq 3\) and every \(l\geq 2\), a distance between the list chromatic number and the chromatic number can be arbitrarily large in the class of \(k\)-uniform hypergraphs with the chromatic number bounded below by \(l\). A similar result for \(k\)-uniform, \(2\)-colorable hypergraphs is known but the proof techniques are different. hypergraph, list colouring 2004 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol24iss2art2 https://www.opuscula.agh.edu.pl/vol24/2/art/opuscula_math_2415.pdf Nearly perfect sets in products of graphs Maria Kwaśnik, Monika Perl The study of nearly perfect sets in graphs was initiated in [Dunbar J. E., Harris F. C., Hedetniemi S. M., Hedetniemi S. T., McRae A. A., Laskar R. C.: Nearly perfect sets in graphs. Discrete Mathematics 138 (1995), 229-246]. Let \(S\subseteq V(G)\). We say that \(S\) is a nearly perfect set (or is nearly perfect) in \(G\) if every vertex in \(V(G)-S\) is adjacent to at most one vertex in \(S\). A nearly perfect set \(S\) in \(G\) is called maximal if for every vertex \(u\in V(G)-S\), \(S\cup \{u\}\) is not nearly perfect in \(G\). The minimum cardinality of a maximal nearly perfect set is denoted by \(n_p(G)\). It is our purpose in this paper to determine maximal nearly perfect sets in two well-known products of two graphs; i.e., in the Cartesian product and in the strong product. Lastly, we give upper bounds of \(n_p(G_1\times G_2)\) and \(n_p(G_1\otimes G_2)\), for some special graphs \(G_1\), \(G_2\). dominating sets, product of graphs 2004 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol24iss2art3 https://www.opuscula.agh.edu.pl/vol24/2/art/opuscula_math_2416.pdf Weakly convex and convex domination numbers Magdalena Lemańska Two new domination parameters for a connected graph \(G\): the weakly convex domination number of \(G\) and the convex domination number of \(G\) are introduced. Relations between these parameters and the other domination parameters are derived. In particular, we study for which cubic graphs the convex domination number equals the connected domination number. dominating set, connected domination number, distance, isometric set, convex set 2004 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol24iss2art4 https://www.opuscula.agh.edu.pl/vol24/2/art/opuscula_math_2417.pdf NP-completeness of weakly convex and convex dominating set decision problems Joanna Raczek The convex domination number and the weakly convex domination number are new domination parameters. In this paper we show that the decision problems of convex and weakly convex dominating sets are \(NP\)-complete for bipartite and split graphs. Using a modified version of Warshall algorithm we can verify in polynomial time whether a given subset of vertices of a graph is convex or weakly convex. dominating set, \(NP\)-completeness, distance, convex set 2004 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol24iss2art5 https://www.opuscula.agh.edu.pl/vol24/2/art/opuscula_math_2418.pdf The minimum exponent of the primitive digraphs on the given number of arcs Jolanta Rosiak Primitive digraphs on \(n\) vertices, \(k\) arcs and girth \(s\) are considered. By \(a(n,k,s)\) we mean the minimum exponent taken over all such digraphs. We estimate the number \(a(n,k,s)\) using the Frobenius number for special values of \(k\) and \(s\). primitive directed graph, exponent, Frobenius number 2004 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol24iss2art6 https://www.opuscula.agh.edu.pl/vol24/2/art/opuscula_math_2419.pdf A note introducing Cayley graphs and group-coset graphs generated by graph packings Robert Rosiek, Mariusz Woźniak The aim of this paper is to construct a class of vertex-transitive graphs that includes the Kneser graphs as a special case. The class will be based on the notion of packing of graphs. Certain families of graphs within this class will be examined more closely, and some of their properties, such as hamiltonicity, will be investigated. Cayley graphs, hamiltonicity, packing of graphs 2004 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol24iss2art7 https://www.opuscula.agh.edu.pl/vol24/2/art/opuscula_math_2420.pdf A note on the vertex-distinguishing index for some cubic graphs Karolina Taczuk, Mariusz Woźniak The vertex-distinguishing index of a graph \(G\) (\(\operatorname{vdi}(G)\)) is the minimum number of colours required to colour properly the edges of a graph in such a way that any two vertices are incident with different sets of colours. We consider this parameter for some families of cubic graphs. edge colouring, vertex-distinguishing colouring, cubic graphs 2004 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol24iss2art8 https://www.opuscula.agh.edu.pl/vol24/2/art/opuscula_math_2421.pdf Domination parameters of a graph with added vertex Maciej Zwierzchowski Let \(G=(V,E)\) be a graph. A subset \(D\subseteq V\) is a total dominating set of \(G\) if for every vertex \(y\in V\) there is a vertex \(x\in D\) with \(xy\in E\). A subset \(D\subseteq V\) is a strong dominating set of \(G\) if for every vertex \(y\in V-D\) there is a vertex \(x\in D\) with \(xy\in E\) and \(\deg _{G}(x)\geq\deg _{G}(y)\). The total domination number \(\gamma _{t}(G)\) (the strong domination number \(\gamma_{S}(G)\)) is defined as the minimum cardinality of a total dominating set (a strong dominating set) of \(G\). The concept of total domination was first defined by Cockayne, Dawes and Hedetniemi in 1980 [Cockayne E. J., Dawes R. M., Hedetniemi S. T.: Total domination in graphs. Networks 10 (1980), 211–219], while the strong domination was introduced by Sampathkumar and Pushpa Latha in 1996 [Pushpa Latha L., Sampathkumar E.: Strong weak domination and domination balance in a graph. Discrete Mathematics 161 (1996), 235–242]. By a subdivision of an edge \(uv\in E\) we mean removing edge \(uv\), adding a new vertex \(x\), and adding edges \(ux\) and \(vx\). A graph obtained from \(G\) by subdivision an edge \(uv\in E\) is denoted by \(G\oplus u_{x}v_{x}\). The behaviour of the total domination number and the strong domination number of a graph \(G\oplus u_{x}v_{x}\) is developed. the total domination number, the strong domination number, subdivision 2004 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol25iss1 https://www.opuscula.agh.edu.pl/om-vol25iss1art1 https://www.opuscula.agh.edu.pl/vol25/1/art/opuscula_math_2501.pdf Solution of the Stieltjes truncated matrix moment problem Vadim M. Adamyan, Igor M. Tkachenko The truncated Stieltjes matrix moment problem consisting in the description of all matrix distributions \(\boldsymbol{\sigma}(t)\) on \([0,\infty)\) with given first \(2n+1\) power moments \((\mathbf{C}_j)_{n=0}^j\) is solved using known results on the corresponding Hamburger problem for which \(\boldsymbol{\sigma}(t)\) are defined on \((-\infty,\infty)\). The criterion of solvability of the Stieltjes problem is given and all its solutions in the non-degenerate case are described by selection of the appropriate solutions among those of the Hamburger problem for the same set of moments. The results on extensions of non-negative operators are used and a purely algebraic algorithm for the solution of both Hamburger and Stieltjes problems is proposed. Stieltjes power moments, canonical solutions, Nevanlinna's formula 2005 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol25iss1art2 https://www.opuscula.agh.edu.pl/vol25/1/art/opuscula_math_2502.pdf A sufficient condition for Schur stability of the convex combination of the polynomials Stanisław Białas In this paper is given a simple sufficient condition for Schur stability of the convex combination of the real polynomials. convex sets of polynomials, stability of polynomial sets, Schur stability 2005 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol25iss1art3 https://www.opuscula.agh.edu.pl/vol25/1/art/opuscula_math_2503.pdf Monotone iterative methods for infinite systems of reaction-diffusion-convection equations with functional dependence Stanisław Brzychczy We consider the Fourier first initial-boundary value problem for an infinite system of semilinear parabolic differential-functional equations of reaction-diffusion-convection type of the form \[\mathcal{F}^i[z^i](t,x)=f^i(t,x,z),\quad i \in S,\] where \[\mathcal{F}^i:=\mathcal{D}_t-\mathcal{L}^i,\quad \mathcal{L}^i:=\sum_{j,k=1}^m a_{jk}^i(t,x)\mathcal{D}^2_{x_jx_k}+\sum_{j=1}^m b_j^i(t,x)\mathcal{D}_{x_j}\] in a bounded cylindrical domain \((0,T] \times G:=D \subset \mathbb{R}^{m+1}\). The right-hand sides of the system are Volterra type functionals of the unknown function \(z\). In the paper, we give methods of the construction of the monotone iterative sequences converging to the unique classical solution of the problem considered in partially ordered Banach spaces with various convergence rates of iterations. We also give remarks on monotone iterative methods in connection with numerical methods, remarks on methods for the construction of lower and upper solutions and remarks concerning the possibility of extending these methods to more general parabolic equations. All monotone iterative methods are based on differential inequalities and, in this paper, we use the theorem on weak partial differential-functional inequalities for infinite systems of parabolic equations, the comparison theorem and the maximum principle. A part of the paper is based on the results of our previous papers. These results generalize the results obtained by several authors in numerous papers for finite systems of semilinear parabolic differential equations to encompass the case of infinite systems of semilinear parabolic differential-functional equations. The monotone iterative schemes can be used for the computation of numerical solutions. infinite systems, reaction-diffusion-convection equations, semilinear parabolic differential-functional equations, Volterra functionals, monotone iterative methods, method of upper and lower solutions 2005 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol25iss1art4 https://www.opuscula.agh.edu.pl/vol25/1/art/opuscula_math_2504.pdf The Abel summation of the Kontorovich-Lebedev integral representation Petru A. Cojuhari, Alexander M. Gomilko A new result on the summation of the Kontorovich–Lebedev integral representation in the sense of Abel mean is given. integral transforms, transforms of special functions, special functions, operational calculus 2005 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol25iss1art5 https://www.opuscula.agh.edu.pl/vol25/1/art/opuscula_math_2505.pdf Numerical approximations of difference functional equations and applications Zdzisław Kamont We give a theorem on the error estimate of approximate solutions for difference functional equations of the Volterra type. We apply this general result in the investigation of the stability of difference schemes generated by nonlinear first order partial differential functional equations and by parabolic problems. We show that all known results on difference methods for initial or initial boundary value problems can be obtained as particular cases of this general and simple result. We assume that the right hand sides of equations satisfy nonlinear estimates of the Perron type with respect to functional variables. functional differential equations, stability and convergence, interpolating operators, nonlinear estimates of the Perron type 2005 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol25iss1art6 https://www.opuscula.agh.edu.pl/vol25/1/art/opuscula_math_2506.pdf Recovering a part of potential by partial information on spectra of boundary problems Vyacheslav Pivovarchik Under additional conditions uniqueness of the solution is proved for the following problem. Given 1) the spectrum of the Dirichlet problem for the Sturm-Liouville equation on \([0,a]\) with real potential \(q(x)\in L_2(0,a)\), 2) a certain part of the spectrum of the Dirichlet problem for the same equation on \([\frac{a}{3},a]\) and 3) the potential on \([0,\frac{a}{3}]\). The aim is to find the potential on \([\frac{a}{3},a]\). sine-type function, Lagrange interpolation series, Dirichlet boundary value problem, Dirichlet-Neumann boundary value problem 2005 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol25iss1art7 https://www.opuscula.agh.edu.pl/vol25/1/art/opuscula_math_2507.pdf A note on inductive limit model of Bargmann space of infinite order Jerzy Stochel It is shown that the generalized creation and annihilation operators on Bargmann space of infinite order in a direction \(a=(a_1,a_2,\ldots) \in l^2\) are inductive limits of the creation and annihilation operator acting on Bargmann space of \(n\)-th order. Hilbert space, Bargmann space, creation operator, annihilation operator, inductive limit 2005 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol25iss1art8 https://www.opuscula.agh.edu.pl/vol25/1/art/opuscula_math_2508.pdf On some application of biorthogonal spline systems to integral equations Zygmunt Wronicz We consider an operator \(P_N: L_p(I) \to S_n(\Delta_N)\), such that \(P_Nf=f\) for \(f\in S_n(\Delta_N)\), where \(S_n(\Delta_N)\) is the space of splines of degree \(n\) with respect to a given partition \(\Delta_N\) of the interval \(I\). This operator is defined by means of a system of step functions biorthogonal to \(B\)-splines. Then we use this operator to approximation to the solution of the Fredholm integral equation of the second kind. Convergence rates for the approximation of the solution of this equation are given. operator associated with step functions, \(B\)-splines, integral equation, approximation 2005 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol25iss2 https://www.opuscula.agh.edu.pl/om-vol25iss2art1 https://www.opuscula.agh.edu.pl/vol25/2/art/opuscula_math_2509.pdf A necessary and sufficient condition for σ-Hurwitz stability of the convex combination of the polynomials Stanisław Białas In the paper are given a necessary and sufficient condition for \(\sigma\)-Hurwitz stability of the convex combination of the polynomials. convex sets of polynomials, stability of polynomial, Hurwitz stability, \(\sigma\)-stability 2005 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol25iss2art2 https://www.opuscula.agh.edu.pl/vol25/2/art/opuscula_math_2510.pdf Calculation of distribution of temperature in three-dimensional solid changing its shape during the process Bogusław Bożek, Czesław Mączka The present paper suplements and continues [Bożek B., Filipek R., Holly K., Mączka C.: Distribution of temperature in three-dimensional solids. Opuscula Mathematica 20 (2000), 27-40]. Galerkin method for the Fourier–Kirchhoff equation in the case when \(\Omega(t)\) – equation domain, dependending on time \(t\), is constructed. For special case \(\Omega(t) \subset \mathbb{R}^2\) the computer program for above method is written. Binaries and sources of this program are available on http://wms.mat.agh.edu.pl/~bozek. parabolic partial differential equations, non-stationary distribution of heat, finite element method, Galerkin method 2005 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol25iss2art3 https://www.opuscula.agh.edu.pl/vol25/2/art/opuscula_math_2511.pdf A note on geodesic and almost geodesic mappings of homogeneous Riemannian manifolds Stanisław Formella Let \(M\) be a differentiable manifold and denote by \(\nabla\) and \(\tilde{\nabla}\) two linear connections on \(M\). \(\nabla\) and \(\tilde{\nabla}\) are said to be geodesically equivalent if and only if they have the same geodesics. A Riemannian manifold \((M,g)\) is a naturally reductive homogeneous manifold if and only if \(\nabla\) and \(\tilde{\nabla}=\nabla-T\) are geodesically equivalent, where \(T\) is a homogeneous structure on \((M,g)\) ([Tricerri F., Vanhecke L., Homogeneous Structure on Riemannian Manifolds. London Math. Soc. Lecture Note Series, vol. 83, Cambridge Univ. Press 1983]). In the present paper we prove that if it is possible to map geodesically a homogeneous Riemannian manifold \((M,g)\) onto \((M,\tilde{\nabla})\), then the map is affine. If a naturally reductive manifold \((M,g)\) admits a nontrivial geodesic mapping onto a Riemannian manifold \((\overline{M},\overline{g})\) then both manifolds are of constant cutvature. We also give some results for almost geodesic mappings \((M,g) \to (M,\tilde{\nabla})\). homogeneous Riemannian manifold, geodesic, almost geodesic, geodesic mapping, almost geodesic mapping 2005 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol25iss2art4 https://www.opuscula.agh.edu.pl/vol25/2/art/opuscula_math_2512.pdf Convex compact family of polynomials and its stability Michał Góra Let \(P\) be a set of real polynomials of degree \(n\). Set \(P\) can be identified with some subset \(P\) of \(\mathbb{R}^n\) consists of vectors of coefficients of \(P\). If \(P\) is a polytope, then to ascertain whether the entire family of polynomials \(P\) is stable or not, it suffices to examine the stability of the one-dimensional boundary sets of \(P\). In present paper, we extend this result to convex compact polynomial families. Examples are presented to illustrate the results. stability, convex set of polynomials, regular set 2005 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol25iss2art5 https://www.opuscula.agh.edu.pl/vol25/2/art/opuscula_math_2513.pdf Solving equations by topological methods Lech Górniewicz In this paper we survey most important results from topological fixed point theory which can be directly applied to differential equations. Some new formulations are presented. We believe that our article will be useful for analysts applying topological fixed point theory in nonlinear analysis and in differential equations. Lefschetz number, fixed points, CAC-maps, condensing maps, ANR-spaces, fixed point index 2005 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol25iss2art6 https://www.opuscula.agh.edu.pl/vol25/2/art/opuscula_math_2514.pdf A singular nonlinear boundary value problem with Neumann conditions Julian Janus We study the existence of solutions for the equations \(x^{\prime\prime}\pm g(t,x)=h(t)\), \(t\in (0,1)\) with Neumann boundary conditions, where \(g:[0,1] \times (0,+\infty) \to [0,+\infty)\) and \(h:[0,1] \to \mathbb{R}\) are continuous and \(g(t,\cdot)\) is singular at \(0\) for each \(t\in [0,1]\). singular nonlinear boundary value problem, Neumann boundary conditions, second order equations, maximal and minimal solutions 2005 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol25iss2art7 https://www.opuscula.agh.edu.pl/vol25/2/art/opuscula_math_2515.pdf The Sturm-Liouville inverse spectral problem with boundary conditions depending on the spectral parameter Cornelis van der Mee, Vjacheslav Pivovarchik We present the complete version including proofs of the results announced in [van der Mee C., Pivovarchik V.: A Sturm-Liouville spectral problem with boundary conditions depending on the spectral parameter. Funct. Anal. Appl. 36 (2002), 315–317 [Funkts. Anal. Prilozh. 36 (2002), 74–77 (Russian)]]. Namely, for the problem of small transversal vibrations of a damped string of nonuniform stiffness with one end fixed we give the description of the spectrum and solve the inverse problem: find the conditions which should be satisfied by a sequence of complex numbers to be the spectrum of a damped string. damped vibrations, inhomogeneous strings, quadratic operator pencil, Hermite-Biehler functions 2005 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol25iss2art8 https://www.opuscula.agh.edu.pl/vol25/2/art/opuscula_math_2516.pdf Topological approach to chain recurrence in continuous dynamical systems Piotr Oprocha In this paper we present equivalent definitions of chain recurrent set for continuous dynamical systems. This definitions allow us to define chain recurrent set in topological spaces. chain-recurrent set, continuous dynamical system, flow, attractor 2005 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol25iss2art9 https://www.opuscula.agh.edu.pl/vol25/2/art/opuscula_math_2517.pdf 2-biplacement without fixed points of (p,q)-bipartite graphs Beata Orchel In this paper we consider \(2\)-biplacement without fixed points of paths and \((p,q)\)-bipartite graphs of small size. We give all \((p,q)\)-bipartite graphs \(G\) of size \(q\) for which the set \(\mathcal{S}^{*}(G)\) of all \(2\)-biplacements of \(G\) without fixed points is empty. bipartite graph, packing, embedding 2005 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol25iss2art10 https://www.opuscula.agh.edu.pl/vol25/2/art/opuscula_math_2518.pdf On intertwining and w-hyponormal operators M. O. Otieno Given \(A, B\in B(H)\), the algebra of operators on a Hilbert Space \(H\), define \(\delta_{A,B}: B(H) \to B(H)\) and \(\Delta_{A,B}: B(H) \to B(H)\) by \(\delta_{A,B}(X)=AX-XB\) and \(\Delta_{A,B}(X)=AXB-X\). In this note, our task is a twofold one. We show firstly that if \(A\) and \(B^{*}\) are contractions with \(C_{.}o\) completely non unitary parts such that \(X \in \ker \Delta_{A,B}\), then \(X \in \ker \Delta_{A*,B*}\). Secondly, it is shown that if \(A\) and \(B^{*}\) are \(w\)-hyponormal operators such that \(X \in \ker \delta_{A,B}\) and \(Y \in \ker \delta_{B,A}\), where \(X\) and \(Y\) are quasi-affinities, then \(A\) and \(B\) are unitarily equivalent normal operators. A \(w\)-hyponormal operator compactly quasi-similar to an isometry is unitary is also proved. \(w\)-hyponormal operators, contraction operators, quasi-similarity 2005 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol25iss2art11 https://www.opuscula.agh.edu.pl/vol25/2/art/opuscula_math_2519.pdf The geometric properties of reduced canonically symplectic spaces with symmetry, their relationship with structures on associated principal fiber bundles and some applications. Part 1 Yarema A. Prykarpatsky, Anatoliy M. Samoilenko, Anatoliy K. Prykarpatsky The canonical reduction method on canonically symplectic manifolds is analized in detail, the relationships with the geometric properties of associated principal fiber bundles endowed with connection structures are stated. Some results devoted to studying geometrical properties of nonabelian Yang-Mills type gauge field quations are presented. Hamiltonian reduction, symplectic structures, connections, principal fiber bundles, Yang-Mills type gauge fields 2005 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol25iss2art12 https://www.opuscula.agh.edu.pl/vol25/2/art/opuscula_math_2520.pdf On the structure of characteristic surfaces related with partial differential equations of first and higher orders. Part 1 Natalia K. Prykarpatska, Marzena Pytel-Kudela The geometric structure of characteristic surfaces related with partial differential equations of first and higher orders is studied making use the vector field technique on hypersurfaces. It is shown, that corresponding characteristics are defined uniquely up to some smooth tensor fields, thereby supplying additional information about the suitable set of their solutions. In particular, it may be very useful for studying asymptotic properties of solutions to our partial differential equations under some boundary conditions. characteristic surface, vector fields, tangency, Monge cone, tensor fields 2005 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol25iss2art13 https://www.opuscula.agh.edu.pl/vol25/2/art/opuscula_math_2521.pdf Monotone iteration for infinite systems of parabolic equations Anna Pudełko In the paper the Cauchy problem for an infinite system of parabolic type equations is studied. The general operators of the parabolic type of second order with variable coefficients are considered and the system is weakly coupled. Among the obtained results there is a theorem on differential inequality as well as the existence and uniqueness theorem in the class of continuous-bounded functions obtained by monotone iterative method. infinite systems, parabolic equations, Cauchy problem, monotone iteration method, differential inequality 2005 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol25iss2art14 https://www.opuscula.agh.edu.pl/vol25/2/art/opuscula_math_2522.pdf A note on self-complementary 4-uniform hypergraphs Artur Szymański We prove that a permutation \(\theta\) is complementing permutation for a \(4\)-uniform hypergraph if and only if one of the following cases is satisfied: (i) the length of every cycle of \(\theta\) is a multiple of \(8\), (ii) \(\theta\) has \(1\), \(2\) or \(3\) fixed points, and all other cycles have length a multiple of \(8\), (iii) \(\theta\) has \(1\) cycle of length \(2\), and all other cycles have length a multiple of \(8\), (iv) \(\theta\) has \(1\) fixed point, \(1\) cycle of length \(2\), and all other cycles have length a multiple of \(8\), (v) \(\theta\) has \(1\) cycle of length \(3\), and all other cycles have length a multiple of \(8\). Moreover, we present algorithms for generating every possible \(3\) and \(4\)-uniform self-complementary hypergraphs. complementing permutation, self-complementary hypergraph, \(k\)-uniform hypergraph 2005 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol25iss2art15 https://www.opuscula.agh.edu.pl/vol25/2/art/opuscula_math_2523.pdf The exact values of nonsquare constants for a class of Orlicz spaces Jincai Wang We extend the \(M_{\triangle}\)-condition from [Han J.,Li X.: On Exact Value of Packing for a Class of Orlicz Spaces. (Chinese), Journal of Tongji Univ. 30 (2002) 7, 895–899] and introduce the \(\Phi_{\triangle}\)-condition at zero. Next we discuss nonsquare constant in Orlicz spaces generated by an \(N\)-function \(\Phi(u)\) which satisfy \(\Phi_{\triangle}\)-condition. We obtain exact value of nonsquare constant in this class of Orlicz spaces equipped with the Luxemburg norm. nonsquare constant, Orlicz space, \(\Phi_{\triangle}\)-condition 2005 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol25iss2art16 https://www.opuscula.agh.edu.pl/vol25/2/art/opuscula_math_2524.pdf Existence of solutions of the Dirichlet problem for an infinite system of nonlinear differential-functional equations of elliptic type Tomasz S. Zabawa The Dirichlet problem for an infinite weakly coupled system of semilinear differential-functional equations of elliptic type is considered. It is shown the existence of solutions to this problem. The result is based on Chaplygin's method of lower and upper functions. infinite systems, elliptic differential-functional equations, monotone iterative technique, Chaplygin's method, Dirichlet problem 2005 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol25iss2art17 https://www.opuscula.agh.edu.pl/vol25/2/art/opuscula_math_2525.pdf Independent set dominating sets in bipartite graphs Bohdan Zelinka The paper continues the study of independent set dominating sets in graphs which was started by E. Sampathkumar. A subset \(D\) of the vertex set \(V(G)\) of a graph \(G\) is called a set dominating set (shortly sd-set) in \(G\), if for each set \(X \subseteq V(G)-D\) there exists a set \(Y \subseteq D\) such that the subgraph \(\langle X \cup Y\rangle\) of \(G\) induced by \(X \cup Y\) is connected. The minimum number of vertices of an sd-set in \(G\) is called the set domination number \(\gamma_s(G)\) of \(G\). An sd-set \(D\) in \(G\) such that \(|D|=\gamma_s(G)\) is called a \(\gamma_s\)-set in \(G\). In this paper we study sd-sets in bipartite graphs which are simultaneously independent. We apply the theory of hypergraphs. set dominating set, set domination number, independent set, bipartite graph, multihypergraph 2005 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol25iss2art18 https://www.opuscula.agh.edu.pl/vol25/2/art/opuscula_math_2526.pdf A note on self-complementary hypergraphs Małgorzata Zwonek In the paper we describe all self-complementary hypergraphs. It turns out that such hypergraphs exist if and only if the number of vertices of the hypergraph is of the form \(n=2^k\). This answers a conjecture posed by A. Szymański (see [Szymański A.: Note on self-complementary 4-uniform hypergraphs.(Preprint)]). self-complementary hypergraphs, complementing permutation 2005 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol26iss1 https://www.opuscula.agh.edu.pl/om-vol26iss1art1 https://www.opuscula.agh.edu.pl/vol26/1/art/opuscula_math_2601.pdf Bounds on the 2-domination number in cactus graphs Mustapha Chellali A \(2\)-dominating set of a graph \(G\) is a set \(D\) of vertices of \(G\) such that every vertex not in \(S\) is dominated at least twice. The minimum cardinality of a \(2\)-dominating set of \(G\) is the \(2\)-domination number \(\gamma_{2}(G)\). We show that if \(G\) is a nontrivial connected cactus graph with \(k(G)\) even cycles (\(k(G)\geq 0\)), then \(\gamma_{2}(G)\geq\gamma_{t}(G)-k(G)\), and if \(G\) is a graph of order \(n\) with at most one cycle, then \(\gamma_{2}(G)\geqslant(n+\ell-s)/2\) improving Fink and Jacobson's lower bound for trees with \(\ell>s\), where \(\gamma_{t}(G)\), \(\ell\) and \(s\) are the total domination number, the number of leaves and support vertices of \(G\), respectively. We also show that if \(T\) is a tree of order \(n\geqslant 3\), then \(\gamma_{2}(T)\leqslant\beta(T)+s-1\), where \(\beta(T)\) is the independence number of \(T\). 2-domination number, total domination number, independence number, cactus graphs, trees 2006 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol26iss1art2 https://www.opuscula.agh.edu.pl/vol26/1/art/opuscula_math_2602.pdf Classical solutions of initial problems for quasilinear partial functional differential equations of the first order Wojciech Czernous We consider the initial problem for a quasilinear partial functional differential equation of the first order \[ \partial_t z(t,x)+\sum_{i=1}^nf_i(t,x,z_{(t,x)})\partial_{x_i} z(t,x)=G(t,x,z_{(t,x)}),\\ z(t,x)=\varphi(t,x)\;\;((t,x)\in[-h_0,0]\times R^n)\] where \(z_{(t,x)}\colon\,[-h_0,0]\times[-h,h]\to R\) is a function defined by \(z_{(t,x)}(\tau,\xi)=z(t+\tau,x+\xi)\) for \((\tau,\xi)\in[-h_0,0]\times[-h,h]\). Using the method of bicharacteristics and the fixed-point theorem we prove, under suitable assumptions, a theorem on the local existence and uniqueness of classical solutions of the problem and its continuous dependence on the initial condition. partial functional differential equations, classical solutions, local existence, bicharacteristics 2006 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol26iss1art3 https://www.opuscula.agh.edu.pl/vol26/1/art/opuscula_math_2603.pdf Equitable coloring of graph products Hanna Furmańczyk A graph is equitably \(k\)-colorable if its vertices can be partitioned into \(k\) independent sets in such a way that the number of vertices in any two sets differ by at most one. The smallest \(k\) for which such a coloring exists is known as the equitable chromatic number of \(G\) and denoted by \(\chi_{=}(G)\). It is interesting to note that if a graph \(G\) is equitably \(k\)-colorable, it does not imply that it is equitably \((k+1)\)-colorable. The smallest integer \(k\) for which \(G\) is equitably \(k'\)-colorable for all \(k'\geq k\) is called the equitable chromatic threshold of \(G\) and denoted by \(\chi_{=}^{*}(G)\). In the paper we establish the equitable chromatic number and the equitable chromatic threshold for certain products of some highly-structured graphs. We extend the results from [Chen B.-L., Lih K.-W., Yan J.-H., Equitable coloring of graph products, manuscript, 1998] for Cartesian, weak and strong tensor products. equitable coloring, graph product 2006 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol26iss1art4 https://www.opuscula.agh.edu.pl/vol26/1/art/opuscula_math_2604.pdf Well-posedness and stability analysis of hybrid feedback systems using Shkalikov's theory Piotr Grabowski The modern method of analysis of the distributed parameter systems relies on the transformation of the dynamical model to an abstract differential equation on an appropriately chosen Banach or, if possible, Hilbert space. A linear dynamical model in the form of a first order abstract differential equation is considered to be well-posed if its right-hand side generates a strongly continuous semigroup. Similarly, a dynamical model in the form of a second order abstract differential equation is well-posed if its right-hand side generates a strongly continuous cosine family of operators. Unfortunately, the presence of a feedback leads to serious complications or even excludes a direct verification of assumptions of the Hille-Phillips-Yosida and/or the Sova-Fattorini Theorems. The class of operators which are similar to a normal discrete operator on a Hilbert space describes a wide variety of linear operators. In the papers [Grabowski P., Well–posedness and stability analysis of hybrid feedback systems, Journal of Mathematical Systems, Estimation and Control 6 (1996), 121–124 (summary), full electronic manuscript – retrieval code 15844, Grabowski P., Spectral approach to well–posedness and stability analysis of hybrid feedback systems, In: Wajs W., Grabowski P. (Eds.), Studies in Automatics, 1996, Kraków, Wydawnictwa AGH, 104–139] two groups of similarity criteria for a given hybrid closed-lop system operator are given. The criteria of the first group are based on some perturbation results, and of the second, on the application of Shkalikov's theory of the Sturm-Liouville eigenproblems with a spectral parameter in the boundary conditions. In the present paper we continue those investigations showing certain advanced applications of the Shkalikov's theory. The results are illustrated by feedback control systems examples governed by wave and beam equations with increasing degree of complexity of the boundary conditions. infinite-dimensional control systems, semigroups, spectral methods, Riesz bases 2006 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol26iss1art5 https://www.opuscula.agh.edu.pl/vol26/1/art/opuscula_math_2605.pdf Rates of convergence for the maximum likelihood estimator in the convolution model Piotr Majerski Rates of convergence for the maximum likelihood estimator in the convolution model, obtained recently by S. van de Geer, are reconsidered and corrected. maximum likelihood estimator, entropy, Hellinger distance, rate of convergence 2006 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol26iss1art6 https://www.opuscula.agh.edu.pl/vol26/1/art/opuscula_math_2606.pdf A note on arbitrarily vertex decomposable graphs Antoni Marczyk A graph \(G\) of order \(n\) is said to be arbitrarily vertex decomposable if for each sequence \((n_{1},\ldots,n_k)\) of positive integers such that \(n_{1}+\ldots+n_{k}=n\) there exists a partition \((V_{1},\ldots,V_{k})\) of the vertex set of \(G\) such that for each \(i \in \{1,\ldots,k\}\), \(V_{i}\) induces a connected subgraph of \(G\) on \(n_i\) vertices. In this paper we show that if \(G\) is a two-connected graph on \(n\) vertices with the independence number at most \(\lceil n/2\rceil\) and such that the degree sum of any pair of non-adjacent vertices is at least \(n-3\), then \(G\) is arbitrarily vertex decomposable. We present another result for connected graphs satisfying a similar condition, where the bound \(n-3\) is replaced by \(n-2\). arbitrarily vertex decomposable graphs, traceable graphs, independence number, perfect matching 2006 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol26iss1art7 https://www.opuscula.agh.edu.pl/vol26/1/art/opuscula_math_2607.pdf Bipartite embedding of (p,q)-trees Beata Orchel A bipartite graph \(G=(L,R;E)\) where \(V(G)=L\cup R\), \(|L|=p\), \(|R| =q\) is called a \((p,q)\)-tree if \(|E(G)|=p+q-1\) and \(G\) has no cycles. A bipartite graph \(G=(L,R;E)\) is a subgraph of a bipartite graph \(H=(L',R';E')\) if \(L\subseteq L'\), \(R\subseteq R'\) and \(E\subseteq E'\). In this paper we present sufficient degree conditions for a bipartite graph to contain a \((p,q)\)-tree. bipartite graph, tree, embedding graph 2006 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol26iss1art8 https://www.opuscula.agh.edu.pl/vol26/1/art/opuscula_math_2608.pdf Proof of Milman's theorem on extension of M-basic sequence Anatolij Plichko We prove Milman's theorem on the extension, in a given direction, of M-basic sequence to M-basis in a separable Banach space. quasicomplement, Markushevich basis 2006 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol26iss1art9 https://www.opuscula.agh.edu.pl/vol26/1/art/opuscula_math_2609.pdf Some analytical properties of dissolving operators related with the Cauchy problem for a class of nonautonomous partial differential equations. Part 1 Marzena Pytel-Kudela, Anatoliy K. Prykarpatsky The analytical properties of dissolving operators related with the Cauchy problem for a class of nonautonomous partial differential equations in Hilbert spaces are studied using theory of bi-linear forms in respectively rigged Hilbert spaces triples. Theorems specifying the existence of a dissolving operator for a class of adiabatically perturbed nonautonomous partial differential equations are stated. Some applications of the results obtained are discussed. dissolving operators, bilinear forms, Cauchy problem, semigroups, evolution equations 2006 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol26iss1art10 https://www.opuscula.agh.edu.pl/vol26/1/art/opuscula_math_2610.pdf The Delsarte-Darboux type binary transformations, their differential-geometric and operator structure with applications. Part 1 Yarema A. Prykarpatsky, Anatoliy M. Samoilenko The structure properties of multidimensional Delsarte-Darboux transmutation operators in parametric functional spaces are studied by means of differential-geometric and topological tools. It is shown that kernels of the corresponding integral operator expressions depend on the topological structure of related homological cycles in the coordinate space. As a natural realization of the construction presented we build pairs of Lax type commutative differential operator expressions related via a Delsarte-Darboux transformation and having a lot of applications in soliton theory. Delsarte transmutation operators, parametric functional spaces, Darboux transformations, inverse spectral transform problem, soliton equations, Zakharov-Shabat equations 2006 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol26iss1art11 https://www.opuscula.agh.edu.pl/vol26/1/art/opuscula_math_2611.pdf A remark on generalized commutation relation and subnormality Jerzy Bartłomiej Stochel Tillmann [Tillmann H. G., Zur Eindeutigkeit der Losungen der quanten mechanischen vertauschungrelationen, Acta Sci. Math. (Szeged) 24 (1963), 258-270] proved that every operator \(A\) which fulfils the canonical commutation relation \(A^{*}A - AA^{*} = Id\) is an orthogonal sum of canonical creation operators. We extend this result for operators which fulfil generalized commutation relation \[A^{*}A - AA^{*}= E^2,\text{ where }EA = AE.\] In addition, some inequalities which are helpful in describing analytic vectors of operators \(A^{*}A\), where \(A\) fulfils the generalized commutation relation, are established. Hilbert space, generalized commutation relation, creation operator, analytic vectors 2006 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol26iss1art12 https://www.opuscula.agh.edu.pl/vol26/1/art/opuscula_math_2612.pdf A distribution associated with the Kontorovich-Lebedev transform Semyon B. Yakubovich We show that in a sense of distributions \[\lim_{\varepsilon\to 0+} {1\over \pi^2} \tau\sinh\pi\tau \int_{\varepsilon}^{\infty} K_{i\tau}(y)K_{ix}(y){dy\over y} =\delta(\tau-x),\] where \(\delta\) is the Dirac distribution, \(\tau\), \(x\in\mathbb{R}\) and \(K_{\nu}(x)\) is the modified Bessel function. The convergence is in \(\mathcal{E}^{\prime}(\mathbb{R})\) for any even \(\varphi(x)\in\mathcal{E}(\mathbb{R})\) being a restriction to \(\mathbb{R}\) of a function \(\varphi(z)\) analytic in a horizontal open strip \(G_a=\{z\in\mathbb{C}\colon\,|\text{Im}\,z|\lt a, \ a\gt 0\}\) and continuous in the strip closure. Moreover, it satisfies the condition \(\varphi(z)=O\bigl(|z|^{-\text{Im}\,z-\alpha}e^{-\pi|\text{Re}\,z|/2}\bigr)\), \(|\text{Re}\,z|\to\infty\), \(\alpha\gt 1\) uniformly in \(\overline{G_a}\). The result is applied to prove the representation theorem for the inverse Kontorovich-Lebedev transformation on distributions. Kontorovich-Lebedev transform, distributions, modified Bessel functions 2006 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol26iss1art13 https://www.opuscula.agh.edu.pl/vol26/1/art/opuscula_math_2613.pdf Stability of solutions of infinite systems of nonlinear differential-functional equations of parabolic type Tomasz S. Zabawa A parabolic initial boundary value problem and an associated elliptic Dirichlet problem for an infinite weakly coupled system of semilinear differential-functional equations are considered. It is shown that the solutions of the parabolic problem is asymptotically stable and the limit of the solution of the parabolic problem as \(t\to\infty\) is the solution of the associated elliptic problem. The result is based on the monotone methods. stability of solutions, infinite systems, parabolic equations, elliptic equations, semilinear differential-functional equations, monotone iterative method 2006 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol26iss1art14 https://www.opuscula.agh.edu.pl/vol26/1/art/opuscula_math_2614.pdf On arc-coloring of digraphs Małgorzata Zwonek In the paper we deal with the problem of the arc-colouring of some classes of digraphs (tournaments, complete digraphs and products of digraphs). arc coloring, digraph 2006 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol26iss2 https://www.opuscula.agh.edu.pl/om-vol26iss2art1 https://www.opuscula.agh.edu.pl/vol26/2/art/opuscula_math_2615.pdf A uniform quantitative stiff stability estimate for BDF schemes Winfried Auzinger, Wolfgang Herfort The concepts of stability regions, \(A\)- and \(A(\alpha)\)-stability - albeit based on scalar models - turned out to be essential for the identification of implicit methods suitable for the integration of stiff ODEs. However, for multistep methods, knowledge of the stability region provides no information on the quantitative stability behavior of the scheme. In this paper we fill this gap for the important class of Backward Differentiation Formulas (BDF). Quantitative stability bounds are derived which are uniformly valid in the stability region of the method. Our analysis is based on a study of the separation of the characteristic roots and a special similarity decomposition of the associated companion matrix. BDF schemes, stiff ODEs, stability, companion matrix, univalence 2006 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol26iss2art2 https://www.opuscula.agh.edu.pl/vol26/2/art/opuscula_math_2616.pdf Collocation methods for the solution of eigenvalue problems for singular ordinary differential equations Winfried Auzinger, Ernst Karner, Othmar Koch, Ewa Weinmüller We demonstrate that eigenvalue problems for ordinary differential equations can be recast in a formulation suitable for the solution by polynomial collocation. It is shown that the well-posedness of the two formulations is equivalent in the regular as well as in the singular case. Thus, a collocation code equipped with asymptotically correct error estimation and adaptive mesh selection can be successfully applied to compute the eigenvalues and eigenfunctions efficiently and with reliable control of the accuracy. Numerical examples illustrate this claim. polynomial collocation, singular boundary value problems, linear and nonlinear eigenvalue problems 2006 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol26iss2art3 https://www.opuscula.agh.edu.pl/vol26/2/art/opuscula_math_2617.pdf Application of Green's operator to quadratic variational problems Nikolay V. Azbelev, Vadim Z. Tsalyuk We use Green's function of a suitable boundary value problem to convert the variational problem with quadratic functional and linear constraints to the equivalent unconstrained extremal problem in some subspace of the space \(L_2\) of quadratically summable functions. We get the necessary and sufficient criterion for unique solvability of the variational problem in terms of the spectrum of some integral Hilbert-Schmidt operator in \(L_2\) with symmetric kernel. The numerical technique is proposed to estimate this criterion. The results are demonstrated on examples: 1) a variational problem with deviating argument, and 2) the problem of the critical force for the vertical pillar with additional support point (the qualities of the pillar may vary discontinuously along the pillar's axis). quadratic variational problem, Sobolev space, boundary value problem, Hilbert space, Green's operator, Fredholm integral operator, spectrum 2006 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol26iss2art4 https://www.opuscula.agh.edu.pl/vol26/2/art/opuscula_math_2618.pdf On reconstruction of structure of a linear system with time delay Marina Blizorukova, Nina Fedina, Vyacheslav Maksimov The problem of reconstruction of a structure of a linear system with delay is considered. A solution algorithm stable with respect to the informational noise and computational errors is specified. The algorithm is based on the method of auxiliary positionally controlled models. time delay systems, reconstruction of structure 2006 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol26iss2art5 https://www.opuscula.agh.edu.pl/vol26/2/art/opuscula_math_2619.pdf The numerical solution of nonlinear two-point boundary value problems using iterated deferred correction - a survey Jeff R. Cash The use of iterated deferred correction has proved to be a very efficient approach to the numerical solution of general first order systems of nonlinear two-point boundary value problems. In particular the two high order codes TWPBVP.f, based on mono-implicit Runge-Kutta (MIRK) formulae, and TWPBVPL.f based on Lobatto Runge-Kutta formulae as well as the continuation codes ACDC.f and COLMOD.f are now widely used. In this survey we describe some of the problems involved in the derivation of efficient deferred correction schemes. In particular we consider the construction of high order methods which preserve the stability of the underlying formulae, the choice of the mesh choosing algorithm which is based both on local accuracy and conditioning, and the computation of continuous solutions. deferred correction, boundary value problems, conditioning, mesh selection 2006 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol26iss2art6 https://www.opuscula.agh.edu.pl/vol26/2/art/opuscula_math_2620.pdf The reactance wave diffraction problem by a strip in a scale of Bessel potential spaces Luís P. Castro, David Natroshvili We consider a boundary-transmission problem for the Helmholtz equation, in a Bessel potential space setting, which arises within the context of wave diffraction theory. The boundary under consideration consists of a strip, and certain reactance conditions are assumed on it. Operator theoretical methods are used to deal with the problem and, as a consequence, several convolution type operators are constructed and associated to the problem. At the end, the well-posedness of the problem is shown for a range of regularity orders of the Bessel potential spaces, and for a set of possible reactance numbers (dependent on the wave number). Helmholtz equation, boundary-transmission problem, wave diffraction, convolution type operator, Wiener-Hopf operator, Fredholm property, factorization 2006 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol26iss2art7 https://www.opuscula.agh.edu.pl/vol26/2/art/opuscula_math_2621.pdf A remark on the linearization technique in half-linear oscillation theory Ondřej Došlý We show that oscillatory properties of the half-linear second order differential equation \[(r(t)\Phi(x'))'+c(t)\Phi(x)=0,\qquad\Phi(x)=|x|^{p-2}x,\quad p\gt 1,\] can be investigated via oscillatory properties of a certain associated second order linear differential equation. In contrast to paper [O. Došlý, S. Peňa, A linearization method in oscillation theory of half-linear differential equations, J. Inequal. Appl. 2005 (2005), 235–245], we do not need to distinguish between the cases \(p\ge 2\) and \(p\in (1,2]\). Our results also improve the oscillation and nonoscillation criteria given in [O. Došlý, A. Lomtatidze, Oscillation and nonoscillation criteria for half-linear second order differential equations, to appear in Hiroshima Math. J.]. half-linear oscillation theory, oscillation and nonoscillation criteria, Riccati technique, perturbation principle 2006 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol26iss2art8 https://www.opuscula.agh.edu.pl/vol26/2/art/opuscula_math_2622.pdf On oscillatory solutions of certain difference equations Grzegorz Grzegorczyk, Jarosław Werbowski Some difference equations with deviating arguments are discussed in the context of the oscillation problem. The aim of this paper is to present the sufficient conditions for oscillation of solutions of the equations discussed. difference equations, oscillation 2006 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol26iss2art9 https://www.opuscula.agh.edu.pl/vol26/2/art/opuscula_math_2623.pdf A dynamical inverse problem for a parabolic equation Vyacheslav Maksimov A problem of dynamical reconstruction of unknown distributed or boundary disturbances acting upon nonlinear parabolic equations is discussed. A regularized algorithm which allows us to reconstruct disturbances synchro with the process under consideration is designed. This algorithm is stable with respect to informational noises and computational errors. nonlinear parabolic equations, inverse problem 2006 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol26iss2art10 https://www.opuscula.agh.edu.pl/vol26/2/art/opuscula_math_2624.pdf Comparison of properties of solutions of differential equations and recurrence equations with the same characteristic equation (on example of third order linear equations with constant coefficients) Jarosław Mikołajski, Ewa Schmeidel Third order linear homogeneous differential and recurrence equations with constant coefficients are considered. We take the both equations with the same characteristic equation. We show that these equations (differential and recurrence) can have solutions with different properties concerning oscillation and boundedness. Especially the numbers of suitable types of solutions taken out from fundamental sets are presented. We give conditions under which the asymptotic properties considered are the same for the both equations. differential equation, recurrence, linear, third order, oscillatory solution, bounded solution 2006 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol26iss2art11 https://www.opuscula.agh.edu.pl/vol26/2/art/opuscula_math_2625.pdf Continuous dependence of solutions of elliptic BVPs on parameters Aleksandra Orpel The continuous dependence of solutions for a certain class of elliptic PDE on functional parameters is studied in this paper. The main result is as follow: the sequence \(\{x_k\}_{k\in N}\) of solutions of the Dirichlet problem discussed here (corresponding to parameters \(\{u_k\}_{k\in N}\)) converges weakly to \(x_0\) (corresponding to \(u_0\)) in \(W^{1,q}_0(\Omega,R)\), provided that \(\{u_k\}_{k\in N}\) tends to \(u_0\) a.e. in \(\Omega\). Our investigation covers both sub and superlinear cases. We apply this result to some optimal control problems. continuous dependence on parameters, elliptic Dirichlet problems, optimal control problem 2006 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol26iss2art12 https://www.opuscula.agh.edu.pl/vol26/2/art/opuscula_math_2626.pdf Oscillatory and asymptotically zero solutions of third order difference equations with quasidifferences Ewa Schmeidel In this paper, third order difference equations are considered. We study the nonlinear third order difference equation with quasidifferences. Using Riccati transformation techniques, we establish some sufficient conditions for each solution of this equation to be either oscillatory or converging to zero. The result is illustrated with examples. linear, nonlinear, difference equations, third order, oscillatory solution, quasidifferences 2006 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol26iss2art13 https://www.opuscula.agh.edu.pl/vol26/2/art/opuscula_math_2627.pdf Oscillatory properties of fourth order nonlinear difference equations with quasidifferences Ewa Schmeidel, Małgorzata Migda, Anna Musielak In this paper we present the oscillation criterion for a class of fourth order nonlinear difference equations with quasidifferences. nonlinear difference equation, oscillatory solution, nonoscillatory solution, fourth order 2006 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol26iss3 https://www.opuscula.agh.edu.pl/om-vol26iss3art1 https://www.opuscula.agh.edu.pl/vol26/3/art/opuscula_math_2628.pdf Further properties of the rational recursive sequence x_{n+1}=\frac{ax_{n-1}}{b+cx_{n}x_{n-1}} Anna Andruch-Sobiło, Małgorzata Migda In this paper we consider the difference equation \[x_{n+1}=\frac{ax_{n-1}}{b+cx_{n}x_{n-1}}, \quad n=0,1,...\tag{E}\] with positive parameters \(a\) and \(c\), negative parameter \(b\) and nonnegative initial conditions. We investigate the asymptotic behavior of solutions of equation \(\text{(E)}\). difference equation, explicit formula, positive solutions, asymptotic stability 2006 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol26iss3art2 https://www.opuscula.agh.edu.pl/vol26/3/art/opuscula_math_2629.pdf Initial data generating bounded solutions of linear discrete equations Jaromír Baštinec, Josef Diblík, Miroslava Růžičková A lot of papers are devoted to the investigation of the problem of prescribed behavior of solutions of discrete equations and in numerous results sufficient conditions for existence of at least one solution of discrete equations having prescribed asymptotic behavior are indicated. Not so much attention has been paid to the problem of determining corresponding initial data generating such solutions. We fill this gap for the case of linear equations in this paper. The initial data mentioned are constructed with use of two convergent monotone sequences. An illustrative example is considered, too. linear discrete equation, bounded solutions, initial data 2006 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol26iss3art3 https://www.opuscula.agh.edu.pl/vol26/3/art/opuscula_math_2630.pdf New efficient time integrators for non-linear parabolic problems Blanca Bujanda, Juan Carlos Jorge In this work a new numerical method is constructed for time-integrating multidimensional parabolic semilinear problems in a very efficient way. The method reaches the fourth order in time and it can be combined with standard spatial discretizations of any order to obtain unconditinally convergent numerical algorithms. The main theoretical results which guarantee this property are explained here, as well as the method characteristics which guarantee a very strong reduction of computational cost in comparison with classical discretization methods. fractional step methods, non-linear parabolic problems, convergence 2006 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol26iss3art4 https://www.opuscula.agh.edu.pl/vol26/3/art/opuscula_math_2631.pdf The asymptotic properties of the dynamic equation with a delayed argument Jan Čermák, Miroslav Urbánek In this paper, we present some asymptotic results related to the scalar dynamic equation with a delayed argument. Using the time scale calculus we generalize some results known in the differential and difference case to the more general dynamic case. dynamic equation, time scale, asymptotic behavior 2006 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol26iss3art5 https://www.opuscula.agh.edu.pl/vol26/3/art/opuscula_math_2632.pdf Polynomial quasisolutions of linear differential-difference equations Valery B. Cherepennikov, Polina G. Ermolaeva The paper discusses a linear differential-difference equation of neutral type with linear coefficients, when at the initial time moment \(t=0\) the value of the desired function \(x(t)\) is known. The authors are not familiar with any results which would state the solvability conditions for the given problem in the class of analytical functions. A polynomial of some degree \(N\) is introduced into the investigation. Then the term "polynomial quasisolution" (PQ-solution) is understood in the sense of appearance of the residual \(\Delta (t)=O(t^N)\), when this polynomial is substituted into the initial problem. The paper is devoted to finding PQ-solutions for the initial-value problem under analysis. differential-difference equations, neutral type, initial value problem, polynomial quasisolution 2006 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol26iss3art6 https://www.opuscula.agh.edu.pl/vol26/3/art/opuscula_math_2633.pdf A retract principle on discrete time scales Josef Diblík, Miroslava Růžičková, Barbora Václavíková In this paper we discuss asymptotic behavior of solutions of a class of scalar discrete equations on discrete real time scales. A powerful tool for the investigation of various qualitative problems in the theory of ordinary differential equations as well as delayed differential equations is the retraction method. The development of this method is discussed in the case of the equation mentioned above. Conditions for the existence of a solution with its graph remaining in a predefined set are formulated. Examples are given to illustrate the results obtained. discrete equation, discrete time scale, asymptotic behavior of solution, retract, retraction 2006 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol26iss3art7 https://www.opuscula.agh.edu.pl/vol26/3/art/opuscula_math_2634.pdf Special functions in Fuzzy Analysis Angel Garrido In the treatment of Fuzzy Logic an useful tool appears: the membership function, with the information about the degree of completion of a condition which defines the respective Fuzzy Set or Fuzzy Relation. With their introduction, it is possible to prove some results on the foundations of Fuzzy Logic and open new ways in Fuzzy Analysis. logics in A. I., Fuzzy Set Theory, Fuzzy Real Analysis, Artificial Intelligence 2006 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol26iss3art8 https://www.opuscula.agh.edu.pl/vol26/3/art/opuscula_math_2635.pdf Construction of an integral manifold for linear differential-difference equations Klara R. Janglajew, Kim G. Valeev In this paper we establish sufficient conditions for the existence of an asymptotic integral manifold of solutions of a linear system of differential-difference equations with a small parameter. This integral manifold is described by a linear system of differential equations without deviating argument. system with deviating argument, integral manifold of solutions, fundamental matrix, exponential dichotomy 2006 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol26iss3art9 https://www.opuscula.agh.edu.pl/vol26/3/art/opuscula_math_2636.pdf Fundamental solution of the problem describing ship motion in waves Jan Jankowski The problem describing a ship motion in waves comprises the Laplace equation, boundary condition on wetted surface of the ship, condition on the free surface of the sea in the form of a differential equation, the radiation condition, and a condition at infinity. This problem can be transformed to a Fredholm equation of second kind, and then numerically solved using the boundary element method, if the fundamental solution of the problem is known. This paper presents the derivation of the fundamental solution. In physical interpretation, the fundamental solution represents the moving and pulsating source under free surface of the sea. The free surface elevation, generated by the source for different forward speed and frequency of pulsation, is presented in this paper. ship hydrodynamics, boundary value problem, free surface potential flows, fundamental solution 2006 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol26iss3art10 https://www.opuscula.agh.edu.pl/vol26/3/art/opuscula_math_2637.pdf Efficient computation of the MCTDHF approximation to the time-dependent Schrödinger equation Othmar Koch We discuss analytical and numerical properties of the multi-configuration time-dependent Hartree-Fock method for the approximate solution of the time-dependent multi-particle (electronic) Schrödinger equation which are relevant for an efficient implementation of this model reduction technique. Particularly, we focus on a discretization and low rank approximation in the evaluation of the meanfield terms occurring in the MCTDHF equations of motion, which is crucial for the computational tractability of the problem. We give error bounds for this approximation and demonstrate the achieved gain in performance. multi-configuration time-dependent Hartree-Fock method, time-dependent multi-particle Schrödinger equation, Coulomb potential, finite elements 2006 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol26iss3art11 https://www.opuscula.agh.edu.pl/vol26/3/art/opuscula_math_2638.pdf On the asymptotics of the difference equation with a proportional delay Petr Kundrát This paper deals with asymptotic properties of a vector difference equation with delayed argument \[\Delta x_k=Ax_k+Bx_{\lfloor\lambda k\rfloor},\qquad 0\lt\lambda\lt 1,\quad k=0,1,2,\dots,\] where \(A\), \(B\) are constant matrices and the term \(\lfloor\lambda k\rfloor\) is the integer part of \(\lambda k\). Our aim is to emphasize some resemblances between the asymptotic behaviour of this delay difference equation and its continuous counterpart. asymptotics of difference equations, approximation methods for dynamical systems 2006 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol26iss3art12 https://www.opuscula.agh.edu.pl/vol26/3/art/opuscula_math_2639.pdf Asymptotic properties of nonoscillatory solutions of higher order neutral difference equations Małgorzata Migda In this paper we study asymptotic behavior of solutions of a higher order neutral difference equation of the form \[\Delta^m(x_n+p_nx_{n-\tau})+f(n,x_{\sigma (n)})=h_n.\] We present conditions under which all nonoscillatory solutions of the above equation have the property \(x_n = cn^{m-1}+o(n^{m-1})\) for some \(c\in R\). neutral difference equation, asymptotic behavior, nonoscillatory solution 2006 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol26iss3art13 https://www.opuscula.agh.edu.pl/vol26/3/art/opuscula_math_2640.pdf Asymptotic stability of a neutral integro-differential equation Gen-qiang Wang, Sui Sun Cheng The global stability behavior of a non-autonomous neutral functional integro-differential equation is studied. A sufficient condition for every solution of this equation to tend to zero is given. asymptotic behavior, nonlinear neutral integro-differential equation 2006 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol26iss3art14 https://www.opuscula.agh.edu.pl/vol26/3/art/opuscula_math_2641.pdf Numerical simulation of liquid motion in a partly filled tank Monika Warmowska The paper presents the problem of liquid motion in a 2D partly filled tank. It is assumed that the flow of liquid in tank is a potential, hence it can be described by Laplace equations with appropriate boundary conditions. The problem is solved using the boundary element method. The developed numerical algorithm makes it possible to determine the free surface elevation, the velocity field and the pressure field during the liquid motion in the tank. The area occupied by liquid is represented by a mesh changing in time. Numerical computations are performed for translatory and rotational motion of the tank. The results of numerical computations are verified by experiment. nonlinear boundary value problems, linear elliptic equations, sloshing, free-surface potential flows 2006 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol27iss1 https://www.opuscula.agh.edu.pl/om-vol27iss1art1 https://www.opuscula.agh.edu.pl/vol27/1/art/opuscula_math_2701.pdf Tree domatic number in graphs Xue-gang Chen A dominating set \(S\) in a graph \(G\) is a tree dominating set of \(G\) if the subgraph induced by \(S\) is a tree. The tree domatic number of \(G\) is the maximum number of pairwise disjoint tree dominating sets in \(V(G)\). First, some exact values of and sharp bounds for the tree domatic number are given. Then, we establish a sharp lower bound for the number of edges in a connected graph of given order and given tree domatic number, and we characterize the extremal graphs. Finally, we show that a tree domatic number of a planar graph is at most \(4\) and give a characterization of planar graphs with the tree domatic number \(3\). tree domatic number, regular graph, planar graph, Cartesian product 2007 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol27iss1art2 https://www.opuscula.agh.edu.pl/vol27/1/art/opuscula_math_2702.pdf On Lipschitzian operators of substitution generated by set-valued functions Jakub Jan Ludew We consider the Nemytskii operator, i.e., the operator of substitution, defined by \((N \phi)(x):=G(x,\phi(x))\), where \(G\) is a given multifunction. It is shown that if \(N\) maps a Hölder space \(H_{\alpha}\) into \(H_{\beta}\) and \(N\) fulfils the Lipschitz condition then \[G(x,y)=A(x,y)+B(x),\tag{1}\] where \(A(x,\cdot)\) is linear and \(A(\cdot ,y),\, B \in H_{\beta}\). Moreover, some conditions are given under which the Nemytskii operator generated by \((1)\) maps \(H_{\alpha}\) into \(H_{\beta}\) and is Lipschitzian. Nemytskii operator, Hölder functions, set-valued functions, Jensen equation 2007 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol27iss1art3 https://www.opuscula.agh.edu.pl/vol27/1/art/opuscula_math_2703.pdf Construction of algebraic-analytic discrete approximations for linear and nonlinear hyperbolic equations in R^{2}. Part I Mirosław Luśtyk, Mykola Prytula An algebraic-analytic method for constructing discrete approximations of linear hyperbolic equations based on a generalized d'Alembert formula of the Lytvyn and Riemann expressions for Cauchy data is proposed. The problem is reduced to some special case of the fixed point problem. algebraic-analytic approximation, d'Alembert type formula, Riemann functions, fixed point problem 2007 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol27iss1art4 https://www.opuscula.agh.edu.pl/vol27/1/art/opuscula_math_2704.pdf Approximation of eigenvalues of some unbounded self-adjoint discrete Jacobi matrices by eigenvalues of finite submatrices Maria Malejki We investigate the problem of approximation of eigenvalues of some self-adjoint operator in the Hilbert space \(l^2(\mathbb{N})\) by eigenvalues of suitably chosen principal finite submatrices of an infinite Jacobi matrix that defines the operator considered. We assume the Jacobi operator is bounded from below with compact resolvent. In our research we estimate the asymptotics (with \(n\to \infty\)) of the joint error of approximation for the first \(n\) eigenvalues and eigenvectors of the operator by the eigenvalues and eigenvectors of the finite submatrix of order \(n \times n\). The method applied in our research is based on the Rayleigh-Ritz method and Volkmer's results included in [H. Volkmer, Error Estimates for Rayleigh-Ritz Approximations of Eigenvalues and Eigenfunctions of the Mathieu and Spheroidal Wave Equation, Constr. Approx. 20 (2004), 39-54]. We extend the method to cover a class of infinite symmetric Jacobi matrices with three diagonals satisfying some polynomial growth estimates. self-adjoint unbounded Jacobi matrix, asymptotics, point spectrum, tridiagonal matrix, eigenvalue 2007 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol27iss1art5 https://www.opuscula.agh.edu.pl/vol27/1/art/opuscula_math_2705.pdf k-perfect geodominating sets in graphs Doost Ali Mojdeh, Nader Jafari Rad A perfect geodominating set in a graph \(G\) is a geodominating set \(S\) such that any vertex \(v \in V(G)\setminus S\) is geodominated by exactly one pair of vertices of \(S\). A \(k\)-perfect geodominating set is a geodominating set \(S\) such that any vertex \(v \in V(G)\setminus S\) is geodominated by exactly one pair \(x\), \(y\) of vertices of \(S\) with \(d(x,y)=k\). We study perfect and \(k\)-perfect geodomination numbers of a graph \(G\). geodominating set, perfect geodomination number, pendant vertex, pendant edge 2007 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol27iss1art6 https://www.opuscula.agh.edu.pl/vol27/1/art/opuscula_math_2706.pdf Approximation properties of some two-layer feedforward neural networks Michał A. Nowak In this article, we present a multivariate two-layer feedforward neural networks that approximate continuous functions defined on \([0,1]^d\). We show that the \(L_1\) error of approximation is asymptotically proportional to the modulus of continuity of the underlying function taken at \(\sqrt{d}/n\), where \(n\) is the number of function values used. neural networks, approximation of functions, sigmoidal function 2007 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol27iss1art7 https://www.opuscula.agh.edu.pl/vol27/1/art/opuscula_math_2707.pdf On quasi-similarity and ω-hyponormal operators Mourice Ouma Otieno In this paper, it is shown that a Putnam-Fuglede type commutativity theorem holds for \(\omega\)-hyponormal operators, the normal parts of quasi-similar \(\omega\)-hyponormal operators are unitarily equivalent and a \(\omega\)-hyponormal spectral operator is normal. \(\omega\)-hyponormal operators, quasi-similarity 2007 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol27iss1art8 https://www.opuscula.agh.edu.pl/vol27/1/art/opuscula_math_2708.pdf Nearly perfect sets in the n-fold products of graphs Monika Perl The study of nearly perfect sets in graphs was initiated in [J. E. Dunbar, F. C. Harris, S. M. Hedetniemi, S. T. Hedetniemi, A. A. McRae, R. C. Laskar, Nearly perfect sets in graphs, Discrete Mathematics 138 (1995), 229-246]. Let \(S \subseteq V(G)\). We say that \(S\) is a nearly perfect set (or is nearly perfect) in \(G\) if every vertex in \(V(G)-S\) is adjacent to at most one vertex in \(S\). A nearly perfect set \(S\) in \(G\) is called \(1\)-maximal if for every vertex \(u \in V(G)-S\), \(S \cup \{u\}\) is not nearly perfect in $G$. We denote the minimum cardinality of a \(1\)-maximal nearly perfect set in \(G\) by \(n_p(G)\). We will call the \(1\)-maximal nearly perfect set of the cardinality \(n_p(G)\) an \(n_p(G)\)-set. In this paper, we evaluate the parameter \(n_p(G)\) for some \(n\)-fold products of graphs. To this effect, we determine \(1\)-maximal nearly perfect sets in the \(n\)-fold Cartesian product of graphs and in the \(n\)-fold strong product of graphs. dominating sets, product of graphs 2007 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol27iss1art9 https://www.opuscula.agh.edu.pl/vol27/1/art/opuscula_math_2709.pdf On the geometric structure of characteristic vector fields related with nonlinear equations of the Hamilton-Jacobi type Natalia K. Prykarpatska, Eugeniusz Wachnicki The Cartan-Monge geometric approach to the characteristics method for Hamilton-Jacobi type equations and nonlinear partial differential equations of higher orders is analyzed. The Hamiltonian structure of characteristic vector fields related with nonlinear partial differential equations of first order is analyzed, the tensor fields of special structure are constructed for defining characteristic vector fields naturally related with nonlinear partial differential equations of higher orders. The generalized characteristics method is developed in the framework of the symplectic theory within geometric Monge and Cartan pictures. The related characteristic vector fields are constructed making use of specially introduced tensor fields, carrying the symplectic structure. Based on their inherited geometric properties, the related functional-analytic Hopf-Lax type solutions to a wide class of boundary and Cauchy problems for nonlinear partial differential equations of Hamilton-Jacobi type are studied. For the non-canonical Hamilton-Jacobi equations there is stated a relationship between their solutions and a good specified functional-analytic fixed point problem, related with Hopf-Lax type solutions to specially constructed dual canonical Hamilton-Jacobi equations. Hamilton-Jacobi equations, the Cartan-Monge geometric approach, Hopf-Lax type representation 2007 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol27iss1art10 https://www.opuscula.agh.edu.pl/vol27/1/art/opuscula_math_2710.pdf The study of Delsarte-Lions type binary transformations, their differential-geometric and operator structure with applications. Part 2 Yarema A. Prykarpatsky, Anatoliy M. Samoilenko The Gelfand-Levitan integral equations for Delsarte-Lions type transformations in multidimension are studied. The corresponding spectral and analytical properties of Delsarte-Lions transformed operators are analyzed by means of the differential-geometric and topological tools. An approach for constructing Delsarte-Lions type transmutation operators for multidimensional differential expressions is devised. Delsarte transmutation operators, generalized de Rham-Hodge differential complex, Delsarte-Lions type transformations, Gelfand-Levitan-Marchenko type integral equations, multidimensional differential operator pencils 2007 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol27iss1art11 https://www.opuscula.agh.edu.pl/vol27/1/art/opuscula_math_2711.pdf [r,s,t]-colourings of paths Marta Salvador Villà, Ingo Schiermeyer The concept of \([r,s,t]\)-colourings was recently introduced by Hackmann, Kemnitz and Marangio [A. Kemnitz, M. Marangio, \([r,s,t]\)-Colorings of Graphs, Discrete Math., to appear] as follows: Given non-negative integers \(r\), \(s\) and \(t\), an \([r,s,t]\)-colouring of a graph \(G=(V(G),E(G))\) is a mapping \(c\) from \(V(G) \cup E(G)\) to the colour set \(\{1,2,\ldots ,k\}\) such that \(|c(v_i)-c(v_j)| \geq r\) for every two adjacent vertices \(v_i\), \(v_j\), \(|c(e_i)-c(e_j)| \geq s\) for every two adjacent edges \(e_i\), \(e_j\), and \(|c(v_i)-c(e_j)| \geq t\) for all pairs of incident vertices and edges, respectively. The \([r,s,t]\)-chromatic number \(\chi_{r,s,t}(G)\) of \(G\) is defined to be the minimum \(k\) such that \(G\) admits an \([r,s,t]\)-colouring. In this paper, we determine the \([r,s,t]\)-chromatic number for paths. total colouring, paths 2007 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol27iss1art12 https://www.opuscula.agh.edu.pl/vol27/1/art/opuscula_math_2712.pdf Unfolding spheres size distribution from linear sections with B-splines and EMDS algorithm Zbigniew Szkutnik The stereological problem of unfolding spheres size distribution from linear sections is formulated as a problem of inverse estimation of a Poisson process intensity function. A singular value expansion of the corresponding integral operator is given. The theory of recently proposed \(B\)-spline sieved quasi-maximum likelihood estimators is modified to make it applicable to the current problem. Strong \(L^2\)-consistency is proved and convergence rates are given. The estimators are implemented with the recently proposed EMDS algorithm. Promising performance of this new methodology in finite samples is illustrated with a numerical example. Data grouping effects are also discussed. inverse problem, singular value expansion, stereology, discretization, quasi-maximum likelihood estimator 2007 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol27iss1art13 https://www.opuscula.agh.edu.pl/vol27/1/art/opuscula_math_2713.pdf J-convexity constants Jincai Wang We introduce the \(J\)-convexity constants on Banach spaces and give some properties of the constants. We give the relations between the \(J\)-convexity constants and the \(n\)-th von Neumann-Jordan constants. Using the quantitative indices we estimate the value of \(J\)-convexity constants in Orlicz spaces. new quantitative index, \(J\)-convexity constants, \(n\)-th von Neumann-Jordan constants, Orlicz spaces 2007 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol27iss2 https://www.opuscula.agh.edu.pl/om-vol27iss2art1 https://www.opuscula.agh.edu.pl/vol27/2/art/opuscula_math_2714.pdf A note on a Vizing's generalized conjecture Mostafa Blidia, Mustapha Chellali In this note we give a generalized version of Vizing's conjecture concerning the distance domination number for the cartesian product of two graphs. graph, dominating sets, Vizing's conjecture 2007 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol27iss2art2 https://www.opuscula.agh.edu.pl/vol27/2/art/opuscula_math_2715.pdf On kinetic Boltzmann equations and related hydrodynamic flows with dry viscosity Nikolai N. Bogoliubov (Jr.), Denis L. Blackmore, Valeriy Hr. Samoylenko, Anatoliy K. Prykarpatsky A two-component particle model of Boltzmann-Vlasov type kinetic equations in the form of special nonlinear integro-differential hydrodynamic systems on an infinite-dimensional functional manifold is discussed. We show that such systems are naturally connected with the nonlinear kinetic Boltzmann-Vlasov equations for some one-dimensional particle flows with pointwise interaction potential between particles. A new type of hydrodynamic two-component Benney equations is constructed and their Hamiltonian structure is analyzed. kinetic Boltzmann-Vlasov equations, hydrodynamic model, Hamiltonian systems, invariants, dynamical equivalence 2007 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol27iss2art3 https://www.opuscula.agh.edu.pl/vol27/2/art/opuscula_math_2716.pdf Transmission problems for the Helmholtz equation for a rectilinear-circular lune Volodymyr Denysenko The question related to the construction of the solution of plane transmission problem for the Helmholtz equation in a rectilinear-circular lune is considered. An approach is proposed based on the method of partial domains and the principle of reflection for the solutions of the Helmholtz equation through the segment. Helmholtz equation, transmission problem, infinite system of linear algebraic equations 2007 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol27iss2art4 https://www.opuscula.agh.edu.pl/vol27/2/art/opuscula_math_2717.pdf The use of integral information in the solution of a two-point boundary value problem Tomasz Drwięga We study the worst-case \(\varepsilon\)-complexity of a two-point boundary value problem \(u^{\prime\prime}(x)=f(x)u(x)\), \(x \in [0,T]\), \(u(0)=c\), \(u^{\prime}(T)=0\), where \(c,T \in \mathbb{R}\) (\(c \neq 0\), \(T \gt 0\)) and \(f\) is a nonnegative function with \(r\) (\(r\geq 0\)) continuous bounded derivatives. We prove an upper bound on the complexity for linear information showing that a speed-up by two orders of magnitude can be obtained compared to standard information. We define an algorithm based on integral information and analyze its error, which provides an upper bound on the \(\varepsilon\)-complexity. boundary value problem, complexity, worst case setting, linear information 2007 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol27iss2art5 https://www.opuscula.agh.edu.pl/vol27/2/art/opuscula_math_2718.pdf Polynomials on the space of ω-ultradifferentiable functions Katarzyna Grasela The space of polynomials on the the space \(D_{\omega}\) of \(\omega\)-ultradifferentiable functions is represented as the direct sum of completions of symmetric tensor powers of \(D^{\prime}_{\omega}\). \(\omega\)-ultradifferentiable functions, polynomial ultradistributions 2007 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol27iss2art6 https://www.opuscula.agh.edu.pl/vol27/2/art/opuscula_math_2719.pdf Integrable three-dimensional coupled nonlinear dynamical systems related with centrally extended operator Lie algebras Oksana Ye. Hentosh, Anatoliy K. Prykarpatsky A hierarchy of Lax-type flows on a dual space to the centrally extended Lie algebra of integral-differential operators with matrix-valued coefficients is considered. By means of a specially constructed Backlund transformation the Hamiltonian representations for these flows coupled with suitable eigenfunctions and adjoint eigenfunctions evolutions of associated spectral problems are obtained. The Hamiltonian description of the corresponding set of additional symmetry hierarchies is represented. The relation of these hierarchies with Lax integrable \((3+1)\)-dimensional nonlinear dynamical systems and their triple Lax-type linearizations is analysed. centrally extended operator Lie algebra, Lax-type flows, Backlund transformation, "ghost" symmetries 2007 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol27iss2art7 https://www.opuscula.agh.edu.pl/vol27/2/art/opuscula_math_2720.pdf Differential equation of transverse vibrations of a beam with local stroke change of stiffness Stanisław Kasprzyk, Margareta Wiciak The aim of this paper is to derive a differential equation of transverse vibrations of a beam with a local, stroke change of stiffness, and to solve it. The presented method is based on the theory of distributions. equation of a beam, joint point, distribution 2007 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol27iss2art8 https://www.opuscula.agh.edu.pl/vol27/2/art/opuscula_math_2721.pdf On the equivalence of pre-Schröder equations Józef Kalinowski In the paper the equivalence of the system of two pre-Schröder functional equations (equations \((S_n)\), \((S_m)\) for \(m \gt n \geq 3\), \(n, m \in \mathbb{N}\)) and the whole system \((S)\), is considered. The results solve the problem of Gy. Targonski [Gy. Targonski, Problem P 63, Aequationes Math. 4 (1970), 251] in a particular case. pre-Schröder equations, Targonski's problem, torsion free semigroups 2007 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol27iss2art9 https://www.opuscula.agh.edu.pl/vol27/2/art/opuscula_math_2722.pdf A finite difference method for nonlinear parabolic-elliptic systems of second order partial differential equations Marian Malec, Lucjan Sapa This paper deals with a finite difference method for a wide class of weakly coupled nonlinear second-order partial differential systems with initial condition and weakly coupled nonlinear implicit boundary conditions. One part of each system is of the parabolic type (degenerated parabolic equations) and the other of the elliptic type (equations with a parameter) in a cube in \(\mathbf{R}^{1+n}\). A suitable finite difference scheme is constructed. It is proved that the scheme has a unique solution, and the numerical method is consistent, convergent and stable. The error estimate is given. Moreover, by the method, the differential problem has at most one classical solution. The proof is based on the Banach fixed-point theorem, the maximum principle for difference functional systems of the parabolic type and some new difference inequalities. It is a new technique of studying the mixed-type systems. Examples of physical applications and numerical experiments are presented. partial differential equation, parabolic-elliptic system, finite difference method, finite difference scheme, consistence, convergence, stability, error estimate, uniqueness 2007 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol27iss2art10 https://www.opuscula.agh.edu.pl/vol27/2/art/opuscula_math_2723.pdf Algebraic construction of a coboundary of a given cycle Marcin Mazur, Jacek Szybowski We present an algebraic construction of the coboundary of a given cycle as a simpler alternative to the geometric one introduced in [M. Allili, T. Kaczyński, Geometric construction of a coboundary of a cycle, Discrete Comput. Geom. 25 (2001), 125–140, T. Kaczyński, Recursive coboundary formula for cycles in acyclic chain complexes, Topol. Methods Nonlinear Anal. 18 (2001), 351–371]. algorithm, homology theory, cycle, coboundary 2007 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol27iss2art11 https://www.opuscula.agh.edu.pl/vol27/2/art/opuscula_math_2724.pdf Spectra of subnormal pairs Krzysztof Rudol In this short note we present an example related to joint spectra of subnormal pairs of bounded operators. A counterexample to the equality between Taylor's spectrum and the closure of the defect spectrum is given. This example is related to the author's modification of N. Sibony's counterexample to Corona Theorem on domains that fail to be strictly pseudoconvex. subnormal operator, corona, joint spectrum 2007 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol27iss2art12 https://www.opuscula.agh.edu.pl/vol27/2/art/opuscula_math_2725.pdf A general boundary value problem and its Weyl function Vladimir Ryzhov We study the abstract boundary value problem defined in terms of the Green identity and introduce the concept of Weyl operator function \(M(\cdot)\) that agrees with other definitions found in the current literature. In typical cases of problems arising from the multidimensional partial equations of mathematical physics the function \(M(\cdot)\) takes values in the set of unbounded densely defined operators acting on the auxiliary boundary space. Exact formulae are obtained and essential properties of \(M(\cdot)\) are studied. In particular, we consider boundary problems defined by various boundary conditions and justify the well known procedure that reduces such problems to the "equation on the boundary" involving the Weyl function, prove an analogue of the Borg-Levinson theorem, and link our results to the classical theory of extensions of symmetric operators abstract boundary value problem, symmetric operators, Green formula, Weyl function 2007 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol28iss1 https://www.opuscula.agh.edu.pl/om-vol28iss1art1 https://www.opuscula.agh.edu.pl/vol28/1/art/opuscula_math_2801.pdf Exponential type vectors in Wiener algebras on a Banach ball Adam Bednarz We consider Wiener type algebras on an open Banach ball. In particular, we prove that such algebras consist of functions analytic in this ball. We also consider a property of one-parameter groups generated by an isometric group acting on a Banach ball. We establish that the subspace of exponential type vectors of its generators form a dense subalgebra in a Wiener algebra and a generator is a derivation on this subspace. unitary one-parametric group, Wiener type algebras 2008 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol28iss1art2 https://www.opuscula.agh.edu.pl/vol28/1/art/opuscula_math_2802.pdf Deformation minimal bending of compact manifolds: case of simple closed curves Oksana Bihun, Carmen Chicone The problem of minimal distortion bending of smooth compact embedded connected Riemannian \(n\)-manifolds \(M\) and \(N\) without boundary is made precise by defining a deformation energy functional \(\Phi\) on the set of diffeomorphisms \(\text{Diff}(M,N)\). We derive the Euler-Lagrange equation for \(\Phi\) and determine smooth minimizers of \(\Phi\) in case \(M\) and \(N\) are simple closed curves. minimal deformation, distortion minimal, geometric optimization 2008 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol28iss1art3 https://www.opuscula.agh.edu.pl/vol28/1/art/opuscula_math_2803.pdf Numerical methods for hyperbolic differential functional problems Roman Ciarski The paper deals with the initial boundary value problem for quasilinear first order partial differential functional systems. A general class of difference methods for the problem is constructed. Theorems on the error estimate of approximate solutions for difference functional systems are presented. The convergence results are proved by means of consistency and stability arguments. A numerical example is given. functional differential equations, stability and convergence 2008 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol28iss1art4 https://www.opuscula.agh.edu.pl/vol28/1/art/opuscula_math_2804.pdf Classical and weak solutions for semilinear parabolic equations with Preisach hysteresis Mathias Jais We consider the solvability of the semilinear parabolic differential equation \[\frac{\partial u}{\partial t}(x,t)- \Delta u(x,t) + c(x,t)u(x,t) = \mathcal{P}(u) + \gamma (x,t)\] in a cylinder \(D=\Omega \times (0,T)\), where \(\mathcal{P}\) is a hysteresis operator of Preisach type. We show that the corresponding initial boundary value problems have unique classical solutions. We further show that using this existence and uniqueness result, one can determine the properties of the Preisach operator \(\mathcal{P}\) from overdetermined boundary data. hysteresis, parabolic, inverse problem, uniqueness 2008 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol28iss1art5 https://www.opuscula.agh.edu.pl/vol28/1/art/opuscula_math_2805.pdf Singular integral equation with a multiplicative Cauchy kernel in the half-plane Paweł Karczmarek In this paper the explicit solutions of singular integral equation with a multiplicative Cauchy kernel in the half-plane are presented. singular integral equation, exact solution, Cauchy kernel, multiplicative kernel 2008 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol28iss1art6 https://www.opuscula.agh.edu.pl/vol28/1/art/opuscula_math_2806.pdf Porous sets for mutually nearest points in Banach spaces Chong Li, Józef Myjak Let \(\mathfrak{B}(X)\) denote the family of all nonempty closed bounded subsets of a real Banach space \(X\), endowed with the Hausdorff metric. For \(E, F \in \mathfrak{B}(X)\) we set \(\lambda_{EF} = \inf \{\|z - x\| : x \in E, z \in F \}\). Let \(\mathfrak{D}\) denote the closure (under the maximum distance) of the set of all \((E, F) \in \mathfrak{B}(X) \times \mathfrak{B}(X)\) such that \(\lambda_{EF} \gt 0\). It is proved that the set of all \((E, F) \in \mathfrak{D}\) for which the minimization problem \(\min_{x \in E, z\in F}\|x - z\|\) fails to be well posed in a \(\sigma\)-porous subset of \(\mathfrak{D}\). minimization problem, well-posedness, \(H_{\rho}\)-topology, \(\sigma\)-porous set 2008 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol28iss1art7 https://www.opuscula.agh.edu.pl/vol28/1/art/opuscula_math_2807.pdf Stability of the equation of homomorphism and completeness of the underlying space Zenon Moszner We prove that all assumptions of a Theorem of Forti and Schwaiger (cf. [G. L. Forti, J. Schwaiger, Stability of homomorphisms and completeness, C. R. Math. Rep. Acad. Sci. Canada 11 (1989), 215–220]) on the coherence of stability of the equation of homomorphism with the completeness of the space of values of all these homomorphisms, are essential. We give some generalizations of this theorem and certain examples of applications. functional equation, stability of equations of homomorphisms, superstability, complete vector spaces 2008 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol28iss1art8 https://www.opuscula.agh.edu.pl/vol28/1/art/opuscula_math_2808.pdf Sign-changing Lyapunov functions in regularity of linear extensions of dynamical systems on a torus Ewa Tkocz-Piszczek In this paper we consider some sign-changing Lyapunov function in research on regularity of sets of linear extensions of dynamical systems on a torus. invariant torus, Green's function of a problem of invariant toruses, regularity of linear extensions of dynamical systems 2008 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol28iss2 https://www.opuscula.agh.edu.pl/om-vol28iss2art1 https://www.opuscula.agh.edu.pl/vol28/2/art/opuscula_math_2809.pdf A note on a family of quadrature formulas and some applications Bogusław Bożek, Wiesław Solak, Zbigniew Szydełko In this paper a construction of a one-parameter family of quadrature formulas is presented. This family contains the classical quadrature formulas: trapezoidal rule, midpoint rule and two-point Gauss rule. One can prove that for any continuous function there exists a parameter for which the value of quadrature formula is equal to the integral. Some applications of this family to the construction of cubature formulas, numerical solution of ordinary differential equations and integral equations are presented. quadrature and cubature formulas, numerical integration 2008 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol28iss2art2 https://www.opuscula.agh.edu.pl/vol28/2/art/opuscula_math_2810.pdf On a complete lattice of retracts of a free monoid generated by three elements Wit Foryś We prove that the family of retracts of a free monoid generated by three elements, partially ordered with respect to the inclusion, is a complete lattice. free monoid, retract, lattice, complete lattice 2008 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol28iss2art3 https://www.opuscula.agh.edu.pl/vol28/2/art/opuscula_math_2811.pdf Application of Chebyshev and trigonometric polynomials to the approximation of a solution of a singular integral equation with a multiplicative Cauchy kernel in the half-plane Paweł Karczmarek In this article Chebyshev and trigonometric polynomials are used to construct an approximate solution of a singular integral equation with a multiplicative Cauchy kernel in the half-plane. singular integral equation, Cauchy kernel, multiplicative kernel, approximate solution, Chebyshev polynomials, trigonometric polynomials 2008 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol28iss2art4 https://www.opuscula.agh.edu.pl/vol28/2/art/opuscula_math_2812.pdf A first-order spectral phase transition in a class of periodically modulated Hermitian Jacobi matrices Irina Pchelintseva We consider self-adjoint unbounded Jacobi matrices with diagonal \(q_n = b_{n}n\) and off-diagonal entries \(\lambda_n = n\), where \(b_{n}\) is a \(2\)-periodical sequence of real numbers. The parameter space is decomposed into several separate regions, where the spectrum of the operator is either purely absolutely continuous or discrete. We study the situation where the spectral phase transition occurs, namely the case of \(b_{1}b_{2} = 4\). The main motive of the paper is the investigation of asymptotics of generalized eigenvectors of the Jacobi matrix. The pure point part of the spectrum is analyzed in detail. Jacobi matrices, spectral phase transition, absolutely continuous spectrum, pure point spectrum, discrete spectrum, subordinacy theory, asymptotics of generalized eigenvectors 2008 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol28iss2art5 https://www.opuscula.agh.edu.pl/vol28/2/art/opuscula_math_2813.pdf On a multivalued second order differential problem with Hukuhara derivative Magdalena Piszczek Let \(K\) be a closed convex cone with the nonempty interior in a real Banach space and let \(cc(K)\) denote the family of all nonempty convex compact subsets of \(K\). Assume that continuous linear multifunctions \(H,\Psi : K \to cc(K)\) are given. We consider the following problem \[\begin{aligned}D^2\Phi(t,x) =& \Phi(t,H(x)),\\ D\Phi(t,x)|_{t=0} =& \{0\},\\ \Phi(0,x) =& \Psi(x)\end{aligned}\] for \(t \geq 0\) and \(x \in K\), where \(D\Phi(t,x)\) denotes the Hukuhara derivative of \(\Phi(t,x)\) with respect to \(t\). Hukuhara's derivative, multivalued cosine families, Riemann integral for multifunctions, Cauchy problem for a set-valued differential equation 2008 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol28iss2art6 https://www.opuscula.agh.edu.pl/vol28/2/art/opuscula_math_2814.pdf On the Chaplyghin method for first order partial differential equations Elżbieta Puźniakowska Classical solutions of initial problems for nonlinear first order partial differential equations are considered. It is shown that under natural assumptions on given functions, there exist Chaplyghin sequences and they are convergent. Error estimates for approximate solutions are given. The method of characteristics is used for the construction of approximate solutions. characteristics, Newton method, Chaplyghin sequences, initial problems 2008 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol28iss2art7 https://www.opuscula.agh.edu.pl/vol28/2/art/opuscula_math_2815.pdf Approximate solutions of a singular integral equation with Cauchy kernels in the quarter plane Dorota Pylak In the paper, we present explicit formulae for the solution of the singular integral equation with Cauchy kernels in the quarter plane. Next, Jacobi and Chebyshev polynomials are used to derive approximate solutions of this equation. singular integral equation, Cauchy kernel, Chebyshev and Jacobi polynomials 2008 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol28iss2art8 https://www.opuscula.agh.edu.pl/vol28/2/art/opuscula_math_2816.pdf Ergodic conditions and spectral properties for A-contractions Laurian Suciu, Nicolae Suciu In this paper the canonical representation of an \(A\)-contraction \(T\) on a Hilbert space \(\mathcal{H}\) is used to obtain some conditions concerning the concept of \(A\)-ergodicity studied in [L. Suciu, Orthogonal decompositions induced by generalized contractions, Acta Sci. Math. (Szeged) 70 (2004), 751–765; L. Suciu, On the ergodic \(A\)-contractions, Analele Universitaţii de Vest din Timişoara, Ser. Mat.-Inf. 2 (2004), 115–136; L. Suciu, Ergodic properties for regular \(A\)-contractions, Integral Equations and Operator Theory 56 (2006) 2, 285–299; L. Suciu, Ergodic properties and saturation for \(A\)-contractions, Operator Theory: Advances and Applications; Proceeding of 20th Conference on Operator Theory, Timişoara 2004, Theta 2006, 225–242]. The regular case and the case of \(\mathcal{R}(A)\) closed are considered, and specifically, the \(TT^{*}\)-contractions are studied. Some spectral properties are also given for certain particular class of \(A\)-isometries. mean ergodic operator, \(A\)-contraction, isometry, spectrum 2008 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol28iss3 https://www.opuscula.agh.edu.pl/om-vol28iss3art1 https://www.opuscula.agh.edu.pl/vol28/3/art/opuscula_math_2817.pdf 3-biplacement of bipartite graphs Lech Adamus, Edyta Leśniak, Beata Orchel Let \(G=(L,R;E)\) be a bipartite graph with color classes \(L\) and \(R\) and edge set \(E\). A set of two bijections \(\{\varphi_1 , \varphi_2\}\), \(\varphi_1 , \varphi_2 :L \cup R \to L \cup R\), is said to be a \(3\)-biplacement of \(G\) if \(\varphi_1(L)= \varphi_2(L) = L\) and \(E \cap \varphi_1^*(E)=\emptyset\), \(E \cap \varphi_2^*(E)=\emptyset\), \(\varphi_1^*(E) \cap \varphi_2^*(E)=\emptyset\), where \(\varphi_1^*\), \(\varphi_2^*\) are the maps defined on \(E\), induced by \(\varphi_1\), \(\varphi_2\), respectively. We prove that if \(|L| = p\), \(|R| = q\), \(3 \leq p \leq q\), then every graph \(G=(L,R;E)\) of size at most \(p\) has a \(3\)-biplacement. bipartite graph, packing of graphs, placement, biplacement 2008 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol28iss3art2 https://www.opuscula.agh.edu.pl/vol28/3/art/opuscula_math_2818.pdf Functional models for Nevanlinna families Jussi Behrndt, Seppo Hassi, Henk de Snoo The class of Nevanlinna families consists of \(\mathbb{R}\)-symmetric holomorphic multivalued functions on \(\mathbb{C} \setminus \mathbb{R}\) with maximal dissipative (maximal accumulative) values on \(\mathbb{C}_{+}\) (\(\mathbb{C}_{-}\), respectively) and is a generalization of the class of operator-valued Nevanlinna functions. In this note Nevanlinna families are realized as Weyl families of boundary relations induced by multiplication operators with the independent variable in reproducing kernel Hilbert spaces. symmetric operator, selfadjoint extension, boundary relation, Weyl family, functional model, reproducing kernel Hilbert space 2008 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol28iss3art3 https://www.opuscula.agh.edu.pl/vol28/3/art/opuscula_math_2819.pdf Randomized and quantum algorithms for solving initial-value problems in ordinary differential equations of order k Maciej Goćwin, Marek Szczęsny The complexity of initial-value problems is well studied for systems of equations of first order. In this paper, we study the \(\varepsilon\)-complexity for initial-value problems for scalar equations of higher order. We consider two models of computation, the randomized model and the quantum model. We construct almost optimal algorithms adjusted to scalar equations of higher order, without passing to systems of first order equations. The analysis of these algorithms allows us to establish upper complexity bounds. We also show (almost) matching lower complexity bounds. The \(\varepsilon\)-complexity in the randomized and quantum setting depends on the regularity of the right-hand side function, but is independent of the order of equation. Comparing the obtained bounds with results known in the deterministic case, we see that randomized algorithms give us a speed-up by \(1/2\), and quantum algorithms by \(1\) in the exponent. Hence, the speed-up does not depend on the order of equation, and is the same as for the systems of equations of first order. We also include results of some numerical experiments which confirm theoretical results. \(k\)-th order initial-value problems, randomized computing, quantum computing, optimal algorithms, complexity 2008 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol28iss3art4 https://www.opuscula.agh.edu.pl/vol28/3/art/opuscula_math_2820.pdf Strong geodomination in graphs Nader Jafari Rad, Doost Ali Mojdeh A pair \(x\), \(y\) of vertices in a nontrivial connected graph \(G\) is said to geodominate a vertex \(v\) of \(G\) if either \(v \in \{x, y\}\) or \(v\) lies in an \(x - y\) geodesic of \(G\). A set \(S\) of vertices of \(G\) is a geodominating set if every vertex of \(G\) is geodominated by some pair of vertices of \(S\). In this paper we study strong geodomination in a graph \(G\). geodomination, \(k\)-geodomination, open geodomination 2008 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol28iss3art5 https://www.opuscula.agh.edu.pl/vol28/3/art/opuscula_math_2821.pdf Generalized characteristic singular integral equation with Hilbert kernel Paweł Karczmarek In this paper an explicit solution of a generalized singular integral equation with a Hilbert kernel depending on indices of characteristic operators is presented. singular integral equation, characteristic equation, exact solution, Hilbert kernel 2008 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol28iss3art6 https://www.opuscula.agh.edu.pl/vol28/3/art/opuscula_math_2822.pdf Local subdifferentials and multivariational inequalities in Banach and Frechet spaces Pavlo O. Kasyanov, Valery S. Mel'nik, Anna M. Piccirillo Some functional-topological concepts of subdifferential and locally subdifferential maps in Frechet spaces are established. Multivariational inequalities with an operator of the pseudo-monotone type, connected with subdifferential maps, are considered. local subdifferential, multi-variational inequality, Frechet space 2008 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol28iss3art7 https://www.opuscula.agh.edu.pl/vol28/3/art/opuscula_math_2823.pdf On the perfectness of C^{∞,s}-diffeomorphism groups on a foliated manifold Jacek Lech The notion of \(C^{r,s}\) and \(C^{\infty,s}\)-diffeomorphisms is introduced. It is shown that the identity component of the group of leaf preserving \(C^{\infty,s}\)-diffeomorphisms with compact supports is perfect. This result is a modification of the Mather and Epstein perfectness theorem. group of \(C^{\infty}\)-diffeomorphisms, perfectness, commutator, foliation 2008 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol28iss3art8 https://www.opuscula.agh.edu.pl/vol28/3/art/opuscula_math_2824.pdf Weakly connected domination critical graphs Magdalena Lemańska, Agnieszka Patyk A dominating set \(D \subset V(G)\) is a weakly connected dominating set in \(G\) if the subgraph \(G[D]_w = (N_{G}[D],E_w)\) weakly induced by \(D\) is connected, where \(E_w\) is the set of all edges with at least one vertex in \(D\). The weakly connected domination number \(\gamma_w(G)\) of a graph \(G\) is the minimum cardinality among all weakly connected dominating sets in \(G\). The graph is said to be weakly connected domination critical (\(\gamma_w\)-critical) if for each \(u, v \in V(G)\) with \(v\) not adjacent to \(u\), \(\gamma_w(G + vu) \lt \gamma_w (G)\). Further, \(G\) is \(k\)-\(\gamma_w\)-critical if \(\gamma_w(G) = k\) and for each edge \(e \not\in E(G)\), \(\gamma_w(G + e) \lt k\). In this paper we consider weakly connected domination critical graphs and give some properties of \(3\)-\(\gamma_w\)-critical graphs. weakly connected domination number, tree, critical graphs 2008 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol28iss3art9 https://www.opuscula.agh.edu.pl/vol28/3/art/opuscula_math_2825.pdf On equality in an upper bound for the acyclic domination number Vladimir Samodivkin A subset \(A\) of vertices in a graph \(G\) is acyclic if the subgraph it induces contains no cycles. The acyclic domination number \(\gamma_a(G)\) of a graph \(G\) is the minimum cardinality of an acyclic dominating set of \(G\). For any graph \(G\) with \(n\) vertices and maximum degree \(\Delta(G)\), \(\gamma_a(G) \leq n - \Delta(G)\). In this paper we characterize the connected graphs and the connected triangle-free graphs which achieve this upper bound. dominating set, acyclic set, independent set, acyclic domination number 2008 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol28iss4 https://www.opuscula.agh.edu.pl/vol28/4/art/opuscula_math_2826.pdf https://www.opuscula.agh.edu.pl/vol28/4/art/opuscula_math_2827.pdf https://www.opuscula.agh.edu.pl/om-vol28iss4art3 https://www.opuscula.agh.edu.pl/vol28/4/art/opuscula_math_2828.pdf Andrzej Lasota's selected results Józef Myjak In this article we recall Andrzej Lasota's selected results which either indicated new directions of research, or layed the foundations for new approaches, or solved interesting problems. The area of mathematical interests of Professor Andrzej Lasota was very large: ordinary differential equations, partial differential equations, dynamical systems, multifunctions, differential inclusions, functional differential equations, equations with retarded arguments, ergodic theory, invariant measures, chaos, stochastic differential equations, control theory, fixed point theory, theory of Markov operators, theory of fractals, theory of dimensions, biomathematics. In all these branches he obtained original and essential results. 2008 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol28iss4art4 https://www.opuscula.agh.edu.pl/vol28/4/art/opuscula_math_2829.pdf The work of Professor Andrzej Lasota on asymptotic stability and recent progress Wojciech Bartoszek The paper is devoted to Professor Andrzej Lasota's contribution to the ergodic theory of stochastic operators. We have selected some of his important papers and shown their influence on the evolution of this topic. We emphasize the role A. Lasota played in promoting abstract mathematical theories by showing their applications. The article is focused exclusively on ergodic properties of discrete stochastic semigroups \(\{P^n : n \geq 0\}\). Nevertheless, almost all of Lasota's results presented here have their one-parameter continuous semigroup analogs. invariant measure, asymptotic stability, asymptotic periodicity, compact attractor, lower function, smoothing, cell cycle, sweeping, genericity 2008 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol28iss4art5 https://www.opuscula.agh.edu.pl/vol28/4/art/opuscula_math_2830.pdf Professor Andrzej Lasota's theories in engineering Piotr Rusek Professor Lasota's achievements in the field of engineering are based on two principal aspects: profound theoretical backgrounds and perfect knowledge of phenomenological processes. That allowed him to solve many different problems of engineering. 2008 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol28iss4art6 https://www.opuscula.agh.edu.pl/vol28/4/art/opuscula_math_2831.pdf Existence of solutions of some quadratic integral equations Giuseppe Anichini, Giuseppe Conti In this paper we study the existence of continuous solutions of quadratic integral equations. The theory of quadratic integral equations has many useful applications in mathematical physics, economics, biology, as well as in describing real world problems. The main tool used in our investigations is a fixed point result for the multivalued solution's map with acyclic values. fixed point property, measure of noncompactness, compact mappings, acyclic valued, quadratic integral equation 2008 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol28iss4art7 https://www.opuscula.agh.edu.pl/vol28/4/art/opuscula_math_2832.pdf Asymptotic properties of solutions of some iterative functional inequalities Dobiesław Brydak, Bogdan Choczewski, Marek Czerni Continuous solutions of iterative linear inequalities of the first and second order are considered, belonging to a class \(\mathcal{F}_T\) of functions behaving at the origin as a prescribed function \(T\). functional inequalities, continuous solutions, test function, asymptotic behavior 2008 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol28iss4art8 https://www.opuscula.agh.edu.pl/vol28/4/art/opuscula_math_2833.pdf On periodic and stable solutions of the Lasota equation in different phase spaces Antoni Leon Dawidowicz, Anna Poskrobko We study properties of the Lasota partial differential equation in two different spaces: \(V_{\alpha}\) (Hölder continuous functions) and \(L^p\). The aim of this paper is to generalize the results of [Z. Brzeźniak, A. L. Dawidowicz, On the periodic solution to the von Foerster-Lasota equation, to appear in Semigroup Forum]. partial differential equations, periodic solutions, stable solutions 2008 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol28iss4art9 https://www.opuscula.agh.edu.pl/vol28/4/art/opuscula_math_2834.pdf Fractal sets satisfying the strong open set condition in complete metric spaces Gerald S. Goodman Let \(K\) be a Hutchinson fractal in a complete metric space \(X\), invariant under the action \(S\) of the union of a finite number of Lipschitz contractions. The Open Set Condition states that \(X\) has a non-empty subinvariant bounded open subset \(V\), whose images under the maps are disjoint. It is said to be strong if \(V\) meets \(K\). We show by a category argument that when \(K \not\subset V\) and the restrictions of the contractions to \(V\) are open, the strong condition implies that \(\check{V}=\bigcap_{n=0}^{\infty} S^n(V)\), termed the core of \(V\) , is non-empty. In this case, it is an invariant, proper, dense, subset of \(K\), made up of points whose addresses are unique. Conversely, \(\check{V}\neq \emptyset\) implies the SOSC, without any openness assumption. address, Baire category, fractal, scaling function, scaling operator, strong open set condition 2008 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol28iss4art10 https://www.opuscula.agh.edu.pl/vol28/4/art/opuscula_math_2835.pdf Invariant measures whose supports possess the strong open set property Gerald S. Goodman Let \(X\) be a complete metric space, and \(S\) the union of a finite number of strict contractions on it. If \(P\) is a probability distribution on the maps, and \(K\) is the fractal determined by \(S\), there is a unique Borel probability measure \(\mu _P\) on \(X\) which is invariant under the associated Markov operator, and its support is \(K\). The Open Set Condition (OSC) requires that a non-empty, subinvariant, bounded open set \(V \subset X\) exists whose images under the maps are disjoint; it is strong if \(K \cap V \neq \emptyset\). In that case, the core of \(V\), \(\check{V}=\bigcap_{n=0}^{\infty} S^n (V)\), is non-empty and dense in \(K\). Moreover, when \(X\) is separable, \(\check{V}\) has full \(\mu _P\)-measure for every \(P\). We show that the strong condition holds for \(V\) satisfying the OSC iff \(\mu_P (\partial V) =0\), and we prove a zero-one law for it. We characterize the complement of \(\check{V}\) relative to \(K\), and we establish that the values taken by invariant measures on cylinder sets defined by \(K\), or by the closure of \(V\), form multiplicative cascades. core, fractal, fractal measure, invariant measure, scaling function, scaling operator, strong open set condition, zero-one law 2008 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol28iss4art11 https://www.opuscula.agh.edu.pl/vol28/4/art/opuscula_math_2836.pdf The motion planning problem and exponential stabilization of a heavy chain. Part II Piotr Grabowski This is the second part of paper [P. Grabowski, The motion planning problem and exponential stabilization of a heavy chain. Part I, to appear in International Journal of Control], where a model of a heavy chain system with a punctual load (tip mass) in the form of a system of partial differential equations was interpreted as an abstract semigroup system and then analysed on a Hilbert state space. In particular, in [P. Grabowski, The motion planning problem and exponential stabilization of a heavy chain. Part I, to appear in International Journal of Control] we have formulated the problem of exponential stabilizability of a heavy chain in a given position. It was also shown that the exponential stability can be achieved by applying a stabilizer of the colocated-type. The proof used the method of Lyapunov functionals. In the present paper, we give other two proofs of the exponential stability, which provides an additional intrinsic insight into the exponential stabilizability mechanism. The first proof makes use of some spectral properties of the system. In the second proof, we employ some relationships between exponential stability and exact observability. infinite-dimensional control systems, semigroups, motion planning problem, exponential stabilization, spectral methods, Riesz bases, exact observability 2008 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol28iss4art12 https://www.opuscula.agh.edu.pl/vol28/4/art/opuscula_math_2837.pdf Some optimal control problems for partial differential inclusions Michał Kisielewicz Partial differential inclusions are considered. In particular, basing on diffusions properties of weak solutions to stochastic differential inclusions, some existence theorems and some properties of solutions to partial differential inclusions are given. partial differential inclusions, diffusion processes, existence theorems 2008 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol28iss4art13 https://www.opuscula.agh.edu.pl/vol28/4/art/opuscula_math_2838.pdf Remarks on the stability of some quadratic functional equations Zygfryd Kominek Stability problems concerning the functional equations of the form \[f(2x+y)=4f(x)+f(y)+f(x+y)-f(x-y),\tag{1}\] and \[f(2x+y)+f(2x-y)=8f(x)+2f(y)\tag{2}\] are investigated. We prove that if the norm of the difference between the LHS and the RHS of one of equations \((1)\) or \((2)\), calculated for a function \(g\) is say, dominated by a function \(\varphi\) in two variables having some standard properties then there exists a unique solution \(f\) of this equation and the norm of the difference between \(g\) and \(f\) is controlled by a function depending on \(\varphi\). quadratic functional equations, stability 2008 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol28iss4art14 https://www.opuscula.agh.edu.pl/vol28/4/art/opuscula_math_2839.pdf Iteration groups, commuting functions and simultaneous systems of linear functional equations Janusz Matkowski Let \(( f^t )_{t \in \mathbb{R}}\) be a measurable iteration group on an open interval \(I\). Under some conditions, we prove that the inequalies \(g\circ f^a \leq f^a \circ g\) and \(g\circ f^b \leq f^b\circ g\) for some \(a,b \in \mathbb{R}\) imply that \(g\) must belong to the iteration group. Some weak conditions under which two iteration groups have to consist of the same elements are given. An extension theorem of a local solution of a simultaneous system of iterative linear functional equations is presented and applied to prove that, under some conditions, if a function \(g\) commutes in a neighbourhood of \(f\) with two suitably chosen elements \(f^a\) and \(f^b\) of an iteration group of \(f\) then, in this neighbourhood, \(g\) coincides with an element of the iteration group. Some weak conditions ensuring equality of iteration groups are considered. iteration group, commuting functions, functional equation, functional inequalities 2008 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol28iss4art15 https://www.opuscula.agh.edu.pl/vol28/4/art/opuscula_math_2840.pdf Reduction and continuation theorems for Brouwer degree and applications to nonlinear difference equations Jean Mawhin The aim of this note is to describe the continuation theorem of [J. Mawhin, Equivalence theorems for nonlinear operator equations and coincidence degree theory for some mappings in locally convex topological vector spaces, J. Differential Equations 12 (1972), 610–636, J. Mawhin, Topological Degree Methods in Nonlinear Boundary Value Problems, CBMS Reg. Conf. in Math., No 40, American Math. Soc., Providence, RI, 1979] directly in the context of Brouwer degree, providing in this way a simple frame for multiple applications to nonlinear difference equations, and to show how the corresponding reduction property can be seen as an extension of the well-known reduction formula of Leray and Schauder [J. Leray, J. Schauder, Topologie et équations fonctionnelles, Ann. Scient. Ecole Normale Sup. (3) 51 (1934), 45–78], which is fundamental for their construction of Leray-Schauder's degree in normed vector spaces. Brouwer degree, nonlinear difference equations 2008 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol28iss4art16 https://www.opuscula.agh.edu.pl/vol28/4/art/opuscula_math_2841.pdf Some remarks on the optimization of eigenvalue problems involving the p-Laplacian Wacław Pielichowski Given a bounded domain \(\Omega \subset \mathbb{R}^n\), numbers \(p \gt 1\), \(\alpha \geq 0\) and \(A \in [0,|\Omega |]\), consider the optimization problem: find a subset \(D \subset \Omega \), of measure \(A\), for which the first eigenvalue of the operator \(u\mapsto -\text{div} (|\nabla u|^{p-2}\nabla u)+ \alpha \chi_D |u|^{p-2}u \) with the Dirichlet boundary condition is as small as possible. We show that the optimal configuration \(D\) is connected with the corresponding positive eigenfunction \(u\) in such a way that there exists a number \(t\geq 1\) for which \(D=\{u \leq t\}\). We also give a new proof of symmetry of optimal solutions in the case when \(\Omega \) is Steiner symmetric and \(p = 2\). \(p\)-Laplacian, the first eigenvalue, Steiner symmetry 2008 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol28iss4art17 https://www.opuscula.agh.edu.pl/vol28/4/art/opuscula_math_2842.pdf Chaotic dynamics in the Volterra predator-prey model via linked twist maps Marina Pireddu, Fabio Zanolin We prove the existence of infinitely many periodic solutions and complicated dynamics, due to the presence of a topological horseshoe, for the classical Volterra predator-prey model with a periodic harvesting. The proof relies on some recent results about chaotic planar maps combined with the study of geometric features which are typical of linked twist maps. Volterra predator-prey system, harvesting, periodic solutions, subharmonics, chaotic-like dynamics, topological horseshoes, linked twist maps 2008 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol29iss1 https://www.opuscula.agh.edu.pl/om-vol29iss1art1 https://www.opuscula.agh.edu.pl/vol29/1/art/opuscula_math_2901.pdf Vertices belonging to all or to no minimum locating dominating sets of trees Mostafa Blidia, Rahma Lounes A set \(D\) of vertices in a graph \(G\) is a locating-dominating set if for every two vertices \(u\), \(v\) of \(G \setminus D\) the sets \(N(u) \cap D\) and \(N(v) \cap D\) are non-empty and different. In this paper, we characterize vertices that are in all or in no minimum locating dominating sets in trees. The characterization guarantees that the \(\gamma_L\)-excellent tree can be recognized in a polynomial time. domination, locating domination 2009 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol29iss1art2 https://www.opuscula.agh.edu.pl/vol29/1/art/opuscula_math_2902.pdf Strong maximum principles for implicit parabolic functional-differential problems together with nonlocal inequalities with functionals Ludwik Byszewski The aim of the paper is to give strong maximum principles for implicit parabolic functional-differential problems together with nonlocal inequalities with functionals in relatively arbitrary \((n+1)\)-dimensional time-space sets more general than the cylindrical domain. strong maximum principles, implicit parabolic problems, nonlocal inequalities 2009 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol29iss1art3 https://www.opuscula.agh.edu.pl/vol29/1/art/opuscula_math_2903.pdf The Cartan-Monge geometric approach to the generalized characteristics method and its application to the heat equation u_{t}-u_{xx}=0 Jolanta Golenia, Yarema A. Prykarpatsky, Eugeniusz Wachnicki The generalized Cartan-Monge type approach to the characteristics method is discussed from the geometric point of view. Its application to the classical one-dimensional linear heat equation \(u_t-u_{xx}=0\) is presented. It is shown that the corresponding exact solution of the Cauchy problem can be represented in a classical functional-analytic Gauss type form. characteristics method, heat equation 2009 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol29iss1art4 https://www.opuscula.agh.edu.pl/vol29/1/art/opuscula_math_2904.pdf Multivariate kernel density estimation with a parametric support Jolanta Jarnicka We consider kernel density estimation in the multivariate case, focusing on the use of some elements of parametric estimation. We present a two-step method, based on a modification of the EM algorithm and the generalized kernel density estimator, and compare this method with a couple of well known multivariate kernel density estimation methods. density estimation, kernel, bandwidth, kernel density estimator, EM algorithm 2009 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol29iss1art5 https://www.opuscula.agh.edu.pl/vol29/1/art/opuscula_math_2905.pdf Best approximation in Chebyshev subspaces of L(l_{1}^{n},l_{1}^{n}) Joanna Kowynia Chebyshev subspaces of \(\mathcal{L}(l_1^n,l_1^n)\) are studied. A construction of a \(k\)-dimensional Chebyshev (not interpolating) subspace is given. interpolating subspace, Chebyshev subspace, strongly unique best approximation 2009 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol29iss1art6 https://www.opuscula.agh.edu.pl/vol29/1/art/opuscula_math_2906.pdf Uniqueness of solutions of a generalized Cauchy problem for a system of first order partial functional differential equations Milena Netka The paper is concerned with weak solutions of a generalized Cauchy problem for a nonlinear system of first order differential functional equations. A theorem on the uniqueness of a solution is proved. Nonlinear estimates of the Perron type are assumed. A method of integral functional inequalities is used. functional differential equations, comparison methods, estimates of the Perron type 2009 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol29iss1art7 https://www.opuscula.agh.edu.pl/vol29/1/art/opuscula_math_2907.pdf On the continuity of the integrable multifunctions Bożena Piątek The generalization of the Polovinkin theorem is studied. multifunctions, Riemann integral, Aumann integral, Hausdorff metric 2009 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol29iss1art8 https://www.opuscula.agh.edu.pl/vol29/1/art/opuscula_math_2908.pdf Extremal traceable graphs with non-traceable edges Adam Paweł Wojda By \(\text{NT}(n)\) we denote the set of graphs of order \(n\) which are traceable but have non-traceable edges, i.e. edges which are not contained in any hamiltonian path. The class \(\text{NT}(n)\) has been considered by Balińska and co-authors in a paper published in 2003, where it was proved that the maximum size \(t_{\max}(n)\) of a graph in \(\text{NT}(n)\) is at least \((n^2-5n+14)/2\) (for \(n \geq 12\)). The authors also found \(t_{\max}(n)\) for \(5 \leq n \leq 11\). We prove that, for \(n \geq 5\), \(t_{\max}(n) = max\left\{ {{n-2}\choose{2}}+4, {{n-\lfloor\frac{n-1}{2}\rfloor}\choose{2}}+\lfloor\frac{n-1}{2}\rfloor^2\right\}\) and, moreover, we characterize the extremal graphs (in fact we prove that these graphs are exactly those already described in the paper by Balinska et al.). We also prove that a traceable graph of order \(n \geq 5\) may have at most \( \lceil\frac{n-3}{2}\rceil \lfloor\frac{n-3}{2}\rfloor\) non traceable edges (this result was conjectured in the mentioned paper by Balinska and co-authors). traceable graph, non-traceable edge 2009 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol29iss1art9 https://www.opuscula.agh.edu.pl/vol29/1/art/opuscula_math_2909.pdf Beurling's theorems and inversion formulas for certain index transforms Semyon B. Yakubovich The familiar Beurling theorem (an uncertainty principle), which is known for the Fourier transform pairs, has recently been proved by the author for the Kontorovich-Lebedev transform. In this paper analogs of the Beurling theorem are established for certain index transforms with respect to a parameter of the modified Bessel functions. In particular, we treat the generalized Lebedev-Skalskaya transforms, the Lebedev type transforms involving products of the Macdonald functions of different arguments and an index transform with the Nicholson kernel function. We also find inversion formulas for the Lebedev-Skalskaya operators of an arbitrary index and the Nicholson kernel transform. Beurling theorem, Kontorovich-Lebedev transform, Lebedev-Skalskaya transforms, Fourier transform, Laplace transform, modified Bessel functions, uncertainty principle, the Nicholson function 2009 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol29iss2 https://www.opuscula.agh.edu.pl/om-vol29iss2art1 https://www.opuscula.agh.edu.pl/vol29/2/art/opuscula_math_2910.pdf On some quadrature rules with Gregory end corrections Bogusław Bożek, Wiesław Solak, Zbigniew Szydełko How can one compute the sum of an infinite series \(s := a_1 + a_2 + \ldots\)? If the series converges fast, i.e., if the term \(a_n\) tends to \(0\) fast, then we can use the known bounds on this convergence to estimate the desired sum by a finite sum \(a_1 + a_2 + \ldots + a_n\). However, the series often converges slowly. This is the case, e.g., for the series \(a_n = n^{-t}\) that defines the Riemann zeta-function. In such cases, to compute \(s\) with a reasonable accuracy, we need unrealistically large values \(n\), and thus, a large amount of computation. Usually, the \(n\)-th term of the series can be obtained by applying a smooth function \(f(x)\) to the value \(n\): \(a_n = f(n)\). In such situations, we can get more accurate estimates if instead of using the upper bounds on the remainder infinite sum \(R = f(n + 1) + f(n + 2) + \ldots\), we approximate this remainder by the corresponding integral \(I\) of \(f(x)\) (from \(x = n + 1\) to infinity), and find good bounds on the difference \(I - R\). First, we derive sixth order quadrature formulas for functions whose 6th derivative is either always positive or always negative and then we use these quadrature formulas to get good bounds on \(I - R\), and thus good approximations for the sum \(s\) of the infinite series. Several examples (including the Riemann zeta-function) show the efficiency of this new method. This paper continues the results from [W. Solak, Z. Szydełko, Quadrature rules with Gregory-Laplace end corrections, Journal of Computational and Applied Mathematics 36 (1991), 251–253] and [W. Solak, A remark on power series estimation via boundary corrections with parameter, Opuscula Mathematica 19 (1999), 75–80]. numerical integration, quadrature formulas, summation of series 2009 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol29iss2art2 https://www.opuscula.agh.edu.pl/vol29/2/art/opuscula_math_2911.pdf On an evolution inclusion in non-separable Banach spaces Aurelian Cernea We consider a Cauchy problem for a class of nonconvex evolution inclusions in non-separable Banach spaces under Filippov-type assumptions. We prove the existence of solutions. Lusin measurable multifunctions, selection, mild solution 2009 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol29iss2art3 https://www.opuscula.agh.edu.pl/vol29/2/art/opuscula_math_2912.pdf A note on Radon-Nikodým derivatives and similarity for completely bounded maps Aurelian Gheondea, Ali Şamil Kavruk We point out a relation between the Arveson's Radon-Nikodým derivative and known similarity results for completely bounded maps. We also consider Jordan type decompositions coming out from Wittstock's Decomposition Theorem and illustrate, by an example, the nonuniqueness of these decompositions. Radon-Nikodým derivative, \(C^*\)-algebra, completely positive map, similarity 2009 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol29iss2art4 https://www.opuscula.agh.edu.pl/vol29/2/art/opuscula_math_2913.pdf Continuous solutions of iterative equations of infinite order Rafał Kapica, Janusz Morawiec Given a probability space \((\Omega,\mathcal{A}, P)\) and a complete separable metric space \(X\), we consider continuous and bounded solutions \(\varphi: X \to \mathbb{R}\) of the equations \(\varphi(x) = \int_{\Omega} \varphi(f(x,\omega))P(d\omega)\) and \(\varphi(x) = 1-\int_{\Omega} \varphi(f(x,\omega))P(d\omega)\), assuming that the given function \(f:X \times \Omega \to X\) is controlled by a random variable \(L: \Omega \to (0,\infty)\) with \(-\infty \lt \int_{\Omega} \log L(\omega)P(d\omega) \lt 0\). An application to a refinement type equation is also presented. random-valued vector functions, sequences of iterates, iterative equations, continuous solutions 2009 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol29iss2art5 https://www.opuscula.agh.edu.pl/vol29/2/art/opuscula_math_2914.pdf A note on the p-domination number of trees You Lu, Xinmin Hou, Jun-Ming Xu Let \(p\) be a positive integer and \(G =(V(G),E(G))\) a graph. A \(p\)-dominating set of \(G\) is a subset \(S\) of \(V(G)\) such that every vertex not in \(S\) is dominated by at least \(p\) vertices in \(S\). The \(p\)-domination number \(\gamma_p(G)\) is the minimum cardinality among the \(p\)-dominating sets of \(G\). Let \(T\) be a tree with order \(n \geq 2\) and \(p \geq 2\) a positive integer. A vertex of \(V(T)\) is a \(p\)-leaf if it has degree at most \(p-1\), while a \(p\)-support vertex is a vertex of degree at least \(p\) adjacent to a \(p\)-leaf. In this note, we show that \(\gamma_p(T) \geq (n + |L_p(T)|-|S_p(T)|)/2\), where \(L_p(T)\) and \(S_p(T)\) are the sets of \(p\)-leaves and \(p\)-support vertices of \(T\), respectively. Moreover, we characterize all trees attaining this lower bound. \(p\)-domination number, trees 2009 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol29iss2art6 https://www.opuscula.agh.edu.pl/vol29/2/art/opuscula_math_2915.pdf On the diameter of dot-critical graphs Doost Ali Mojdeh, Somayeh Mirzamani A graph G is \(k\)-dot-critical (totaly \(k\)-dot-critical) if \(G\) is dot-critical (totaly dot-critical) and the domination number is \(k\). In the paper [T. Burtona, D. P. Sumner, Domination dot-critical graphs, Discrete Math, 306 (2006), 11-18] the following question is posed: What are the best bounds for the diameter of a \(k\)-dot-critical graph and a totally \(k\)-dot-critical graph \(G\) with no critical vertices for \(k \geq 4\)? We find the best bound for the diameter of a \(k\)-dot-critical graph, where \(k \in\{4,5,6\}\) and we give a family of \(k\)-dot-critical graphs (with no critical vertices) with sharp diameter \(2k-3\) for even \(k \geq 4\). dot-critical graph, diameter, 2009 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol29iss2art7 https://www.opuscula.agh.edu.pl/vol29/2/art/opuscula_math_2916.pdf Vague BCK/BCI-algebras Arsham Borumand Saeid In this note, by using the concept of vague sets, the notion of vague \(BCK/BCI\)-algebra is introduced. And the notions of \(\alpha\)-cut and vague-cut are introduced and the relationships between these notions and crisp subalgebras are studied. vague sets, vague \(BCK/BCI\)-algebra, vague-cut 2009 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol29iss2art8 https://www.opuscula.agh.edu.pl/vol29/2/art/opuscula_math_2917.pdf Weyl-Titchmarsh type formula for Hermite operator with small perturbation Sergey Simonov Small perturbations of the Jacobi matrix with weights \(\sqrt{n}\) and zero diagonal are considered. A formula relating the asymptotics of polynomials of the first kind to the spectral density is obtained, which is an analogue of the classical Weyl-Titchmarsh formula for the Schrödinger operator on the half-line with summable potential. Additionally, a base of generalized eigenvectors for "free" Hermite operator is studied and asymptotics of Plancherel-Rotach type are obtained. Jacobi matrices, absolutely continuous spectrum, subordinacy theory, Weyl-Titchmarsh theory 2009 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol29iss2art9 https://www.opuscula.agh.edu.pl/vol29/2/art/opuscula_math_2918.pdf On the property of the BBGKY hierarchy solution in cumulant representation Mykhailo O. Stashenko, Halyna M. Hubal We consider a one-dimensional nonsymmetric system of particles interacting via the hard-core potential. For this system, we prove that the BBGKY hierarchy solution in a cumulant representation is an equilibrium in the case of equilibrium initial data. BBGKY hierarchy of equations, nonsymmetric system, cumulant (semi-invariant), equilibrium distribution functions 2009 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol29iss3 https://www.opuscula.agh.edu.pl/om-vol29iss3art1 https://www.opuscula.agh.edu.pl/vol29/3/art/opuscula_math_2919.pdf On the global offensive alliance number of a tree Mohamed Bouzefrane, Mustapha Chellali For a graph \(G=(V,E)\), a set \(S \subseteq V\) is a dominating set if every vertex in \(V-S\) has at least a neighbor in \(S\). A dominating set \(S\) is a global offensive alliance if for every vertex \(v\) in \(V-S\), at least half of the vertices in its closed neighborhood are in \(S\). The domination number \(\gamma(G)\) is the minimum cardinality of a dominating set of \(G\) and the global offensive alliance number \(\gamma_o(G)\) is the minimum cardinality of a global offensive alliance of \(G\). We first show that every tree of order at least three with \(l\) leaves and \(s\) support vertices satisfies \(\gamma_o(T) \geq (n-l+s+1)/3\) and we characterize extremal trees attaining this lower bound. Then we give a constructive characterization of trees with equal domination and global offensive alliance numbers. global offensive alliance number, domination number, trees 2009 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol29iss3art2 https://www.opuscula.agh.edu.pl/vol29/3/art/opuscula_math_2920.pdf Smoothed estimator of the periodic hazard function Anna Dudek A smoothed estimator of the periodic hazard function is considered and its asymptotic probability distribution and bootstrap simultaneous confidence intervals are derived. Moreover, consistency of the bootstrap method is proved and some applications of the developed theory are presented. The bootstrap method is based on the phase-consistent resampling scheme developed in Dudek and Leśkow [A. Dudek, J. Leśkow, Bootstrap algorithm in periodic multiplicative intensity model, to appear]. bootstrap, consistency, multiplicative intensity model, periodic hazard function 2009 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol29iss3art3 https://www.opuscula.agh.edu.pl/vol29/3/art/opuscula_math_2921.pdf Sensitivity analysis in piecewise linear fractional programming problem with non-degenerate optimal solution Behrouz Kheirfam In this paper, we study how changes in the coefficients of objective function and the right-hand-side vector of constraints of the piecewise linear fractional programming problems affect the non-degenerate optimal solution. We consider separate cases when changes occur in different parts of the problem and derive bounds for each perturbation, while the optimal solution is invariant. We explain that this analysis is a generalization of the sensitivity analysis for \(LP\), \(LFP\) and \(PLP\). Finally, the results are described by some numerical examples. piecewise linear fractional programming, fractional programming, piecewise linear programming, sensitivity analysis 2009 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol29iss3art4 https://www.opuscula.agh.edu.pl/vol29/3/art/opuscula_math_2922.pdf Approximation methods for a class of discrete Wiener-Hopf equations Michał A. Nowak In this paper, we consider approximation methods for operator equations of the form \[Au + Bu = f,\] where \(A\) is a discrete Wiener-Hopf operator on \(l_p\) (\(1 \leq p \lt \infty\)) which symbol has roots on the unit circle with arbitrary multiplicities (not necessary integers). Conditions on perturbation \(B\) and \(f\) are given in order to guarantee the applicability of projection-iterative methods. Effective error estimates, and simultaneously, decaying properties for solutions are obtained in terms of some smooth spaces. projection methods, iterative methods, discrete Wiener-Hopf equations, Toeplitz operators 2009 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol29iss3art5 https://www.opuscula.agh.edu.pl/vol29/3/art/opuscula_math_2923.pdf Monotone iterative technique for fractional differential equations with periodic boundary conditions J. D. Ramírez, A. S. Vatsala In this paper we develop Monotone Method using upper and lower solutions for fractional differential equations with periodic boundary conditions. Initially we develop a comparison result and prove that the solution of the linear fractional differential equation with periodic boundary condition exists and is unique. Using this we develop iterates which converge uniformly monotonically to minimal and maximal solutions of the nonlinear fractional differential equations with periodic boundary conditions in the weighted norm. Riemann-Liouville fractional derivative, monotone method, periodic boundary value problem 2009 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol29iss3art6 https://www.opuscula.agh.edu.pl/vol29/3/art/opuscula_math_2924.pdf A note on the maximum likelihood estimator in the gamma regression model Jerzy P. Rydlewski This paper considers a nonlinear regression model, in which the dependent variable has the gamma distribution. A model is considered in which the shape parameter of the random variable is the sum of continuous and algebraically independent functions. The paper proves that there is exactly one maximum likelihood estimator for the gamma regression model. gamma regression, nonlinear regression, maximum likelihood estimator, shape parameter 2009 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol29iss3art7 https://www.opuscula.agh.edu.pl/vol29/3/art/opuscula_math_2925.pdf A double index transform with a product of Macdonald's functions revisited Semyon B. Yakubovich We prove an inversion theorem for a double index transform, which is associated with the product of Macdonald's functions \(K_{i \tau}(\sqrt{x^2+y^2}-y) K_{i \tau}(\sqrt{x^2+y^2}+y)\), where \((x, y) \in \mathbb{R}_+ \times \mathbb{R}_+\) and \(i \tau, \tau \in \mathbb{R}_+\) is a pure imaginary index. The results obtained in the sequel are applied to find particular solutions of integral equations involving the square and the cube of the Macdonald function \(K_{i \tau}(t)\) as a kernel. Macdonald function, index transform, Kontorovich-Lebedev transform, double Mellin transform, Plancherel theorem, Parseval equality 2009 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol29iss4 https://www.opuscula.agh.edu.pl/om-vol29iss4art1 https://www.opuscula.agh.edu.pl/vol29/4/art/opuscula_math_2926.pdf Edge condition for hamiltonicity in balanced tripartite graphs Janusz Adamus A well-known theorem of Entringer and Schmeichel asserts that a balanced bipartite graph of order \(2n\) obtained from the complete balanced bipartite \(K_{n,n}\) by removing at most \(n-2\) edges, is bipancyclic. We prove an analogous result for balanced tripartite graphs: If \(G\) is a balanced tripartite graph of order \(3n\) and size at least \(3n^2-2n+2\), then \(G\) contains cycles of all lengths. Hamilton cycle, pancyclicity, tripartite graph, edge condition 2009 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol29iss4art2 https://www.opuscula.agh.edu.pl/vol29/4/art/opuscula_math_2927.pdf Cyclability in bipartite graphs Denise Amar, Evelyne Flandrin, Grzegorz Gancarzewicz Let \(G=(X,Y,E)\) be a balanced \(2\)-connected bipartite graph and \(S \subset V(G)\). We will say that \(S\) is cyclable in \(G\) if all vertices of \(S\) belong to a common cycle in \(G\). We give sufficient degree conditions in a balanced bipartite graph \(G\) and a subset \(S \subset V(G)\) for the cyclability of the set \(S\). graphs, cycles, bipartite graphs 2009 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol29iss4art3 https://www.opuscula.agh.edu.pl/vol29/4/art/opuscula_math_2928.pdf Matrices defined by frames Zbigniew Ambroziński, Krzysztof Rudol Matrix representations of bounded Hilbert space operators are considered. The matrices in question are defined with respect to frames, rather than bases. The frames, a generalisation of bases, used extensively in applied harmonic analysis, are overcomplete sequences. We consider some properties related to tight frames, where, up to some multiplicative constant, a form of Parseval Identity takes place. We also describe parts of spectra of operators in terms of their matrices. frames, operators, spectrum 2009 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol29iss4art4 https://www.opuscula.agh.edu.pl/vol29/4/art/opuscula_math_2929.pdf On elliptic problems with a nonlinearity depending on the gradient Jan Chabrowski We investigate the solvability of the Neumann problem \((1.1)\) involving the nonlinearity depending on the gradient. We prove the existence of a solution when the right hand side \(f\) of the equation belongs to \(L^m(\Omega )\) with \(1 \leq m \lt 2\). Neumann problem, nonlinearity depending on the gradient, \(L^1\) data 2009 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol29iss4art5 https://www.opuscula.agh.edu.pl/vol29/4/art/opuscula_math_2930.pdf α_{2}-labeling of graphs Dalibor Fronček We show that if a graph \(G\) on \(n\) edges allows certain special type of rosy labeling (a.k.a. \(\rho\)-labeling), called \(\alpha_2\)-labeling, then for any positive integer \(k\) the complete graph \(K_{2nk+1}\) can be decomposed into copies of \(G\). This notion generalizes the \(\alpha\)-labeling introduced in 1967 by A. Rosa. graph decomposition, graph labeling 2009 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol29iss4art6 https://www.opuscula.agh.edu.pl/vol29/4/art/opuscula_math_2931.pdf Multipoint normal differential operators of first order Zameddin I. Ismailov In this paper we discuss all normal extensions of a minimal operator generated by a linear multipoint differential-operator expression of first order in the Hilbert space of vector-functions on the finite interval in terms of boundary and interior point values. Later on, we investigate the structure of the spectrum, its discreteness and the asymptotic behavior of the eigenvalues at infinity for these extensions. differential operator, formally normal and normal operator, multipoint minimal and maximal operators, extension, selfadjoint, accretive and unitary operators, class of compact operators, spectrum of an operators and its discreteness, asymptotics of eigenvalues, direct sum of spaces and operators 2009 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol29iss4art7 https://www.opuscula.agh.edu.pl/vol29/4/art/opuscula_math_2932.pdf On extension of solutions of a simultaneous system of iterative functional equations Janusz Matkowski Some sufficient conditions which allow to extend every local solution of a simultaneous system of equations in a single variable of the form \[ \varphi(x) = h (x, \varphi[f_1(x)],\ldots,\varphi[f_m(x)]),\] \[\varphi(x) = H (x, \varphi[F_1(x)],\ldots,\varphi[F_m(x)]),\] to a global one are presented. Extensions of solutions of functional equations, both in single and in several variables, play important role (cf. for instance [M. Kuczma, Functional equations in a single variable, Monografie Mat. 46, Polish Scientific Publishers, Warsaw, 1968, M. Kuczma, B. Choczewski, R. Ger, Iterative functional equations, Encyclopedia of Mathematics and Its Applications v. 32, Cambridge, 1990, J. Matkowski, Iteration groups, commuting functions and simultaneous systems of linear functional equations, Opuscula Math. 28 (2008) 4, 531-541]). functional equation, simultaneous system of equations, local solution, extension theorem 2009 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol29iss4art8 https://www.opuscula.agh.edu.pl/vol29/4/art/opuscula_math_2933.pdf Extensions of solutions of a functional equation in two variables Janusz Matkowski An extension theorem for the functional equation of several variables \[f(M(x,y))=N(f(x),f(y)),\] where the given functions \(M\) and \(N\) are left-side autodistributive, is presented. functional equation, autodistributivity, strict mean, extension theorem 2009 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol29iss4art9 https://www.opuscula.agh.edu.pl/vol29/4/art/opuscula_math_2934.pdf The upper edge geodetic number and the forcing edge geodetic number of a graph A. P. Santhakumaran, J. John An edge geodetic set of a connected graph \(G\) of order \(p \geq 2\) is a set \(S \subseteq V(G)\) such that every edge of \(G\) is contained in a geodesic joining some pair of vertices in \(S\). The edge geodetic number \(g_1(G)\) of \(G\) is the minimum cardinality of its edge geodetic sets and any edge geodetic set of cardinality \(g_1(G)\) is a minimum edge geodetic set of \(G\) or an edge geodetic basis of \(G\). An edge geodetic set \(S\) in a connected graph \(G\) is a minimal edge geodetic set if no proper subset of \(S\) is an edge geodetic set of \(G\). The upper edge geodetic number \(g_1^+(G)\) of \(G\) is the maximum cardinality of a minimal edge geodetic set of \(G\). The upper edge geodetic number of certain classes of graphs are determined. It is shown that for every two integers \(a\) and \(b\) such that \(2 \leq a \leq b\), there exists a connected graph \(G\) with \(g_1(G)=a\) and \(g_1^+(G)=b\). For an edge geodetic basis \(S\) of \(G\), a subset \(T \subseteq S\) is called a forcing subset for \(S\) if \(S\) is the unique edge geodetic basis containing \(T\). A forcing subset for \(S\) of minimum cardinality is a minimum forcing subset of \(S\). The forcing edge geodetic number of \(S\), denoted by \(f_1(S)\), is the cardinality of a minimum forcing subset of \(S\). The forcing edge geodetic number of \(G\), denoted by \(f_1(G)\), is \(f_1(G) = min\{f_1(S)\}\), where the minimum is taken over all edge geodetic bases \(S\) in \(G\). Some general properties satisfied by this concept are studied. The forcing edge geodetic number of certain classes of graphs are determined. It is shown that for every pair \(a\), \(b\) of integers with \(0 \leq a \lt b\) and \(b \geq 2\), there exists a connected graph \(G\) such that \(f_1(G)=a\) and \(g_1(G)=b\). geodetic number, edge geodetic basis, edge geodetic number, upper edge geodetic number, forcing edge geodetic number 2009 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol29iss4art10 https://www.opuscula.agh.edu.pl/vol29/4/art/opuscula_math_2935.pdf Topological classification of conformal actions on p-hyperelliptic and (q,n)-gonal Riemann surfaces Ewa Tyszkowska A compact Riemann surface \(X\) of genus \(g \gt 1\) is said to be \(p\)-hyperelliptic if \(X\) admits a conformal involution \(\rho\) for which \(X / \rho\) has genus \(p\). A conformal automorphism \(\delta\) of prime order \(n\) such that \(X / \delta\) has genus \(q\) is called a \((q,n)\)-gonal automorphism. Here we study conformal actions on \(p\)-hyperelliptic Riemann surface with \((q,n)\)-gonal automorphism. \(p\)-hyperelliptic Riemann surface, automorphism of a Riemann surface 2009 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol30iss1 https://www.opuscula.agh.edu.pl/om-vol30iss1art1 https://www.opuscula.agh.edu.pl/vol30/1/art/opuscula_math_3001.pdf A study of chaos for processes under small perturbations II: rigorous proof of chaos Piotr Oprocha, Paweł Wilczyński In the present paper we prove distributional chaos for the Poincaré map in the perturbed equation \[\dot{z}=\left(1 + e^{i\kappa t} |z|^2\right)\bar{z}^2 - N e^{-i\frac{\pi}{3}}.\] Heteroclinic and homoclinic connections between two periodic solutions bifurcating from the stationary solution \(0\) present in the system when \(N = 0\) are also discussed. distributional chaos, isolating segments, fixed point index, bifurcation 2010 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol30iss1art2 https://www.opuscula.agh.edu.pl/vol30/1/art/opuscula_math_3002.pdf Dominating sets and domination polynomials of certain graphs, II Saeid Alikhani, Yee-hock Peng The domination polynomial of a graph \(G\) of order \(n\) is the polynomial \(D(G,x) = \sum _{i=\gamma(G)}^n d(G,i)x^i\), where \(d(G,i)\) is the number of dominating sets of \(G\) of size \(i\), and \(\gamma (G)\) is the domination number of \(G\). In this paper, we obtain some properties of the coefficients of \(D(G,x)\). Also, by study of the dominating sets and the domination polynomials of specific graphs denoted by \(G^{\prime}(m)\), we obtain a relationship between the domination polynomial of graphs containing an induced path of length at least three, and the domination polynomial of related graphs obtained by replacing the path by shorter path. As examples of graphs \(G^{\prime}(m)\), we study the dominating sets and domination polynomials of cycles and generalized theta graphs. Finally, we show that, if \(n \equiv 0,2(mod\, 3)\) and \(D(G,x) = D(C_n, x)\), then \(G = C_n\). domination polynomial, dominating set, cycle, theta graph 2010 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol30iss1art3 https://www.opuscula.agh.edu.pl/vol30/1/art/opuscula_math_3003.pdf Uniformly continuous set-valued composition operators in the spaces of functions of bounded variation in the sense of Wiener A. Azócar, J. A. Guerrero, J. Matkowski, N. Merentes We show that the one-sided regularizations of the generator of any uniformly continuous and convex compact valued composition operator, acting in the spaces of functions of bounded variation in the sense of Wiener, is an affine function. \(\varphi\)-variation in the sense of Wiener, set-valued functions, left and right regularizations, uniformly continuous composition (Nemytskii) operator, Jensen equation 2010 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol30iss1art4 https://www.opuscula.agh.edu.pl/vol30/1/art/opuscula_math_3004.pdf 2-splittable and cordial graphs Sylwia Cichacz E. Miller and G. E. Stevens proved in [E. Miller, G. E. Stevens, Some graphs for which even size is sufficient for splittability, Congressus Numerantium 173 (2005), 137–147] the existence of certain families of \(2\)-splittable caterpillars. In this paper we characterize other families of \(2\)-splittable caterpillars. Moreover, we show that for some of them there exists a friendly labeling inducing two isomorphic subgraphs. cordial graphs, \(2\)-splittable graphs 2010 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol30iss1art5 https://www.opuscula.agh.edu.pl/vol30/1/art/opuscula_math_3005.pdf On some properties of the superposition operator on topological manifolds Janusz Dronka In this paper the superposition operator in the space of vector-valued, bounded and continuous functions on a topological manifold is considered. The acting conditions and criteria of continuity and compactness are established. As an application, an existence result for the nonlinear Hammerstein integral equation is obtained. superposition operator, continuous function, topological manifold, Hammerstein integral equation 2010 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol30iss1art6 https://www.opuscula.agh.edu.pl/vol30/1/art/opuscula_math_3006.pdf Hyponormal differential operators with discrete spectrum Zameddin I. Ismailov, Erdal Unluyol In this work, we first describe all the maximal hyponormal extensions of a minimal operator generated by a linear differential-operator expression of the first-order in the Hilbert space of vector-functions in a finite interval. Next, we investigate the discreteness of the spectrum and the asymptotical behavior of the modules of the eigenvalues for these maximal hyponormal extensions. hyponormal operators, differential operators, minimal and maximal operators, extension of operators, compact operators, eigenvalues, asymptotes of eigenvalues 2010 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol30iss1art7 https://www.opuscula.agh.edu.pl/vol30/1/art/opuscula_math_3007.pdf Differential difference inequalities related to parabolic functional differential equations Milena Netka Initial boundary value problems for nonlinear parabolic functional differential equations are transformed by discretization in space variables into systems of ordinary functional differential equations. A comparison theorem for differential difference inequalities is proved. Sufficient conditions for the convergence of the method of lines is given. Nonlinear estimates of the Perron type for given operators with respect to functional variables are used. Results obtained in the paper can be applied to differential integral problems and to equations with deviated variables. parabolic functional differential equations, method of lines, stability and convergence 2010 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol30iss2 https://www.opuscula.agh.edu.pl/om-vol30iss2art1 https://www.opuscula.agh.edu.pl/vol30/2/art/opuscula_math_3008.pdf On chromatic equivalence of a pair of K_{4}-homeomorphs S. Catada-Ghimire, H. Roslan, Y. H. Peng Let \(P(G, \lambda)\) be the chromatic polynomial of a graph \(G\). Two graphs \(G\) and \(H\) are said to be chromatically equivalent, denoted \(G \sim H\), if \(P(G, \lambda) = P(H, \lambda)\). We write \([G] = \{H| H \sim G\}\). If \([G] = \{G\}\), then \(G\) is said to be chromatically unique. In this paper, we discuss a chromatically equivalent pair of graphs in one family of \(K_4\)-homeomorphs, \(K_4(1, 2, 8, d, e, f)\). The obtained result can be extended in the study of chromatic equivalence classes of \(K_4(1, 2, 8, d, e, f)\) and chromatic uniqueness of \(K_4\)-homeomorphs with girth \(11\). chromatic polynomial, chromatic equivalence, \(K_4\)-homeomorphs 2010 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol30iss2art2 https://www.opuscula.agh.edu.pl/vol30/2/art/opuscula_math_3009.pdf Pseudospectral method for semilinear partial functional differential equations Wojciech Czernous We present a convergence result for two spectral methods applied to an initial boundary value problem with functional dependence of Volterra type. Explicit condition of Courant-Friedrichs-Levy type is assumed on time step \(\tau \) and the number \(N\) of collocation points. Stability statements and error estimates are written using continuous norms in weighted Jacobi spaces. pseudospectral collocation, CFS condition, convergence, error estimates 2010 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol30iss2art3 https://www.opuscula.agh.edu.pl/vol30/2/art/opuscula_math_3010.pdf On some families of arbitrarily vertex decomposable spiders Tomasz Juszczyk, Irmina A. Zioło A graph \(G\) of order \(n\) is called arbitrarily vertex decomposable if for each sequence \((n_1, ..., n_k)\) of positive integers such that \(\sum _{i=1}^{k} n_i = n\), there exists a partition \((V_1, ..., V_k)\) of the vertex set of \(G\) such that for every \(i \in \{1, ...., k\}\) the set \(V_i\) induces a connected subgraph of \(G\) on \(n_i\) vertices. A spider is a tree with one vertex of degree at least \(3\). We characterize two families of arbitrarily vertex decomposable spiders which are homeomorphic to stars with at most four hanging edges. arbitrarily vertex decomposable graph, trees 2010 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol30iss2art4 https://www.opuscula.agh.edu.pl/vol30/2/art/opuscula_math_3011.pdf Fréchet differential of a power series in Banach algebras Benedetto Silvestri We present two new forms in which the Fréchet differential of a power series in a unitary Banach algebra can be expressed in terms of absolutely convergent series involving the commutant \(C(T) : A \mapsto [A,T]\). Then we apply the results to study series of vector-valued functions on domains in Banach spaces and to the analytic functional calculus in a complex Banach space. Fréchet differentiation in Banach algebras, functional calculus 2010 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol30iss2art5 https://www.opuscula.agh.edu.pl/vol30/2/art/opuscula_math_3012.pdf Domination hypergraphs of certain digraphs Martin Sonntag, Hanns-Martin Teichert If \(D = (V,A)\) is a digraph, its domination hypergraph \(\mathcal{DH}(D) = (V,\mathcal{E})\) has the vertex set \(V\) and \(e \subseteq V\) is an edge of \(\mathcal{DH}(D)\) if and only if \(e\) is a minimal dominating set of \(D\). We investigate domination hypergraphs of special classes of digraphs, namely tournaments, paths and cycles. Finally, using a special decomposition/composition method we construct edge sets of domination hypergraphs of certain digraphs. hypergraph, dominating set, directed graph 2010 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol30iss2art6 https://www.opuscula.agh.edu.pl/vol30/2/art/opuscula_math_3013.pdf A note on the discrete Schrödinger operator with a perturbed periodic potential Beata Strack The aim of this paper is to study the spectrum of the one-dimensional discrete Schrödinger operator with a perturbed periodic potential. We obtain natural conditions under which this perturbation preserves the essential spectrum of the considered operator. Conditions on the number of isolated eigenvalues are given. one-dimensional Schrödinger operator, Jacobi operator, perturbation of periodic potential, essential spectrum, discrete part of the spectrum 2010 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol30iss2art7 https://www.opuscula.agh.edu.pl/vol30/2/art/opuscula_math_3014.pdf A note on minimax rates of convergence in the Spektor-Lord-Willis problem Zbigniew Szkutnik In this note, attainable lower bounds are constructed for the convergence rates in a stereological problem of unfolding spheres size distribution from linear sections, which shows that a spectral type estimator is strictly rate minimax over some Sobolev-type classes of functions. Poisson inverse problem, rate minimaxity, singular value decomposition, stereology 2010 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol30iss2art8 https://www.opuscula.agh.edu.pl/vol30/2/art/opuscula_math_3015.pdf On a property of ϕ-variational modular spaces Jincai Wang, Chunyan Wu Maligranda pointed out whether condition (B.1) is satisfied in the variational modular space \(X_{\rho}^{*}\) is an open problem. We will answer this open problem in \(X_{\rho}^{*\prime}\), a subspace of \(X_{\rho}^{*}\). As a consequence this modular space can \(X_{\rho}^{*\prime}\) be \(F\)-normed. condition (B.1), modular, \(\phi\) -function, \(\phi\) -variation 2010 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol30iss2art9 https://www.opuscula.agh.edu.pl/vol30/2/art/opuscula_math_3016.pdf Fractional nonlocal integrodifferential equations of mixed type with time-varying generating operators and optimal control JinRong Wang, W. Wei, YanLong Yang In this paper, a class of fractional integrodifferential equations of mixed type with time-varying generating operators and nonlocal conditions is considered. Using a contraction mapping principle and Krasnoselskii's fixed point theorem via Gronwall's inequality, the existence and uniqueness of mild solution are given. The existence of optimal pairs of systems governed by fractional integrodifferential equations of mixed type with time-varying generating operators and nonlocal conditions is also presented. fractional integrodifferential equations of mixed type, time-varying generating operators, nonlocal conditions, fixed point theorem, existence, optimal control 2010 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol30iss3 https://www.opuscula.agh.edu.pl/om-vol30iss3art1 https://www.opuscula.agh.edu.pl/vol30/3/art/opuscula_math_3017.pdf Uniformly continuous set-valued composition operators in the space of total φ-bidimensional variation in the sense of Riesz Wadie Aziz, José Giménez, Nelson Merentes, José Luis Sánchez In this paper we prove that if a Nemytskij composition operator, generated by a function of three variables in which the third variable is a function one, maps a suitable large subset of the space of functions of bounded total \(\varphi\)-bidimensional variation in the sense of Riesz, into another such space, and is uniformly continuous, then its generator is an affine function in the function variable. This extends some previous results in the one-dimensional setting. \(\varphi\)-bidimensional variation, uniformly continuous, Nemytskij operator 2010 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol30iss3art2 https://www.opuscula.agh.edu.pl/vol30/3/art/opuscula_math_3018.pdf Trees with equal global offensive k-alliance and k-domination numbers Mustapha Chellali Let \(k \geq 1\) be an integer. A set \(S\) of vertices of a graph \(G = (V(G),E(G))\) is called a global offensive \(k\)-alliance if \(|N(v) \cap S| \geq |N(v) - S| + k\) for every \(v \in V(G)- S\), where \(N(v)\) is the neighborhood of \(v\). The subset \(S\) is a \(k\)-dominating set of \(G\) if every vertex in \(V(G) - S\) has at least \(k\) neighbors in \(S\). The global offensive \(k\)-alliance number \(\gamma_0^k (G)\) is the minimum cardinality of a global offensive \(k\)-alliance in \(G\) and the \(k\)-domination number \(\gamma _k (G)\) is the minimum cardinality of a \(k\)-dominating set of \(G\). For every integer \(k \geq 1\) every graph \(G\) satisfies \(\gamma_0^k (G) \geq \gamma_k (G)\). In this paper we provide for \(k \geq 2\) a characterization of trees \(T\) with equal \(\gamma_0^k (T)\) and \(\gamma_k (T)\). global offensive \(k\)-alliance number, \(k\)-domination number, trees 2010 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol30iss3art3 https://www.opuscula.agh.edu.pl/vol30/3/art/opuscula_math_3019.pdf On the approximation theorem of Wong-Zakai type for the Lasota operator Antoni Leon Dawidowicz, Krystyna Twardowska We consider in this paper a stochastic evolution equation with Professor A. Lasota's operator as the infinitesimal generator of a strongly continuous semigroup of transformations and with Hammerstein operator connected with a noise being the Wiener process. We show that such evolution equation satisfies the Wong-Zakai type approximation theorem. The idea of the definition of the Lasota operator has the origin in the mathematical model of the creation and differentiation of cells in biology and medicine. stochastic evolution equations, Wong-Zakai approximations, Lasota operator 2010 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol30iss3art4 https://www.opuscula.agh.edu.pl/vol30/3/art/opuscula_math_3020.pdf Graph choosability and double list colorability Hamid-Reza Fanaï In this paper, we give a sufficient condition for graph choosability, based on Combinatorial Nullstellensatz and a specific property, called "double list colorability", which means that there is a list assignment for which there are exactly two admissible colorings. list coloring, choosability 2010 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol30iss3art5 https://www.opuscula.agh.edu.pl/vol30/3/art/opuscula_math_3021.pdf Decomposition of complete graphs into small graphs Dalibor Froncek In 1967, A. Rosa proved that if a bipartite graph \(G\) with \(n\) edges has an \(\alpha\)-labeling, then for any positive integer \(p\) the complete graph \(K_{2np+1}\) can be cyclically decomposed into copies of \(G\). This has become a part of graph theory folklore since then. In this note we prove a generalization of this result. We show that every bipartite graph \(H\) which decomposes \(K_k\) and \(K_m\) also decomposes \(K_{km}\). graph decomposition, graph labeling 2010 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol30iss3art6 https://www.opuscula.agh.edu.pl/vol30/3/art/opuscula_math_3022.pdf Postoptimal analysis in the coefficients matrix of piecewise linear fractional programming problems with non-degenerate optimal solution Behrouz Kheirfam In this paper, we discuss how changes in the coefficients matrix of piecewise linear fractional programming problems affect the non-degenerate optimal solution. We consider separate cases when changes occur in the coefficients of the basic and non-basic variables and derive bounds for each perturbation, while the optimal solution is invariant. We explain that this analysis is a generalization of the sensitivity analysis for \(LP\), \(LFP\) and \(PLP\). Finally, the results are described by some numerical examples. piecewise linear fractional programming, degeneracy, optimal basis, fractional programming, piecewise linear programming, sensitivity analysis 2010 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol30iss3art7 https://www.opuscula.agh.edu.pl/vol30/3/art/opuscula_math_3023.pdf Geometric properties of quantum graphs and vertex scattering matrices Pavel Kurasov, Marlena Nowaczyk Differential operators on metric graphs are investigated. It is proven that vertex matching (boundary) conditions can be successfully parameterized by the vertex scattering matrix. Two new families of matching conditions are investigated: hyperplanar Neumann and hyperplanar Dirichlet conditions. Using trace formula it is shown that the spectrum of the Laplace operator determines certain geometric properties of the underlying graph. scattering theory, quantum graphs, matching (boundary) conditions 2010 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol30iss3art8 https://www.opuscula.agh.edu.pl/vol30/3/art/opuscula_math_3024.pdf Asymptotic behaviour and approximation of eigenvalues for unbounded block Jacobi matrices Maria Malejki The research included in the paper concerns a class of symmetric block Jacobi matrices. The problem of the approximation of eigenvalues for a class of a self-adjoint unbounded operators is considered. We estimate the joint error of approximation for the eigenvalues, numbered from \(1\) to \(N\), for a Jacobi matrix \(J\) by the eigenvalues of the finite submatrix \(J_n\) of order \(pn \times pn\), where \(N = \max \{k \in \mathbb{N}: k \leq rpn\}\) and \(r \in (0,1)\) is suitably chosen. We apply this result to obtain the asymptotics of the eigenvalues of \(J\) in the case \(p=3\). symmetric unbounded Jacobi matrix, block Jacobi matrix, tridiagonal matrix, point spectrum, eigenvalue, asymptotics 2010 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol30iss3art9 https://www.opuscula.agh.edu.pl/vol30/3/art/opuscula_math_3025.pdf Convergence of an implicit iteration process for a finite family of asymptotically quasi-nonexpansive mappings in convex metric spaces Gurucharan Singh Saluja, Hemant Kumar Nashine In this paper, we give some necessary and sufficient conditions for an implicit iteration process with errors for a finite family of asymptotically quasi-nonexpansive mappings converging to a common fixed of the mappings in convex metric spaces. Our results extend and improve some recent results of Sun, Wittmann, Xu and Ori, and Zhou and Chang. implicit iteration process, finite family of asymptotically quasi-nonexpansive mappings, common fixed point, convex metric space 2010 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol30iss3art10 https://www.opuscula.agh.edu.pl/vol30/3/art/opuscula_math_3026.pdf Strong convergence theorem of a hybrid projection algorithm for a family of quasi-ϕ-asymptotically nonexpansive mappings J. F. Tang, S. S. Chang, M. Liu, J. A. Liu The main purpose of this paper is by using a new hybrid projection iterative algorithm to prove some strong convergence theorems for a family of quasi-\(\phi\)-asymptotically nonexpansive mappings. The results presented in the paper improve and extend the corresponding results announced by some authors. quasi-\(\phi\)-asymptotically nonexpansive mapping, asymptotically regular mapping, hybrid projection iterative algorithm, strong convergence theorem 2010 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol30iss3art11 https://www.opuscula.agh.edu.pl/vol30/3/art/opuscula_math_3027.pdf Bifurcation in a nonlinear steady state system Gen-Qiang Wang, Sui Sun Cheng The steady state solutions of a nonlinear digital cellular neural network with \(\omega\) neural units and a nonnegative variable parameter \(\lambda\) are sought. We show that \(\lambda = 1\) is a critical value such that the qualitative behavior of our network changes. More specifically, when \(\omega\) is odd, then for \(\lambda \in [0,1)\), there is one positive and one negative steady state, and for \(\lambda \in [1,\infty)\), steady states cannot exist; while when \(\omega\) is even, then for \(\lambda \in [0,1)\), there is one positive and one negative steady state, and for \(\lambda = 1\), there are no nontrivial steady states, and for \(\lambda \in (1,\infty)\), there are two fully oscillatory steady states. Furthermore, the number of existing nontrivial solutions cannot be improved. It is hoped that our results are of interest to digital neural network designers. bifurcation, cellular neural network, steady state, Krasnoselsky fixed point theorem 2010 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol30iss3art12 https://www.opuscula.agh.edu.pl/vol30/3/art/opuscula_math_3028.pdf Nonlocal impulsive problems for fractional differential equations with time-varying generating operators in Banach spaces JinRong Wang, YanLong Yang, W. Wei In this paper, we study the existence and uniqueness of the \(PC\)-mild solution for a class of impulsive fractional differential equations with time-varying generating operators and nonlocal conditions. By means of the generalized Ascoli-Arzela Theorem given by us and the fixed point theorem, some existence and uniqueness results are obtained. Finally, an example is given to illustrate the theory. nonlocal conditions, impulsive equations, fractional differential equations, time-varying generating operators, \(PC\)-mild solution 2010 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol30iss4 https://www.opuscula.agh.edu.pl/om-vol30iss4art1 https://www.opuscula.agh.edu.pl/vol30/4/art/opuscula_math_3029.pdf Necessary optimality conditions for predator-prey system with a hunter population Narcisa C. Apreutesei An optimal control problem is studied for a predator-prey reaction-diffusion system. A hunter population is introduced in the ecosystem and it is interpreted as a control variable. One finds necessary optimality conditions in order that, in the end of a given time interval, the total density of the two populations is maximal. predator-prey system, optimal control, adjoint system 2010 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol30iss4art2 https://www.opuscula.agh.edu.pl/vol30/4/art/opuscula_math_3030.pdf On some dynamical reconstruction problems for a nonlinear system of the second-order Marina Blizorukova, Vyacheslav Maksimov The problem of reconstruction of unknown characteristics of a nonlinear system is considered. Solution algorithms stable with respect to the informational noise and computational errors are specified. These algorithms are based on the method of auxiliary positionally controlled models. ordinary differential equations, reconstruction 2010 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol30iss4art3 https://www.opuscula.agh.edu.pl/vol30/4/art/opuscula_math_3031.pdf Existence and attractivity results for nonlinear first order random differential equations Bapurao C. Dhage, Sotiris K. Ntouyas In this paper, the existence and attractivity results are proved for nonlinear first order ordinary random differential equations. Two examples are provided to demonstrate the realization of the abstract developed theory. random differential equations, locally attractive, globally attractive, local asymptotic attractive, fixed point theorem 2010 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol30iss4art4 https://www.opuscula.agh.edu.pl/vol30/4/art/opuscula_math_3032.pdf On the global attractivity and the periodic character of a recursive sequence E. M. Elsayed In this paper we investigate the global convergence result, boundedness, and periodicity of solutions of the recursive sequence \[x_{n+1} = ax_n + \frac{bx_{n-1}+cx_{n+2}}{dx_{n-1}+ex_{n+2}}, \quad n=0,1,\ldots,\] where the parameters \(a\), \(b\), \(c\), \(d\) and \(e\) are positive real numbers and the initial conditions \(x_{-2}\), \(x_{-1}\), and \(x_0\) are positive real numbers. stability, periodic solutions, boundedness, difference equations 2010 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol30iss4art5 https://www.opuscula.agh.edu.pl/vol30/4/art/opuscula_math_3033.pdf Right focal boundary value problems for difference equations Johnny Henderson, Xueyan Liu, Jeffrey W. Lyons, Jeffrey T. Neugebauer An application is made of a new Avery et al. fixed point theorem of compression and expansion functional type in the spirit of the original fixed point work of Leggett and Williams, to obtain positive solutions of the second order right focal discrete boundary value problem. In the application of the fixed point theorem, neither the entire lower nor entire upper boundary is required to be mapped inward or outward. A nontrivial example is also provided. difference equation, boundary value problem, right focal, fixed point theorem, positive solution 2010 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol30iss4art6 https://www.opuscula.agh.edu.pl/vol30/4/art/opuscula_math_3034.pdf Minimal and co-minimal projections in spaces of continuous functions Dominik Mielczarek Minimal and co-minimal projections in the space \(C[0,1]\) are studied. We construct a minimal and co-minimal projection from \(C[0,1]\) onto a subspace \(Y\) defined in the introduction. minimal projection, co-minimal projection 2010 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol30iss4art7 https://www.opuscula.agh.edu.pl/vol30/4/art/opuscula_math_3035.pdf An algorithm for finding a common solution for a system of mixed equilibrium problem, quasi-variational inclusion problem and fixed point problem of nonexpansive semigroup Liu Min, Shih-sen Chang, Ping Zuo In this paper, we introduce a hybrid iterative scheme for finding a common element of the set of solutions for a system of mixed equilibrium problems, the set of common fixed points for a nonexpansive semigroup and the set of solutions of the quasi-variational inclusion problem with multi-valued maximal monotone mappings and inverse-strongly monotone mappings in a Hilbert space. Under suitable conditions, some strong convergence theorems are proved. Our results extend some recent results in the literature. nonexpansive semigroup, mixed equilibrium problem, viscosity approximation method, quasi-variational inclusion problem, multi-valued maximal monotone mappings, \(\alpha\)-inverse-strongly monotone mapping 2010 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol30iss4art8 https://www.opuscula.agh.edu.pl/vol30/4/art/opuscula_math_3036.pdf Convergence theorems for strictly asymptotically pseudocontractive mappings in Hilbert spaces Gurucharan Singh Saluja In this paper, we establish the weak and strong convergence theorems for a \(k\)-strictly asymptotically pseudo-contractive mapping in the framework of Hilbert spaces. Our result improve and extend the corresponding result of Acedo and Xu, Liu, Marino and Xu, Osilike and Akuchu, and some others. strictly asymptotically pseudo-contractive mapping, implicit iteration scheme, common fixed point , strong convergence, weak convergence, Hilbert space 2010 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol30iss4art9 https://www.opuscula.agh.edu.pl/vol30/4/art/opuscula_math_3037.pdf On the relative equilibrium configurations in the planar five-body problem Agnieszka Siluszyk The number of central configurations in the Grebenicov-Elmabsout model of the planar five-body problem is estimated. An appropriate rational parameterization is used to reduce the equations defining such configurations to some polynomial ones. For the restricted five-body problem a sharp estimation is given by using the Sturm separation theorem. planar five-body problem, relative equilibrium, central configuration, Grebenicov-Elmabsout model 2010 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol30iss4art10 https://www.opuscula.agh.edu.pl/vol30/4/art/opuscula_math_3038.pdf On some impulsive fractional differential equations in Banach spaces JinRong Wang, W. Wei, YanLong Yang This paper deals with some impulsive fractional differential equations in Banach spaces. Utilizing the Leray-Schauder fixed point theorem and the impulsive nonlinear singular version of the Gronwall inequality, the existence of \(PC\)-mild solutions for some fractional differential equations with impulses are obtained under some easily checked conditions. At last, an example is given for demonstration. fractional differential equations with impulses, nonlinear impulsive singular version of the Gronwall inequality, \(PC\)-mild solutions, existence 2010 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol30iss4art11 https://www.opuscula.agh.edu.pl/vol30/4/art/opuscula_math_3039.pdf Research problems from the 18th workshop ‘3in1’ 2009 (edited by Mariusz Meszka) Zdeněk Ryjáček, Carol T. Zamfirescu A collection of open problems that were posed at the 18th Workshop ‘3in1’, held on November 26-28, 2009 in Krakow, Poland. The problems are presented by Zdenek Ryjacek in "Does the Thomassen's conjecture imply N=NP?" and "Dominating cycles and hamiltonian prisms", and by Carol T. Zamfirescu in "Two problems on bihomogeneously traceable digraphs". Hamilton-connected graph, hamiltonian graph, dominating cycle, bihomogeneously traceble graph 2010 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol31iss1 https://www.opuscula.agh.edu.pl/om-vol31iss1art1 https://www.opuscula.agh.edu.pl/vol31/1/art/opuscula_math_3101.pdf A Meir-Keeler type common fixed point theorem for four mappings Mohamed Akkouchi In this paper, we prove a general common fixed point theorem for two pairs of weakly compatible self-mappings of a metric space satisfying a weak Meir-Keeler type contractive condition by using a class of implicit relations. In particular, our result generalizes and improves a result of K. Jha, R.P. Pant, S.L. Singh, by removing the assumption of continuity, relaxing compatibility to weakly compatibility property and replacing the completeness of the space with a set of four alternative conditions for maps satisfying an implicit relation. Also, our result improves the main result of H. Bouhadjera, A. Djoudi. common fixed point for four mappings, weakly compatible mappings, Meir-Keeler type contractive condition, complete metric spaces 2011 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol31iss1art2 https://www.opuscula.agh.edu.pl/vol31/1/art/opuscula_math_3102.pdf On strongly midconvex functions Antonio Azócar, José Giménez, Kazimierz Nikodem, José Luis Sánchez In this paper we collect some properties of strongly midconvex functions. First, counterparts of the classical theorems of Bernstein-Doetsch, Ostrowski and Sierpinski are presented. A version of Rodé support theorem for strongly midconvex functions and a Kuhn-type result on the relation between strongly midconvex functions and strongly \(t\)-convex functions are obtained. Finally, a connection between strong midconvexity and generalized convexity in the sense of Beckenbach is established. strongly convex functions, strongly midconvex functions, Bernstein-Doetsch-type theorem, Kuhn theorem, Rodé support theorem, Beckenbach convexity 2011 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol31iss1art3 https://www.opuscula.agh.edu.pl/vol31/1/art/opuscula_math_3103.pdf Multi-valued codensing random operators and functional random integral inclusions Bapurao C. Dhage In this paper, some random fixed point theorems for continuous and condensing multi-valued random operators are proved and they are further applied to the random integral inclusions for proving the existence of the solutions via the priori bound method. monotone multi-valued random operator, random differential inclusion, existence 2011 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol31iss1art4 https://www.opuscula.agh.edu.pl/vol31/1/art/opuscula_math_3104.pdf On operators of transition in Krein spaces A. Grod, S. Kuzhel, V. Sudilovskaya The paper is devoted to investigation of operators of transition and the corresponding decompositions of Krein spaces. The obtained results are applied to the study of relationship between solutions of operator Riccati equations and properties of the associated operator matrix \(L\). In this way, we complete the known result (see Theorem 5.2 in the paper of S. Albeverio, A. Motovilov, A. Skhalikov, Integral Equ. Oper. Theory 64 (2004), 455-486) and show the equivalence between the existence of a strong solution \(K\) (\(\|K\|\lt 1\)) of the Riccati equation and similarity of the \(J\)-self-adjoint operator \(L\) to a self-adjoint one. Krein spaces, indefinite metrics, operator of transition, operator Riccati equation 2011 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol31iss1art5 https://www.opuscula.agh.edu.pl/vol31/1/art/opuscula_math_3105.pdf Existence and uniqueness of anti-periodic solutions for a class of nonlinear n-th order functional differential equations Ling Liu, Yongkun Li In this paper, we use the method of coincide degree theory to establish new results on the existence and uniqueness of anti-periodic solutions for a class of nonlinear \(n\)-th order functional differential equations of the form \[x^{(n)}(t)=F(t, x_t, x^{(n-1)}_t, x(t), x^{(n-1)}(t), x(t-\tau(t)), x^{(n-1)}(t-\sigma(t))).\] anti-periodic solution, coincidence degree, nonlinear \(n\)-th-order equation, delay 2011 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol31iss1art6 https://www.opuscula.agh.edu.pl/vol31/1/art/opuscula_math_3106.pdf Subadditive periodic functions Janusz Matkowski Some conditions under which any subadditive function is periodic are presented. It is shown that the boundedness from below in a neighborhood of a point of a subadditive periodic (s.p.) function implies its nonnegativity, and the boundedness from above in a neighborhood of a point implies it nonnegativity and global boundedness from above. A necessary and sufficient condition for existence of a subadditive periodic extension of a function \(f_{0}:[0,1)\rightarrow \mathbb{R}\) is given. The continuity, differentiability of a s.p. function is discussed, and an example of a continuous nowhere differentiable s.p. function is presented. The functions which are the sums of linear functions and s.p. functions are characterized. The refinements of some known results on the continuity of subadditive functions are presented. subadditive function, periodic function, periodic extension, concave function, continuity, continuous nowhere differentiable function 2011 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol31iss1art7 https://www.opuscula.agh.edu.pl/vol31/1/art/opuscula_math_3107.pdf Probabilistic characterization of strong convexity Teresa Rajba, Szymon Wąsowicz Strong convexity is considered for real functions defined on a real interval. Probabilistic characterization is given and its geometrical sense is explained. Using this characterization some inequalities of Jensen-type are obtained. convexity, strong convexity, Jensen's inequality, Jensen gap of a function, distribution of a random variable, variance 2011 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol31iss1art8 https://www.opuscula.agh.edu.pl/vol31/1/art/opuscula_math_3108.pdf Existence and stabilizability of steady-state for semilinear pulse-width sampler controlled system JinRong Wang In this paper, we study the steady-state of a semilinear pulse-width sampler controlled system on infinite dimensional spaces. Firstly, by virtue of Schauder's fixed point theorem, the existence of periodic solutions is given. Secondly, utilizing a generalized Gronwall inequality given by us and the Banach fixed point theorem, the existence and stabilizability of a steady-state for the semilinear control system with pulse-width sampler is also obtained. At last, an example is given for demonstration. pulse-width sampler system, compact semigroup, steady-state, existence, stabilizability 2011 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol31iss1art9 https://www.opuscula.agh.edu.pl/vol31/1/art/opuscula_math_3109.pdf A class of nonlocal integrodifferential equations via fractional derivative and its mild solutions JinRong Wang, X. Yan, X.-H. Zhang, T.-M. Wang, X.-Z. Li In this paper, we discuss a class of integrodifferential equations with nonlocal conditions via a fractional derivative of the type: \[\begin{aligned}D_{t}^{q}x(t)=Ax(t)+\int\limits_{0}^{t}B(t-s)x(s)ds+t^{n}f\left(t,x(t)\right),&\;t\in [0,T],\;n\in Z^{+},\\&q\in(0,1],\;x(0)=g(x)+x_{0}.\end{aligned}\] Some sufficient conditions for the existence of mild solutions for the above system are given. The main tools are the resolvent operators and fixed point theorems due to Banach's fixed point theorem, Krasnoselskii's fixed point theorem and Schaefer's fixed point theorem. At last, an example is given for demonstration. integrodifferential equations, fractional derivative, nonlocal conditions, resolvent operator and their norm continuity, fixed point theorem, mild solutions 2011 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol31iss1art10 https://www.opuscula.agh.edu.pl/vol31/1/art/opuscula_math_3110.pdf A radial version of the Kontorovich-Lebedev transform in the unit ball Semyon B. Yakubovich, Nelson Vieira In this paper we introduce a radial version of the Kontorovich-Lebedev transform in the unit ball. Mapping properties and an inversion formula are proved in \(L_p\) Kontorovich-Lebedev transform, modified Bessel function, index transforms, Fourier integrals 2011 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol31iss2 https://www.opuscula.agh.edu.pl/om-vol31iss2art1 https://www.opuscula.agh.edu.pl/vol31/2/art/opuscula_math_3111.pdf A note on global alliances in trees Mohamed Bouzefrane, Mustapha Chellali For a graph \(G=(V,E)\), a set \(S\subseteq V\) is a dominating set if every vertex in \(V-S\) has at least a neighbor in \(S\). A dominating set \(S\) is a global offensive (respectively, defensive) alliance if for each vertex in \(V-S\) (respectively, in \(S\)) at least half the vertices from the closed neighborhood of \(v\) are in \(S\). The domination number \(\gamma(G)\) is the minimum cardinality of a dominating set of \(G\), and the global offensive alliance number \(\gamma_{o}(G)\) (respectively, global defensive alliance number \(\gamma_{a}(G)\)) is the minimum cardinality of a global offensive alliance (respectively, global deffensive alliance) of \(G\). We show that if \(T\) is a tree of order \(n\), then \(\gamma_{o}(T)\leq 2\gamma(T)-1\) and if \(n\geq 3\), then \(\gamma_{o}(T)\leq \frac{3}{2}\gamma_{a}(T)-1\). Moreover, all extremal trees attaining the first bound are characterized. global defensive alliance, global offensive alliance, domination, trees 2011 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol31iss2art2 https://www.opuscula.agh.edu.pl/vol31/2/art/opuscula_math_3112.pdf Convolution algebras for topological groupoids with locally compact fibres Mădălina Roxana Buneci The aim of this paper is to introduce various convolution algebras associated with a topological groupoid with locally compact fibres. Instead of working with continuous functions on \(G\), we consider functions having a uniformly continuity property on fibres. We assume that the groupoid is endowed with a system of measures (supported on its fibres) subject to the "left invariance" condition in the groupoid sense. convolution algebra, groupoid, uniform continuity, Haar system 2011 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol31iss2art3 https://www.opuscula.agh.edu.pl/vol31/2/art/opuscula_math_3113.pdf The Hardy potential and eigenvalue problems Jan Chabrowski We establish the existence of principal eigenfunctions for the Laplace operator involving weighted Hardy potentials. We consider the Dirichlet and Neumann boundary conditions. Dirichlet and Neumann problems, Hardy potential, principal eigenfuctions 2011 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol31iss2art4 https://www.opuscula.agh.edu.pl/vol31/2/art/opuscula_math_3114.pdf On the Bochner subordination of exit laws Mohamed Hmissi, Wajdi Maaouia Let \(\mathbb{P}=(P_t)_{t\ge 0}\) be a sub-Markovian semigroup on \(L^2(m)\), let \(\beta=(\beta_t)_{t\ge 0}\) be a Bochner subordinator and let \(\mathbb{P}^{\beta}=(P_t^{\beta})_{t\ge 0}\) be the subordinated semigroup of \(\mathbb{P}\) by means of \(\beta\), i.e. \(P^{\beta}_s:=\int_0^{\infty} P_r\,\beta_s(dr)\). Let \(\varphi:=(\varphi_t)_{t\gt 0}\) be a \(\mathbb{P}\)-exit law, i.e. \[ P_t\varphi_s= \varphi_{s+t}, \qquad s,t\gt 0\] and let \(\varphi^{\beta}_t:=\int_0^{\infty} \varphi_s\,\beta_t(ds)\). Then \(\varphi^{\beta}:=(\varphi_t^{\beta})_{t\gt 0}\) is a \(\mathbb{P}^{\beta}\)-exit law whenever it lies in \(L^2(m)\). This paper is devoted to the converse problem when \(\beta\) is without drift. We prove that a \(\mathbb{P}^{\beta}\)-exit law \(\psi:=(\psi_t)_{t\gt 0}\) is subordinated to a (unique) \(\mathbb{P}\)-exit law \(\varphi\) (i.e. \(\psi=\varphi^{\beta}\)) if and only if \((P_tu)_{t\gt 0}\subset D(A^{\beta})\), where \(u=\int_0^{\infty} e^{-s} \psi_s ds\) and \(A^{\beta}\) is the \(L^2(m)\)-generator of \(\mathbb{P}^{\beta}\). sub-Markovian semigroup, exit law, subordinator, Bernstein function, Bochner subordination 2011 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol31iss2art5 https://www.opuscula.agh.edu.pl/vol31/2/art/opuscula_math_3115.pdf A sampling theory for infinite weighted graphs Palle E. T. Jorgensen We prove two sampling theorems for infinite (countable discrete) weighted graphs \(G\); one example being "large grids of resistors" i.e., networks and systems of resistors. We show that there is natural ambient continuum \(X\) containing \(G\), and there are Hilbert spaces of functions on \(X\) that allow interpolation by sampling values of the functions restricted only on the vertices in \(G\). We sample functions on \(X\) from their discrete values picked in the vertex-subset \(G\). We prove two theorems that allow for such realistic ambient spaces \(X\) for a fixed graph \(G\), and for interpolation kernels in function Hilbert spaces on \(X\), sampling only from points in the subset of vertices in \(G\). A continuum is often not apparent at the outset from the given graph \(G\). We will solve this problem with the use of ideas from stochastic integration. weighted graph, Hilbert space, Laplace operator, sampling, Shannon, white noise, Wiener transform, interpolation 2011 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol31iss2art6 https://www.opuscula.agh.edu.pl/vol31/2/art/opuscula_math_3116.pdf Operator representations of function algebras and functional calculus Adina Juratoni, Nicolae Suciu This paper deals with some operator representations \(\Phi\) of a weak*-Dirichlet algebra \(A\), which can be extended to the Hardy spaces \(H^{p}(m)\), associated to \(A\) and to a representing measure \(m\) of \(A\), for \(1\leq p\leq\infty\). A characterization for the existence of an extension \(\Phi_p\) of \(\Phi\) to \(L^p(m)\) is given in the terms of a semispectral measure \(F_\Phi\) of \(\Phi\). For the case when the closure in \(L^p(m)\) of the kernel in \(A\) of \(m\) is a simply invariant subspace, it is proved that the map \(\Phi_p|H^p(m)\) can be reduced to a functional calculus, which is induced by an operator of class \(C_\rho\) in the Nagy-Foiaş sense. A description of the Radon-Nikodym derivative of \(F_\Phi\) is obtained, and the log-integrability of this derivative is proved. An application to the scalar case, shows that the homomorphisms of \(A\) which are bounded in \(L^p(m)\) norm, form the range of an embedding of the open unit disc into a Gleason part of \(A\). weak*-Dirichlet algebra, Hardy space, operator representation, semispectral measure 2011 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol31iss2art7 https://www.opuscula.agh.edu.pl/vol31/2/art/opuscula_math_3117.pdf Four positive periodic solutions of a discrete time Lotka-Volterra competitive system with harvesting terms Xinggui Liu, Yaping Ren, Yongkun Li In this paper, by using Mawhin's continuation theorem of coincidence degree theory, we establish the existence of at least four positive periodic solutions for a discrete time Lotka-Volterra competitive system with harvesting terms. An example is given to illustrate the effectiveness of our results. discrete systems, Lotka-Volterra competitive models, coincidence degree, harvesting terms 2011 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol31iss2art8 https://www.opuscula.agh.edu.pl/vol31/2/art/opuscula_math_3118.pdf The equality case in some recent convexity inequalities Gyula Maksa, Zsolt Páles In this paper, we investigate a functional equation related to some recently introduced and investigated convexity type inequalities. generalized convexity, affine functions, functional equations, extension theorem 2011 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol31iss2art9 https://www.opuscula.agh.edu.pl/vol31/2/art/opuscula_math_3119.pdf Polynomial stability of evolution operators in Banach spaces Megan Mihail, Traian Ceauşu, Magda Luminiţa Ramneanţu The paper considers three concepts of polynomial stability for linear evolution operators which are defined in a general Banach space and whose norms can increase not faster than exponentially. Our approach is based on the extension of techniques for exponential stability to the case of polynomial stability. Some illustrating examples clarify the relations between the stability concepts considered in paper. The obtained results are generalizations of well-known theorems about the uniform and nonuniform exponential stability. evolution operator, polynomial stability, exponential stability 2011 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol31iss2art10 https://www.opuscula.agh.edu.pl/vol31/2/art/opuscula_math_3120.pdf Matrices related to some Fock space operators Krzysztof Rudol Matrices of operators with respect to frames are sometimes more natural and easier to compute than the ones related to bases. The present work investigates such operators on the Segal-Bargmann space, known also as the Fock space. We consider in particular some properties of matrices related to Toeplitz and Hankel operators. The underlying frame is provided by normalised reproducing kernel functions at some lattice points. frames, operators, Fock space, reproducing kernel 2011 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol31iss3 https://www.opuscula.agh.edu.pl/om-vol31iss3art1 https://www.opuscula.agh.edu.pl/vol31/3/art/opuscula_math_3121.pdf Investigating the numerical range and q-numerical range of non square matrices Aikaterini Aretaki, John Maroulas A presentation of numerical ranges for rectangular matrices is undertaken in this paper, introducing two different definitions and elaborating basic properties. Further, we extend to the \(q\)-numerical range. numerical range, projectors, matrix norms, singular values 2011 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol31iss3art2 https://www.opuscula.agh.edu.pl/vol31/3/art/opuscula_math_3122.pdf Stability of the Popoviciu type functional equations on groups Małgorzata Chudziak We consider the stability problem for a class of functional equations related to the Popoviciu equation. stability, Popoviciu equation, quadratic equation, additive function 2011 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol31iss3art3 https://www.opuscula.agh.edu.pl/vol31/3/art/opuscula_math_3123.pdf Monotone iterative technique for finite systems of nonlinear Riemann-Liouville fractional differential equations Z. Denton, A. S. Vatsala Comparison results of the nonlinear scalar Riemann-Liouville fractional differential equation of order \(q\), \(0 \lt q \leq 1\), are presented without requiring Hölder continuity assumption. Monotone method is developed for finite systems of fractional differential equations of order \(q\), using coupled upper and lower solutions. Existence of minimal and maximal solutions of the nonlinear fractional differential system is proved. fractional differential systems, coupled lower and upper solutions, mixed quasimonotone property 2011 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol31iss3art4 https://www.opuscula.agh.edu.pl/vol31/3/art/opuscula_math_3124.pdf On nonlocal problems for fractional differential equations in Banach spaces XiWang Dong, JinRong Wang, Yong Zhou In this paper, we study the existence and uniqueness of solutions to the nonlocal problems for the fractional differential equation in Banach spaces. New sufficient conditions for the existence and uniqueness of solutions are established by means of fractional calculus and fixed point method under some suitable conditions. Two examples are given to illustrate the results. nonlocal problems, fractional differential equations, existence, generalized singular Gronwall inequality, fixed point method 2011 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol31iss3art5 https://www.opuscula.agh.edu.pl/vol31/3/art/opuscula_math_3125.pdf Existence of solutions for a four-point boundary value problem of a nonlinear fractional differential equation Xiaoyan Dou, Yongkun Li, Ping Liu In this paper, we discuss a four-point boundary value problem for a nonlinear differential equation of fractional order. The differential operator is the Riemann-Liouville derivative and the inhomogeneous term depends on the fractional derivative of lower order. We obtain the existence of at least one solution for the problem by using the Schauder fixed-point theorem. Our analysis relies on the reduction of the problem considered to the equivalent Fredholm integral equation. four-point boundary value problem, Riemann-Liouville fractional derivative, Green's function, Schauder fixed-point theorem 2011 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol31iss3art6 https://www.opuscula.agh.edu.pl/vol31/3/art/opuscula_math_3126.pdf Oscillation theorems concerning non-linear differential equations of the second order E. M. Elabbasy, Sh. R. Elzeiny This paper concerns the oscillation of solutions of the differential eq. \[ \left[ r\left( t\right) \psi \left(x\left( t\right) \right) f\text{ }( \overset{\cdot }{x}(t))\right]^{\cdot }+q\left( t\right) \varphi (g\left( x\left( t\right) \right), r\left( t\right) \psi \left( x\left( t\right) \right) f(\overset{\cdot }{x}(t)))=0,\] where \(u\varphi \left( u,v\right) \gt 0\) for all \(u\neq 0\), \(xg\left( x\right) \gt 0\), \(xf\left( x\right)\gt 0\) for all \(x\neq 0\), \(\psi \left( x\right) \gt 0\) for all \(x\in \mathbb{R}\), \(r\left( t\right) \gt 0\) for \(t\geq t_{0}\gt 0\) and \(q\) is of arbitrary sign. Our results complement the results in [A.G. Kartsatos, On oscillation of nonlinear equations of second order, J. Math. Anal. Appl. 24 (1968), 665–668], and improve a number of existing oscillation criteria. Our main results are illustrated with examples. second order, nonlinear, differential equations, oscillation 2011 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol31iss3art7 https://www.opuscula.agh.edu.pl/vol31/3/art/opuscula_math_3127.pdf Existence and uniqueness theorem for a Hammerstein nonlinear integral equation A. Kh. Khachatryan, Kh. A. Khachatryan The existence of a solution, as well as some properties of the obtained solution for a Hammerstein type nonlinear integral equation have been investigated. For a certain class of functions the uniqueness theorem has also been proved. iteration, Wiener-Hopf operator, pointwise convergence, Hammerstein type equation 2011 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol31iss3art8 https://www.opuscula.agh.edu.pl/vol31/3/art/opuscula_math_3128.pdf Some constructions of Lyapunov function for linear extensions of dynamical systems Viktor Kulik, Ewa Tkocz-Piszczek In this note we consider some sets of linear extensions of dynamical systems and research regularity by means of the sign-changing Lyapunov function. We examine some constructions of Lyapunov functions for given systems. Lyapunov function, invariant manifold, invariant torus 2011 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol31iss3art9 https://www.opuscula.agh.edu.pl/vol31/3/art/opuscula_math_3129.pdf Existence and asymptotic behavior of solutions for Hénon type equations Wei Long, Jianfu Yang This paper is concerned with ground state solutions for the Hénon type equation \(-\Delta u(x)=|y|^{\alpha} u^{p-1}(x)\) in \(\Omega\), where \(\Omega=B^k(0,1)\times B^{n-k}(0,1)\subset \mathbb{R}^n\) and \(x=(y,z) \in \mathbb{R}^k \times \mathbb{R}^{n-k}\). We study the existence of cylindrically symmetric and non-cylindrically symmetric ground state solutions for the problem. We also investigate asymptotic behavior of the ground state solution when \(p\) tends to the critical exponent \(2^*=\frac {2n}{n-2}\) if \(n\geq 3\). Hénon equation, cylindrical symmetry, non-cylindrical symmetry, asymptotic behavior 2011 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol31iss3art10 https://www.opuscula.agh.edu.pl/vol31/3/art/opuscula_math_3130.pdf A note on invariant measures Piotr Niemiec The aim of the paper is to show that if \(\mathcal{F}\) is a family of continuous transformations of a nonempty compact Hausdorff space \(\Omega\), then there is no \(\mathcal{F}\)-invariant probabilistic Borel measures on \(\Omega\) iff there are \(\varphi_1,\ldots,\varphi_p \in \mathcal{F}\) (for some \(p \geq 2\)) and a continuous function \(u:\, \Omega^p \to \mathbb{R}\) such that \(\sum_{\sigma \in S_p} u(x_{\sigma(1)},\ldots ,x_{\sigma(p)}) = 0\) and \(\liminf_{n\to\infty} \frac1n \sum_{k=0}^{n-1} (u \circ \Phi^k)(x_1,\ldots,x_p) \geq 1\) for each \(x_1,\ldots,x_p \in \Omega\), where \(\Phi:\, \Omega^p \ni (x_1,\ldots,x_p) \mapsto (\varphi_1(x_1),\ldots,\varphi_p(x_p)) \in \Omega^p\) and \(\Phi^k\) is the \(k\)-th iterate of \(\Phi\). A modified version of this result in case the family \(\mathcal{F}\) generates an equicontinuous semigroup is proved. invariant measures, equicontinuous semigroups, compact spaces 2011 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol31iss3art11 https://www.opuscula.agh.edu.pl/vol31/3/art/opuscula_math_3131.pdf Positive solutions of arbitrary order nonlinear fractional differential equations with advanced arguments Sotiris K. Ntouyas, Guotao Wang, Lihong Zhang In this paper, we investigate the existence and uniqueness of positive solutions to arbitrary order nonlinear fractional differential equations with advanced arguments. By applying some known fixed point theorems, sufficient conditions for the existence and uniqueness of positive solutions are established. positive solution, advanced arguments, fractional differential equations, superlinear (sublinear) condition, uniqueness 2011 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol31iss3art12 https://www.opuscula.agh.edu.pl/vol31/3/art/opuscula_math_3132.pdf On some existence results of mild solutions for nonlocal integrodifferential Cauchy problems in Banach spaces YanLong Yang, JinRong Wang In this paper, we study a class of integrodifferential evolution equations with nonlocal initial conditions in Banach spaces. Existence results of mild solutions are proved for a class of integrodifferential evolution equations with nonlocal initial conditions in Banach spaces. The main results are obtained by using the Schaefer fixed point theorem and semigroup theory. Finally, an example is given for demonstration. integrodifferential equations, nonlocal initial condition, completely continuous operator, Schaefer fixed point theorem, mild solutions 2011 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol31iss3art13 https://www.opuscula.agh.edu.pl/vol31/3/art/opuscula_math_3133.pdf A new composition theorem for S^{p}-weighted pseudo almost periodic functions and applications to semilinear differential equations Zhi-Han Zhao, Yong-Kui Chang, G. M. N'Guérékata In this paper, we establish a new composition theorem for \(S^p\)-weighted pseudo almost periodic functions under weaker conditions than the Lipschitz ones currently encountered in the literatures. We apply this new composition theorem along with the Schauder's fixed point theorem to obtain new existence theorems for weighted pseudo almost periodic mild solutions to a semilinear differential equation in a Banach space. \(S^p\)-weighted pseudo almost periodic, weighted pseudo almost periodicity, semilinear differential equations 2011 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol31iss4 https://www.opuscula.agh.edu.pl/om-vol31iss4art1 https://www.opuscula.agh.edu.pl/vol31/4/art/opuscula_math_3134.pdf A mixed integer nonlinear programming formulation for the problem of fitting positive exponential sums to empirical data Adalys Alvarez, Hugo Lara In this work we deal with exponential sum models coming from data acquisition in the empirical sciences. We present a two step approach based on Tikhonov regularization and combinatorial optimization, to obtain stable parameter estimations, which fit the data. We develop properties of the solutions, based on their optimality conditions. Some numerical experiments are shown to illustrate our approach. mixed integer nonlinear programming, regularization, nonlinear least squares 2011 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol31iss4art2 https://www.opuscula.agh.edu.pl/vol31/4/art/opuscula_math_3135.pdf Operators in divergence form and their Friedrichs and Kreĭn extensions Yury Arlinskiĭ, Yury Kovalev For a densely defined nonnegative symmetric operator \(\mathcal{A} = L_2^*L_1 \) in a Hilbert space, constructed from a pair \(L_1 \subset L_2\) of closed operators, we give expressions for the Friedrichs and Kreĭn nonnegative selfadjoint extensions. Some conditions for the equality \((L_2^* L_1)^* = L_1^* L_2\) are obtained. Applications to 1D nonnegative Hamiltonians, corresponding to point interactions, are given. symmetric operator, divergence form, Friedrichs extension, Kreĭn extension 2011 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol31iss4art3 https://www.opuscula.agh.edu.pl/vol31/4/art/opuscula_math_3136.pdf Neighbourhood total domination in graphs S. Arumugam, C. Sivagnanam Let \(G = (V,E)\) be a graph without isolated vertices. A dominating set \(S\) of \(G\) is called a neighbourhood total dominating set (ntd-set) if the induced subgraph \(\langle N(S)\rangle\) has no isolated vertices. The minimum cardinality of a ntd-set of \(G\) is called the neighbourhood total domination number of \(G\) and is denoted by \(\gamma _{nt}(G)\). The maximum order of a partition of \(V\) into ntd-sets is called the neighbourhood total domatic number of \(G\) and is denoted by \(d_{nt}(G)\). In this paper we initiate a study of these parameters. neighbourhood total domination, total domination, connected domination, paired domination, neighbourhood total domatic number 2011 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol31iss4art4 https://www.opuscula.agh.edu.pl/vol31/4/art/opuscula_math_3137.pdf Recursively arbitrarily vertex-decomposable suns Olivier Baudon, Frédéric Gilbert, Mariusz Woźniak A graph \(G = (V,E)\) is arbitrarily vertex decomposable if for any sequence \(\tau\) of positive integers adding up to \(|V|\), there is a sequence of vertex-disjoint subsets of \(V\) whose orders are given by \(\tau\), and which induce connected graphs. The aim of this paper is to study the recursive version of this problem on a special class of graphs called suns. This paper is a complement of [O. Baudon, F. Gilbert, M. Woźniak, Recursively arbitrarily vertex-decomposable graphs, research report, 2010]. arbitrarily vertex-decomposable graphs (AVD), recursively AVD graphs 2011 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol31iss4art5 https://www.opuscula.agh.edu.pl/vol31/4/art/opuscula_math_3138.pdf Free probability induced by electric resistance networks on energy Hilbert spaces Ilwoo Cho, Palle E. T. Jorgensen We show that a class of countable weighted graphs arising in the study of electric resistance networks (ERNs) are naturally associated with groupoids. Starting with a fixed ERN, it is known that there is a canonical energy form and a derived energy Hilbert space \(H_{\mathcal{E}}\). From \(H_{\mathcal{E}}\), one then studies resistance metrics and boundaries of the ERNs. But in earlier research, there does not appear to be a natural algebra of bounded operators acting on \(H_{\mathcal{E}}\). With the use of our ERN-groupoid, we show that \(H_{\mathcal{E}}\) may be derived as a representation Hilbert space of a universal representation of a groupoid algebra \(\mathfrak{A}_G\), and we display other representations. Among our applications, we identify a free structure of \(\mathfrak{A}_G\) in terms of the energy form. directed graphs, graph groupoids, electric resistance networks, ERN-groupoids, energy Hilbert spaces, ERN-algebras, free moments, free cumulants 2011 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol31iss4art6 https://www.opuscula.agh.edu.pl/vol31/4/art/opuscula_math_3139.pdf Open trails in digraphs Sylwia Cichacz, Agnieszka Görlich It has been shown in [S. Cichacz, A. Görlich, Decomposition of complete bipartite graphs into open trails, Preprint MD 022, (2006)] that any bipartite graph \(K_{a,b}\), is decomposable into open trails of prescribed even lengths. In this article we consider the corresponding question for directed graphs. We show that the complete directed graphs \(\overleftrightarrow{K}_n\) and \(\overleftrightarrow{K}_{a,b}\) are arbitrarily decomposable into directed open trails. trail, graphdecomposition, bipartite graph 2011 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol31iss4art7 https://www.opuscula.agh.edu.pl/vol31/4/art/opuscula_math_3140.pdf The Fejer-Riesz type result for some weighted Hilbert spaces of analytic functions in the unit disc Piotr Jakóbczak In this note we prove Fejer-Riesz inequality type results for some weighted Hilbert spaces of analytic functions in the unit disc. We describe also a class of such spaces for which Fejer-Riesz inequality type results do not hold. Fejer-Riesz inequality, Hilbert spaces of analytic functions 2011 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol31iss4art8 https://www.opuscula.agh.edu.pl/vol31/4/art/opuscula_math_3141.pdf Singular continuous spectrum of half-line Schrödinger operators with point interactions on a sparse set Vladimir Lotoreichik We say that a discrete set \(X = \{ x_n \}_{n\in \mathbb{N}_0}\) on the half-line \[0 = x_0 \lt x_1 \lt x_2 \lt x_3 \lt ... \lt x_n \lt ... \lt +\infty \] is sparse if the distances \(\Delta x_n = x_{n+1}- x_n\) between neighbouring points satisfy the condition \(\frac{\Delta x_n}{\Delta x_{n-1}} \to +\infty\). In this paper half-line Schrödinger operators with point \(\delta\)- and \(\delta'\)-interactions on a sparse set are considered. Assuming that strengths of point interactions tend to \(\infty\) we give simple sufficient conditions for such Schrödinger operators to have non-empty singular continuous spectrum and to have purely singular continuous spectrum, which coincides with \(\mathbb{R}_+\). half-line Schrödinger operators, \(\delta\)-interactions, \(\delta '\)-interactions, singular continuous spectrum 2011 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol31iss4art9 https://www.opuscula.agh.edu.pl/vol31/4/art/opuscula_math_3142.pdf Existence and uniqueness results for fractional differential equations with boundary value conditions LinLi Lv, JinRong Wang, Wei Wei In this paper, we study the existence and uniqueness of fractional differential equations with boundary value conditions. A new generalized singular type Gronwall inequality is given to obtain important a priori bounds. Existence and uniqueness results of solutions are established by virtue of fractional calculus and fixed point method under some weak conditions. An example is given to illustrate the results. fractional differential equations, boundary value conditions, singular Gronwall inequality, existence, uniqueness 2011 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol31iss4art10 https://www.opuscula.agh.edu.pl/vol31/4/art/opuscula_math_3143.pdf Strengthened Stone-Weierstrass type theorem Piotr Niemiec The aim of the paper is to prove that if \(L\) is a linear subspace of the space \(\mathcal{C}(K)\) of all real-valued continuous functions defined on a nonempty compact Hausdorff space \(K\) such that \(\min(|f|, 1) \in L\) whenever \(f \in L\), then for any nonzero \(g \in \overline{L}\) (where \(\overline{L}\) denotes the uniform closure of \(L\) in \(\mathcal{C}(K)\)) and for any sequence \((b_n)_{n=1}^{\infty}\) of positive numbers satisfying the relation \(\sum_{n=1}^{\infty} b_n = \|g\|\) there exists a sequence \((f_n)_{n=1}^{\infty}\) of elements of \(L\) such that \(\|f_n \|= b_n\) for each \(n \geq 1\), \(g = \sum _{n=1}^{\infty} f_n \) and \(|g|= \sum _{n=1}^{\infty} |f_n| \). Also the formula for \(\overline{L}\) is given. Stone-Weierstrass theorem, function lattices 2011 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol31iss4art11 https://www.opuscula.agh.edu.pl/vol31/4/art/opuscula_math_3144.pdf On some classes of meromorphic functions defined by subordination and superordination Alina Totoi Let \(p\in \mathbb{N}^*\) and \(\beta,\gamma\in \mathbb{C}\) with \(\beta\neq 0\) and let \(\Sigma_p\) denote the class of meromorphic functions of the form \(g(z)=\frac{a_{-p}}{z^p}+a_0+a_1 z+\ldots,\,z\in \dot U\), \(a_{-p}\neq 0\). We consider the integral operator \(J_{p,\beta,\gamma}:K_{p,\beta,\gamma}\subset\Sigma_p\to \Sigma_p\) defined by \[J_{p,\beta,\gamma}(g)(z)=\left[\frac{\gamma-p\beta}{z^\gamma }\int_0^zg^{\beta}(t) t^{\gamma-1}dt\right]^{\frac{1}{\beta}},\,g\in K_{p,\beta,\gamma},\,z\in \dot U.\] We introduce some new subclasses of the class \(\Sigma_p\), associated with subordination and superordination, such that, in some particular cases, these new subclasses are the well-known classes of meromorphic starlike functions and we study the properties of these subclasses with respect to the operator \(J_{p,\beta,\gamma}\). meromorphic functions, integral operators, subordination, superordination 2011 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol31iss4art12 https://www.opuscula.agh.edu.pl/vol31/4/art/opuscula_math_3145.pdf Non symmetric random walk on infinite graph Marcin J. Zygmunt We investigate properties of a non symmetric Markov's chain on an infinite graph. We show the connection with matrix valued random walk polynomials which satisfy the orthogonality formula with respect to non a symmetric matrix valued measure. random walk on an infinite graph, block tridiagonal transition matrix, spectral measure matrix orthogonal polynomials 2011 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol32iss1 https://www.opuscula.agh.edu.pl/om-vol32iss1art1 https://www.opuscula.agh.edu.pl/vol32/1/art/opuscula_math_3201.pdf Fixed points and stability in neutral nonlinear differential equations with variable delays Abdelouaheb Ardjouni, Ahcene Djoudi By means of Krasnoselskii's fixed point theorem we obtain boundedness and stability results of a neutral nonlinear differential equation with variable delays. A stability theorem with a necessary and sufficient condition is given. The results obtained here extend and improve the work of C. H. Jin and J. W. Luo [Nonlinear Anal. 68 (2008), 3307-3315], and also those of T. A. Burton [Fixed Point Theory 4 (2003), 15-32; Dynam. Systems Appl. 11 (2002), 499-519] and B. Zhang [Nonlinear Anal. 63 (2005), e233-e242]. In the end we provide an example to illustrate our claim. fixed points, stability, nonlinear neutral differential equation, integral equation, variable delays 2012 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol32iss1art2 https://www.opuscula.agh.edu.pl/vol32/1/art/opuscula_math_3202.pdf Compactly supported multi-wavelets Wojciech Banaś In this paper we show some construction of compactly supported multi-wavelets in \(L^2(\mathbb{R}^d)\), \(d \geq 2\) which is based on the one-dimensional case, when \(d=1\). We also demonstrate that some methods, which are useful in the construction of wavelets with a compact support at \(d=1\), can be adapted to higher-dimensional cases if \(A \in M_{d \times d}(\mathbb{Z})\) is an expansive matrix of a special form. compactly supported multi-wavelet, compactly supported scaling function, multiresolution analysis, expansive matrix 2012 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol32iss1art3 https://www.opuscula.agh.edu.pl/vol32/1/art/opuscula_math_3203.pdf Weak solutions for nonlinear fractional differential equations with integral boundary conditions in Banach spaces Mouffak Benchohra, Fatima-Zohra Mostefai The aim of this paper is to investigate a class of boundary value problems for fractional differential equations involving nonlinear integral conditions. The main tool used in our considerations is the technique associated with measures of weak noncompactness. boundary value problem, Caputo fractional derivative, measure of weak noncompactness, Pettis integrals, weak solution 2012 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol32iss1art4 https://www.opuscula.agh.edu.pl/vol32/1/art/opuscula_math_3204.pdf Isospectral integrability analysis of dynamical systems on discrete manifolds Denis Blackmore, Anatoliy K. Prykarpatsky, Yarema A. Prykarpatsky It is shown how functional-analytic gradient-holonomic structures can be used for an isospectral integrability analysis of nonlinear dynamical systems on discrete manifolds. The approach developed is applied to obtain detailed proofs of the integrability of the discrete nonlinear Schrödinger, Ragnisco-Tu and Riemann-Burgers dynamical systems. gradient holonomic algorithm, conservation laws, asymptotic analysis, Poissonian structures, Lax representation, finite-dimensional reduction, Liouville integrability, nonlinear discrete dynamical systems 2012 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol32iss1art5 https://www.opuscula.agh.edu.pl/vol32/1/art/opuscula_math_3205.pdf Note on the stability of first order linear differential equations Florin Bojor In this paper, we will prove the generalized Hyers-Ulam stability of the linear differential equation of the form \(y'(x)+f(x)y(x)+g(x)=0\) under some additional conditions. fixed point method, differential equation, Hyers-Ulam stability 2012 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol32iss1art6 https://www.opuscula.agh.edu.pl/vol32/1/art/opuscula_math_3206.pdf An abstract nonlocal second order evolution problem Ludwik Byszewski, Teresa Winiarska The aim of the paper is to prove theorems on the existence and uniqueness of mild and classical solutions of a semilinear evolution second order equation together with nonlocal conditions. The theory of strongly continuous cosine families of linear operators in a Banach space is applied. nonlocal, second order, evolution problem, Banach space 2012 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol32iss1art7 https://www.opuscula.agh.edu.pl/vol32/1/art/opuscula_math_3207.pdf Global offensive k-alliance in bipartite graphs Mustapha Chellali, Lutz Volkmann Let \(k \geq 0\) be an integer. A set \(S\) of vertices of a graph \(G=(V(G),E(G))\) is called a global offensive \(k\)-alliance if \(|N(v) \cap S| \geq |N(v) \cap S|+k\) for every \(v \in V(G)-S\), where \(0 \leq k \leq \Delta\) and \(\Delta\) is the maximum degree of \(G\). The global offensive \(k\)-alliance number \(\gamma^k_o(G)\) is the minimum cardinality of a global offensive \(k\)-alliance in \(G\). We show that for every bipartite graph \(G\) and every integer \(k \geq 2\), \(\gamma^k_o(G) \leq \frac{n(G)+|L_k(G)|}{2}\), where \(L_k(G)\) is the set of vertices of degree at most \(k-1\). Moreover, extremal trees attaining this upper bound are characterized. global offensive \(k\)-alliance number, bipartite graphs, trees 2012 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol32iss1art8 https://www.opuscula.agh.edu.pl/vol32/1/art/opuscula_math_3208.pdf On the existence of positive continuous solutions for some polyharmonic elliptic systems on the half space Zagharide Zine El Abidine We study the existence of positive continuous solutions of the nonlinear polyharmonic system \((-\Delta)^m u + \lambda q g(v) = 0\); \((-\Delta)^m v + \mu p f(u) = 0\) in the half space \(\mathbb{R}^n_+:=\{x = (x_1,...,x_n) \in \mathbb{R}^n : x_n \gt 0\}\), where \(m \geq 1\) and \(n \gt 2m\). The nonlinear term is required to satisfy some conditions related to the Kato class \(K^{\infty}_{m,n}(\mathbb{R}^n_+)\). Our arguments are based on potential theory tools associated to \((-\Delta)^m\) and properties of functions belonging to \(K^{\infty}_{m,n}(\mathbb{R}^n_+)\). polyharmonic elliptic system, Green function, Kato class, positive continuous solution, Schauder fixed point theorem 2012 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol32iss1art9 https://www.opuscula.agh.edu.pl/vol32/1/art/opuscula_math_3209.pdf A note on a fourth order discrete boundary value problem Marek Galewski, Joanna Smejda Using variational methods we investigate the existence of solutions and their dependence on parameters for certain fourth order difference equations. discrete boundary value problem, variational method, coercivity, continuous dependence on parameters 2012 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol32iss1art10 https://www.opuscula.agh.edu.pl/vol32/1/art/opuscula_math_3210.pdf Weyl's theorem for algebraically k-quasiclass A operators Fugen Gao, Xiaochun Fang If \(T\) or \(T^*\) is an algebraically \(k\)-quasiclass \(A\) operator acting on an infinite dimensional separable Hilbert space and \(F\) is an operator commuting with \(T\), and there exists a positive integer \(n\) such that \(F^n\) has a finite rank, then we prove that Weyl's theorem holds for \(f(T)+F\) for every \(f \in H(\sigma(T))\), where \(H(\sigma(T))\) denotes the set of all analytic functions in a neighborhood of \(\sigma(T)\). Moreover, if \(T^*\) is an algebraically \(k\)-quasiclass \(A\) operator, then \(\alpha\)-Weyl's theorem holds for \(f(T)\). Also, we prove that if \(T\) or \(T^*\) is an algebraically \(k\)-quasiclass \(A\) operator then both the Weyl spectrum and the approximate point spectrum of \(T\) obey the spectral mapping theorem for every \(f \in H(\sigma(T))\). algebraically \(k\)-quasiclass \(A\) operator, Weyl's theorem, \(\alpha\)-Weyl's theorem 2012 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol32iss1art11 https://www.opuscula.agh.edu.pl/vol32/1/art/opuscula_math_3211.pdf Integral representation of functions of bounded second Φ-variation in the sense of Schramm José Giménez, Nelson Merentes, Sergio Rivas In this article we introduce the concept of second \(\Phi\)-variation in the sense of Schramm for normed-space valued functions defined on an interval \([a,b] \subset \mathbb{R}\). To that end we combine the notion of second variation due to de la Vallée Poussin and the concept of \(\varphi\)-variation in the sense of Schramm for real valued functions. In particular, when the normed space is complete we present a characterization of the functions of the introduced class by means of an integral representation. Indeed, we show that a function \(f \in \mathbb{X}^{[a,b]}\) (where \(\mathbb{X}\) is a reflexive Banach space) is of bounded second \(\Phi\)-variation in the sense of Schramm if and only if it can be expressed as the Bochner integral of a function of (first) bounded variation in the sense of Schramm. Young function, \(\Phi\)-variation, second \(\Phi\)-variation of a function 2012 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol32iss1art12 https://www.opuscula.agh.edu.pl/vol32/1/art/opuscula_math_3212.pdf An upper bound on the total outer-independent domination number of a tree Marcin Krzywkowski A total outer-independent dominating set of a graph \(G=(V(G),E(G))\) is a set \(D\) of vertices of \(G\) such that every vertex of \(G\) has a neighbor in \(D\), and the set \(V(G) \setminus D\) is independent. The total outer-independent domination number of a graph \(G\), denoted by \(\gamma_t^{oi}(G)\), is the minimum cardinality of a total outer-independent dominating set of \(G\). We prove that for every tree \(T\) of order \(n \geq 4\), with \(l\) leaves and \(s\) support vertices we have \(\gamma_t^{oi}(T) \leq (2n + s - l)/3\), and we characterize the trees attaining this upper bound. total outer-independent domination, total domination, tree 2012 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol32iss1art13 https://www.opuscula.agh.edu.pl/vol32/1/art/opuscula_math_3213.pdf Some properties of set-valued sine families Ewelina Mainka-Niemczyk Let \(\{F_t : t \geq 0\}\) be a family of continuous additive set-valued functions defined on a convex cone \(K\) in a normed linear space \(X\) with nonempty convex compact values in \(X\). It is shown that (under some assumptions) a regular sine family associated with \(\{F_t : t \geq 0\}\) is continuous and \(\{F_t : t \geq 0\}\) is a continuous cosine family. set-valued sine and cosine families, continuity of sine families, Hukuhara differences, concave set-valued functions 2012 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol32iss1art14 https://www.opuscula.agh.edu.pl/vol32/1/art/opuscula_math_3214.pdf A characterization of convex φ-functions Bartosz Micherda The properties of four elements \((LPFE)\) and \((UPFE)\), introduced by Isac and Persson, have been recently examined in Hilbert spaces, \(L^p\)-spaces and modular spaces. In this paper we prove a new theorem showing that a modular of form \(\rho_{\Phi}(f)=\int_{\Omega}\Phi(t,|f(t)|)d\mu(t)\) satisfies both \((LPFE)\) and \((UPFE)\) if and only if \(\Phi\) is convex with respect to its second variable. A connection of this result with the study of projections and antiprojections onto latticially closed subsets of the modular space \(L^{\Phi}\) is also discussed. inequalities, modulars, Orlicz-Musielak spaces, convexity, isotonicity, antiprojections 2012 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol32iss1art15 https://www.opuscula.agh.edu.pl/vol32/1/art/opuscula_math_3215.pdf Analysis of integrodifferential control system with pulse-width modulated sampler on Banach spaces JinRong Wang This paper studies steady-state control and stability for a class of integrodifferential control system with pulse-width modulated sampler on Banach spaces. The existence and stability of the steady-state for the integrodifferential control system with pulse-width modulated sampler are given. An example is given to illustrate the theory. integrodifferential system, pulse-width modulated sampler, steady-state control, steady-state stability 2012 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol32iss2 https://www.opuscula.agh.edu.pl/om-vol32iss2art1 https://www.opuscula.agh.edu.pl/vol32/2/art/opuscula_math_3216.pdf Boundary value problems for n-th order differential inclusions with four-point integral boundary conditions Bashir Ahmad, Sotiris K. Ntouyas In this paper, we discuss the existence of solutions for a four-point integral boundary value problem of \(n\)-th order differential inclusions involving convex and non-convex multivalued maps. The existence results are obtained by applying the nonlinear alternative of Leray Schauder type and some suitable theorems of fixed point theory. differential inclusions, four-point integral boundary conditions, existence, nonlinear alternative of Leray Schauder type, fixed point theorems 2012 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol32iss2art2 https://www.opuscula.agh.edu.pl/vol32/2/art/opuscula_math_3217.pdf On the extended and Allan spectra and topological radii Hugo Arizmendi-Peimbert, Angel Carrillo-Hoyo, Jairo Roa-Fajardo In this paper we prove that the extended spectrum \(\Sigma(x)\), defined by W. Żelazko, of an element \(x\) of a pseudo-complete locally convex unital complex algebra \(A\) is a subset of the spectrum \(\sigma_A(x)\), defined by G.R. Allan. Furthermore, we prove that they coincide when \(\Sigma(x)\) is closed. We also establish some order relations between several topological radii of \(x\), among which are the topological spectral radius \(R_t(x)\) and the topological radius of boundedness \(\beta_t(x)\). topological algebra, bounded element, spectrum, pseudocomplete algebra, topologically invertible element, extended spectral radius, topological spectral radius 2012 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol32iss2art3 https://www.opuscula.agh.edu.pl/vol32/2/art/opuscula_math_3218.pdf A note on a relation between the weak and strong domination numbers of a graph Razika Boutrig, Mustapha Chellali In a graph \(G=(V,E)\) a vertex is said to dominate itself and all its neighbors. A set \(D \subset V\) is a weak (strong, respectively) dominating set of \(G\) if every vertex \(v \in V-S\) is adjacent to a vertex \(u \in D\) such that \(d_G(v) \geq d_G(u)\) (\(d_G(v) \leq d_G(u)\), respectively). The weak (strong, respectively) domination number of \(G\), denoted by \(\gamma_w(G)\) (\(\gamma_s(G)\), respectively), is the minimum cardinality of a weak (strong, respectively) dominating set of \(G\). In this note we show that if \(G\) is a connected graph of order \(n \geq 3\), then \(\gamma_w(G) + t\gamma_s(G) \leq n\), where \(t=3/(\Delta+1)\) if \(G\) is an arbitrary graph, \(t=3/5\) if \(G\) is a block graph, and \(t=2/3\) if \(G\) is a claw free graph. weak domination, strong domination 2012 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol32iss2art4 https://www.opuscula.agh.edu.pl/vol32/2/art/opuscula_math_3219.pdf Uniformly continuous composition operators in the space of bounded Φ-variation functions in the Schramm sense Tomás Ereú, Nelson Merentes, José L. Sánchez, Małgorzata Wróbel We prove that any uniformly continuous Nemytskii composition operator in the space of functions of bounded generalized \(\Phi\)-variation in the Schramm sense is affine. A composition operator is locally defined. We show that every locally defined operator mapping the space of continuous functions of bounded (in the sense of Jordan) variation into the space of continous monotonic functions is constant. \(\Phi\)-variation in the sense of Schramm, uniformly continuous operator, regularization, Jensen equation, locally defined operators 2012 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol32iss2art5 https://www.opuscula.agh.edu.pl/vol32/2/art/opuscula_math_3220.pdf Existence and solution sets of impulsive functional differential inclusions with multiple delay Mohmed Helal, Abdelghani Ouahab In this paper, we present some existence results of solutions and study the topological structure of solution sets for the following first-order impulsive neutral functional differential inclusions with initial condition: \[ \begin{cases}\frac{d}{dt}[y(t)-g(t,y_t)] \in F(t,y_t) + \sum_{i=1}^{n_*} y(t-Ti), & a.e.\, t \in J\setminus\{t_1,...,t_m\} \\ y(t_k^+)-y(t_k^-)=I_k(y(t_k^-)), & k=1,...,m, \\ y(t)=\phi(t), & t \in [-r,0],\end{cases} \] where \(J:=[0,b]\) and \(0=t_0\lt t_1 \lt ...\lt t_m\lt t_{m+1}=b\) (\(m \in \mathbb{N}^*\)), \(F\) is a set-valued map and \(g\) is single map. The functions \(I_k\) characterize the jump of the solutions at impulse points \(t_k\) (\(k=1,...,m\)). Our existence result relies on a nonlinear alternative for compact u.s.c. maps. Then, we present some existence results and investigate the compactness of solution sets, some regularity of operator solutions and absolute retract (in short AR). The continuousdependence of solutions on parameters in the convex case is also examined. Applications to a problem from control theory are provided. impulsive functional differential inclusions, decomposable set, parameter differential inclusions, AR-set, control theory 2012 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol32iss2art6 https://www.opuscula.agh.edu.pl/vol32/2/art/opuscula_math_3221.pdf On the hat problem on a graph Marcin Krzywkowski The topic of this paper is the hat problem in which each of \(n\) players is uniformly and independently fitted with a blue or red hat. Then everybody can try to guess simultaneously his own hat color by looking at the hat colors of the other players. The team wins if at least one player guesses his hat color correctly, and no one guesses his hat color wrong; otherwise the team loses. The aim is to maximize the probability of winning. In this version every player can see everybody excluding himself. We consider such a problem on a graph, where vertices correspond to players, and a player can see each player to whom he is connected by an edge. The solution of the hat problem on a graph is known for trees and for cycles on four or at least nine vertices. In this paper first we give an upper bound on the maximum chance of success for graphs with neighborhood-dominated vertices. Next we solve the problem on unicyclic graphs containing a cycle on at least nine vertices. We prove that the maximum chance of success is one by two. Then we consider the hat problem on a graph with a universal vertex. We prove that there always exists an optimal strategy such that in every case some vertex guesses its color. Moreover, we prove that there exists a graph with a universal vertex for which there exists an optimal strategy such that in some case no vertex guesses its color. We also give some Nordhaus-Gaddum type inequalities. hat problem, graph, degree, neighborhood, neighborhood-dominated, unicyclic, universal vertex, Nordhaus-Gaddum 2012 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol32iss2art7 https://www.opuscula.agh.edu.pl/vol32/2/art/opuscula_math_3222.pdf On self-adjoint operators in Krein spaces constructed by Clifford algebra Cl_{2} Sergii Kuzhel, Olexiy Patsyuck Let \(J\) and \(R\) be anti-commuting fundamental symmetries in a Hilbert space \(\mathfrak{H}\). The operators \(J\) and \(R\) can be interpreted as basis (generating) elements of the complex Clifford algebra \(Cl_2(J,R):=\text{span}\{I,J,R,iJR\}\). An arbitrary non-trivial fundamental symmetry from \(Cl_2(J,R)\) is determined by the formula \(J_{\vec{\alpha}}=\alpha_1 J +\alpha_2 R+\alpha_3 iJR\), where \(\vec{\alpha} \in \mathbb{S}^2\). Let \(S\) be a symmetric operator that commutes with \(Cl_2(J,R)\). The purpose of this paper is to study the sets \(\Sigma_{J_{\vec{\alpha}}}\) (\(\forall \vec{\alpha} \in \mathbb{S}^2\)) of self-adjoint extensions of \(S\) in Krein spaces generated by fundamental symmetries \(J_{\vec{\alpha}}\) (\(J_{\vec{\alpha}}\)-self-adjoint extensions). We show that the sets \(\Sigma_{J_{\vec{\alpha}}}\) and \(\Sigma_{J_{\vec{\beta}}}\) are unitarily equivalent for different \(\vec{\alpha}, \vec{\beta} \in \mathbb{S}^2\) and describe in detail the structure of operators \(A \in \Sigma_{J_{\vec{\alpha}}}\) with empty resolvent set. Krein spaces, extension theory of symmetric operators, operators with empty resolvent set, \(J\)-self-adjoint operators, Clifford algebra \(Cl_2\) 2012 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol32iss2art8 https://www.opuscula.agh.edu.pl/vol32/2/art/opuscula_math_3223.pdf An application of the Choquet theorem to the study of randomly-superinvariant measures Teresa Rajba Given a real valued random variable \(\Theta\) we consider Borel measures \(\mu\) on \(\mathcal{B}(\mathbb{R})\), which satisfy the inequality \(\mu(B) \geq E\mu(B-\Theta)\) (\(B \in \mathcal{B}(\mathbb{R})\)) (or the integral inequality \(\mu(B) \geq \int_{-\infty}^{+\infty} \mu(B-h)\gamma (dh)\)). We apply the Choquet theorem to obtain an integral representation of measures \(\mu\) satisfying this inequality. We give integral representations of these measures in the particular cases of the random variable \(\Theta\). backward translation operator, backward difference operator, integral inequality, extreme point 2012 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol32iss2art9 https://www.opuscula.agh.edu.pl/vol32/2/art/opuscula_math_3224.pdf Implicit random iteration process with errors for asymptotically quasi-nonexpansive in the intermediate sense random operators Gurucharan Singh Saluja In this paper, we give a necessary and sufficient condition for the strong convergence of an implicit random iteration process with errors to a common fixed point for a finite family of asymptotically quasi-nonexpansive in the intermediate sense random operators and also prove some strong convergence theorems using condition (\(\overline{C}\)) and the semi-compact condition for said iteration scheme and operators. The results presented in this paper extend and improve the recent ones obtained by S. Plubtieng, P. Kumam and R. Wangkeeree, and also by the author. asymptotically quasi-nonexpansive in the intermediate sense random operator, implicit random iteration process with errors, common random fixed point, strong convergence, separable uniformly convex Banach space 2012 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol32iss2art10 https://www.opuscula.agh.edu.pl/vol32/2/art/opuscula_math_3225.pdf On controllability for fractional differential inclusions in Banach spaces JinRong Wang, XueZhu Li, Wei Wei In this paper, we investigate the controllability for systems governed by fractional differential inclusions in Banach spaces. The techniques rely on fractional calculus, multivalue mapping on a bounded set and Bohnenblust-Karlin's fixed point theorem controllability, fractional differential inclusions, Bohnenblust-Karlin's fixed point theorem 2012 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol32iss2art11 https://www.opuscula.agh.edu.pl/vol32/2/art/opuscula_math_3226.pdf Planar nonautonomous polynomial equations IV. Nonholomorphic case Paweł Wilczyński We give a few sufficient conditions for the existence of periodic solutions of the equation \(\dot{z}=\sum_{j=0}^n a_j(t)z^j-\sum_{k=1}^r c_k(t)\overline{z}^k\) where \(n \gt r\) and \(a_j\)'s, \(c_k\)'s are complex valued. We prove the existence of one up to two periodic solutions. periodic orbits, polynomial equations 2012 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol32iss2art12 https://www.opuscula.agh.edu.pl/vol32/2/art/opuscula_math_3227.pdf On a nonlinear integrodifferential evolution inclusion with nonlocal initial conditions in Banach spaces Zuomao Yan In this paper, we discuss the existence results for a class of nnlinear integrodifferential evolution inclusions with nonlocal initial conditions in Banach spaces. Our results are based on a fixed point theorem for condensing maps due to Martelli and the resolvent operators combined with approximation techniques. nonlinear integrodifferential evolution inclusions, fixed point, resolvent operator, nonlocal initial condition 2012 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol32iss3 https://www.opuscula.agh.edu.pl/om-vol32iss3art1 https://www.opuscula.agh.edu.pl/vol32/3/art/opuscula_math_3228.pdf White noise based stochastic calculus associated with a class of Gaussian processes Daniel Alpay, Haim Attia, David Levanony Using the white noise space setting, we define and study stochastic integrals with respect to a class of stationary increment Gaussian processes. We focus mainly on continuous functions with values in the Kondratiev space of stochastic distributions, where use is made of the topology of nuclear spaces. We also prove an associated Ito formula. white noise space, Wick product, stochastic integral 2012 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol32iss3art2 https://www.opuscula.agh.edu.pl/vol32/3/art/opuscula_math_3229.pdf Trees whose 2-domination subdivision number is 2 M. Atapour, S. M. Sheikholeslami, Abdollah Khodkar A set \(S\) of vertices in a graph \(G = (V,E)\) is a \(2\)-dominating set if every vertex of \(V\setminus S\) is adjacent to at least two vertices of \(S\). The \(2\)-domination number of a graph \(G\), denoted by \(\gamma_2(G)\), is the minimum size of a \(2\)-dominating set of \(G\). The \(2\)-domination subdivision number \(sd_{\gamma_2}(G)\) is the minimum number of edges that must be subdivided (each edge in \(G\) can be subdivided at most once) in order to increase the \(2\)-domination number. The authors have recently proved that for any tree \(T\) of order at least \(3\), \(1 \leq sd_{\gamma_2}(T)\leq 2\). In this paper we provide a constructive characterization of the trees whose \(2\)-domination subdivision number is \(2\). \(2\)-dominating set, \(2\)-domination number, \(2\)-domination subdivision number 2012 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol32iss3art3 https://www.opuscula.agh.edu.pl/vol32/3/art/opuscula_math_3230.pdf Existence result for hemivariational inequality involving p(x)-Laplacian Sylwia Barnaś In this paper we study the nonlinear elliptic problem with \(p(x)\)-Laplacian (hemivariational inequality).We prove the existence of a nontrivial solution. Our approach is based on critical point theory for locally Lipschitz functionals due to Chang [J. Math. Anal. Appl. 80 (1981), 102–129]. \(p(x)\)-Laplacian, Palais-Smale condition, mountain pass theorem, variable exponent Sobolev space 2012 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol32iss3art4 https://www.opuscula.agh.edu.pl/vol32/3/art/opuscula_math_3231.pdf Stepanov-like C^{(n)}-pseudo almost automorphy and applications to some nonautonomous higher-order differential equations Toka Diagana, Valerie Nelson, Gaston M. N'Guérékata In this paper we introduce and study a new concept called Stepanov-like \(C^{(n)}\)-pseudo almost automorphy, which generalizes in a natural fashion both the notions of \(C^{(n)}\)-pseudo almost periodicity and that of \(C^{(n)}\)-pseudo almost automorphy recently introduced in the literature by the authors. Basic properties of these new functions are investigated. Furthermore, we study and obtain the existence of \(C^{(N+m)}\)-pseudo almost automorphic solutions to some nonautonomous higher-order systems of differential equations with Stepanov-like \(C^{(m)}\)-pseudo almost automorphic coefficients. pseudo almost automorphic \(C^{(n)}\)-pseudo almost automorphy, Stepanov-like \(C^{(n)}\)-pseudo almost automorphy, exponential dichotomy 2012 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol32iss3art5 https://www.opuscula.agh.edu.pl/vol32/3/art/opuscula_math_3232.pdf On the existence of three solutions for quasilinear elliptic problem Paweł Goncerz We consider a quasilinear elliptic problem of the type \(-\Delta_p u = \lambda (f(u)+\mu g(u))\) in \(\Omega\), \(u|_{\partial \Omega} =0\), where \(\Omega \in \mathbb{R}^N\) is an open and bounded set, \(f\), \(g\) are continuous real functions on \(\mathbb{R}\) and \(\lambda , \mu \in \mathbb{R}\). We prove the existence of at least three solutions for this problem using the so called three critical points theorem due to Ricceri. critical point, elliptic problem, minimax inequality, \(p\)-Laplacian, three critical points theorem, weak solution 2012 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol32iss3art6 https://www.opuscula.agh.edu.pl/vol32/3/art/opuscula_math_3233.pdf Global well-posedness and scattering for the focusing nonlinear Schrödinger equation in the nonradial case Pigong Han The energy-critical, focusing nonlinear Schrödinger equation in the nonradial case reads as follows: \[i\partial_t u = -\Delta u -|u|^{\frac{4}{N-2}}u,\quad (x,0)=u_0 \in H^1 (\mathbb{R}^N),\quad N\geq 3.\] Under a suitable assumption on the maximal strong solution, using a compactness argument and a virial identity, we establish the global well-posedness and scattering in the nonradial case, which gives a positive answer to one open problem proposed by Kenig and Merle [Invent. Math. 166 (2006), 645–675]. critical energy, focusing Schrödinger equation, global well-posedness, scattering 2012 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol32iss3art7 https://www.opuscula.agh.edu.pl/vol32/3/art/opuscula_math_3234.pdf Rank-one perturbation of Toeplitz operators and reflexivity Kamila Kliś-Garlicka It was shown that rank-one perturbation of the space of Toeplitz operators preserves \(2\)-hyperreflexivity. Toeplitz operators, reflexivity, hyperreflexivity 2012 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol32iss3art8 https://www.opuscula.agh.edu.pl/vol32/3/art/opuscula_math_3235.pdf On the structure of certain nontransitive diffeomorphism groups on open manifolds Agnieszka Kowalik, Jacek Lech, Ilona Michalik It is shown that in some generic cases the identity component of the group of leaf preserving diffeomorphisms (with not necessarily compact support) on a foliated open manifold is perfect. Next, it is proved that it is also bounded, i.e. bounded with respect to any bi-invariant metric. It follows that the group is uniformly perfect as well. foliated manifold, bounded group, conjugation-invariant norm, group of diffeomorphisms, commutator, perfectness, uniform perfectness 2012 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol32iss3art9 https://www.opuscula.agh.edu.pl/vol32/3/art/opuscula_math_3236.pdf Integrability of trigonometric series with generalized semi-convex coefficients Xhevat Z. Krasniqi In this paper we deal with cosine and sine trigonometric series with generalized semi-convex coefficients. Integrability conditions for them are obtained. trigonometric series, generalized semi-convex sequences 2012 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol32iss3art10 https://www.opuscula.agh.edu.pl/vol32/3/art/opuscula_math_3237.pdf Estimates of solutions for parabolic differential and difference functional equations and applications Lucjan Sapa The theorems on the estimates of solutions for nonlinear second-order partial differential functional equations of parabolic type with Dirichlet's condition and for suitable implicit finite difference functional schemes are proved. The proofs are based on the comparison technique. The convergent and stable difference method is considered without the assumption of the global generalized Perron condition posed on the functional variable but with the local one only. It is a consequence of our estimates theorems. In particular, these results cover quasi-linear equations. However, such equations are also treated separately. The functional dependence is of the Volterra type. parabolic differential and discrete functional equations, estimate of solution, implicit difference method 2012 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol32iss3art11 https://www.opuscula.agh.edu.pl/vol32/3/art/opuscula_math_3238.pdf Boundary value problems for second order delay differential equations Lidia Skóra We present some existence and uniqueness result for a boundary value problem for functional differential equations of second order. functional differential equation, existence, uniqueness, fixed point theorem 2012 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol32iss3art12 https://www.opuscula.agh.edu.pl/vol32/3/art/opuscula_math_3239.pdf On the asymptotic behaviour of solutions to a linear functional equation Dariusz Sokołowski We investigate the asymptotic behaviour at infinity of solutions of the equation \[\varphi (x) = \int_S \varphi (x+M(s))\sigma(d s).\] We show among others that, under some assumptions, any positive solution of the equation which is integrable on a vicinity of infinity or vanishes at \(+\infty\) tends on some sequence to zero faster than some exponential function, but it does not vanish faster than another such function. linear functional equations and inequalities, solutions with a constant sign, asymptotic behaviour of solutions 2012 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol32iss3art13 https://www.opuscula.agh.edu.pl/vol32/3/art/opuscula_math_3240.pdf On the uniqueness of minimal projections in Banach spaces Ewa Szlachtowska, Dominik Mielczarek Let \(X\) be a uniformly convex Banach space with a continuous semi-inner product. We investigate the relation of orthogonality in \(X\) and generalized projections acting on \(X\). We prove uniqueness of orthogonal and co-orthogonal projections. minimal projection, orthogonal projection, co-orthogonal projection, uniqueness of norm-one projection 2012 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol32iss3art14 https://www.opuscula.agh.edu.pl/vol32/3/art/opuscula_math_3241.pdf On some inequality of Hermite-Hadamard type Szymon Wąsowicz, Alfred Witkowski It is well-known that the left term of the classical Hermite-Hadamard inequality is closer to the integral mean value than the right one. We show that in the multivariate case it is not true. Moreover, we introduce some related inequality comparing the methods of the approximate integration, which is optimal. We also present its counterpart of Fejér type. convex function, Hermite-Hadamard inequality, Fejér inequality, simplex, approximate integration 2012 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol32iss3art15 https://www.opuscula.agh.edu.pl/vol32/3/art/opuscula_math_3242.pdf Existence results for mild solutions of impulsive periodic systems YanLong Yang, JinRong Wang By applying the Horn's fixed point theorem, we prove the existence of \(T_0\)-periodic \(PC\)-mild solution of impulsive periodic systems when \(PC\)-mild solutions are ultimate bounded. impulsive periodic systems, \(T_0\)-periodic \(PC\)-mild solution, Horn's fixed point theorem, existence 2012 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol32iss3art16 https://www.opuscula.agh.edu.pl/vol32/3/art/opuscula_math_3243.pdf Existence of periodic solutions for totally nonlinear neutral differential equations with functional delay Ernest Yankson We use a variant of Krasnoselskii's fixed point theorem by T. A. Burton to show that the nonlinear neutral differential equation with functional delay \[x'(t) = -a(t)h(x(t)) +c(t)x'(t-g(t)) + q(t,x(t) x(t-g(t)))\] has a periodic solution. fixed point, large contraction, periodic solution, totally nonlinear neutral equation 2012 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol32iss4 https://www.opuscula.agh.edu.pl/om-vol32iss4art1 https://www.opuscula.agh.edu.pl/vol32/4/art/opuscula_math_3244.pdf Convolutions, integral transforms and integral equations by means of the theory of reproducing kernels Luis P. Castro, Saburou Saitoh, Nguyen Minh Tuan This paper introduces a general concept of convolutions by means of the theory of reproducing kernels which turns out to be useful for several concrete examples and applications. Consequent properties are exposed (including, in particular, associated norm inequalities). Hilbert space, linear transform, reproducing kernel, linear mapping, convolution, norminequality, integral equation, Tikhonov regularization 2012 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol32iss4art2 https://www.opuscula.agh.edu.pl/vol32/4/art/opuscula_math_3245.pdf Energy integral of the Stokes flow in a singularly perturbed exterior domain Matteo Dalla Riva We consider a pair of domains \(\Omega ^b\) and \(\Omega ^s\) in \(\mathbb{R}^n\) and we assume that the closure of \(\Omega ^b\) does not intersect the closure of \(\epsilon \Omega ^s\) for \(\epsilon \in (0,\epsilon _0)\). Then for a fixed \(\epsilon \in (0,\epsilon_0)\) we consider a boundary value problem in \(\mathbb{R}^n \setminus (\Omega ^b \cup \epsilon \Omega ^s)\) which describes the steady state Stokes flow of an incompressible viscous fluid past a body occupying the domain \(\Omega ^b\) and past a small impurity occupying the domain \(\epsilon \Omega ^s\). The unknown of the problem are the velocity field \(u\) and the pressure field \(p\), and we impose the value of the velocity field \(u\) on the boundary both of the body and of the impurity. We assume that the boundary velocity on the impurity displays an arbitrarily strong singularity when \(\epsilon\) tends to 0. The goal is to understand the behaviour of the strain energy of \( (u, p)\) for \(\epsilon\) small and positive. The methods developed aim at representing the limiting behaviour in terms of analytic maps and possibly singular but completely known functions of \(\epsilon\), such as \(\epsilon ^{-1}\), \(\log \epsilon\). boundary value problem for the Stokes system, singularly perturbed exterior domain, real analytic continuation in Banach space 2012 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol32iss4art3 https://www.opuscula.agh.edu.pl/vol32/4/art/opuscula_math_3246.pdf A note on a one-parameter family of non-symmetric number triangles Maria Irene Falcão, Helmuth R. Malonek The recent growing interest in special Clifford algebra valued polynomial solutions of generalized Cauchy-Riemann systems in \((n + 1)\)-dimensional Euclidean spaces suggested a detailed study of the arithmetical properties of their coefficients, due to their combinatoric relevance. This concerns, in particular, a generalized Appell sequence of homogeneous polynomials whose coefficient set can be treated as a one-parameter family of non-symmetric triangles of fractions. The discussion of its properties, similar to those of the ordinary Pascal triangle (which itself does not belong to the family), is carried out in this paper. Clifford analysis, generalized Appell polynomials, number triangle, central binomial coefficient, binomial identity 2012 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol32iss4art4 https://www.opuscula.agh.edu.pl/vol32/4/art/opuscula_math_3247.pdf Symbolic approach to the general cubic decomposition of polynomial sequences. Results for several orthogonal and symmetric cases Teresa A. Mesquita, Z. Da Rocha We deal with a symbolic approach to the cubic decomposition (CD) of polynomial sequences - presented in a previous article referenced herein - which allows us to compute explicitly the first elements of the nine component sequences of a CD. Properties are investigated and several experimental results are discussed, related to the CD of some widely known orthogonal sequences. Results concerning the symmetric character of the component sequences are established. symmetric polynomials, orthogonal polynomials, cubic decomposition, symbolic computations, Mathematica 8.0.1.0 2012 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol32iss4art5 https://www.opuscula.agh.edu.pl/vol32/4/art/opuscula_math_3248.pdf Recursively arbitrarily vertex-decomposable graphs Olivier Baudon, Frédéric Gilbert, Mariusz Woźniak A graph \(G = (V;E)\) is arbitrarily vertex decomposable if for any sequence \(\tau\) of positive integers adding up to \(|V|\), there is a sequence of vertex-disjoint subsets of \(V\) whose orders are given by \(\tau\), and which induce connected graphs. The main aim of this paper is to study the recursive version of this problem. We present a solution for trees, suns, and partially for a class of 2-connected graphs called balloons. arbitrary vertex decomposable (AVD) graph, recursively AVD graphs 2012 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol32iss4art6 https://www.opuscula.agh.edu.pl/vol32/4/art/opuscula_math_3249.pdf Bounds on perfect k-domination in trees: an algorithmic approach B. Chaluvaraju, K. A. Vidya Let \(k\) be a positive integer and \(G = (V;E)\) be a graph. A vertex subset \(D\) of a graph \(G\) is called a perfect \(k\)-dominating set of \(G\) if every vertex \(v\) of \(G\) not in \(D\) is adjacent to exactly \(k\) vertices of \(D\). The minimum cardinality of a perfect \(k\)-dominating set of \(G\) is the perfect \(k\)-domination number \(\gamma_{kp}(G)\). In this paper, a sharp bound for \(\gamma_{kp}(T)\) is obtained where \(T\) is a tree. \(k\)-domination, perfect domination, perfect \(k\)-domination 2012 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol32iss4art7 https://www.opuscula.agh.edu.pl/vol32/4/art/opuscula_math_3250.pdf A note on the independent roman domination in unicyclic graphs Mustapha Chellali, Nader Jafari Rad A Roman dominating function (RDF) on a graph \(G = (V;E)\) is a function \(f : V \to \{0, 1, 2\}\) satisfying the condition that every vertex \(u\) for which \(f(u) = 0\) is adjacent to at least one vertex \(v\) for which \(f(v) = 2\). The weight of an RDF is the value \(f(V(G)) = \sum _{u \in V (G)} f(u)\). An RDF \(f\) in a graph \(G\) is independent if no two vertices assigned positive values are adjacent. The Roman domination number \(\gamma _R (G)\) (respectively, the independent Roman domination number \(i_{R}(G)\)) is the minimum weight of an RDF (respectively, independent RDF) on \(G\). We say that \(\gamma _R (G)\) strongly equals \(i_R (G)\), denoted by \(\gamma _R (G) \equiv i_R (G)\), if every RDF on \(G\) of minimum weight is independent. In this note we characterize all unicyclic graphs \(G\) with \(\gamma _R (G) \equiv i_R (G)\). Roman domination, independent Roman domination, strong equality 2012 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol32iss4art8 https://www.opuscula.agh.edu.pl/vol32/4/art/opuscula_math_3251.pdf Oscillation behavior of second order nonlinear neutral differential equations with deviating arguments Elmetwally M. Elabbasy, T. S. Hassan, O. Moaaz Oscillation criteria are established for second order nonlinear neutral differential equations with deviating arguments of the form \[r(t)\psi(x(t))|z'(t)|^{\alpha -1} z'(t)+ \int_a^b q(t,\xi)f(x(g(t,\phi)))d\sigma (\xi) =0,\quad t\gt t_0,\] where \(\alpha \gt 0\) and \(z(t)= x(t)+p(t)x(t-\tau)\). Our results improve and extend some known results in the literature. Some illustrating examples are also provided to show the importance of our results. oscillation, second order, neutral differential equations, deviating arguments 2012 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol32iss4art9 https://www.opuscula.agh.edu.pl/vol32/4/art/opuscula_math_3252.pdf On a class of nonhomogenous quasilinear problems in Orlicz-Sobolev spaces Asma Karoui Souayah We study the nonlinear boundary value problem \(-div ((a_1(|\nabla u(x)|)+a_2(|\nabla u(x)|))\nabla u(x))=\lambda |u|^{q(x)-2}u-\mu |u|^{\alpha(x)-2}u\) in \(\Omega\), \(u = 0\) on \(\partial \Omega\) , where \(\Omega\) is a bounded domain in \(\mathbb{R}^N\) with smooth boundary, \(\lambda\), \(\mu\) are positive real numbers, \(q\) and \(\alpha\) are continuous functions and \(a_1\), \(a_2\) are two mappings such that \(a_1(|t|)t\), \(a_2(|t|)t\) are increasing homeomorphisms from \(\mathbb{R}\) to \(\mathbb{R}\). The problem is analyzed in the context of Orlicz-Soboev spaces. First we show the existence of infinitely many weak solutions for any \(\lambda,\mu \gt 0\). Second we prove that for any \(\mu \gt 0\), there exists \(\lambda_*\) sufficiently small, and \(\lambda ^*\) large enough such that for any \(\lambda \in (0,\lambda_*)\cup(\lambda^*,\infty)\), the above nonhomogeneous quasilinear problem has a non-trivial weak solution. variable exponent Lebesgue space, Orlicz-Sobolev space, critical point, weak solution 2012 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol32iss4art10 https://www.opuscula.agh.edu.pl/vol32/4/art/opuscula_math_3253.pdf False alarms in fault-tolerant dominating sets in graphs Mateusz Nikodem We develop the problem of fault-tolerant dominating sets (liar's dominating sets) in graphs. Namely, we consider a new kind of fault - a false alarm. Characterization of such fault-tolerant dominating sets in three different cases (dependent on the classification of the types of the faults) are presented. liar's dominating set, fault-tolerant dominating set, false alarm, Hamming distance 2012 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol32iss4art11 https://www.opuscula.agh.edu.pl/vol32/4/art/opuscula_math_3254.pdf On the maximum likelihood estimator in the generalized beta regression model Jerzy P. Rydlewski, Dominik Mielczarek The subject of this article is to present the beta - regression model, where we assume that one parameter in the model is described as a combination of algebraically independent continuous functions. The proposed beta model is useful when the dependent variable is continuous and restricted to the bounded interval. The parameters are obtained by maximum likelihood estimation. We prove that estimators are consistent and asymptotically normal. nonlinear regression, beta distribution, scale parameter, shape parameter, maximum likelihood estimator 2012 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol32iss4art12 https://www.opuscula.agh.edu.pl/vol32/4/art/opuscula_math_3255.pdf On the solvability of Dirichlet problem for the weighted p-Laplacian Ewa Szlachtowska The paper investigates the existence and uniqueness of weak solutions for a non-linear boundary value problem involving the weighted \(p\)-Laplacian. Our approach is based on variational principles and representation properties of the associated spaces. \(p\)-Laplacian, weak solutions, solvability 2012 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol32iss4art13 https://www.opuscula.agh.edu.pl/vol32/4/art/opuscula_math_3256.pdf Existence and asymptotic behavior of positive continuous solutions for a nonlinear elliptic system in the half space Sameh Turki This paper deals with the existence and the asymptotic behavior of positive continuous solutions of the nonlinear elliptic system \(\Delta u=p(x)u^{\alpha}v^r\), \(\Delta v = q(x)u^s v^{\beta}\), in the half space \(\mathbb{R}^n_+ :=\{x=(x_1,..., x_n)\in \mathbb{R}^n : x_n \gt 0\}\), \(n \geq 2\), where \(\alpha, \beta \gt 1\) and \(r, s \geq 0\). The functions \(p\) and \(q\) are required to satisfy some appropriate conditions related to the Kato class \(K^{\infty}(\mathbb{R}^n_+)\). Our approach is based on potential theory tools and the use of Schauder's fixed point theorem. asymptotic behavior, elliptic system, regular equation 2012 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol33iss1 https://www.opuscula.agh.edu.pl/om-vol33iss1art1 https://www.opuscula.agh.edu.pl/vol33/1/art/opuscula_math_3301.pdf A note on Blasius type boundary value problems Grzegorz Andrzejczak, Magdalena Nockowska-Rosiak, Bogdan Przeradzki The existence and uniqueness of a solution to a generalized Blasius equation with asymptotic boundary conditions is proved. A new numerical approximation method is proposed. Blasius equation, shooting method 2013 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol33iss1art2 https://www.opuscula.agh.edu.pl/vol33/1/art/opuscula_math_3302.pdf On vertex b-critical trees Mostafa Blidia, Noureddine Ikhlef Eschouf, Frédéric Maffray A b-coloring is a proper coloring of the vertices of a graph such that each color class has a vertex that has neighbors of all other colors. The b-chromatic number of a graph \(G\) is the largest \(k\) such that \(G\) admits a b-coloring with \(k\) colors. A graph \(G\) is b-critical if the removal of any vertex of \(G\) decreases the b-chromatic number. We prove various properties of b-critical trees. In particular, we characterize b-critical trees. b-coloring, b-critical graphs, b-critical trees 2013 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol33iss1art3 https://www.opuscula.agh.edu.pl/vol33/1/art/opuscula_math_3303.pdf The lq-controller synthesis problem for infinite-dimensional systems in factor form Piotr Grabowski The general lq-problem with infinite time horizon for well-posed infinite-dimensional systems has been investigated by George Weiss and Martin Weiss and by Olof Staffans with a complement by Kalle Mikkola and Olof Staffans. Our aim in this paper is to present a solution of a general lq-optimal controller synthesis problem for infinite-dimensional systems in factor form. The systems in factor form are an alternative to additive models, used in the theory of well-posed systems, which rely on leading the analysis exclusively within the basic state space. As a result of applying the simplified analysis in terms of the factor systems and an another derivation technique, we obtain an equivalent, however, astonishingly not the same formulae expressing the optimal controller in the time-domain and the method of spectral factorization. The results are illustrated by two examples of the construction of both the optimal control and optimal controller for some standard lq-problems met in literature: a control problem for a class of boundary controlled hyperbolic equations initiated by Chapelon and Xu, to which we give full solution and an example of the synthesis of the optimal control/controller for the standard lq-problem with infinite-time horizon met in the problem of improving a river water quality by artificial aeration, proposed by Żołopa and the author. control of infinite-dimensional systems, semigroups, infinite-time lq-control problem 2013 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol33iss1art4 https://www.opuscula.agh.edu.pl/vol33/1/art/opuscula_math_3304.pdf Eigenvalue-eigenfunction problem for Steklov's smoothing operator and differential-difference equations of mixed type Serguei I. Iakovlev, Valentina Iakovleva It is shown that any \(\mu \in \mathbb{C}\) is an infinite multiplicity eigenvalue of the Steklov smoothing operator \(S_h\) acting on the space \(L^1_{loc}(\mathbb{R})\). For \(\mu \neq 0\) the eigenvalue-eigenfunction problem leads to studying a differential-difference equation of mixed type. An existence and uniqueness theorem is proved for this equation. Further a transformation group is defined on a countably normed space of initial functions and the spectrum of the generator of this group is studied. Some possible generalizations are pointed out. Steklov's smoothing operator, spectrum, eigenvalues, eigenfunctions, mixed-type differential-difference equations, initial function, method of steps, countably normed space, transformation group, generator 2013 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol33iss1art5 https://www.opuscula.agh.edu.pl/vol33/1/art/opuscula_math_3305.pdf The maximum principle for viscosity solutions of elliptic differential functional equations Adrian Karpowicz This paper is devoted to the study of the maximum principle for the elliptic equation with a deviated argument. We will consider viscosity solutions of this equation. maximum principle, viscosity solution, elliptic equations 2013 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol33iss1art6 https://www.opuscula.agh.edu.pl/vol33/1/art/opuscula_math_3306.pdf Notes on topological indices of graph and its complement Tomáš Madaras, Martina Mockovčiaková In this note, we derive the lower bound on the sum for Wiener index of bipartite graph and its bipartite complement, as well as the lower and upper bounds on this sum for the Randić index and Zagreb indices. We also discuss the quality of these bounds. Wiener index, Zagreb index, Randić index, bipartite graph, bipartite complement 2013 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol33iss1art7 https://www.opuscula.agh.edu.pl/vol33/1/art/opuscula_math_3307.pdf Boundary value problems for nonlinear fractional differential equations and inclusions with nonlocal and fractional integral boundary conditions Sotiris K. Ntouyas This paper studies the boundary value problem of nonlinear fractional differential equations and inclusions of order \(q \in (1,2]\) with nonlocal and integral boundary conditions. Some new existence and uniqueness results are obtained by using fixed point theorems. fractional differential equations, fractional differential inclusions, nonlocal conditions, fractional integral boundary conditions, existence, contraction principle, nonlinear contraction 2013 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol33iss1art8 https://www.opuscula.agh.edu.pl/vol33/1/art/opuscula_math_3308.pdf A vertex operator representation of solutions to the Gurevich-Zybin hydrodynamical equation Yarema A. Prykarpatsky, Denis Blackmore, Jolanta Golenia, Anatoliy K. Prykarpatsky An approach based on the spectral and Lie-algebraic techniques for constructing vertex operator representation for solutions to a Riemann type hydrodynamical hierarchy is devised. A functional representation generating an infinite hierarchy of dispersive Lax type integrable flows is obtained. Lax type integrability, vertex operator representation, Lax integrability, Lie-algebraic approach 2013 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol33iss1art9 https://www.opuscula.agh.edu.pl/vol33/1/art/opuscula_math_3309.pdf Generating the exponentially stable C_{0}-semigroup in a nonhomogeneous string equation with damping at the end Łukasz Rzepnicki Small vibrations of a nonhomogeneous string of length one with left end fixed and right one moving with damping are described by the one-dimensional wave equation \[\begin{cases} v_{tt}(x,t) - \frac{1}{\rho}v_{xx}(x,t) = 0, x \in [0,1], t \in [0, \infty),\\ v(0,t) = 0, v_x(1,t) + hv_t(1,t) = 0, \\ v(x,0) = v_0(x), v_t(x,0) = v_1(x),\end{cases}\] where \(\rho\) is the density of the string and \(h\) is a complex parameter. This equation can be rewritten in an operator form as an abstract Cauchy problem for the closed, densely defined operator \(B\) acting on a certain energy space \(H\). It is proven that the operator \(B\) generates the exponentially stable \(C_0\)-semigroup of contractions in the space \(H\) under assumptions that \(\text{Re}\; h \gt 0\) and the density function is of bounded variation satisfying \(0 \lt m \leq \rho(x)\) for a.e. \(x \in [0, 1]\). nonhomogeneous string, one-dimensional wave equation, exponentially stable \(C_0\)-semigroup, Hilbert space 2013 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol33iss1art10 https://www.opuscula.agh.edu.pl/vol33/1/art/opuscula_math_3310.pdf Some efficient seventh-order derivative-free families in root-finding Fazlollah Soleymani The interest in efficient root-finding iterations is nowadays growing and influenced by the widespread use of high-speed computers. On the other hand, the calculation of derivatives is often hard, when the problems are formulated in terms of nonlinear equations and as a result, the importance of derivative-free methods emerges. For these reasons, some efficient three-step families of iterations for solving nonlinear equations are suggested, where the analytical proofs show their seventh-order error equations consuming only four function evaluations per iteration. We employ hard numerical test problems to illustrate the accuracy of the new methods from the families. numerical analysis, derivative-free families, order, iterative methods 2013 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol33iss1art11 https://www.opuscula.agh.edu.pl/vol33/1/art/opuscula_math_3311.pdf Planar nonautonomous polynomial equations V. The Abel equation Paweł Wilczyński We give a full description of the dynamics of the Abel equation \(\dot{z}=z^3+f(t)\) for some special complex valued \(f\). We also prove the existence of at least three periodic solutions for equations of the form \(\dot{z}=z^n+f(t)\) for odd \(n \geq 5\). periodic orbits, polynomial equations 2013 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol33iss1art12 https://www.opuscula.agh.edu.pl/vol33/1/art/opuscula_math_3312.pdf On the multiplicative Zagreb coindex of graphs Kexiang Xu, Kinkar Ch. Das, Kechao Tang For a (molecular) graph \(G\) with vertex set \(V(G)\) and edge set \(E(G)\), the first and second Zagreb indices of \(G\) are defined as \(M_1(G) = \sum_{v \in V(G)} d_G(v)^2\) and \(M_2(G) = \sum_{uv \in E(G)} d_G(u)d_G(v)\), respectively, where \(d_G(v)\) is the degree of vertex \(v\) in \(G\). The alternative expression of \(M_1(G)\) is \(\sum_{uv \in E(G)}(d_G(u) + d_G(v))\). Recently Ashrafi, Došlić and Hamzeh introduced two related graphical invariants \(\overline{M}_1(G) = \sum_{uv \notin E(G)}(d_G(u)+d_G(v))\) and \(\overline{M}_2(G) = \sum_{uv \notin E(G)} d_G(u)d_G(v)\) named as first Zagreb coindex and second Zagreb coindex, respectively. Here we define two new graphical invariants \(\overline{\Pi}_1(G) = \prod_{uv \notin E(G)}(d_G(u)+d_G(v))\) and \(\overline{\Pi}_2(G) = \prod_{uv \notin E(G)} d_G(u)d_G(v)\) as the respective multiplicative versions of \(\overline{M}_i\) for \(i = 1, 2\). In this paper, we have reported some properties, especially upper and lower bounds, for these two graph invariants of connected (molecular) graphs. Moreover, some corresponding extremal graphs have been characterized with respect to these two indices. vertex degree, tree, upper or lower bound 2013 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol33iss2 https://www.opuscula.agh.edu.pl/om-vol33iss2art1 https://www.opuscula.agh.edu.pl/vol33/2/art/opuscula_math_3313.pdf Fractional order Riemann-Liouville integral inclusions with two independent variables and multiple delay Saïd Abbas, Mouffak Benchohra In the present paper we investigate the existence of solutions for a system of integral inclusions of fractional order with multiple delay. Our results are obtained upon suitable fixed point theorems, namely the Bohnenblust-Karlin fixed point theorem for the convex case and the Covitz-Nadler for the nonconvex case. integral inclusion, left-sided mixed Riemann-Liouville integral, time delay solution, fixed point 2013 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol33iss2art2 https://www.opuscula.agh.edu.pl/vol33/2/art/opuscula_math_3314.pdf Some generalized method for constructing nonseparable compactly supported wavelets in L^{2}(R^{2}) Wojciech Banaś In this paper we show some construction of nonseparable compactly supported bivariate wavelets. We deal with the dilation matrix \(A = \tiny{\left[\begin{matrix}0 & 2 \cr 1 & 0 \cr \end{matrix} \right]}\) and some three-row coefficient mask; that is a scaling function satisfies a dilation equation with scaling coefficients which can be contained in the set \(\{c_{n}\}_{n \in\mathcal{S}},\) where \(\mathcal{S}=S_{1} \times \{0,1,2\},\) \(S_{1} \subset \mathbb{N},\) \(\sharp S_{1} \lt \infty.\) compactly supported wavelet, compactly supported scaling function, multiresolution analysis, dilation matrix, orthonormality, accuracy 2013 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol33iss2art3 https://www.opuscula.agh.edu.pl/vol33/2/art/opuscula_math_3315.pdf Existence of solution of sub-elliptic equations on the Heisenberg group with critical growth and double singularities Jianqing Chen, Eugénio M. Rocha For a class of sub-elliptic equations on Heisenberg group \(\mathbb{H}^N\) with Hardy type singularity and critical nonlinear growth, we prove the existence of least energy solutions by developing new techniques based on the Nehari constraint. This result extends previous works, e.g., by Han et al. [Hardy-Sobolev type inequalities on the H-type group, Manuscripta Math. 118 (2005), 235–252]. sub-elliptic equations, Heisenberg group, Least energy solutions 2013 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol33iss2art4 https://www.opuscula.agh.edu.pl/vol33/2/art/opuscula_math_3316.pdf Stability by Krasnoselskii's theorem in totally nonlinear neutral differential equations Ishak Derrardjia, Abdelouaheb Ardjouni, Ahcene Djoudi In this paper we use fixed point methods to prove asymptotic stability results of the zero solution of a class of totally nonlinear neutral differential equations with functional delay. The study concerns \[x'(t)=a(t)x^3(t)+c(t)x'(t-r(t))+b(t)x^3(t-r(t)).\] The equation has proved very challenging in the theory of Liapunov's direct method. The stability results are obtained by means of Krasnoselskii-Burton's theorem and they improve on the work of T.A. Burton (see Theorem 4 in [Liapunov functionals, fixed points, and stability by Krasnoselskii's theorem, Nonlinear Studies 9 (2001), 181-190]) in which he takes \(c=0\) in the above equation. fixed point, stability, nonlinear neutral equation, Krasnoselskii-Burton theorem 2013 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol33iss2art5 https://www.opuscula.agh.edu.pl/vol33/2/art/opuscula_math_3317.pdf A unified representation of some starlike and convex harmonic functions with negative coefficients R. M. El-Ashwah, M. K. Aouf, A. A. M. Hassan, A. H. Hassan In this paper we introduce a unified representation of starlike and convex harmonic functions with negative coefficients, related to uniformly starlike and uniformly convex analytic functions. We obtain extreme points, distortion bounds, convolution conditions and convex combinations for this family. harmonic, analytic, univalent, sense-preserving, extreme points 2013 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol33iss2art6 https://www.opuscula.agh.edu.pl/vol33/2/art/opuscula_math_3318.pdf Inequalities for regularized determinants of operators with the Nakano type modulars Michael Gil' Let \(\{p_k\}\) be a nondecreasing sequence of integers, and \(A\) be a compact operator in a Hilbert space whose eigenvalues and singular values are \(\lambda_k(A)\) and \(s_k(A)\) \((k=1, 2, .... )\), respectively. We establish upper and lower bounds for the regularized determinant \[\prod_{k=1}^\infty (1-\lambda_k(A)){\rm exp}\;[\sum_{m=1}^{p_k-1} \frac{\lambda_k^m(A)}{m}],\mbox{ assuming that } \sum_{j=1}^{\infty} \frac{s_j^{p_j}(A/c)}{p_j}\lt \infty\] for a constant \(c\in (0,1)\). Hilbert space, compact operators, regularized determinant, Nakano type modular 2013 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol33iss2art7 https://www.opuscula.agh.edu.pl/vol33/2/art/opuscula_math_3319.pdf Multiple solutions for systems of multi-point boundary value problems John R. Graef, Shapour Heidarkhani, Lingju Kong In this paper, we establish the existence of at least three solutions of the multi-point boundary value system \[\left\{\begin{array}{ll} -(\phi_{p_i}(u'_{i}))'=\lambda F_{u_{i}}(x,u_{1},\ldots,u_{n}),\ t\in(0,1),\\ u_{i}(0)=\sum_{j=1}^m a_ju_i(x_j),\ u_{i}(1)=\sum_{j=1}^m b_ju_i(x_j), \end{array}\right. i=1,\ldots,n.\] The approaches used are based on variational methods and critical point theory. multiple solutions, multi-point boundary value problem, critical point theory 2013 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol33iss2art8 https://www.opuscula.agh.edu.pl/vol33/2/art/opuscula_math_3320.pdf Variational characterizations for eigenfunctions of analytic self-adjoint operator functions Georgios Katsouleas, John Maroulas In this paper we consider Rellich's diagonalization theorem for analytic self-adjoint operator functions and investigate variational principles for their eigenfunctions and interlacing statements. As an application, we present a characterization for the eigenvalues of hyperbolic operator polynomials. operator functions, eigenfunctions, eigenvalues, variational principles 2013 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol33iss2art9 https://www.opuscula.agh.edu.pl/vol33/2/art/opuscula_math_3321.pdf Random integral equations on time scales Vasile Lupulescu, Cristina Lungan In this paper, we present the existence and uniqueness of random solution of a random integral equation of Volterra type on time scales. We also study the asymptotic properties of the unique random solution. random integral equations, time scale, existence, uniqueness, stability 2013 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol33iss2art10 https://www.opuscula.agh.edu.pl/vol33/2/art/opuscula_math_3322.pdf A new method for solving ill-conditioned linear systems Fazlollah Soleymani An accurate numerical method is established for matrix inversion. It is shown theoretically that the scheme possesses the high order of convergence of seven. Subsequently, the method is taken into account for solving linear systems of equations. The accuracy of the contributed iterative method is clarified on solving numerical examples when the coefficient matrices are ill-conditioned. All of the computations are performed on a PC using several programs written in Mathematica 7. matrix inversion, linear systems, Hilbert matrix, ill-conditioned, approximate inverse 2013 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol33iss2art11 https://www.opuscula.agh.edu.pl/vol33/2/art/opuscula_math_3323.pdf The forwarding indices of graphs - a survey Jun-Ming Xu, Min Xu A routing \(R\) of a connected graph \(G\) of order \(n\) is a collection of \(n(n-1)\) simple paths connecting every ordered pair of vertices of \(G\). The vertex-forwarding index \(\xi(G,R)\) of \(G\) with respect to a routing \(R\) is defined as the maximum number of paths in \(R\) passing through any vertex of \(G\). The vertex-forwarding index \(\xi(G)\) of \(G\) is defined as the minimum \(\xi(G,R)\) over all routings \(R\) of \(G\). Similarly, the edge-forwarding index \(\pi(G,R)\) of \(G\) with respect to a routing \(R\) is the maximum number of paths in \(R\) passing through any edge of \(G\). The edge-forwarding index \(\pi(G)\) of \(G\) is the minimum \(\pi(G,R)\) over all routings \(R\) of \(G\). The vertex-forwarding index or the edge-forwarding index corresponds to the maximum load of the graph. Therefore, it is important to find routings minimizing these indices and thus has received much research attention for over twenty years. This paper surveys some known results on these forwarding indices, further research problems and several conjectures, also states some difficulty and relations to other topics in graph theory. graph theory, vertex-forwarding index, edge-forwarding index, routing, networks 2013 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol33iss2art12 https://www.opuscula.agh.edu.pl/vol33/2/art/opuscula_math_3324.pdf Existence of critical elliptic systems with boundary singularities Jianfu Yang, Yimin Zhou In this paper, we are concerned with the existence of positive solutions of the following nonlinear elliptic system involving critical Hardy-Sobolev exponent \begin{equation*}\label{eq:1}(*) \left\{ \begin{array}{lll} -\Delta u= \frac{2\alpha}{\alpha+\beta}\frac{u^{\alpha-1}v^\beta}{|x|^s}-\lambda u^p, & \quad {\rm in}\quad \Omega,\\[2mm] -\Delta v= \frac{2\beta}{\alpha+\beta}\frac{u^\alpha v^{\beta-1}}{|x|^s}-\lambda v^p, & \quad {\rm in}\quad \Omega,\\[2mm] u\gt 0, v\gt 0, &\quad {\rm in}\quad \Omega,\\[2mm] u=v=0, &\quad {\rm on}\quad \partial\Omega, \end{array} \right. \end{equation*} where \(N\geq 4\) and \(\Omega\) is a \(C^1\) bounded domain in \(\mathbb{R}^N\) with \(0\in\partial\Omega\). \(0\lt s \lt 2\), \(\alpha+\beta=2^*(s)=\frac{2(N-s)}{N-2}\), \(\alpha,\beta\gt 1\), \(\lambda\gt 0\) and \(1 \lt p\lt \frac{N+2}{N-2}\). The case when 0 belongs to the boundary of \(\Omega\) is closely related to the mean curvature at the origin on the boundary. We show in this paper that problem \((*)\) possesses at least a positive solution. existence, compactness, critical Hardy-Sobolev exponent, nonlinear system 2013 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol33iss3 https://www.opuscula.agh.edu.pl/om-vol33iss3art1 https://www.opuscula.agh.edu.pl/vol33/3/art/opuscula_math_3325.pdf A generalized white noise space approach to stochastic integration for a class of Gaussian stationary increment processes Daniel Alpay, Alon Kipnis Given a Gaussian stationary increment processes, we show that a Skorokhod-Hitsuda stochastic integral with respect to this process, which obeys the Wick-Itô calculus rules, can be naturally defined using ideas taken from Hida's white noise space theory. We use the Bochner-Minlos theorem to associate a probability space to the process, and define the counterpart of the S-transform in this space. We then use this transform to define the stochastic integral and prove an associated Itô formula. stochastic integral, white noise space, fractional Brownian motion 2013 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol33iss3art2 https://www.opuscula.agh.edu.pl/vol33/3/art/opuscula_math_3326.pdf On the decomposition of families of quasinormal operators Zbigniew Burdak The canonical injective decomposition of a jointly quasinormal family of operators is given. Relations between the decomposition of a quasinormal operator T and the decomposition of a partial isometry in the polar decomposition of T are described. The decomposition of pairs of commuting quasinormal partial isometries and its applications to pairs of commuting quasinormal operators is shown. Examples are given. multiple canonical decomposition, quasinormal operators, partial isometry 2013 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol33iss3art3 https://www.opuscula.agh.edu.pl/vol33/3/art/opuscula_math_3327.pdf Existence results for Dirichlet problems with degenerated p-Laplacian Albo Carlos Cavalheiro In this article, we prove the existence of entropy solutions for the Dirichlet problem \[(P)\left\{ \begin{array}{ll} & -{\rm div}[{\omega}(x){\vert{\nabla}u\vert}^{p-2}{\nabla}u]= f(x) - {\rm div}(G(x)),\ \ {\rm in} \ \ {\Omega} \\ & u(x)=0, \ \ {\rm in} \ \ {\partial\Omega} \end{array} \right.\] where \(\Omega\) is a bounded open set of \(\mathbb{R}^N\) \( (N \geq 2)\), \(f \in L^1(\Omega)\) and \(G/\omega \in [L^p(\Omega,\omega)]^N\). degenerate elliptic equations, entropy solutions, weighted Sobolev spaces 2013 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol33iss3art4 https://www.opuscula.agh.edu.pl/vol33/3/art/opuscula_math_3328.pdf Periodic solutions in multivariate invariance arguments Jacek Chudziak, Sebastian Wójcik Inspired by the recent results of A. E. Abbas we determine continuous multivariate utility functions invariant with respect to a wide family of transformations related to the shift transformations. multivariate utility function, invariance, periodic solution, additive function, exponential function 2013 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol33iss3art5 https://www.opuscula.agh.edu.pl/vol33/3/art/opuscula_math_3329.pdf Weyl-Titchmarsh theory for Sturm-Liouville operators with distributional potentials Jonathan Eckhardt, Fritz Gesztesy, Roger Nichols, Gerald Teschl We systematically develop Weyl-Titchmarsh theory for singular differential operators on arbitrary intervals \((a,b) \subseteq \mathbb{R}\) associated with rather general differential expressions of the type \begin{equation*}\tau f = \frac{1}{\tau} (-(p[f'+sf])'+sp[f'+sf]+qf),\end{equation*} where the coefficients \(p, q, r, s\) are real-valued and Lebesgue measurable on \((a,b)\), with \(p \neq 0\), \(r \gt 0\) a.e. on \((a,b)\), and \(p^{-1}, q, r, s \in L_{loc}^1((a,b),dx)\), and \(f\) is supposed to satisfy \begin{equation*} f \in AC_{loc}((a,b)), p[f'+sf] \in AC_{loc}((a,b)). \end{equation*} In particular, this setup implies that \(\tau\) permits a distributional potential coefficient, including potentials in \(H_{loc}^{-1}((a,b))\). We study maximal and minimal Sturm-Liouville operators, all self-adjoint restrictions of the maximal operator \(T_{max}\), or equivalently, all self-adjoint extensions of the minimal operator \(T_{min}\), all self-adjoint boundary conditions (separated and coupled ones), and describe the resolvent of any self-adjoint extension of \(T_{min}\). In addition, we characterize the principal object of this paper, the singular Weyl-Titchmarsh-Kodaira m-function corresponding to any self-adjoint extension with separated boundary conditions and derive the corresponding spectral transformation, including a characterization of spectral multiplicities and minimal supports of standard subsets of the spectrum. We also deal with principal solutions and characterize the Friedrichs extension of \(T_{min}\). Finally, in the special case where \(\tau\) is regular, we characterize the Krein-von Neumann extension of \(T_{min}\) and also characterize all boundary conditions that lead to positivity preserving, equivalently, improving, resolvents (and hence semigroups). Sturm-Liouville operators, distributional coefficients, Weyl-Titchmarsh theory, Friedrichs and Krein extensions, positivity preserving and improving semigroups 2013 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol33iss3art6 https://www.opuscula.agh.edu.pl/vol33/3/art/opuscula_math_3330.pdf The Putnam-Fuglede property for paranormal and ∗-paranormal operators Patryk Pagacz An operator \(T \in {\cal B}(H)\) is said to have the Putnam-Fuglede commutativity property (PF property for short) if \(T^*X = XJ\) for any \(X \in {\cal B}(K,H)\) and any isometry \(J \in {\cal B}(K)\) such that \(TX = XJ^*\). The main purpose of this paper is to examine if paranormal operators have the PF property. We prove that \(k*\)-paranormal operators have the PF property. Furthermore, we give an example of a paranormal without the PF property. power-bounded operators, paranormal operators, \(*\)-paranormal operators, \(k\)-paranormal operators, \(k*\)-paranormal operators, the Putnam-Fuglede theorem 2013 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol33iss3art7 https://www.opuscula.agh.edu.pl/vol33/3/art/opuscula_math_3331.pdf On intersections of Cantor sets: Hausdorff measure Steen Pedersen, Jason D. Phillips We establish formulas for bounds on the Haudorff measure of the intersection of certain Cantor sets with their translates. As a consequence we obtain a formula for the Hausdorff dimensions of these intersections. Hausdorff measure, fractal, Cantor set, translation, intersection, digit expansion 2013 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol33iss4 https://www.opuscula.agh.edu.pl/om-vol33iss4art1 https://www.opuscula.agh.edu.pl/vol33/4/art/opuscula_math_3332.pdf Concavity of solutions of a 2n-th order problem with symmetry Abdulmalik Al Twaty, Paul W. Eloe In this article we apply an extension of a Leggett-Williams type fixed point theorem to a two-point boundary value problem for a \(2n\)-th order ordinary differential equation. The fixed point theorem employs concave and convex functionals defined on a cone in a Banach space. Inequalities that extend the notion of concavity to \(2n\)-th order differential inequalities are derived and employed to provide the necessary estimates. Symmetry is employed in the construction of the appropriate Banach space. Fixed-point theorems, concave and convex functionals, differential inequalities, symmetry 2013 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol33iss4art2 https://www.opuscula.agh.edu.pl/vol33/4/art/opuscula_math_3333.pdf Global existence and asymptotic behavior for a nonlinear degenerate SIS model Tarik Ali Ziane In this paper we investigate the global existence and asymptotic behavior of a reaction diffusion system with degenerate diffusion arising in the modeling and the spatial spread of an epidemic disease. reaction diffusion systems, degenerate diffusion, global existence, asymptotic behavior, population dynamics 2013 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol33iss4art3 https://www.opuscula.agh.edu.pl/vol33/4/art/opuscula_math_3334.pdf On the longest path in a recursively partitionable graph Julien Bensmail A connected graph \(G\) with order \(n \geq 1\) is said to be recursively arbitrarily partitionable (R-AP for short) if either it is isomorphic to \(K_1\), or for every sequence \((n_1, \ldots , n_p)\) of positive integers summing up to \(n\) there exists a partition \((V_1, \ldots , V_p)\) of \(V(G)\) such that each \(V_i\) induces a connected R-AP subgraph of \(G\) on \(n_i\) vertices. Since previous investigations, it is believed that a R-AP graph should be 'almost traceable' somehow. We first show that the longest path of a R-AP graph on \(n\) vertices is not constantly lower than \(n\) for every \(n\). This is done by exhibiting a graph family \(\mathcal{C}\) such that, for every positive constant \(c \geq 1\), there is a R-AP graph in \(\mathcal{C}\) that has arbitrary order \(n\) and whose longest path has order \(n-c\). We then investigate the largest positive constant \(c' \lt 1\) such that every R-AP graph on \(n\) vertices has its longest path passing through \(n \cdot c'\) vertices. In particular, we show that \(c' \leq \frac{2}{3}\). This result holds for R-AP graphs with arbitrary connectivity. recursively partitionable graph, longest path 2013 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol33iss4art4 https://www.opuscula.agh.edu.pl/vol33/4/art/opuscula_math_3335.pdf A note on k-Roman graphs Ahmed Bouchou, Mostafa Blidia, Mustapha Chellali Let \(G=\left(V,E\right)\) be a graph and let \(k\) be a positive integer. A subset \(D\) of \(V\left( G\right) \) is a \(k\)-dominating set of \(G\) if every vertex in \(V\left( G\right) \backslash D\) has at least \(k\) neighbours in \(D\). The \(k\)-domination number \(\gamma_{k}(G)\) is the minimum cardinality of a \(k\)-dominating set of \(G.\) A Roman \(k\)-dominating function on \(G\) is a function \(f\colon V(G)\longrightarrow\{0,1,2\}\) such that every vertex \(u\) for which \(f(u)=0\) is adjacent to at least \(k\) vertices \(v_{1},v_{2},\ldots ,v_{k}\) with \(f(v_{i})=2\) for \(i=1,2,\ldots ,k.\) The weight of a Roman \(k\)-dominating function is the value \(f(V(G))=\sum_{u\in V(G)}f(u)\) and the minimum weight of a Roman \(k\)-dominating function on \(G\) is called the Roman \(k\)-domination number \(\gamma_{kR}\left( G\right)\) of \(G\). A graph \(G\) is said to be a \(k\)-Roman graph if \(\gamma_{kR}(G)=2\gamma_{k}(G).\) In this note we study \(k\)-Roman graphs. Roman \(k\)-domination, \(k\)-Roman graph 2013 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol33iss4art5 https://www.opuscula.agh.edu.pl/vol33/4/art/opuscula_math_3336.pdf Chaotic expansion in the G-expectation space Hacène Boutabia, Imen Grabsia In this paper, we are motivated by uncertainty problems in volatility. We prove the equivalent theorem of Wiener chaos with respect to \(G\)-Brownian motion in the framework of a sublinear expectation space. Moreover, we establish some relationship between Hermite polynomials and \(G\)-stochastic multiple integrals. An equivalent of the orthogonality of Wiener chaos was found. \(G\)-expectation, \(G\)-Brownian motion, \(G\)-multiple integrals, Hermite polynomials, \(G\)-Wiener chaos 2013 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol33iss4art6 https://www.opuscula.agh.edu.pl/vol33/4/art/opuscula_math_3337.pdf Contractive and optimal sets in Musielak-Orlicz spaces with a smoothness condition Anna Denkowska In this paper we use our recent generalization of a theorem of Jamison-Kamińska-Lewicki (characterizing one-complemented subspaces in Musielak-Orlicz sequence spaces defined by Musielak-Orlicz functions satisfying a general smoothness condition) in order to compare contractive and optimal sets in finite-dimensional Musielak-Orlicz \(\ell^{(n)}_\Phi\) spaces in the spirit of Kamińska-Lewicki. We also give an example illustrating the importance of the smoothness assumptions in our theorem. Musielak-Orlicz sequence spaces, one-complemented subspaces, contractive and optimal sets 2013 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol33iss4art7 https://www.opuscula.agh.edu.pl/vol33/4/art/opuscula_math_3338.pdf Universal third parts of any complete 2-graph and none of DK_{5} Artur Fortuna, Zdzisław Skupień It is shown that there is no digraph \(F\) which could decompose the complete digraph on 5 vertices minus any 2-arc remainder into three parts isomorphic to \(F\) for each choice of the remainder. On the other hand, for each \(n\ge3\) there is a universal third part \(F\) of the complete 2-graph \(^2K_n\) on \(n\) vertices, i.e., for each edge subset \(R\) of size \(|R|=\|^2K_n\| \bmod 3\), there is an \(F\)-decomposition of \(^2K_n-R\). Using an exhaustive computer-aided search, we find all, exactly six, mutually nonisomorphic universal third parts of the 5-vertex 2-graph. Nevertheless, none of their orientations is a universal third part of the corresponding complete digraph. decomposition, remainder, universal parts, isomorphic parts 2013 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol33iss4art8 https://www.opuscula.agh.edu.pl/vol33/4/art/opuscula_math_3339.pdf A note on bounded harmonic functions over homogeneous trees Francisco Javier González Vieli Let \(\mathcal{T}_k\) be the homogeneous tree of degree \(k\geq 3\). J.M. Cohen and F. Colonna have proved that if \(f\) is a bounded harmonic function on \(\mathcal{T}_k\), then \(|f(x)-f(y)|\leq \|f\|_\infty\cdot 2(k-2)/k\) for any adjacent vertices \(x\) and \(y\) in \(\mathcal{T}_k\). We give here a new and very simple proof of this inequality. bounded harmonic function, homogeneous tree 2013 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol33iss4art9 https://www.opuscula.agh.edu.pl/vol33/4/art/opuscula_math_3340.pdf Picone-type identity and comparison results for a class of partial differential equations of order 4m Jaroslav Jaroš In the paper, a Picone-type identity for the weighted \(p\)-polyharmonic operator is established and comparison theorems and other qualitative results for a class of half-linear partial differential equations of the \(4m\)th order based on this identity are derived. \(p\)-polyharmonic operator, Picone's identity 2013 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol33iss4art10 https://www.opuscula.agh.edu.pl/vol33/4/art/opuscula_math_3341.pdf Sobolev norm estimates of solutions for the sublinear Emden-Fowler equation Ryuji Kajikiya We study the sublinear Emden-Fowler equation in small domains. As the domain becomes smaller, so does any solution. We investigate the convergence rate of the Sobolev norm of solutions as the volume of the domain converges to zero. The result is obtained by estimating the first eigenvalue of the Laplacian with the help of the variational method. Emden-Fowler equation, sign-changing solution, positive solution, variational method, norm estimate 2013 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol33iss4art11 https://www.opuscula.agh.edu.pl/vol33/4/art/opuscula_math_3342.pdf Solvability of functional quadratic integral equations with perturbation Mohamed M. A. Metwali We study the existence of solutions of the functional quadratic integral equation with a perturbation term in the space of Lebesgue integrable functions on an unbounded interval by using the Krasnoselskii fixed point theory and the measure of weak noncompactness. quadratic integral equation, measure of noncompactness, Krasnoselskii fixed point theorem, superposition operators 2013 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol33iss4art12 https://www.opuscula.agh.edu.pl/vol33/4/art/opuscula_math_3343.pdf Vulnerability parameters of tensor product of complete equipartite graphs P. Paulraja, V. Sheeba Agnes Let \(G_{1}\) and \(G_{2}\) be two simple graphs. The tensor product of \(G_{1}\) and \(G_{2}\), denoted by \(G_{1}\times G_{2}\), has vertex set \(V(G_{1}\times G_{2})=V(G_{1})\times V(G_{2})\) and edge set \(E(G_{1}\times G_{2})=\{(u_{1},v_{1})(u_{2},v_{2}):u_{1}u_{2}\in E(G_{1})\) and \(v_{1}v_{2}\in E(G_{2})\}\). In this paper, we determine vulnerability parameters such as toughness, scattering number, integrity and tenacity of the tensor product of the graphs \(K_{r(s)}\times K_{m(n)}\) for \(r\geq 3, m\geq 3, s\geq 1\) and \(n\geq 1,\) where \(K_{r(s)}\) denotes the complete \(r\)-partite graph in which each part has \(s\) vertices. Using the results obtained here the theorems proved in [Aygul Mamut, Elkin Vumar, Vertex Vulnerability Parameters of Kronecker Products of Complete Graphs, Information Processing Letters 106 (2008), 258-262] are obtained as corollaries. fault tolerance, tensor product, vulnerability parameters 2013 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol33iss4art13 https://www.opuscula.agh.edu.pl/vol33/4/art/opuscula_math_3344.pdf On Gelfand pairs associated to transitive groupoids Ibrahima Toure, Kinvi Kangni Let \(G\) be a topological locally compact, Hausdorff and second countable groupoid with a Haar system and \(K\) a compact subgroupoid of \(G\) with a Haar system too. \((G,K)\) is a Gelfand pair if the algebra of bi-\(K\)-invariant functions is commutative under convolution. In this paper, we give a characterization of Gelfand pairs associated to transitive groupoids which generalize a well-known result in the groups case. Using this result, we prove that the study of Gelfand pairs associated to transitive groupoids is equivalent to that of Gelfand pairs associated to its isotropy groups. transitive groupoids, groupoid representation, Gelfand pairs 2013 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol33iss4art14 https://www.opuscula.agh.edu.pl/vol33/4/art/opuscula_math_3345.pdf All graphs with paired-domination number two less than their order Włodzimierz Ulatowski Let \(G=(V,E)\) be a graph with no isolated vertices. A set \(S\subseteq V\) is a paired-dominating set of \(G\) if every vertex not in \(S\) is adjacent with some vertex in \(S\) and the subgraph induced by \(S\) contains a perfect matching. The paired-domination number \(\gamma_{p}(G)\) of \(G\) is defined to be the minimum cardinality of a paired-dominating set of \(G\). Let \(G\) be a graph of order \(n\). In [Paired-domination in graphs, Networks 32 (1998), 199-206] Haynes and Slater described graphs \(G\) with \(\gamma_{p}(G)=n\) and also graphs with \(\gamma_{p}(G)=n-1\). In this paper we show all graphs for which \(\gamma_{p}(G)=n-2\). paired-domination, paired-domination number 2013 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol34iss1 https://www.opuscula.agh.edu.pl/om-vol34iss1art1 https://www.opuscula.agh.edu.pl/vol34/1/art/opuscula_math_3401.pdf Asymptotic behavior of nonoscillatory solutions of higher-order integro-dynamic equations Martin Bohner, Said Grace, Nasrin Sultana In this paper, we establish some new criteria on the asymptotic behavior of nonoscillatory solutions of higher-order integro-dynamic equations on time scales. dynamic equations, time scales, nonoscillation, asymptotics 2014 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol34iss1art2 https://www.opuscula.agh.edu.pl/vol34/1/art/opuscula_math_3402.pdf Existence and uniqueness of the solutions of some degenerate nonlinear elliptic equations Albo Carlos Cavalheiro In this paper we are interested in the existence of solutions for the Dirichlet problem associated with degenerate nonlinear elliptic equations \[\begin{split}&-\sum_{j=1}^n D_j{\bigl[}{\omega}(x) {\cal A}_j(x, u, {\nabla}u){\bigr]} + b(x, u, {\nabla}u)\,{\omega}(x) + g(x)\,u(x)=\\&= f_0(x) - \sum_{j=1}^nD_jf_j(x) \quad{\rm on}\quad {\Omega}\end{split}\] in the setting of the weighted Sobolev spaces \({\rm W}_0^{1,p}(\Omega, \omega)\). degenerate nonlinear elliptic equations, weighted Sobolev spaces 2014 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol34iss1art3 https://www.opuscula.agh.edu.pl/vol34/1/art/opuscula_math_3403.pdf p-adic Banach space operators and adelic Banach space operators Ilwoo Cho In this paper, we study non-Archimedean Banach \(*\)-algebras \(\frak{M}_{p}\) over the \(p\)-adic number fields \(\mathbb{Q}_{p}\), and \(\frak{M}_{\mathbb{Q}}\) over the adele ring \(\mathbb{A}_{\mathbb{Q}}\). We call elements of \(\frak{M}_{p}\), \(p\)-adic operators, for all primes \(p\), respectively, call those of \(\frak{M}_{\mathbb{Q}}\), adelic operators. We characterize \(\frak{M}_{ \mathbb{Q}}\) in terms of \(\frak{M}_{p}\)'s. Based on such a structure theorem of \(\frak{M}_{\mathbb{Q}}\), we introduce some interesting \(p\)-adic operators and adelic operators. prime fields, \(p\)-adic number fields, adele ring, \(p\)-adic Banach spaces, adelic Banach space, \(p\)-adic operators, adelic operators 2014 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol34iss1art4 https://www.opuscula.agh.edu.pl/vol34/1/art/opuscula_math_3404.pdf Approximating fixed points of a countable family of strict pseudocontractions in Banach spaces Prasit Cholamjiak We prove the strong convergence of the modified Mann-type iterative scheme for a countable family of strict pseudocontractions in \(q\)-uniformly smooth Banach spaces. Our results mainly improve and extend the results announced in [Y. Yao, H. Zhou, Y.-C. Liou, Strong convergence of a modified Krasnoselski-Mann iterative algorithm for non-expansive mappings, J. Appl. Math. Comput. 29 (2009), 383-389]. common fixed points, convergence theorem, modified Mann iteration, strict pseudocontractions, \(q\)-uniformly smooth Banach spaces 2014 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol34iss1art5 https://www.opuscula.agh.edu.pl/vol34/1/art/opuscula_math_3405.pdf Problem of detecting inclusions by topological optimization I. Faye, M. Ndiaye, I. Ly, D. Seck In this paper we propose a new method to detect inclusions. The proposed method is based on shape and topological optimization tools. In fact after presenting the problem, we use topologication optimization tools to detect inclusions in the domain. Numerical results are presented. topological optimization, topological gradient, shape optimization, detection of inclusions, numerical simulations 2014 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol34iss1art6 https://www.opuscula.agh.edu.pl/vol34/1/art/opuscula_math_3406.pdf Comparison and oscillation theorems for singular Sturm-Liouville operators Monika Homa, Rostyslav Hryniv We prove analogues of the classical Sturm comparison and oscillation theorems for Sturm-Liouville operators on a finite interval with real-valued distributional potentials. Sturm-Liouville equation, distributional potential, Sturm comparison and oscillation theorem, Prüfer angle 2014 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol34iss1art7 https://www.opuscula.agh.edu.pl/vol34/1/art/opuscula_math_3407.pdf A totally magic cordial labeling of one-point union of n copies of a graph P. Jeyanthi, N. Angel Benseera A graph \(G\) is said to have a totally magic cordial (TMC) labeling with constant \(C\) if there exists a mapping \(f: V(G)\cup E(G)\rightarrow \left\{0,1\right\}\) such that \(f(a) + f(b) + f(ab) \equiv C(\mbox{mod 2})\) for all \(ab\in E(G)\) and \(\left|n_f(0)-n_f(1)\right|\leq1\), where \(n_f(i)\) \((i = 0, 1)\) is the sum of the number of vertices and edges with label \(i\). In this paper, we establish the totally magic cordial labeling of one-point union of \(n\)-copies of cycles, complete graphs and wheels. totally magic cordial labeling, one-point union of graphs 2014 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol34iss1art8 https://www.opuscula.agh.edu.pl/vol34/1/art/opuscula_math_3408.pdf On the dimension of Archimedean solids Tomáš Madaras, Pavol Široczki We study the dimension of graphs of the Archimedean solids. For most of these graphs we find the exact value of their dimension by finding unit-distance embeddings in the euclidean plane or by proving that such an embedding is not possible. Archimedean solid, unit-distance graph, dimension of a graph 2014 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol34iss1art9 https://www.opuscula.agh.edu.pl/vol34/1/art/opuscula_math_3409.pdf Asymptotics of the discrete spectrum for complex Jacobi matrices Maria Malejki The spectral properties and the asymptotic behaviour of the discrete spectrum for a special class of infinite tridiagonal matrices are given. We derive the asymptotic formulae for eigenvalues of unbounded complex Jacobi matrices acting in \(l^2(\mathbb{N})\). tridiagonal matrix, complex Jacobi matrix, discrete spectrum, eigenvalue, asymptotic formula, unbounded operator, Riesz projection 2014 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol34iss1art10 https://www.opuscula.agh.edu.pl/vol34/1/art/opuscula_math_3410.pdf Asymptotically isometric copies of c_{0} in Musielak-Orlicz spaces Agata Narloch, Lucjan Szymaszkiewicz Criteria in order that a Musielak-Orlicz function space \(L^\Phi\) as well as Musielak-Orlicz sequence space \(l^\Phi\) contains an asymptotically isometric copy of \(c_0\) are given. These results extend some results of [Y.A. Cui, H. Hudzik, G. Lewicki, Order asymptotically isometric copies of \(c_0\) in the subspaces of order continuous elements in Orlicz spaces, Journal of Convex Analysis 21 (2014)] to Musielak-Orlicz spaces. Musielak-Orlicz space, Luxemburg norm, condition \(\Delta_2\), asymptotically isometric copy of \(c_0\) 2014 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol34iss1art11 https://www.opuscula.agh.edu.pl/vol34/1/art/opuscula_math_3411.pdf A Neumann boundary value problem for a class of gradient systems Wen-Wu Pan, Lin Li In this paper we study a class of two-point boundary value systems. Using very recent critical points theorems, we establish the existence of one non-trivial solution and infinitely many solutions of this problem, respectively. Neumann problems, weak solutions, critical points, \((p_1,\ldots, p_n)\)-Laplacian 2014 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol34iss1art12 https://www.opuscula.agh.edu.pl/vol34/1/art/opuscula_math_3412.pdf Fixed point theorems for a semigroup of total asymptotically nonexpansive mappings in uniformly convex Banach spaces Suthep Suantai, Withun Phuengrattana In this paper, we provide existence and convergence theorems of common fixed points for left (or right) reversible semitopological semigroups of total asymptotically nonexpansive mappings in uniformly convex Banach spaces. The results presented in this paper extend and improve some recent results announced by other authors. fixed point, semitopological semigroup, reversible semigroup, total asymptotically nonexpansive semigroup, uniformly convex Banach spaces 2014 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol34iss2 https://www.opuscula.agh.edu.pl/vol34/2/art/opuscula_math_3413.pdf https://www.opuscula.agh.edu.pl/om-vol34iss2art2 https://www.opuscula.agh.edu.pl/vol34/2/art/opuscula_math_3414.pdf Existence and regularity of solutions for hyperbolic functional differential problems Zdzisław Kamont A generalized Cauchy problem for quasilinear hyperbolic functional differential systems is considered. A theorem on the local existence of weak solutions is proved. The initial problem is transformed into a system of functional integral equations for an unknown function and for their partial derivatives with respect to spatial variables. The existence of solutions for this system is proved by using a method of successive approximations. We show a theorem on the differentiability of solutions with respect to initial functions which is the main result of the paper. functional differential equations, weak solutions, Haar pyramid, differentiability with respect to initial functions 2014 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol34iss2art3 https://www.opuscula.agh.edu.pl/vol34/2/art/opuscula_math_3415.pdf Local error structures and order conditions in terms of Lie elements for exponential splitting schemes Winfried Auzinger, Wolfgang Herfort We discuss the structure of the local error of exponential operator splitting methods. In particular, it is shown that the leading error term is a Lie element, i.e., a linear combination of higher-degree commutators of the given operators. This structural assertion can be used to formulate a simple algorithm for the automatic generation of a minimal set of polynomial equations representing the order conditions, for the general case as well as in symmetric settings. exponential splitting schemes, local error, defect, order conditions, free Lie algebra 2014 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol34iss2art4 https://www.opuscula.agh.edu.pl/vol34/2/art/opuscula_math_3416.pdf On reconstructing an unknown coordinate of a nonlinear system of differential equations Marina Blizorukova, Alexander Kuklin, Vyacheslav Maksimov The paper discusses a method of auxiliary controlled models and the application of this method to solving problems of dynamical reconstruction of an unknown coordinate in a nonlinear system of differential equations. The solving algorithm, which is stable with respect to informational noises and computational errors, is presented. ordinary differential equations, inverse problems 2014 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol34iss2art5 https://www.opuscula.agh.edu.pl/vol34/2/art/opuscula_math_3417.pdf On a singular nonlinear Neumann problem Jan Chabrowski We investigate the solvability of the Neumann problem involving two critical exponents: Sobolev and Hardy-Sobolev. We establish the existence of a solution in three cases: \(\text{(i)}\;\ 2\lt p+1\lt 2^*(s),\) \(\text{(ii)}\;\ p+1=2^*(s)\) and \(\text{(iii)}\;\ 2^*(s)\lt p+1 \leq 2^*,\) where \(2^*(s)=\frac{2(N-s)}{N-2},\) \(0\lt s\lt 2,\) and \(2^*=\frac{2N}{N-2}\) denote the critical Hardy-Sobolev exponent and the critical Sobolev exponent, respectively. Neumann problem, critical Sobolev exponent, Hardy-Sobolev exponent Neumann problem 2014 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol34iss2art6 https://www.opuscula.agh.edu.pl/vol34/2/art/opuscula_math_3418.pdf Classical solutions of mixed problems for quasilinear first order PFDEs on a cylindrical domain Wojciech Czernous We abandon the setting of the domain as a Cartesian product of real intervals, customary for first order PFDEs (partial functional differential equations) with initial boundary conditions. We give a new set of conditions on the possibly unbounded domain \(\Omega\) with Lipschitz differentiable boundary. Well-posedness is then reliant on a variant of the normal vector condition. There is a neighbourhood of \(\partial\Omega\) with the property that if a characteristic trajectory has a point therein, then its every earlier point lies there as well. With local assumptions on coefficients and on the free term, we prove existence and Lipschitz dependence on data of classical solutions on \((0,c)\times\Omega\) to the initial boundary value problem, for small \(c\). Regularity of solutions matches this domain, and the proof uses the Banach fixed-point theorem. Our general model of functional dependence covers problems with deviating arguments and integro-differential equations. partial functional differential equations, classical solutions, local existence, characteristics, cylindrical domain, a priori estimates 2014 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol34iss2art7 https://www.opuscula.agh.edu.pl/vol34/2/art/opuscula_math_3419.pdf Difference problems generated by infinite systems of nonlinear parabolic functional differential equations with the Robin conditions Wojciech Czernous, Danuta Jaruszewska-Walczak We consider the classical solutions of mixed problems for infinite, countable systems of parabolic functional differential equations. Difference methods of two types are constructed and convergence theorems are proved. In the first type, we approximate the exact solutions by solutions of infinite difference systems. Methods of second type are truncation of the infinite difference system, so that the resulting difference problem is finite and practically solvable. The proof of stability is based on a comparison technique with nonlinear estimates of the Perron type for the given functions. The comparison system is infinite. Parabolic problems with deviated variables and integro-differential problems can be obtained from the general model by specifying the given operators. nonlinear parabolic equations, functional difference equations, infinite systems, Volterra type operators, nonlinear estimates of the Perron type, truncation methods 2014 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol34iss2art8 https://www.opuscula.agh.edu.pl/vol34/2/art/opuscula_math_3420.pdf Global convergence of successive approximations of the Darboux problem for partial functional differential equations with infinite delay Tomasz Człapiński We consider the Darboux problem for the hyperbolic partial functional differential equation with infinite delay. We deal with generalized (in the "almost everywhere" sense) solutions of this problem. We prove a theorem on the global convergence of successive approximations to a unique solution of the Darboux problem. successive approximations, Darboux problem, infinite delay 2014 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol34iss2art9 https://www.opuscula.agh.edu.pl/vol34/2/art/opuscula_math_3421.pdf About sign-constancy of Green's functions for impulsive second order delay equations Alexander Domoshnitsky, Guy Landsman, Shlomo Yanetz We consider the following second order differential equation with delay \[\begin{cases} (Lx)(t)\equiv{x''(t)+\sum_{j=1}^p {b_{j}(t)x(t-\theta_{j}(t))}}=f(t), \quad t\in[0,\omega],\\ x(t_j)=\gamma_{j}x(t_j-0), x'(t_j)=\delta_{j}x'(t_j-0), \quad j=1,2,\ldots,r. \end{cases}\] In this paper we find necessary and sufficient conditions of positivity of Green's functions for this impulsive equation coupled with one or two-point boundary conditions in the form of theorems about differential inequalities. By choosing the test function in these theorems, we obtain simple sufficient conditions. For example, the inequality \(\sum_{i=1}^p{b_i(t)\left(\frac{1}{4}+r\right)}\lt \frac{2}{\omega^2}\) is a basic one, implying negativity of Green's function of two-point problem for this impulsive equation in the case \(0\lt \gamma_i\leq{1}\), \(0\lt \delta_i\leq{1}\) for \(i=1,\ldots ,p\). impulsive equations, Green's functions, positivity/negativity of Green's functions, boundary value problem, second order 2014 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol34iss2art10 https://www.opuscula.agh.edu.pl/vol34/2/art/opuscula_math_3422.pdf Uniqueness and parameter dependence of positive doubly periodic solutions of nonlinear telegraph equations John R. Graef, Lingju Kong, Min Wang The authors study a type of second order nonlinear telegraph equation. The existence and uniqueness of positive doubly periodic solutions are discussed. The parametric dependence of the solutions is also investigated. Two examples are given as applications of the results. telegraph equation, doubly periodic solution, Green's function 2014 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol34iss2art11 https://www.opuscula.agh.edu.pl/vol34/2/art/opuscula_math_3423.pdf Stability of finite difference schemes for generalized von Foerster equations with renewal Henryk Leszczyński, Piotr Zwierkowski We consider a von Foerster-type equation describing the dynamics of a population with the production of offsprings given by the renewal condition. We construct a finite difference scheme for this problem and give sufficient conditions for its stability with respect to \(l^1\) and \(l^\infty\) norms. structured model, renewal, finite differences, stability 2014 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol34iss2art12 https://www.opuscula.agh.edu.pl/vol34/2/art/opuscula_math_3424.pdf Solving boundary value problems in the open source software R: package bvpSolve Francesca Mazzia, Jeff R. Cash, Karline Soetaert The R package bvpSolve for the numerical solution of Boundary Value Problems (BVPs) is presented. This package is free software which is distributed under the GNU General Public License, as part of the R open source software project. It includes some well known codes to solve boundary value problems of ordinary differential equations (ODEs) and differential algebraic equations (DAEs). In addition to the packages already available for solving initial value problems, the new package now allows non expert users to efficiently solve boundary value problems in the problem solving environment R. ordinary differential equations, boundary value problems, singular perturbation problems, test problems, R, Fortran 2014 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol34iss2art13 https://www.opuscula.agh.edu.pl/vol34/2/art/opuscula_math_3425.pdf Difference functional inequalities and applications Anna Szafrańska The paper deals with the difference inequalities generated by initial boundary value problems for hyperbolic nonlinear differential functional systems. We apply this result to investigate the stability of constructed difference schemes. The proof of the convergence of the difference method is based on the comparison technique, and the result for difference functional inequalities is used. Numerical examples are presented. initial boundary value problems, difference functional inequalities, difference methods, stability and convergence, interpolating operators, error estimates 2014 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol34iss2art14 https://www.opuscula.agh.edu.pl/vol34/2/art/opuscula_math_3426.pdf On the Tonelli method for the degenerate parabolic Cauchy problem with functional argument Krzysztof A. Topolski The degenerate parabolic Cauchy problem is considered. A functional argument in the equation is of the Hale type. As a limit of piecewise classical solutions we obtain a viscosity solution of the main problem. Presented method is an adaptation of Tonelli's constructive method to the partial differential-functional equation. It is also shown that this approach can be improved by the vanishing viscosity method and regularisation process. viscosity solutions, parabolic equation, differential-functional equation 2014 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol34iss2art15 https://www.opuscula.agh.edu.pl/vol34/2/art/opuscula_math_3427.pdf Method of lines for parabolic stochastic functional partial differential equations Maria Ziemlańska We approximate parabolic stochastic functional differential equations substituting the derivatives in the space variable by finite differences. We prove the stability of the method of lines corresponding to a parabolic SPDE driven by Brownian motion. stochastic partial differential equations, stability of the method of lines, white noise, Volterra stochastic equations 2014 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol34iss3 https://www.opuscula.agh.edu.pl/om-vol34iss3art1 https://www.opuscula.agh.edu.pl/vol34/3/art/opuscula_math_3428.pdf A note on on-line Ramsey numbers for quadrilaterals Joanna Cyman, Tomasz Dzido We consider on-line Ramsey numbers defined by a game played between two players, Builder and Painter. In each round Builder draws an the edge and Painter colors it either red or blue, as it appears. Builder's goal is to force Painter to create a monochromatic copy of a fixed graph \(H\) in as few rounds as possible. The minimum number of rounds (assuming both players play perfectly) is the on-line Ramsey number \(\widetilde{r}(H)\) of the graph \(H\). An asymmetric version of the on-line Ramsey numbers \(\widetilde{r}(G,H)\) is defined accordingly. In 2005, Kurek and Ruciński computed \(\widetilde{r}(C_3)\). In this paper, we compute \(\widetilde{r}(C_4,C_k)\) for \(3 \le k \le 7\). Most of the results are based on computer algorithms but we obtain the exact value \(\widetilde{r}(C_4)\) and do so without the help of computer algorithms. Ramsey theory, on-line games 2014 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol34iss3art2 https://www.opuscula.agh.edu.pl/vol34/3/art/opuscula_math_3429.pdf On the existence of positive periodic solutions for totally nonlinear neutral differential equations of the second-order with functional delay Emmanuel K. Essel, Ernest Yankson We prove that the totally nonlinear second-order neutral differential equation \[\frac{d^2}{dt^2}x(t)+p(t)\frac{d}{dt}x(t)+q(t)h(x(t))\] \[=\frac{d}{dt}c(t,x(t-\tau(t)))+f(t,\rho(x(t)),g(x(t-\tau(t))))\] has positive periodic solutions by employing the Krasnoselskii-Burton hybrid fixed point theorem. Krasnoselskii, neutral, positive periodic solution 2014 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol34iss3art3 https://www.opuscula.agh.edu.pl/vol34/3/art/opuscula_math_3430.pdf On reflectionless equi-transmitting matrices Pavel Kurasov, Rao Ogik, Amar Rauf Reflectionless equi-transmitting unitary matrices are studied in connection to matching conditions in quantum graphs. All possible such matrices of size 6 are described explicitly. It is shown that such matrices form 30 six-parameter families intersected along 12 five-parameter families closely connected to conference matrices. quantum graphs, vertex scattering matrix, equi-transmitting matrices 2014 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol34iss3art4 https://www.opuscula.agh.edu.pl/vol34/3/art/opuscula_math_3431.pdf Existence and controllability results for damped second order impulsive functional differential systems with state-dependent delay M. Mallika Arjunan, N. Y. Nadaf In this paper, we investigate the existence and controllability of mild solutions for a damped second order impulsive functional differential equation with state-dependent delay in Banach spaces. The results are obtained by using Sadovskii's fixed point theorem combined with the theories of a strongly continuous cosine family of bounded linear operators. Finally, an example is provided to illustrate the main results. damped second order differential equations, impulsive differential equations, controllability, state-dependent delay, cosine function, mild solution, fixed point 2014 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol34iss3art5 https://www.opuscula.agh.edu.pl/vol34/3/art/opuscula_math_3432.pdf Conjugate functions, L^{p}-norm like functionals, the generalized Hölder inequality, Minkowski inequality and subhomogeneity Janusz Matkowski For \(h:(0,\infty )\rightarrow \mathbb{R}\), the function \(h^{\ast }\left( t\right) :=th(\frac{1}{t})\) is called \((\ast)\)-conjugate to \(h\). This conjugacy is related to the Hölder and Minkowski inequalities. Several properties of \((\ast)\)-conjugacy are proved. If \(\varphi\) and \(\varphi ^{\ast }\) are bijections of \(\left(0,\infty \right)\) then \((\varphi ^{-1}) ^{\ast }=\left( \left[ \left( \varphi ^{\ast }\right) ^{-1}\right] ^{\ast }\right) ^{-1}\). Under some natural rate of growth conditions at \(0\) and \(\infty\), if \(\varphi\) is increasing, convex, geometrically convex, then \(\left[ \left( \varphi^{-1}\right) ^{\ast }\right] ^{-1}\) has the same properties. We show that the Young conjugate functions do not have this property. For a measure space \((\Omega ,\Sigma ,\mu )\) denote by \(S=S(\Omega ,\Sigma ,\mu )\) the space of all \(\mu\)-integrable simple functions \(x:\Omega \rightarrow \mathbb{R}\). Given a bijection \(\varphi :(0,\infty )\rightarrow (0,\infty )\), define \(\mathbf{P}_{\varphi }:S\rightarrow \lbrack 0,\infty )\) by \[\mathbf{P}_{\varphi }(x):=\varphi ^{-1}\bigg( \int\limits_{\Omega (x)}\varphi \circ \left\vert x\right\vert d\mu \bigg),\] where \(\Omega (x)\) is the support of \(x\). Applying some properties of the \((\ast)\) operation, we prove that if \(\int\limits_{\Omega }xy\leq \mathbf{P}_{\varphi }(x)\mathbf{P}_{\psi }(y)\) where \(\varphi ^{-1}\) and \(\psi ^{-1}\) are conjugate, then \(\varphi\) and \(\psi\) are conjugate power functions. The existence of nonpower bijections \(\varphi \) and \(\psi\) with conjugate inverse functions \(\psi =\left[ ( \varphi ^{-1}) ^{\ast}\right] ^{-1}\) such that \(\mathbf{P}_{\varphi }\) and \(\mathbf{P}_{\psi }\) are subadditive and subhomogeneous is considered. \(L^{p}\)-norm like functional, homogeneity, subhomogeneity, subadditivity, the converses of Minkowski and Hölder inequalities, generalization of the Minkowski and Hölder inequalities, conjugate (complementary) functions, Young conjugate functions, convex function, geometrically convex function, Wright convex function, functional equation 2014 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol34iss3art6 https://www.opuscula.agh.edu.pl/vol34/3/art/opuscula_math_3433.pdf On the hyperreflexivity of subspaces of Toeplitz operators on regions in the complex plane Wojciech Młocek, Marek Ptak The results of hyperreflexivity or 2-hyperreflexivity for subspaces of Toeplitz operators on the Hardy spaces on Jordan regions or upper half-plane are given. hyperreflexivity, Toeplitz operator, upper half-plane, simply connected region 2014 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol34iss3art7 https://www.opuscula.agh.edu.pl/vol34/3/art/opuscula_math_3434.pdf Global solutions to the initial-boundary value problem for the quasilinear viscoelastic equation with a derivative nonlinearity Mitsuhiro Nakao We prove the existence and uniqueness of a global decaying solution to the initial boundary value problem for the quasilinear wave equation with Kelvin-Voigt dissipation and a derivative nonlinearity. To derive the required estimates of the solutions we employ a 'loan' method and use a difference inequality on the energy. global solutions, energy decay, quasilinear wave equation, Kelvin-Voigt dissipation, derivative nonlinearity 2014 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol34iss3art8 https://www.opuscula.agh.edu.pl/vol34/3/art/opuscula_math_3435.pdf On Gevrey orders of formal power series solutions to the third and fifth Painlevé equations near infinity Anastasia V. Parusnikova The question under consideration is Gevrey summability of formal power series solutions to the third and fifth Painlevé equations near infinity. We consider the fifth Painlevé equation in two cases: when \(\alpha\beta\gamma\delta \neq 0\) and when \(\alpha\beta\gamma \neq 0\), \(\delta =0\) and the third Painlevé equation when all the parameters of the equation are not equal to zero. In the paper we prove Gevrey summability of the formal solutions to the fifth Painlevé equation and to the third Painlevé equation, respectively. Painlevé equations, Newton polygon, asymptotic expansions, Gevrey orders 2014 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol34iss3art9 https://www.opuscula.agh.edu.pl/vol34/3/art/opuscula_math_3436.pdf Positivity preserving results for a biharmonic equation under Dirichlet boundary conditions Hanen Ben Omrane, Saïma Khenissy We prove a dichotomy result giving the positivity preserving property for a biharmonic equation with Dirichlet boundary conditions arising in MEMS models. We adapt some ideas in [H.-Ch. Grunau, G. Sweers, Positivity for equations involving polyharmonic operators with Dirichlet boundary conditions, Math. Ann. 307 (1997), 589-626]. biharmonic equation, positivity preserving, Dirichlet problem 2014 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol34iss3art10 https://www.opuscula.agh.edu.pl/vol34/3/art/opuscula_math_3437.pdf Signed star (k,k)-domatic number of a graph S. M. Sheikholeslami, L. Volkmann Let \(G\) be a simple graph without isolated vertices with vertex set \(V(G)\) and edge set \(E(G)\) and let \(k\) be a positive integer. A function \(f:E(G)\longrightarrow \{-1, 1\}\) is said to be a signed star \(k\)-dominating function on \(G\) if \(\sum_{e\in E(v)}f(e)\ge k\) for every vertex \(v\) of \(G\), where \(E(v)=\{uv\in E(G)\mid u\in N(v)\}\). A set \(\{f_1,f_2,\ldots,f_d\}\) of signed star \(k\)-dominating functions on \(G\) with the property that \(\sum_{i=1}^df_i(e)\le k\) for each \(e\in E(G)\), is called a signed star \((k,k)\)-dominating family (of functions) on \(G\). The maximum number of functions in a signed star \((k,k)\)-dominating family on \(G\) is the signed star \((k,k)\)-domatic number of \(G\), denoted by \(d^{(k,k)}_{SS}(G)\). In this paper we study properties of the signed star \((k,k)\)-domatic number \(d_{SS}^{(k,k)}(G)\). In particular, we present bounds on \(d_{SS}^{(k,k)}(G)\), and we determine the signed \((k,k)\)-domatic number of some regular graphs. Some of our results extend these given by Atapour, Sheikholeslami, Ghameslou and Volkmann [Signed star domatic number of a graph, Discrete Appl. Math. 158 (2010), 213-218] for the signed star domatic number. signed star \((k,k)\)-domatic number, signed star domatic number, signed star \(k\)-dominating function, signed star dominating function, signed star \(k\)-domination number, signed star domination number, regular graphs 2014 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol34iss3art11 https://www.opuscula.agh.edu.pl/vol34/3/art/opuscula_math_3438.pdf Existence and multiplicity results for nonlinear problems involving the p(x)-Laplace operator Najib Tsouli, Omar Darhouche In this paper we study the following nonlinear boundary-value problem \[-\Delta_{p(x)} u=\lambda f(x,u) \quad \text{ in } \Omega,\] \[|\nabla u|^{p(x)-2}\frac{\partial u}{\partial \nu}+\beta(x)|u|^{p(x)-2}u=\mu g(x,u) \quad \text{ on } \partial\Omega,\] where \(\Omega\subset\mathbb{R}^N\) is a bounded domain with smooth boundary \(\partial\Omega\), \(\frac{\partial u}{\partial\nu}\) is the outer unit normal derivative on \(\partial\Omega\), \(\lambda, \mu\) are two real numbers such that \(\lambda^{2}+\mu^{2}\neq0\), \(p\) is a continuous function on \(\overline{\Omega}\) with \(\inf_{x\in \overline{\Omega}} p(x)\gt 1\), \(\beta\in L^{\infty}(\partial\Omega)\) with \(\beta^{-}:=\inf_{x\in \partial\Omega}\beta(x)\gt 0\) and \(f : \Omega\times\mathbb{R}\rightarrow \mathbb{R}\), \(g : \partial\Omega\times\mathbb{R}\rightarrow \mathbb{R}\) are continuous functions. Under appropriate assumptions on \(f\) and \(g\), we obtain the existence and multiplicity of solutions using the variational method. The positive solution of the problem is also considered. critical points, variational method, \(p(x)\)-Laplacian, generalized Lebesgue-Sobolev spaces 2014 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol34iss3art12 https://www.opuscula.agh.edu.pl/vol34/3/art/opuscula_math_3439.pdf On the stability of first order impulsive evolution equations JinRong Wang, Michal Fečkan, Yong Zhou In this paper, concepts of Ulam-Hyers stability, generalized Ulam-Hyers stability, Ulam-Hyers-Rassias stability and generalized Ulam-Hyers-Rassias stability for impulsive evolution equations are raised. Ulam-Hyers-Rassias stability results on a compact interval and an unbounded interval are presented by using an impulsive integral inequality of the Gronwall type. Two examples are also provided to illustrate our results. Finally, some extensions of the Ulam-Hyers-Rassias stability for the case with infinite impulses are given. first order, impulsive evolution equations, Ulam-Hyers-Rassias stability 2014 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol34iss4 https://www.opuscula.agh.edu.pl/om-vol34iss4art1 https://www.opuscula.agh.edu.pl/vol34/4/art/opuscula_math_3440.pdf Sufficient optimality criteria and duality for multiobjective variational control problems with B-(p,r)-invex functions Tadeusz Antczak, Manuel Arana Jiménez In this paper, we generalize the notion of \(B\)-\((p,r)\)-invexity introduced by Antczak in [A class of \(B\)-\((p; r)\)-invex functions and mathematical programming, J. Math. Anal. Appl. 286 (2003), 187-206] for scalar optimization problems to the case of a multiobjective variational programming control problem. For such nonconvex vector optimization problems, we prove sufficient optimality conditions under the assumptions that the functions constituting them are \(B\)-\((p,r)\)-invex. Further, for the considered multiobjective variational control problem, its dual multiobjective variational control problem in the sense of Mond-Weir is given and several duality results are established under \(B\)-\((p,r)\)-invexity. multiobjective variational control problems, efficient solution, \(B\)-\((p,r)\)-invex functions, optimality conditions, duality 2014 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol34iss4art2 https://www.opuscula.agh.edu.pl/vol34/4/art/opuscula_math_3441.pdf Constant-sign solutions for a nonlinear Neumann problem involving the discrete p-Laplacian Pasquale Candito, Giuseppina D'Aguí In this paper, we investigate the existence of constant-sign solutions for a nonlinear Neumann boundary value problem involving the discrete \(p\)-Laplacian. Our approach is based on an abstract local minimum theorem and truncation techniques. constant-sign solution, difference equations, Neumann problem 2014 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol34iss4art3 https://www.opuscula.agh.edu.pl/vol34/4/art/opuscula_math_3442.pdf Remarks for one-dimensional fractional equations Massimiliano Ferrara, Giovanni Molica Bisci In this paper we study a class of one-dimensional Dirichlet boundary value problems involving the Caputo fractional derivatives. The existence of infinitely many solutions for this equations is obtained by exploiting a recent abstract result. Concrete examples of applications are presented. fractional differential equations, Caputo fractional derivatives, variational methods 2014 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol34iss4art4 https://www.opuscula.agh.edu.pl/vol34/4/art/opuscula_math_3443.pdf Dynamic programming approach to structural optimization problem - numerical algorithm Piotr Fulmański, Andrzej Nowakowski, Jan Pustelnik In this paper a new shape optimization algorithm is presented. As a model application we consider state problems related to fluid mechanics, namely the Navier-Stokes equations for viscous incompressible fluids. The general approach to the problem is described. Next, transformations to classical optimal control problems are presented. Then, the dynamic programming approach is used and sufficient conditions for the shape optimization problem are given. A new numerical method to find the approximate value function is developed. sufficient optimality condition, elliptic equations, optimal shape control, structural optimization, stationary Navier-Stokes equations, dynamic programming, numerical approximation 2014 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol34iss4art5 https://www.opuscula.agh.edu.pl/vol34/4/art/opuscula_math_3444.pdf On a new critical point theorem and some applications to discrete equations Marek Galewski, Elżbieta Galewska Using the Fenchel-Young duality we derive a new critical point theorem. We illustrate our results with solvability for certain discrete BVP. Multiple solutions are also considered. critical point, multiplicity, discrete equation, Fenchel-Young transform 2014 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol34iss4art6 https://www.opuscula.agh.edu.pl/vol34/4/art/opuscula_math_3445.pdf Weak heteroclinic solutions and competition phenomena to anisotropic difference equations with variable exponents Aboudramane Guiro, Blaise Kone, Stanislas Ouaro In this paper, we prove the existence of weak heteroclinic solutions for a family of anisotropic difference equations under competition phenomena between parameters. anisotropic difference equations, anisotropic difference equations, heteroclinic solutions, discrete Hölder type inequality, competition phenomena 2014 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol34iss4art7 https://www.opuscula.agh.edu.pl/vol34/4/art/opuscula_math_3446.pdf Existence of three solutions for perturbed nonlinear difference equations Shapour Heidarkhani, Mohsen Khaleghi Moghadam Using critical point theory, we study the existence of at least three solutions for perturbed nonlinear difference equations with discrete boundary-value condition depending on two positive parameters. nonlinear difference equations, discrete boundary value problem, three solutions, critical point theory, variational methods 2014 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol34iss4art8 https://www.opuscula.agh.edu.pl/vol34/4/art/opuscula_math_3447.pdf Optimization of a fractional Mayer problem - existence of solutions, maximum principle, gradient methods Dariusz Idczak, Stanislaw Walczak In the paper, we study a linear-quadratic optimal control problem of Mayer type given by a fractional control system. First, we prove a theorem on the existence of a solution to such a problem. Next, using the local implicit function theorem, we derive a formula for the gradient of a cost functional under constraints given by a control system and prove a maximum principle in the case of a control constraint set. The formula for the gradient is used to implement the gradient methods for the problem under consideration. fractional Riemann-Liouville derivative, Mayer problem, existence of an optimal solution, maximum principle, gradient method 2014 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol34iss4art9 https://www.opuscula.agh.edu.pl/vol34/4/art/opuscula_math_3448.pdf On some subclasses of the family of Darboux Baire 1 functions Gertruda Ivanova, Elżbieta Wagner-Bojakowska We introduce a subclass of the family of Darboux Baire 1 functions \(f:\mathbb{R}\rightarrow\mathbb{R}\) modifying the Darboux property analogously as it was done by Z. Grande in [On a subclass of the family of Darboux functions, Colloq. Math. 17 (2009), 95-104], and replacing approximate continuity with \(\mathcal{I}\)-approximate continuity, i.e. continuity with respect to the \(\mathcal{I}\)-density topology. We prove that the family of all Darboux quasi-continuous functions from the first Baire class is a strongly porous set in the space \(\mathcal{DB}_1\) of Darboux Baire 1 functions, equipped with the supremum metric. Darboux property, strong Świątkowski property, Baire property, \(\mathcal{I}\)-approximate continuity, quasi-continuity 2014 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol34iss4art10 https://www.opuscula.agh.edu.pl/vol34/4/art/opuscula_math_3449.pdf Oscillatory properties of solutions of the fourth order difference equations with quasidifferences Robert Jankowski, Ewa Schmeidel, Joanna Zonenberg A class of fourth-order neutral type difference equations with quasidifferences and deviating arguments is considered. Our approach is based on studying the considered equation as a system of a four-dimensional difference system. The sufficient conditions under which the considered equation has no quickly oscillatory solutions are given. Finally, the sufficient conditions under which the equation is almost oscillatory are presented. fourth-order difference equation, neutral type, quickly oscillatory solutions, almost oscillatory 2014 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol34iss4art11 https://www.opuscula.agh.edu.pl/vol34/4/art/opuscula_math_3450.pdf On ∞-entropy points in real analysis Ewa Korczak-Kubiak, Anna Loranty, Ryszard J. Pawlak We will consider \(\infty\)-entropy points in the context of the possibilities of approximation mappings by the functions having \(\infty\)-entropy points and belonging to essential (from the point of view of real analysis theory) classes of functions: almost continuous, Darboux Baire one and approximately continuous functions. topological entropy, Darboux function, almost continuity, Baire one function, approximately continuous function, pseudo fixed point, topology of uniform convergence, compact-open topology, \(\infty\)-entropy point 2014 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol34iss4art12 https://www.opuscula.agh.edu.pl/vol34/4/art/opuscula_math_3451.pdf Existence results for random fractional differential equations Vasile Lupulescu, Donal O'Regan, Ghaus ur Rahman In this paper, an existence result for a random fractional differential equation is established under a Carathéodory condition. Existence results for extremal random solutions are also proved. Finally, an existence and uniqueness result is given random fractional differential equations, fractional integral, Caputo fractional derivative, existence 2014 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol34iss4art13 https://www.opuscula.agh.edu.pl/vol34/4/art/opuscula_math_3452.pdf Necessary and sufficient conditions for a Pareto optimal allocation in a discontinuous Gale economic model Anna Michalak, Marcin Studniarski In this paper we examine the concept of Pareto optimality in a simplified Gale economic model without assuming continuity of the utility functions. We apply some existing results on higher-order optimality conditions to get necessary and sufficient conditions for a locally Pareto optimal allocation. Gale model, discontinuous functions, generalized directional derivatives, higher-order conditions, Pareto optimality 2014 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol34iss4art14 https://www.opuscula.agh.edu.pl/vol34/4/art/opuscula_math_3453.pdf A note on the dependence of solutions on functional parameters for nonlinear Sturm-Liouville problems Aleksandra Orpel We deal with the existence and the continuous dependence of solutions on functional parameters for boundary valued problems containing Sturm-Liouville equation. We apply these result to prove the existence of at least one solution for a certain class of optimal control problems. positive solution, continuous dependence of solutions on functional parameters, Sturm-Liouville equation 2014 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol34iss4art15 https://www.opuscula.agh.edu.pl/vol34/4/art/opuscula_math_3454.pdf On the dependence on parameters for second order discrete boundary value problems with the p(k)-Laplacian Joanna Smejda, Renata Wieteska In this paper we study the existence and the nonexistence of solutions for the boundary value problems of a class of nonlinear second-order discrete equations depending on a parameter. Variational (the mountain pass technique) and non-variational methods are applied. discrete boundary value problems, variational methods, mountain pass theorem 2014 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol34iss4art16 https://www.opuscula.agh.edu.pl/vol34/4/art/opuscula_math_3455.pdf Spectral representations for a class of banded Jacobi-type matrices Ewelina Zalot, Witold Majdak We describe some spectral representations for a class of non-self-adjoint banded Jacobi-type matrices. Our results extend those obtained by P.B. Naïman for (two-sided infinite) periodic tridiagonal Jacobi matrices. spectral representation, Laurent operators, Jacobi matrices 2014 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol35iss1 https://www.opuscula.agh.edu.pl/om-vol35iss1art1 https://www.opuscula.agh.edu.pl/vol35/1/art/opuscula_math_3501.pdf Positive solutions with specific asymptotic behavior for a polyharmonic problem on R^{n} Abdelwaheb Dhifli This paper is concerned with positive solutions of the semilinear polyharmonic equation \((-\Delta)^{m} u = a(x){u}^{\alpha}\) on \(\mathbb{R}^{n}\), where \(m\) and \(n\) are positive integers with \(n\gt 2m\), \(\alpha\in (-1,1)\). The coefficient \(a\) is assumed to satisfy \[a(x)\approx{(1+|x|)}^{-\lambda}L(1+|x|)\quad \text{for}\quad x\in \mathbb{R}^{n},\] where \(\lambda\in [2m,\infty)\) and \(L\in C^{1}([1,\infty))\) is positive with \(\frac{tL'(t)}{L(t)}\longrightarrow 0\) as \(t\longrightarrow \infty\); if \(\lambda=2m\), one also assumes that \(\int_{1}^{\infty}t^{-1}L(t)dt\lt \infty\). We prove the existence of a positive solution \(u\) such that \[u(x)\approx{(1+|x|)}^{-\widetilde{\lambda}}\widetilde{L}(1+|x|) \quad\text{for}\quad x\in \mathbb{R}^{n},\] with \(\widetilde{\lambda}:=\min(n-2m,\frac{\lambda-2m}{1-\alpha})\) and a function \(\widetilde{L}\), given explicitly in terms of \(L\) and satisfying the same condition at infinity. (Given positive functions \(f\) and \(g\) on \(\mathbb{R}^{n}\), \(f\approx g\) means that \(c^{-1}g\leq f\leq cg\) for some constant \(c\gt 1\).) asymptotic behavior, Dirichlet problem, Schauder fixed point theorem, positive bounded solutions 2015 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol35iss1art2 https://www.opuscula.agh.edu.pl/vol35/1/art/opuscula_math_3502.pdf Asymptotic behavior of positive solutions of a semilinear Dirichlet problem in the annulus Safa Dridi, Bilel Khamessi In this paper, we establish existence and asymptotic behavior of a positive classical solution to the following semilinear boundary value problem: \[-\Delta u=q(x)u^{\sigma }\;\text{in}\;\Omega,\quad u_{|\partial\Omega}=0.\] Here \(\Omega\) is an annulus in \(\mathbb{R}^{n}\), \(n\geq 3\), \(\sigma \lt 1\) and \(q\) is a positive function in \(\mathcal{C}_{loc}^{\gamma }(\Omega )\), \(0\lt\gamma \lt 1\), satisfying some appropriate assumptions related to Karamata regular variation theory. Our arguments combine a method of sub- and supersolutions with Karamata regular variation theory. asymptotic behavior, Dirichlet problem, Karamata function, subsolution, supersolution 2015 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol35iss1art3 https://www.opuscula.agh.edu.pl/vol35/1/art/opuscula_math_3503.pdf Characterizations and decomposition of strongly Wright-convex functions of higher order Attila Gilányi, Nelson Merentes, Kazimierz Nikodem, Zsolt Páles Motivated by results on strongly convex and strongly Jensen-convex functions by R. Ger and K. Nikodem in [Strongly convex functions of higher order, Nonlinear Anal. 74 (2011), 661-665] we investigate strongly Wright-convex functions of higher order and we prove decomposition and characterization theorems for them. Our decomposition theorem states that a function \(f\) is strongly Wright-convex of order \(n\) if and only if it is of the form \(f(x)=g(x)+p(x)+c x^{n+1}\), where \(g\) is a (continuous) \(n\)-convex function and \(p\) is a polynomial function of degree \(n\). This is a counterpart of Ng's decomposition theorem for Wright-convex functions. We also characterize higher order strongly Wright-convex functions via generalized derivatives. generalized convex function, Wright-convex function of higher order, strongly convex function 2015 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol35iss1art4 https://www.opuscula.agh.edu.pl/vol35/1/art/opuscula_math_3504.pdf Strongly increasing solutions of cyclic systems of second order differential equations with power-type nonlinearities Jaroslav Jaroš, Kusano Takaŝi We consider \(n\)-dimensional cyclic systems of second order differential equations \[(p_i(t)|x_{i}'|^{\alpha_i -1}x_{i}')' = q_{i}(t)|x_{i+1}|^{\beta_i-1}x_{i+1},\] \[\quad i = 1,\ldots,n, \quad (x_{n+1} = x_1) \tag{\(\ast\)}\] under the assumption that the positive constants \(\alpha_i\) and \(\beta_i\) satisfy \(\alpha_1{\ldots}\alpha_n \gt \beta_1{\ldots}\beta_n\) and \(p_i(t)\) and \(q_i(t)\) are regularly varying functions, and analyze positive strongly increasing solutions of system (\(\ast\)) in the framework of regular variation. We show that the situation for the existence of regularly varying solutions of positive indices for (\(\ast\)) can be characterized completely, and moreover that the asymptotic behavior of such solutions is governed by the unique formula describing their order of growth precisely. We give examples demonstrating that the main results for (\(\ast\)) can be applied to some classes of partial differential equations with radial symmetry to acquire accurate information about the existence and the asymptotic behavior of their radial positive strongly increasing solutions. systems of differential equations, positive solutions, asymptotic behavior, regularly varying functions 2015 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol35iss1art5 https://www.opuscula.agh.edu.pl/vol35/1/art/opuscula_math_3505.pdf Controllability of semilinear systems with fixed delay in control Surendra Kumar, N. Sukavanam In this paper, different sufficient conditions for exact controllability of semilinear systems with a single constant point delay in control are established in infinite dimensional space. The existence and uniqueness of mild solution is also proved under suitable assumptions. In particular, local Lipschitz continuity of a nonlinear function is used. To illustrate the developed theory some examples are given. first order delay system, mild solution, fixed point, exact controllability 2015 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol35iss1art6 https://www.opuscula.agh.edu.pl/vol35/1/art/opuscula_math_3506.pdf Growth and oscillation of some polynomials generated by solutions of complex differential equations Zinelâabidine Latreuch, Benharrat Belaïdi In this paper, we continue the study of some properties on the growth and oscillation of solutions of linear differential equations with entire coefficients of the type \[f^{\prime \prime }+A(z) f^{\prime }+B(z) f=0\] and \[f^{\left( k\right) }+A_{k-2}(z) f^{\left( k-2\right) }+\ldots +A_{0}(z) f=0.\] linear differential equations, finite order, exponent of convergence of the sequence of distinct zeros, hyper-exponent of convergence of the sequence of distinct zeros 2015 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol35iss1art7 https://www.opuscula.agh.edu.pl/vol35/1/art/opuscula_math_3507.pdf Diffusion approximation of recurrent schemes for financial markets, with application to the Ornstein-Uhlenbeck process Yuliya Mishura We adapt the general conditions of the weak convergence for the sequence of processes with discrete time to the diffusion process towards the weak convergence for the discrete-time models of a financial market to the continuous-time diffusion model. These results generalize a classical scheme of the weak convergence for discrete-time markets to the Black-Scholes model. We give an explicit and direct method of approximation by a recurrent scheme. As an example, an Ornstein-Uhlenbeck process is considered as a limit model. diffusion approximation, semimartingale, recurrent scheme, financial market, multiplicative scheme, Ornstein-Uhlenbeck process 2015 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol35iss1art8 https://www.opuscula.agh.edu.pl/vol35/1/art/opuscula_math_3508.pdf The generalized sine function and geometrical properties of normed spaces Tomasz Szostok Let \((X,\|\cdot\|)\) be a normed space. We deal here with a function \(s:X\times X\to\mathbb{R}\) given by the formula \[s(x,y):=\inf_{\lambda\in\mathbb{R}}\frac{\|x+\lambda y\|}{\|x\|}\] (for \(x=0\) we must define it separately). Then we take two unit vectors \(x\) and \(y\) such that \(y\) is orthogonal to \(x\) in the Birkhoff-James sense. Using these vectors we construct new functions \(\phi_{x,y}\) which are defined on \(\mathbb{R}\). If \(X\) is an inner product space, then \(\phi_{x,y}=\sin\) and, therefore, one may call this function a generalization of the sine function. We show that the properties of this function are connected with geometrical properties of the normed space \(X\). geometry of normed spaces, smoothness, strict convexity, Birkhoff-James orthogonality, conditional functional equations 2015 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol35iss1art9 https://www.opuscula.agh.edu.pl/vol35/1/art/opuscula_math_3509.pdf The paired-domination and the upper paired-domination numbers of graphs Włodzimierz Ulatowski In this paper we continue the study of paired-domination in graphs. A paired-dominating set, abbreviated PDS, of a graph \(G\) with no isolated vertex is a dominating set of vertices whose induced subgraph has a perfect matching. The paired-domination number of \(G\), denoted by \(\gamma_{p}(G)\), is the minimum cardinality of a PDS of \(G\). The upper paired-domination number of \(G\), denoted by \(\Gamma_{p}(G)\), is the maximum cardinality of a minimal PDS of \(G\). Let \(G\) be a connected graph of order \(n\geq 3\). Haynes and Slater in [Paired-domination in graphs, Networks 32 (1998), 199-206], showed that \(\gamma_{p}(G)\leq n-1\) and they determine the extremal graphs \(G\) achieving this bound. In this paper we obtain analogous results for \(\Gamma_{p}(G)\). Dorbec, Henning and McCoy in [Upper total domination versus upper paired-domination, Questiones Mathematicae 30 (2007), 1-12] determine \(\Gamma_{p}(P_n)\), instead in this paper we determine \(\Gamma_{p}(C_n)\). Moreover, we describe some families of graphs \(G\) for which the equality \(\gamma_{p}(G)=\Gamma_{p}(G)\) holds. paired-domination, paired-domination number, upper paired-domination number 2015 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol35iss2 https://www.opuscula.agh.edu.pl/om-vol35iss2art1 https://www.opuscula.agh.edu.pl/vol35/2/art/opuscula_math_3510.pdf On small vibrations of a damped Stieltjes string Olga Boyko, Vyacheslav Pivovarchik Inverse problem of recovering masses, coefficients of damping and lengths of the intervals between the masses using two spectra of boundary value problems and the total length of the Stieltjes string (an elastic thread bearing point masses) is considered. For the case of point-wise damping at the first counting from the right end mass the problem of recovering the masses, the damping coefficient and the lengths of the subintervals by one spectrum and the total length of the string is solved. damping, Dirichlet boundary condition, point mass, Hermite-Biehler polynomial, continued fraction, eigenvalues 2015 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol35iss2art2 https://www.opuscula.agh.edu.pl/vol35/2/art/opuscula_math_3511.pdf Simple eigenvectors of unbounded operators of the type "normal plus compact" Michael Gil' The paper deals with operators of the form \(A=S+B\), where \(B\) is a compact operator in a Hilbert space \(H\) and \(S\) is an unbounded normal one in \(H\), having a compact resolvent. We consider approximations of the eigenvectors of \(A\), corresponding to simple eigenvalues by the eigenvectors of the operators \(A_n=S+B_n\) (\(n=1,2, \ldots\)), where \(B_n\) is an \(n\)-dimensional operator. In addition, we obtain the error estimate of the approximation. Hilbert space, linear operators, eigenvectors, approximation, integro-differential operators, Schatten-von Neumann operators 2015 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol35iss2art3 https://www.opuscula.agh.edu.pl/vol35/2/art/opuscula_math_3512.pdf On b-vertex and b-edge critical graphs Noureddine Ikhlef Eschouf, Mostafa Blidia A \(b\)-coloring is a coloring of the vertices of a graph such that each color class contains a vertex that has a neighbor in all other color classes, and the \(b\)-chromatic number \(b(G)\) of a graph \(G\) is the largest integer \(k\) such that \(G\) admits a \(b\)-coloring with \(k\) colors. A simple graph \(G\) is called \(b^{+}\)-vertex (edge) critical if the removal of any vertex (edge) of \(G\) increases its \(b\)-chromatic number. In this note, we explain some properties in \(b^{+}\)-vertex (edge) critical graphs, and we conclude with two open problems. \(b\)-coloring, \(b\)-chromatic number, critical graphs 2015 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol35iss2art4 https://www.opuscula.agh.edu.pl/vol35/2/art/opuscula_math_3513.pdf Bounded, asymptotically stable, and L^{1} solutions of Caputo fractional differential equations Muhammad N. Islam The existence of bounded solutions, asymptotically stable solutions, and \(L^1\) solutions of a Caputo fractional differential equation has been studied in this paper. The results are obtained from an equivalent Volterra integral equation which is derived by inverting the fractional differential equation. The kernel function of this integral equation is weakly singular and hence the standard techniques that are normally applied on Volterra integral equations do not apply here. This hurdle is overcomed using a resolvent equation and then applying some known properties of the resolvent. In the analysis Schauder's fixed point theorem and Liapunov's method have been employed. The existence of bounded solutions are obtained employing Schauder's theorem, and then it is shown that these solutions are asymptotically stable by a definition found in [C. Avramescu, C. Vladimirescu, On the existence of asymptotically stable solution of certain integral equations, Nonlinear Anal. 66 (2007), 472-483]. Finally, the \(L^1\) properties of solutions are obtained using Liapunov's method. Caputo fractional differential equations, Volterra integral equations, weakly singular kernel, Schauder fixed point theorem, Liapunov's method 2015 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol35iss2art5 https://www.opuscula.agh.edu.pl/vol35/2/art/opuscula_math_3514.pdf A new characterization of convex φ-functions with a parameter Bartosz Micherda We show that, under some additional assumptions, all projection operators onto latticially closed subsets of the Orlicz-Musielak space generated by \(\Phi\) are isotonic if and only if \(\Phi\) is convex with respect to its second variable. A dual result of this type is also proven for antiprojections. This gives the positive answer to the problem presented in Opuscula Mathematica in 2012. Orlicz-Musielak space, convex function, isotonic operator, projection operator, antiprojection operator 2015 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol35iss2art6 https://www.opuscula.agh.edu.pl/vol35/2/art/opuscula_math_3515.pdf Decisiveness of the spectral gaps of periodic Schrödinger operators on the dumbbell-like metric graph Hiroaki Niikuni In this paper, we consider periodic Schrödinger operators on the dumbbell-like metric graph, which is a periodic graph consisting of lines and rings. Let one line and two rings be in the basic period. We see the relationship between the structure of graph and the band-gap spectrum. quantum graph, spectral gap, band structure, Hill operator 2015 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol35iss2art7 https://www.opuscula.agh.edu.pl/vol35/2/art/opuscula_math_3516.pdf ILU preconditioning based on the FAPINV algorithm Davod Khojasteh Salkuyeh, Amin Rafiei, Hadi Roohani A technique for computing an ILU preconditioner based on the factored approximate inverse (FAPINV) algorithm is presented. We show that this algorithm is well-defined for H-matrices. Moreover, when used in conjunction with Krylov-subspace-based iterative solvers such as the GMRES algorithm, results in reliable solvers. Numerical experiments on some test matrices are given to show the efficiency of the new ILU preconditioner. system of linear equations, preconditioner, FAPINV, ILU preconditioner, H-matrix, GMRES 2015 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol35iss2art8 https://www.opuscula.agh.edu.pl/vol35/2/art/opuscula_math_3517.pdf Steering projections in von Neumann algebras Adam Wegert A steering projection of an arbitrary von Neumann algebra is introduced. It is shown that a steering projection always exists and is unique (up to Murray-von Neumann equivalence). A general decomposition of arbitrary projections with respect to a steering projection is established. Murray-von Neumann order, central projection, steering projection 2015 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol35iss3 https://www.opuscula.agh.edu.pl/om-vol35iss3art1 https://www.opuscula.agh.edu.pl/vol35/3/art/opuscula_math_3518.pdf Hildebrandt's theorem for the essential spectrum Janko Bračič, Cristina Diogo We prove a variant of Hildebrandt's theorem which asserts that the convex hull of the essential spectrum of an operator \(A\) on a complex Hilbert space is equal to the intersection of the essential numerical ranges of operators which are similar to \(A\). As a consequence, it is given a necessary and sufficient condition for zero not being in the convex hull of the essential spectrum of \(A\). essential spectrum, essential numerical range, Hildebrandt's theorem 2015 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol35iss3art2 https://www.opuscula.agh.edu.pl/vol35/3/art/opuscula_math_3519.pdf A note on M_{2}-edge colorings of graphs Július Czap An edge coloring \(\varphi\) of a graph \(G\) is called an \(M_2\)-edge coloring if \(|\varphi(v)|\le2 \) for every vertex \(v\) of \(G\), where \(\varphi(v)\) is the set of colors of edges incident with \(v\). Let \(K_2(G)\) denote the maximum number of colors used in an \(M_2\)-edge coloring of \(G\). Let \(G_1\), \(G_2\) and \(G_3\) be graphs such that \(G_1\subseteq G_2\subseteq G_3\). In this paper we deal with the following question: Assuming that \(K_2(G_1)=K_2(G_3)\), does it hold \(K_2(G_1)=K_2(G_2)=K_2(G_3)\)? edge coloring, graph 2015 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol35iss3art3 https://www.opuscula.agh.edu.pl/vol35/3/art/opuscula_math_3520.pdf Frames and factorization of graph Laplacians Palle Jorgensen, Feng Tian Using functions from electrical networks (graphs with resistors assigned to edges), we prove existence (with explicit formulas) of a canonical Parseval frame in the energy Hilbert space \(\mathscr{H}_{E}\) of a prescribed infinite (or finite) network. Outside degenerate cases, our Parseval frame is not an orthonormal basis. We apply our frame to prove a number of explicit results: With our Parseval frame and related closable operators in \(\mathscr{H}_{E}\) we characterize the Friedrichs extension of the \(\mathscr{H}_{E}\)-graph Laplacian. We consider infinite connected network-graphs \(G=\left(V,E\right)\), \(V\) for vertices, and \(E\) for edges. To every conductance function \(c\) on the edges \(E\) of \(G\), there is an associated pair \(\left(\mathscr{H}_{E},\Delta\right)\) where \(\mathscr{H}_{E}\) in an energy Hilbert space, and \(\Delta\left(=\Delta_{c}\right)\) is the \(c\)-Graph Laplacian; both depending on the choice of conductance function \(c\). When a conductance function is given, there is a current-induced orientation on the set of edges and an associated natural Parseval frame in \(\mathscr{H}_{E}\) consisting of dipoles. Now \(\Delta\) is a well-defined semibounded Hermitian operator in both of the Hilbert \(l^{2}\left(V\right)\) and \(\mathscr{H}_{E}\). It is known to automatically be essentially selfadjoint as an \(l^{2}\left(V\right)\)-operator, but generally not as an \(\mathscr{H}_{E}\) operator. Hence as an \(\mathscr{H}_{E}\) operator it has a Friedrichs extension. In this paper we offer two results for the Friedrichs extension: a characterization and a factorization. The latter is via \(l^{2}\left(V\right)\). unbounded operators, deficiency-indices, Hilbert space, boundary values, weighted graph, reproducing kernel, Dirichlet form, graph Laplacian, resistance network, harmonic analysis, harmonic analysis, frame, Parseval frame, Friedrichs extension, reversible random walk, resistance distance, energy Hilbert space 2015 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol35iss3art4 https://www.opuscula.agh.edu.pl/vol35/3/art/opuscula_math_3521.pdf Ruin probability in a risk model with variable premium intensity and risky investments Yuliya Mishura, Mykola Perestyuk, Olena Ragulina We consider a generalization of the classical risk model when the premium intensity depends on the current surplus of an insurance company. All surplus is invested in the risky asset, the price of which follows a geometric Brownian motion. We get an exponential bound for the infinite-horizon ruin probability. To this end, we allow the surplus process to explode and investigate the question concerning the probability of explosion of the surplus process between claim arrivals. risk process, infinite-horizon ruin probability, variable premium intensity, risky investments, exponential bound, stochastic differential equation, explosion time, existence and uniqueness theorem, supermartingale property 2015 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol35iss3art5 https://www.opuscula.agh.edu.pl/vol35/3/art/opuscula_math_3522.pdf Spectra of some selfadjoint Jacobi operators in the double root case Wojciech Motyka In this paper we prove a mixed spectrum of Jacobi operators defined by \(\lambda_n=s(n)(1+x(n))\) and \(q_n=-2s(n)(1+y(n))\), where \((s(n))\) is a real unbounded sequence, \((x(n))\) and \((y(n))\) are some perturbations. Jacobi matrices, double root case, asymptotic behavior, subordination theory, absolutely continuous spectrum, discrete spectrum 2015 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol35iss3art6 https://www.opuscula.agh.edu.pl/vol35/3/art/opuscula_math_3523.pdf On the eigenvalues of a 2×2 block operator matrix Mukhiddin I. Muminov, Tulkin H. Rasulov A \(2\times2\) block operator matrix \({\mathbf H}\) acting in the direct sum of one- and two-particle subspaces of a Fock space is considered. The existence of infinitely many negative eigenvalues of \(H_{22}\) (the second diagonal entry of \({\bf H}\)) is proved for the case where both of the associated Friedrichs models have a zero energy resonance. For the number \(N(z)\) of eigenvalues of \(H_{22}\) lying below \(z\lt0\), the following asymptotics is found \[\lim\limits_{z\to -0} N(z) |\log|z||^{-1}=\,{\mathcal U}_0 \quad (0\lt {\mathcal U}_0\lt \infty).\] Under some natural conditions the infiniteness of the number of eigenvalues located respectively inside, in the gap, and below the bottom of the essential spectrum of \({\mathbf H}\) is proved. block operator matrix, Fock space, discrete and essential spectra, Birman-Schwinger principle, the Efimov effect, discrete spectrum asymptotics, embedded eigenvalues 2015 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol35iss3art7 https://www.opuscula.agh.edu.pl/vol35/3/art/opuscula_math_3524.pdf Generalized Levinson's inequality and exponential convexity Josip Pečarić, Marjan Praljak, Alfred Witkowski We give a probabilistic version of Levinson's inequality under Mercer's assumption of equal variances for the family of 3-convex functions at a point. We also show that this is the largest family of continuous functions for which the inequality holds. New families of exponentially convex functions and related results are derived from the obtained inequality. Levinson's inequality, exponential convexity 2015 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol35iss3art8 https://www.opuscula.agh.edu.pl/vol35/3/art/opuscula_math_3525.pdf Hermite-Hadamard type inequalities for Wright-convex functions of several variables Dorota Śliwińska, Szymon Wąsowicz We present Hermite-Hadamard type inequalities for Wright-convex, strongly convex and strongly Wright-convex functions of several variables defined on simplices. convex functions, Wright-convex functions, strongly Wright-convex functions, Hermite-Hadamard inequality 2015 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol35iss4 https://www.opuscula.agh.edu.pl/om-vol35iss4art1 https://www.opuscula.agh.edu.pl/vol35/4/art/opuscula_math_3526.pdf More on the behaviors of fixed points sets of multifunctions and applications Boualem Alleche, Khadra Nachi In this paper, we study the behaviors of fixed points sets of non necessarily pseudo-contractive multifunctions. Rather than comparing the images of the involved multifunctions, we make use of some conditions on the fixed points sets to establish general results on their stability and continuous dependence. We illustrate our results by applications to differential inclusions and give stability results of fixed points sets of non necessarily pseudo-contractive multifunctions with respect to the bounded proximal convergence. multifunction, fixed point, Pompeiu-Hausdorff metric, bounded proximal convergence, differential inclusion 2015 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol35iss4art2 https://www.opuscula.agh.edu.pl/vol35/4/art/opuscula_math_3527.pdf On dynamical systems induced by p-adic number fields Ilwoo Cho In this paper, we construct dynamical systems induced by \(p\)-adic number fields \(\mathbb{Q}_{p}\). We study the corresponding crossed product operator algebras induced by such dynamical systems. In particular, we are interested in structure theorems, and free distributional data of elements in the operator algebras. prime fields, \(p\)-adic number fields, the Adele ring, \(p\)-adic von Neumann algebras, \(p\)-adic dynamical systems 2015 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol35iss4art3 https://www.opuscula.agh.edu.pl/vol35/4/art/opuscula_math_3528.pdf Oscillation criteria for third order nonlinear delay differential equations with damping Said R. Grace This note is concerned with the oscillation of third order nonlinear delay differential equations of the form \[\left( r_{2}(t)\left( r_{1}(t)y^{\prime}(t)\right)^{\prime}\right)^{\prime}+p(t)y^{\prime}(t)+q(t)f(y(g(t)))=0.\tag{\(\ast\)}\] In the papers [A. Tiryaki, M. F. Aktas, Oscillation criteria of a certain class of third order nonlinear delay differential equations with damping, J. Math. Anal. Appl. 325 (2007), 54-68] and [M. F. Aktas, A. Tiryaki, A. Zafer, Oscillation criteria for third order nonlinear functional differential equations, Applied Math. Letters 23 (2010), 756-762], the authors established some sufficient conditions which insure that any solution of equation \((\ast)\) oscillates or converges to zero, provided that the second order equation \[\left( r_{2}(t)z^{\prime }(t)\right)^{\prime}+\left(p(t)/r_{1}(t)\right) z(t)=0\tag{\(\ast\ast\)}\] is nonoscillatory. Here, we shall improve and unify the results given in the above mentioned papers and present some new sufficient conditions which insure that any solution of equation \((\ast)\) oscillates if equation \((\ast\ast)\) is nonoscillatory. We also establish results for the oscillation of equation \((\ast)\) when equation \((\ast\ast)\) is oscillatory. oscillation, third order, delay differential equation 2015 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol35iss4art4 https://www.opuscula.agh.edu.pl/vol35/4/art/opuscula_math_3529.pdf On potential kernels associated with random dynamical systems Mohamed Hmissi, Farida Mokchaha, Aya Hmissi Let \((\theta,\varphi)\) be a continuous random dynamical system defined on a probability space \((\Omega,\mathcal{F},\mathbb{P})\) and taking values on a locally compact Hausdorff space \(E\). The associated potential kernel \(V\) is given by \[ Vf(\omega ,x)= \int\limits_{0}^{\infty} f(\theta_{t}\omega,\varphi(t,\omega)x)dt, \quad \omega \in \Omega, x\in E.\] In this paper, we prove the equivalence of the following statements: 1. The potential kernel of \((\theta,\varphi)\) is proper, i.e. \(Vf\) is \(x\)-continuous for each bounded, \(x\)-continuous function \(f\) with uniformly random compact support. 2. \((\theta ,\varphi)\) has a global Lyapunov function, i.e. a function \(L:\Omega\times E \rightarrow (0,\infty)\) which is \(x\)-continuous and \(L(\theta_t\omega, \varphi(t,\omega)x)\downarrow 0\) as \(t\uparrow \infty\). In particular, we provide a constructive method for global Lyapunov functions for gradient-like random dynamical systems. This result generalizes an analogous theorem known for deterministic dynamical systems. dynamical system, random dynamical system, random differential equation, stochastic differential equation, potential kernel, domination principle, Lyapunov function 2015 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol35iss4art5 https://www.opuscula.agh.edu.pl/vol35/4/art/opuscula_math_3530.pdf Notes on the nonlinear dependence of a multiscale coupled system with respect to the interface Fernando A. Morales This work studies the dependence of the solution with respect to interface geometric perturbations, in a multiscaled coupled Darcy flow system in direct variational formulation. A set of admissible perturbation functions and a sense of convergence is presented, as well as sufficient conditions on the forcing terms, in order to conclude strong convergence statements. For the rate of convergence of the solutions we start solving completely the one dimensional case, using orthogonal decompositions on the appropriate subspaces. Finally, the rate of convergence question is analyzed in a simple multi dimensional setting, by studying the nonlinear operators introduced due to the geometric perturbations. multiscale coupled systems, interface geometric perturbations, variational formulations, nonlinear dependence 2015 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol35iss4art6 https://www.opuscula.agh.edu.pl/vol35/4/art/opuscula_math_3531.pdf Optimal consumption problem in the Vasicek model Jakub Trybuła We consider the problem of an optimal consumption strategy on the infinite time horizon based on the hyperbolic absolute risk aversion utility when the interest rate is an Ornstein-Uhlenbeck process. Using the method of subsolution and supersolution we obtain the existence of solutions of the dynamic programming equation. We illustrate the paper with a numerical example of the optimal consumption strategy and the value function. stochastic control, interest rate model, optimal consumption, HJB equation 2015 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol35iss5 https://www.opuscula.agh.edu.pl/om-vol35iss5art1 https://www.opuscula.agh.edu.pl/vol35/5/art/opuscula_math_3532.pdf Rigidity of monodromies for Appell's hypergeometric functions Yoshishige Haraoka, Tatsuya Kikukawa For monodromy representations of holonomic systems, the rigidity can be defined. We examine the rigidity of the monodromy representations for Appell's hypergeometric functions, and get the representations explicitly. The results show how the topology of the singular locus and the spectral types of the local monodromies work for the study of the rigidity. rigidity, monodromy, arrangement of hyperplanes 2015 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol35iss5art2 https://www.opuscula.agh.edu.pl/vol35/5/art/opuscula_math_3533.pdf On the summability of divergent power series solutions for certain first-order linear PDEs Masaki Hibino This article is concerned with the study of the Borel summability of divergent power series solutions for certain singular first-order linear partial differential equations of nilpotent type. Our main purpose is to obtain conditions which coefficients of equations should satisfy in order to ensure the Borel summability of divergent solutions. We will see that there is a close affinity between the Borel summability of divergent solutions and global analytic continuation properties for coefficients of equations. partial differential equation, divergent power series, summability, asymptotic expansion, analytic continuation, integro-differential equation, integral equation 2015 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol35iss5art3 https://www.opuscula.agh.edu.pl/vol35/5/art/opuscula_math_3534.pdf On k-summability of formal solutions for certain partial differential operators with polynomial coefficients Kunio Ichinobe, Masatake Miyake We study the \(k\)-summability of divergent formal solutions for the Cauchy problem of certain linear partial differential operators with coefficients which are polynomial in \(t\). We employ the method of successive approximation in order to construct the formal solutions and to obtain the properties of analytic continuation of the solutions of convolution equations and their exponential growth estimates. \(k\)-summability, Cauchy problem, power series solutions, successive approximation 2015 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol35iss5art4 https://www.opuscula.agh.edu.pl/vol35/5/art/opuscula_math_3535.pdf On mean-value properties for the Dunkl polyharmonic functions Grzegorz Łysik We derive differential relations between the Dunkl spherical and solid means of continuous functions. Next we use the relations to give inductive proofs of mean-value properties for the Dunkl polyharmonic functions and their converses. Dunkl Laplacian, Dunkl polyharmonic functions, mean-values, Pizzetti formula 2015 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol35iss5art5 https://www.opuscula.agh.edu.pl/vol35/5/art/opuscula_math_3536.pdf Katz's middle convolution and Yokoyama's extending operation Toshio Oshima We give a concrete relation between Katz's middle convolution and Yokoyama's extension and show the equivalence of both algorithms using these operations for the reduction of Fuchsian systems on the Riemann sphere. Fuchsian systems, middle convolution 2015 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol35iss5art6 https://www.opuscula.agh.edu.pl/vol35/5/art/opuscula_math_3537.pdf Maillet type theorem for singular first order nonlinear partial differential equations of totally characteristic type. Part II Akira Shirai In this paper, we study the following nonlinear first order partial differential equation: \[f(t,x,u,\partial_t u,\partial_x u)=0\quad\text{with}\quad u(0,x)\equiv 0.\] The purpose of this paper is to determine the estimate of Gevrey order under the condition that the equation is singular of a totally characteristic type. The Gevrey order is indicated by the rate of divergence of a formal power series. This paper is a continuation of the previous papers [Convergence of formal solutions of singular first order nonlinear partial differential equations of totally characteristic type, Funkcial. Ekvac. 45 (2002), 187-208] and [Maillet type theorem for singular first order nonlinear partial differential equations of totally characteristic type, Surikaiseki Kenkyujo Kokyuroku, Kyoto University 1431 (2005), 94-106]. Especially the last-mentioned paper is regarded as part I of this paper. singular partial differential equations, totally characteristic type, nilpotent vector field, formal solution, Gevrey order, Maillet type theorem 2015 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol35iss5art7 https://www.opuscula.agh.edu.pl/vol35/5/art/opuscula_math_3538.pdf q-analogue of summability of formal solutions of some linear q-difference-differential equations Hidetoshi Tahara, Hiroshi Yamazawa Let \(q\gt 1\). The paper considers a linear \(q\)-difference-differential equation: it is a \(q\)-difference equation in the time variable \(t\), and a partial differential equation in the space variable \(z\). Under suitable conditions and by using \(q\)-Borel and \(q\)-Laplace transforms (introduced by J.-P. Ramis and C. Zhang), the authors show that if it has a formal power series solution \(\hat{X}(t,z)\) one can construct an actual holomorphic solution which admits \(\hat{X}(t,z)\) as a \(q\)-Gevrey asymptotic expansion of order \(1\). \(q\)-difference-differential equations, summability, formal power series solutions, \(q\)-Gevrey asymptotic expansions, \(q\)-Laplace transform 2015 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol35iss5art8 https://www.opuscula.agh.edu.pl/vol35/5/art/opuscula_math_3539.pdf Analytic continuation of solutions of some nonlinear convolution partial differential equations Hidetoshi Tahara The paper considers a problem of analytic continuation of solutions of some nonlinear convolution partial differential equations which naturally appear in the summability theory of formal solutions of nonlinear partial differential equations. Under a suitable assumption it is proved that any local holomorphic solution has an analytic extension to a certain sector and its extension has exponential growth when the variable goes to infinity in the sector. convolution equations, partial differential equations, analytic continuation, summability, sector 2015 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol35iss5art9 https://www.opuscula.agh.edu.pl/vol35/5/art/opuscula_math_3540.pdf On the multisummability of WKB solutions of certain singularly perturbed linear ordinary differential equations Yoshitsugu Takei Using two concrete examples, we discuss the multisummability of WKB solutions of singularly perturbed linear ordinary differential equations. Integral representations of solutions and a criterion for the multisummability based on the Cauchy-Heine transform play an important role in the proof. exact WKB analysis, WKB solution, multisummability 2015 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol35iss5art10 https://www.opuscula.agh.edu.pl/vol35/5/art/opuscula_math_3541.pdf Alien derivatives of the WKB solutions of the Gauss hypergeometric differential equation with a large parameter Mika Tanda We compute alien derivatives of the WKB solutions of the Gauss hypergeometric differential equation with a large parameter and discuss the singularity structures of the Borel transforms of the WKB solution expressed in terms of its alien derivatives. hypergeometric differential equation, WKB solution, Voros coefficient, alien derivative, Stokes curve, fixed singularity 2015 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol35iss5art11 https://www.opuscula.agh.edu.pl/vol35/5/art/opuscula_math_3542.pdf Parametric Borel summability for some semilinear system of partial differential equations Hiroshi Yamazawa, Masafumi Yoshino In this paper we study the Borel summability of formal solutions with a parameter of first order semilinear system of partial differential equations with \(n\) independent variables. In [Singular perturbation of linear systems with a regular singularity, J. Dynam. Control. Syst. 8 (2002), 313-322], Balser and Kostov proved the Borel summability of formal solutions with respect to a singular perturbation parameter for a linear equation with one independent variable. We shall extend their results to a semilinear system of equations with general independent variables. Borel summability, singular perturbation, Euler type operator 2015 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol35iss6 https://www.opuscula.agh.edu.pl/om-vol35iss6art1 https://www.opuscula.agh.edu.pl/vol35/6/art/opuscula_math_3543.pdf Continuous spectrum of Steklov nonhomogeneous elliptic problem Mostafa Allaoui By applying two versions of the mountain pass theorem and Ekeland's variational principle, we prove three different situations of the existence of solutions for the following Steklov problem: \[\begin{aligned}\Delta_{p(x)} u&=|u|^{p(x)-2}u \phantom{\lambda} \quad\text{in}\;\Omega, \\ |\nabla u|^{p(x)-2}\frac{\partial u}{\partial \nu}&= \lambda|u|^{q(x)-2}u \quad\text{on}\;\partial\Omega,\end{aligned}\] where \(\Omega \subset \mathbb{R}^N\) \((N\geq 2)\) is a bounded smooth domain and \(p,q: \overline{\Omega}\rightarrow(1,+\infty)\) are continuous functions. \(p(x)\)-Laplacian, Steklov problem, critical point theorem 2015 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol35iss6art2 https://www.opuscula.agh.edu.pl/vol35/6/art/opuscula_math_3544.pdf Inversion of the Riemann-Liouville operator and its dual using wavelets C. Baccar, N. B. Hamadi, H. Herch, F. Meherzi We define and study the generalized continuous wavelet transform associated with the Riemann-Liouville operator that we use to express the new inversion formulas of the Riemann-Liouville operator and its dual. inverse problem, Riemann-Liouville operator, Fourier transform, wavelets 2015 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol35iss6art3 https://www.opuscula.agh.edu.pl/vol35/6/art/opuscula_math_3545.pdf Existence and multiplicity of solutions for a nonhomogeneous Neumann boundary problem Liliana Klimczak We consider a nonlinear Neumann elliptic equation driven by a \(p\)-Laplacian-type operator which is not homogeneous in general. For such an equation the energy functional does not need to be coercive, and we use suitable variational methods to show that the problem has at least two distinct, nontrivial smooth solutions. Our formulation incorporates strongly resonant equations. Palais-Smale condition, noncoercive functional, second deformation theorem 2015 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol35iss6art4 https://www.opuscula.agh.edu.pl/vol35/6/art/opuscula_math_3546.pdf On vertex stability of complete k-partite graphs Mateusz Nikodem Let \(H\) be any graph. We say that graph \(G\) is \(H\)-stable if \(G-u\) contains a subgraph isomorphic to \(H\) for an arbitrary chosen \(u\in V(G)\). We characterize all \(H\)-stable graphs of minimal size where \(H\) is any complete \(k\)-partite graph. Thus, we generalize the results of Dudek and Żak regarding complete bipartite graphs. vertex stability, minimal stable graphs, complete \(k\)-partite graphs 2015 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol35iss6art5 https://www.opuscula.agh.edu.pl/vol35/6/art/opuscula_math_3547.pdf On the quasilinear Cauchy problem for a hyperbolic functional differential equation Elżbieta Puźniakowska-Gałuch The Cauchy problem for hyperbolic functional differential equations is considered. Volterra and Fredholm dependence are considered. A theorem on the local existence of generalized solutions defined on the Haar pyramid is proved. A result on differentiability of a solution with respect to initial data is proved. functional differential equations, Haar pyramid, differentiability of solutions, Fredholm type of equation 2015 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol35iss6art6 https://www.opuscula.agh.edu.pl/vol35/6/art/opuscula_math_3548.pdf Existence, uniqueness and estimates of classical solutions to some evolutionary system Lucjan Sapa The theorem of the local existence, uniqueness and estimates of solutions in Hölder spaces for some nonlinear differential evolutionary system with initial conditions is formulated and proved. This system is composed of one partial hyperbolic second-order equation and an ordinary subsystem with a parameter. In the proof of the theorem we use the Banach fixed-point theorem, the Arzeli-Ascola lemma and the integral form of the differential problem. hyperbolic wave equation, telegraph equation, system of nonlinear equations, existence, uniqueness and estimates of solutions, Hölder space 2015 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol35iss6art7 https://www.opuscula.agh.edu.pl/vol35/6/art/opuscula_math_3549.pdf Nontrivial solutions of linear functional equations: methods and examples Adrienn Varga, Csaba Vincze For a wide class of linear functional equations the solutions are generalized polynomials. The existence of non-trivial monomial terms of the solution strongly depends on the algebraic properties of some related families of parameters. As a continuation of the previous work [A. Varga, Cs. Vincze, G. Kiss, Algebraic methods for the solution of linear functional equations, Acta Math. Hungar.] we are going to present constructive algebraic methods of the solution in some special cases. Explicit examples will be also given. linear functional equations, spectral analysis, field homomorphisms 2015 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol35iss6art8 https://www.opuscula.agh.edu.pl/vol35/6/art/opuscula_math_3550.pdf Affine extensions of functions with a closed graph Marek Wójtowicz, Waldemar Sieg Let \(A\) be a closed \(G_{\delta}\)-subset of a normal space \(X\). We prove that every function \(f_0: A\to\mathbb{R}\) with a closed graph can be extended to a function \(f: X\to\mathbb{R}\) with a closed graph, too. This is a consequence of a more general result which gives an affine and constructive method of obtaining such extensions. real-valued functions with a closed graph, points of discontinuity, affine extensions of functions 2015 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol36iss1 https://www.opuscula.agh.edu.pl/om-vol36iss1art1 https://www.opuscula.agh.edu.pl/vol36/1/art/opuscula_math_3601.pdf Vertex-weighted Wiener polynomials of subdivision-related graphs Mahdieh Azari, Ali Iranmanesh, Tomislav Došlić Singly and doubly vertex-weighted Wiener polynomials are generalizations of both vertex-weighted Wiener numbers and the ordinary Wiener polynomial. In this paper, we show how the vertex-weighted Wiener polynomials of a graph change with subdivision operators, and apply our results to obtain vertex-weighted Wiener numbers. vertex-weighted Wiener numbers, vertex-weighted Wiener polynomials, subdivision graphs 2016 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol36iss1art2 https://www.opuscula.agh.edu.pl/vol36/1/art/opuscula_math_3602.pdf Kernel conditional quantile estimator under left truncation for functional regressors Nacéra Helal, Elias Ould Saïd Let \(Y\) be a random real response which is subject to left-truncation by another random variable \(T\). In this paper, we study the kernel conditional quantile estimation when the covariable \(X\) takes values in an infinite-dimensional space. A kernel conditional quantile estimator is given under some regularity conditions, among which in the small-ball probability, its strong uniform almost sure convergence rate is established. Some special cases have been studied to show how our work extends some results given in the literature. Simulations are drawn to lend further support to our theoretical results and assess the behavior of the estimator for finite samples with different rates of truncation and sizes. almost sure convergence, functional variables, kernel estimator, Lynden-Bell estimator, small-ball probability, truncated data 2016 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol36iss1art3 https://www.opuscula.agh.edu.pl/vol36/1/art/opuscula_math_3603.pdf On a linear-quadratic problem with Caputo derivative Dariusz Idczak, Stanislaw Walczak In this paper, we study a linear-quadratic optimal control problem with a fractional control system containing a Caputo derivative of unknown function. First, we derive the formulas for the differential and gradient of the cost functional under given constraints. Next, we prove an existence result and derive a maximum principle. Finally, we describe the gradient and projection of the gradient methods for the problem under consideration. fractional Caputo derivative, linear-quadratic problem, existence and uniqueness of a solution, maximum principle, gradient method, projection of the gradient method 2016 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol36iss1art4 https://www.opuscula.agh.edu.pl/vol36/1/art/opuscula_math_3604.pdf Positive solutions of boundary value problems with nonlinear nonlocal boundary conditions Seshadev Padhi, Smita Pati, D. K. Hota We consider the existence of at least three positive solutions of a nonlinear first order problem with a nonlinear nonlocal boundary condition given by \[\begin{aligned} x^{\prime}(t)& = r(t)x(t) + \sum_{i=1}^{m} f_i(t,x(t)), \quad t \in [0,1],\\ \lambda x(0)& = x(1) + \sum_{j=1}^{n} \Lambda_j(\tau_j, x(\tau_j)),\quad \tau_j \in [0,1],\end{aligned}\] where \(r:[0,1] \rightarrow [0,\infty)\) is continuous; the nonlocal points satisfy \(0 \leq \tau_1 \lt \tau_2 \lt \ldots \lt \tau_n \leq 1\), the nonlinear function \(f_i\) and \(\tau_j\) are continuous mappings from \([0,1] \times [0,\infty) \rightarrow [0,\infty)\) for \(i = 1,2,\ldots ,m\) and \(j = 1,2,\ldots ,n\) respectively, and \(\lambda \gt 0\) is a positive parameter. positive solutions, Leggett-Williams fixed point theorem, nonlinear boundary conditions 2016 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol36iss1art5 https://www.opuscula.agh.edu.pl/vol36/1/art/opuscula_math_3605.pdf Eigenvalue problems for anisotropic equations involving a potential on Orlicz-Sobolev type spaces Ionela-Loredana Stăncuţ, Iulia Dorotheea Stîrcu In this paper we consider an eigenvalue problem that involves a nonhomogeneous elliptic operator, variable growth conditions and a potential \(V\) on a bounded domain in \(\mathbb{R}^N\) (\(N\geq 3\)) with a smooth boundary. We establish three main results with various assumptions. The first one asserts that any \(\lambda\gt 0\) is an eigenvalue of our problem. The second theorem states the existence of a constant \(\lambda_{*}\gt 0\) such that any \(\lambda\in(0,\lambda_{*}]\) is an eigenvalue, while the third theorem claims the existence of a constant \(\lambda^{*}\gt 0\) such that every \(\lambda\in[\lambda^{*}, \infty)\) is an eigenvalue of the problem. anisotropic Orlicz-Sobolev space, potential, critical point, weak solution, eigenvalue 2016 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol36iss1art6 https://www.opuscula.agh.edu.pl/vol36/1/art/opuscula_math_3606.pdf On triangular (D_{n})-actions on cyclic p-gonal Riemann surfaces Ewa Tyszkowska A compact Riemann surface \(X\) of genus \(g\gt 1\) which has a conformal automorphism \(\rho\) of prime order \(p\) such that the orbit space \(X/ \langle \rho \rangle \) is the Riemann sphere is called cyclic \(p\)-gonal. Exceptional points in the moduli space \(\mathcal{M}_g\) of compact Riemann surfaces of genus \(g\) are unique surface classes whose full group of conformal automorphisms acts with a triangular signature. We study symmetries of exceptional points in the cyclic \(p\)-gonal locus in \(\mathcal{M}_g\) for which \(\text{Aut}(X)/ \langle \rho \rangle\) is a dihedral group \(D_n\). Riemann surface, symmetry, triangle group, Fuchsian group, NEC group 2016 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol36iss1art7 https://www.opuscula.agh.edu.pl/vol36/1/art/opuscula_math_3607.pdf Fractional evolution equation nonlocal problems with noncompact semigroups Xuping Zhang, Pengyu Chen This paper is concerned with the existence results of mild solutions to the nonlocal problem of fractional semilinear integro-differential evolution equations. New existence theorems are obtained by means of the fixed point theorem for condensing maps. The results extend and improve some related results in this direction. fractional evolution equation, mild solution, nonlocal condition, \(C_0\)-semigroup, condensing maps, measure of noncompactness 2016 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol36iss2 https://www.opuscula.agh.edu.pl/om-vol36iss2art1 https://www.opuscula.agh.edu.pl/vol36/2/art/opuscula_math_3608.pdf Bounds on the inverse signed total domination numbers in graphs M. Atapour, S. Norouzian, S. M. Sheikholeslami, L. Volkmann Let \(G=(V,E)\) be a simple graph. A function \(f:V\rightarrow \{-1,1\}\) is called an inverse signed total dominating function if the sum of its function values over any open neighborhood is at most zero. The inverse signed total domination number of \(G\), denoted by \(\gamma_{st}^0(G)\), equals to the maximum weight of an inverse signed total dominating function of \(G\). In this paper, we establish upper bounds on the inverse signed total domination number of graphs in terms of their order, size and maximum and minimum degrees. inverse signed total dominating function, inverse signed total domination number 2016 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol36iss2art2 https://www.opuscula.agh.edu.pl/vol36/2/art/opuscula_math_3609.pdf Free probability on Hecke algebras and certain group C^{*}-algebras induced by Hecke algebras Ilwoo Cho In this paper, by establishing free-probabilistic models on the Hecke algebras \(\mathcal{H}\left(GL_{2}(\mathbb{Q}_{p})\right)\) induced by \(p\)-adic number fields \(\mathbb{Q}_{p}\), we construct free probability spaces for all primes \(p\). Hilbert-space representations are induced by such free-probabilistic structures. We study \(C^{*}\)-algebras induced by certain partial isometries realized under the representations. free probability, free moments, free cumulants, Hecke algebras, normal Hecke subalgebras, representations, groups, group \(C^{*}\)-algebras 2016 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol36iss2art3 https://www.opuscula.agh.edu.pl/vol36/2/art/opuscula_math_3610.pdf Monotone method for Riemann-Liouville multi-order fractional differential systems Zachary Denton In this paper we develop the monotone method for nonlinear multi-order \(N\)-systems of Riemann-Liouville fractional differential equations. That is, a hybrid system of nonlinear equations of orders \(q_i\) where \(0 \lt q_i \lt 1\). In the development of this method we recall any needed existence results along with any necessary changes. Through the method's development we construct a generalized multi-order Mittag-Leffler function that fulfills exponential-like properties for multi-order systems. Further we prove a comparison result paramount for the discussion of fractional multi-order inequalities that utilizes lower and upper solutions of the system. The monotone method is then developed via the construction of sequences of linear systems based on the upper and lower solutions, and are used to approximate the solution of the original nonlinear multi-order system. fractional differential systems, multi-order systems, lower and upper solutions, monotone method 2016 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol36iss2art4 https://www.opuscula.agh.edu.pl/vol36/2/art/opuscula_math_3611.pdf A two cones support theorem Ricardo Estrada We show that if the Radon transform of a distribution \(f\) vanishes outside of an acute cone \(C_{0}\), the support of the distribution is contained in the union of \(C_{0}\) and another acute cone \(C_{1}\), the cones are in a suitable position, and \(f\) vanishes distributionally in the direction of the axis of \(C_{1}\), then actually \(\operatorname*{supp}f\subset C_{0}\). We show by examples that this result is sharp. Radon transforms, support theorems, distributions 2016 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol36iss2art5 https://www.opuscula.agh.edu.pl/vol36/2/art/opuscula_math_3612.pdf Solutions of fractional nabla difference equations - existence and uniqueness Jagan Mohan Jonnalagadda In this article, we discuss existence, uniqueness and dependency of solutions of nonlinear fractional nabla difference equations in a Banach space equipped with a suitable norm, using the contractive mapping approach. As an application of the established results we present and analyse a few examples. nabla difference, exponential function, fixed point, existence, uniqueness, continuous dependence 2016 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol36iss2art6 https://www.opuscula.agh.edu.pl/vol36/2/art/opuscula_math_3613.pdf Pareto optimal control problem and its Galerkin approximation for a nonlinear one-dimensional extensible beam equation Andrzej Just, Zdzislaw Stempień Our goal is to study the Pareto optimal control system for a nonlinear one-dimensional extensible beam equation and its Galerkin approximation. First we consider a mathematical model of the beam equation which was obtained by S. Woinowsky-Krieger in 1950. Next we consider the Pareto optimal control problem based on this equation. Further, we describe the approximation of this system. We use the Galerkin method to approximate the solution of this control problem with respect to a spatial variable. Based on the standard finite dimensional approximation we prove that as the discretization parameters tend to zero then the weak accumulation point of the solutions of the discrete optimal control problems exist and each of these points is the solution of the original Pareto optimal control problem. nonlinear beam equation, Pareto optimal control, Galerkin approximation 2016 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol36iss2art7 https://www.opuscula.agh.edu.pl/vol36/2/art/opuscula_math_3614.pdf Multiple solutions for fourth order elliptic problems with p(x)-biharmonic operators Lingju Kong We study the multiplicity of weak solutions to the following fourth order nonlinear elliptic problem with a \(p(x)\)-biharmonic operator \[\begin{cases}\Delta^2_{p(x)}u+a(x)|u|^{p(x)-2}u=\lambda f(x,u)\quad\text{ in }\Omega,\\ u=\Delta u=0\quad\text{ on }\partial\Omega,\end{cases}\] where \(\Omega\) is a smooth bounded domain in \(\mathbb{R}^N\), \(p\in C(\overline{\Omega})\), \(\Delta^2_{p(x)}u=\Delta(|\Delta u|^{p(x)-2}\Delta u)\) is the \(p(x)\)-biharmonic operator, and \(\lambda\gt 0\) is a parameter. We establish sufficient conditions under which there exists a positive number \(\lambda^{*}\) such that the above problem has at least two nontrivial weak solutions for each \(\lambda\gt\lambda^{*}\). Our analysis mainly relies on variational arguments based on the mountain pass lemma and some recent theory on the generalized Lebesgue-Sobolev spaces \(L^{p(x)}(\Omega)\) and \(W^{k,p(x)}(\Omega)\). critical points, \(p(x)\)-biharmonic operator, weak solutions, mountain pass lemma 2016 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol36iss2art8 https://www.opuscula.agh.edu.pl/vol36/2/art/opuscula_math_3615.pdf Asymptotic behavior of solutions of discrete Volterra equations Janusz Migda, Małgorzata Migda We consider the nonlinear discrete Volterra equations of non-convolution type \[\Delta^m x_n=b_n+\sum\limits_{i=1}^{n}K(n,i)f\left(i,x_i\right), \quad n\geq 1.\] We present sufficient conditions for the existence of solutions with prescribed asymptotic behavior, especially asymptotically polynomial and asymptotically periodic solutions. We use \(\operatorname{o}(n^s)\), for a given nonpositive real \(s\), as a measure of approximation. We also give conditions under which all solutions are asymptotically polynomial. Volterra difference equation, prescribed asymptotic behavior, asymptotically polynomial solution, asymptotically periodic solution, bounded solution 2016 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol36iss2art9 https://www.opuscula.agh.edu.pl/vol36/2/art/opuscula_math_3616.pdf Corrigendum to "Hermite-Hadamard type inequalities for Wright-convex functions of several variables" [Opuscula Math. 35, no. 3 (2015), 411-419] Alfred Witkowski We correct a small mistake made by the authors of the paper [Hermite-Hadamard type inequalities for Wright-convex functions of several variables, Opuscula Math. 35, no. 3 (2015), 411-419]. Write convex function, Hermite-Hadamard inequality, symmetrization, simplex 2016 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol36iss3 https://www.opuscula.agh.edu.pl/om-vol36iss3art1 https://www.opuscula.agh.edu.pl/vol36/3/art/opuscula_math_3617.pdf Multiplicative Zagreb indices and coindices of some derived graphs Bommanahal Basavanagoud, Shreekant Patil In this note, we obtain the expressions for multiplicative Zagreb indices and coindices of derived graphs such as a line graph, subdivision graph, vertex-semitotal graph, edge-semitotal graph, total graph and paraline graph. multiplicative Zagreb indices and coindices, derived graphs 2016 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol36iss3art2 https://www.opuscula.agh.edu.pl/vol36/3/art/opuscula_math_3618.pdf Higher order Nevanlinna functions and the inverse three spectra problem Olga Boyko, Olga Martinyuk, Vyacheslav Pivovarchik The three spectra problem of recovering the Sturm-Liouville equation by the spectrum of the Dirichlet-Dirichlet boundary value problem on \([0,a]\), the Dirichlet-Dirichlet problem on \([0,a/2]\) and the Neumann-Dirichlet problem on \([a/2,a]\) is considered. Sufficient conditions of solvability and of uniqueness of the solution to such a problem are found. spectrum, eigenvalue, Dirichlet boundary condition, Neumann boundary condition, Marchenko equation, Nevanlinna function 2016 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol36iss3art3 https://www.opuscula.agh.edu.pl/vol36/3/art/opuscula_math_3619.pdf Existence and asymptotic behavior of positive solutions of a semilinear elliptic system in a bounded domain Majda Chaieb, Abdelwaheb Dhifli, Samia Zermani Let \(\Omega\) be a bounded domain in \(\mathbb{R}^{n}\) (\(n\geq 2\)) with a smooth boundary \(\partial \Omega\). We discuss in this paper the existence and the asymptotic behavior of positive solutions of the following semilinear elliptic system \[\begin{aligned} -\Delta u&=a_{1}(x)u^{\alpha}v^{r}\quad\text{in}\;\Omega ,\;\;\,u|_{\partial\Omega}=0,\\ -\Delta v&=a_{2}(x)v^{\beta}u^{s}\quad\text{in}\;\Omega ,\;\;\,v|_{\partial\Omega }=0.\end{aligned}\] Here \(r,s\in \mathbb{R}\), \(\alpha,\beta \lt 1\) such that \(\gamma :=(1-\alpha)(1-\beta)-rs\gt 0\) and the functions \(a_{i}\) (\(i=1,2\)) are nonnegative and satisfy some appropriate conditions with reference to Karamata regular variation theory. semilinear elliptic system, asymptotic behavior, Karamata class, sub-super solution 2016 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol36iss3art4 https://www.opuscula.agh.edu.pl/vol36/3/art/opuscula_math_3620.pdf Certain group dynamical systems induced by Hecke algebras Ilwoo Cho In this paper, we study dynamical systems induced by a certain group \(\mathfrak{T}_{N}^{K}\) embedded in the Hecke algebra \(\mathcal{H}(G_{p})\) induced by the generalized linear group \(G_{p} = GL_{2}(\mathbb{Q}_{p})\) over the \(p\)-adic number fields \(\mathbb{Q}_{p}\) for a fixed prime \(p\). We study fundamental properties of such dynamical systems and the corresponding crossed product algebras in terms of free probability on the Hecke algebra \(\mathcal{H}(G_{p})\). free probability, free moments, free cumulants, Hecke algebra, normal Hecke subalgebra, free probability spaces, representations, operators, Hilbert spaces, dynamical systems, crossed product algebras 2016 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol36iss3art5 https://www.opuscula.agh.edu.pl/vol36/3/art/opuscula_math_3621.pdf The hardness of the independence and matching clutter of a graph Sasun Hambardzumyan, Vahan V. Mkrtchyan, Vahe L. Musoyan, Hovhannes Sargsyan A clutter (or antichain or Sperner family) \(L\) is a pair \((V,E)\), where \(V\) is a finite set and \(E\) is a family of subsets of \(V\) none of which is a subset of another. Usually, the elements of \(V\) are called vertices of \(L\), and the elements of \(E\) are called edges of \(L\). A subset \(s_e\) of an edge \(e\) of a clutter is called recognizing for \(e\), if \(s_e\) is not a subset of another edge. The hardness of an edge \(e\) of a clutter is the ratio of the size of \(e\)'s smallest recognizing subset to the size of \(e\). The hardness of a clutter is the maximum hardness of its edges. We study the hardness of clutters arising from independent sets and matchings of graphs. clutter, hardness, independent set, maximal independent set, matching, maximal matching 2016 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol36iss3art6 https://www.opuscula.agh.edu.pl/vol36/3/art/opuscula_math_3622.pdf Existence theorems of nonlinear asymptotic BVP for a homeomorphism Katarzyna Szymańska-Dębowska In this work, we are concerned with the existence of solutions for the following \(\varphi\)-Laplacian boundary value problem on the half-line \[(\varphi (x'))' =f(t,x,x'),\quad x(0)=0,\quad x'(\infty)=0,\] where \(f:\mathbb{R}_+\times\mathbb{R}^k\times\mathbb{R}^k\to\mathbb{R}^k\) is continuous. The results are proved using the properties of the Leray-Schauder topological degree. half-line, nonlinear, asymptotic boundary value problem, \(\varphi\)-Laplacian, Leray-Schauder degree 2016 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol36iss3art7 https://www.opuscula.agh.edu.pl/vol36/3/art/opuscula_math_3623.pdf Matrix polynomials orthogonal with respect to a non-symmetric matrix of measures Marcin J. Zygmunt The paper focuses on matrix-valued polynomials satisfying a three-term recurrence relation with constant matrix coefficients. It is shown that they form an orthogonal system with respect to a matrix of measures, not necessarily symmetric. Moreover, it is stated the condition on the coefficients of the recurrence formula for which the matrix measure is symmetric. matrix orthogonal polynomials, recurrence formula, matrix of measures, block Jacobi matrices 2016 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol36iss4 https://www.opuscula.agh.edu.pl/om-vol36iss4art1 https://www.opuscula.agh.edu.pl/vol36/4/art/opuscula_math_3624.pdf Integral and fractional equations, positive solutions, and Schaefer's fixed point theorem L. C. Becker, T. A. Burton, I. K. Purnaras This is the continuation of four earlier studies of a scalar fractional differential equation of Riemann-Liouville type \[D^qx(t) = -f(t,x(t)), \quad \lim_{t\to 0^+} t^{1-q} x(t) = x^0 \in\Re \quad (0 \lt q \lt 1), \tag {a}\] in which we first invert it as a Volterra integral equation \[x(t)=x^0 t^{q-1} -\frac{1}{\Gamma (q)}\int\limits^t_0 (t-s)^{q-1}f(s,x(s))\,ds \tag {b}\] and then transform it into \[\begin{multline}x(t)=x^0t^{q-1}-\int\limits^t_0 R(t-s)x^0s^{q-1}ds\\+\int\limits^t_0R(t-s) \bigg[x(s)-\frac{f(s,x(s))}{J} \bigg] ds, \tag {c}\end{multline}\] where \(R\) is completely monotone with \(\int^{\infty}_0 R(s)\,ds =1\) and \(J\) is an arbitrary positive constant. Notice that when \(x\) is restricted to a bounded set, then by choosing \(J\) large enough, we can frequently change the sign of the integrand in going from \(\text{(b)}\) to \(\text{(c)}\). Moreover, the same kind of transformation will produce a similar effect in a wide variety of integral equations from applied mathematics. Because of that change in sign, we can obtain an a priori upper bound on solutions of \(\text{(b)}\) with a parameter \(\lambda \in (0,1]\) and then obtain an a priori lower bound on solutions of \(\text{(c)}\). Using this property and Schaefer's fixed point theorem, we obtain positive solutions of an array of fractional differential equations of both Caputo and Riemann-Liouville type as well as problems from turbulence, heat transfer, and equations of logistic growth. Very simple results establishing global existence and uniqueness of solutions are also obtained in the same way. fixed points, fractional differential equations, integral equations, Riemann-Liouville operators 2016 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol36iss4art2 https://www.opuscula.agh.edu.pl/vol36/4/art/opuscula_math_3625.pdf Some stability conditions for scalar Volterra difference equations Leonid Berezansky, Małgorzata Migda, Ewa Schmeidel New explicit stability results are obtained for the following scalar linear difference equation \[x(n+1)-x(n)=-a(n)x(n)+\sum_{k=1}^n A(n,k)x(k)+f(n)\] and for some nonlinear Volterra difference equations. linear and nonlinear Volterra difference equations, boundedness of solutions, exponential and asymptotic stability 2016 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol36iss4art3 https://www.opuscula.agh.edu.pl/vol36/4/art/opuscula_math_3626.pdf A remark on the intersections of subanalytic leaves Maciej P. Denkowski We discuss a new sufficient condition - weaker than the usual transversality condition - for the intersection of two subanalytic leaves to be smooth. It involves the tangent cone of the intersection and, as typically non-transversal, it is of interest in analytic geometry or dynamical systems. We also prove an identity principle for real analytic manifolds and subanalytic functions. transversality conditions, subanalytic sets 2016 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol36iss4art4 https://www.opuscula.agh.edu.pl/vol36/4/art/opuscula_math_3627.pdf On the Baire classification of continuous mappings defined on products of Sorgenfrey lines Olena Karlova, Olga Fodchuk We study the Baire measurability of functions defined on \(\mathbb{R}^T\) which are continuous with respect to the product topology on a power \(\mathbb{S}^T\) of Sorgenfrey lines. Baire-one function, Sorgenfrey line, equiconnected space 2016 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol36iss4art5 https://www.opuscula.agh.edu.pl/vol36/4/art/opuscula_math_3628.pdf Uniform approximation by polynomials with integer coefficients Artur Lipnicki Let \(r\), \(n\) be positive integers with \(n\ge 6r\). Let \(P\) be a polynomial of degree at most \(n\) on \([0,1]\) with real coefficients, such that \(P^{(k)}(0)/k!\) and \(P^{(k)}(1)/k!\) are integers for \(k=0,\dots,r-1\). It is proved that there is a polynomial \(Q\) of degree at most \(n\) with integer coefficients such that \(|P(x)-Q(x)|\le C_1C_2^r r^{2r+1/2}n^{-2r}\) for \(x\in[0,1]\), where \(C_1\), \(C_2\) are some numerical constants. The result is the best possible up to the constants. approximation by polynomials with integer coefficients, lattice, covering radius 2016 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol36iss4art6 https://www.opuscula.agh.edu.pl/vol36/4/art/opuscula_math_3629.pdf Entropy of foliations with leafwise Finsler structure Ilona Michalik, Szymon Walczak We extend the notion of the geometric entropy of foliation to foliated manifolds equipped with leafwise Finsler structure. We study the relation between the geometric entropy and the topological entropy of the holonomy pseudogroup. The case of a foliated manifold with leafwise Randers structure is considered. In this case the estimates for one dimensional foliation defined by a vector field in terms of the topological entropy of a flow are presented. geometric entropy, leafwise Finsler structure 2016 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol36iss4art7 https://www.opuscula.agh.edu.pl/vol36/4/art/opuscula_math_3630.pdf A model for the inverse 1-median problem on trees under uncertain costs Kien Trung Nguyen, Nguyen Thi Linh Chi We consider the problem of justifying vertex weights of a tree under uncertain costs so that a prespecified vertex become optimal and the total cost should be optimal in the uncertainty scenario. We propose a model which delivers the information about the optimal cost which respect to each confidence level \(\alpha \in [0,1]\). To obtain this goal, we first define an uncertain variable with respect to the minimum cost in each confidence level. If all costs are independently linear distributed, we present the inverse distribution function of this uncertain variable in \(O(n^{2}\log n)\) time, where \(n\) is the number of vertices in the tree. location problem, uncertain variable, inverse optimization problem, tree 2016 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol36iss4art8 https://www.opuscula.agh.edu.pl/vol36/4/art/opuscula_math_3631.pdf On solvability of some difference-discrete equations Alexander V. Vasilyev, Vladimir B. Vasilyev Multidimensional difference equations in a discrete half-space are considered. Using the theory of periodic Riemann problems a general solution and solvability conditions in discrete Lebesgue spaces are obtained. Some statements of boundary value problems of discrete type are given. multidimensional difference-discrete equation, symbol, factorization, periodic Riemann problem 2016 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol36iss4art9 https://www.opuscula.agh.edu.pl/vol36/4/art/opuscula_math_3632.pdf On fractional random differential equations with delay Ho Vu, Nguyen Ngoc Phung, Nguyen Phuong In this paper, we consider the existence and uniqueness of solutions of the fractional random differential equations with delay. Moreover, some kind of boundedness of the solution is proven. Finally, the applicability of the theoretical results is illustrated with some real world examples. sample path fractional integral, sample path fractional derivative, fractional differential equations, sample fractional random differential equations, Caputo fractional derivative, delay 2016 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol36iss5 https://www.opuscula.agh.edu.pl/om-vol36iss5art1 https://www.opuscula.agh.edu.pl/vol36/5/art/opuscula_math_3633.pdf Criticality indices of 2-rainbow domination of paths and cycles Ahmed Bouchou, Mostafa Blidia A \(2\)-rainbow dominating function of a graph \(G\left(V(G),E(G)\right)\) is a function \(f\) that assigns to each vertex a set of colors chosen from the set \(\{1,2\}\) so that for each vertex with \(f(v)=\emptyset\) we have \({\textstyle\bigcup_{u\in N(v)}} f(u)=\{1,2\}\). The weight of a \(2\)RDF \(f\) is defined as \(w\left( f\right)={\textstyle\sum\nolimits_{v\in V(G)}} |f(v)|\). The minimum weight of a \(2\)RDF is called the \(2\)-rainbow domination number of \(G\), denoted by \(\gamma_{2r}(G)\). The vertex criticality index of a \(2\)-rainbow domination of a graph \(G\) is defined as \(ci_{2r}^{v}(G)=(\sum\nolimits_{v\in V(G)}(\gamma_{2r}\left(G\right) -\gamma_{2r}\left( G-v\right)))/\left\vert V(G)\right\vert\), the edge removal criticality index of a \(2\)-rainbow domination of a graph \(G\) is defined as \(ci_{2r}^{-e}(G)=(\sum\nolimits_{e\in E(G)}(\gamma_{2r}\left(G\right)-\gamma_{2r}\left( G-e\right)))/\left\vert E(G)\right\vert\) and the edge addition of a \(2\)-rainbow domination criticality index of \(G\) is defined as \(ci_{2r}^{+e}(G)=(\sum\nolimits_{e\in E(\overline{G})}(\gamma_{2r}\left(G\right)-\gamma_{2r}\left( G+e\right)))/\left\vert E(\overline{G})\right\vert\), where \(\overline{G}\) is the complement graph of \(G\). In this paper, we determine the criticality indices of paths and cycles. 2-rainbow domination number, criticality index 2016 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol36iss5art2 https://www.opuscula.agh.edu.pl/vol36/5/art/opuscula_math_3634.pdf Edge subdivision and edge multisubdivision versus some domination related parameters in generalized corona graphs Magda Dettlaff, Joanna Raczek, Ismael G. Yero Given a graph \(G=(V,E)\), the subdivision of an edge \(e=uv\in E(G)\) means the substitution of the edge \(e\) by a vertex \(x\) and the new edges \(ux\) and \(xv\). The domination subdivision number of a graph \(G\) is the minimum number of edges of \(G\) which must be subdivided (where each edge can be subdivided at most once) in order to increase the domination number. Also, the domination multisubdivision number of \(G\) is the minimum number of subdivisions which must be done in one edge such that the domination number increases. Moreover, the concepts of paired domination and independent domination subdivision (respectively multisubdivision) numbers are defined similarly. In this paper we study the domination, paired domination and independent domination (subdivision and multisubdivision) numbers of the generalized corona graphs. domination, paired domination, independent domination, edge subdivision, edge multisubdivision, corona graph 2016 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol36iss5art3 https://www.opuscula.agh.edu.pl/vol36/5/art/opuscula_math_3635.pdf On one oscillatory criterion for the second order linear ordinary differential equations Gevorg Avagovich Grigorian The Riccati equation method is used to establish an oscillatory criterion for second order linear ordinary differential equations. An oscillatory condition is obtained for the generalized Hill's equation. By means of examples the obtained result are compared with some known oscillatory criteria. Riccati equation, normal and extremal solutions, integral and interval oscillatory criteria, the generalized Hill's equation 2016 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol36iss5art4 https://www.opuscula.agh.edu.pl/vol36/5/art/opuscula_math_3636.pdf M_{2}-edge colorings of dense graphs Jaroslav Ivančo An edge coloring \(\varphi\) of a graph \(G\) is called an \(\mathrm{M}_i\)-edge coloring if \(|\varphi(v)|\leq i\) for every vertex \(v\) of \(G\), where \(\varphi(v)\) is the set of colors of edges incident with \(v\). Let \(\mathcal{K}_i(G)\) denote the maximum number of colors used in an \(\mathrm{M}_i\)-edge coloring of \(G\). In this paper we establish some bounds of \(\mathcal{K}_2(G)\), present some graphs achieving the bounds and determine exact values of \(\mathcal{K}_2(G)\) for dense graphs. edge coloring, dominating set, dense graphs 2016 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol36iss5art5 https://www.opuscula.agh.edu.pl/vol36/5/art/opuscula_math_3637.pdf Existence and boundary behavior of positive solutions for a Sturm-Liouville problem Syrine Masmoudi, Samia Zermani In this paper, we discuss existence, uniqueness and boundary behavior of a positive solution to the following nonlinear Sturm-Liouville problem \[\begin{aligned}&\frac{1}{A}(Au^{\prime })^{\prime }+a(t)u^{\sigma}=0\;\;\text{in}\;(0,1),\\ &\lim\limits_{t\to 0}Au^{\prime}(t)=0,\quad u(1)=0,\end{aligned}\] where \(\sigma \lt 1\), \(A\) is a positive differentiable function on \((0,1)\) and \(a\) is a positive measurable function in \((0,1)\) satisfying some appropriate assumptions related to the Karamata class. Our main result is obtained by means of fixed point methods combined with Karamata regular variation theory. nonlinear Sturm-Liouville problem, Green's function, positive solutions, Karamata regular variation theory 2016 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol36iss5art6 https://www.opuscula.agh.edu.pl/vol36/5/art/opuscula_math_3638.pdf Multiplicity results for an impulsive boundary value problem of p(t)-Kirchhoff type via critical point theory A. Mokhtari, T. Moussaoui, D. O'Regan In this paper we obtain existence results of \(k\) distinct pairs nontrivial solutions for an impulsive boundary value problem of \(p(t)\)-Kirchhoff type under certain conditions on the parameter \(\lambda\). genus theory, nonlocal problems, impulsive conditions, Kirchhoff equation, \(p(t)\)-Laplacian, variational methods, critical point theory 2016 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol36iss5art7 https://www.opuscula.agh.edu.pl/vol36/5/art/opuscula_math_3639.pdf On nonoscillatory solutions of two dimensional nonlinear delay dynamical systems Özkan Öztürk, Elvan Akın We study the classification schemes for nonoscillatory solutions of a class of nonlinear two dimensional systems of first order delay dynamic equations on time scales. Necessary and sufficient conditions are also given in order to show the existence and nonexistence of such solutions and some of our results are new for the discrete case. Examples will be given to illustrate some of our results. time scales, oscillation, two-dimensional dynamical system 2016 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol36iss5art8 https://www.opuscula.agh.edu.pl/vol36/5/art/opuscula_math_3640.pdf On a dense minimizer of empirical risk in inverse problems Jacek Podlewski, Zbigniew Szkutnik Properties of estimators of a functional parameter in an inverse problem setup are studied. We focus on estimators obtained through dense minimization (as opposed to minimization over \(\delta\)-nets) of suitably defined empirical risk. At the cost of imposition of a sort of local finite-dimensionality assumption, we fill some gaps in the proofs of results published by Klemelä and Mammen [Ann. Statist. 38 (2010), 482-511]. We also give examples of functional classes that satisfy the modified assumptions. inverse problem, empirical risk minimization 2016 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol36iss5art9 https://www.opuscula.agh.edu.pl/vol36/5/art/opuscula_math_3641.pdf On a problem of Gevorkyan for the Franklin system Zygmunt Wronicz In 1870 G. Cantor proved that if \(\lim_{N\rightarrow\infty}\sum_{n=-N}^N\,c_{n}e^{inx} = 0\) for every real \(x\), where \(\bar{c}_{n}=c_{n}\) (\(n\in \mathbb{Z}\)), then all coefficients \(c_{n}\) are equal to zero. Later, in 1950 V. Ya. Kozlov proved that there exists a trigonometric series for which a subsequence of its partial sums converges to zero, where not all coefficients of the series are zero. In 2004 G. Gevorkyan raised the issue that if Cantor's result extends to the Franklin system. The conjecture remains open until now. In the present paper we show however that Kozlov's version remains true for Franklin's system. Franklin system, orthonormal spline system, trigonometric system, uniqueness of series 2016 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol36iss6 https://www.opuscula.agh.edu.pl/om-vol36iss6art1 https://www.opuscula.agh.edu.pl/vol36/6/art/opuscula_math_3642.pdf Characterizations of rectangular (para)-unitary rational functions Daniel Alpay, Palle Jorgensen, Izchak Lewkowicz We here present three characterizations of not necessarily causal, rational functions which are (co)-isometric on the unit circle: (i) through the realization matrix of Schur stable systems, (ii) the Blaschke-Potapov product, which is then employed to introduce an easy-to-use description of all these functions with dimensions and McMillan degree as parameters, (iii) through the (not necessarily reducible) Matrix Fraction Description (MFD). In cases (ii) and (iii) the poles of the rational functions involved may be anywhere in the complex plane, but the unit circle (including both zero and infinity). A special attention is devoted to exploring the gap between the square and rectangular cases. isometry, coisometry, lossless, all-pass, realization, gramians, matrix fraction description, Blaschke-Potapov product 2016 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol36iss6art2 https://www.opuscula.agh.edu.pl/vol36/6/art/opuscula_math_3643.pdf Eigenvalue estimates for operators with finitely many negative squares Jussi Behrndt, Roland Möws, Carsten Trunk Let \(A\) and \(B\) be selfadjoint operators in a Krein space. Assume that the resolvent difference of \(A\) and \(B\) is of rank one and that the spectrum of \(A\) consists in some interval \(I\subset\mathbb{R}\) of isolated eigenvalues only. In the case that \(A\) is an operator with finitely many negative squares we prove sharp estimates on the number of eigenvalues of \(B\) in the interval \(I\). The general results are applied to singular indefinite Sturm-Liouville problems. selfadjoint operator, Krein space, finitely many negative squares, eigenvalue estimate, indefinite Sturm-Liouville operator 2016 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol36iss6art3 https://www.opuscula.agh.edu.pl/vol36/6/art/opuscula_math_3644.pdf On locally Hilbert spaces Aurelian Gheondea This is an investigation of some basic properties of strictly inductive limits of Hilbert spaces, called locally Hilbert spaces, with respect to their topological properties, the geometry of their subspaces, linear functionals and dual spaces. locally Hilbert space, inductive limit, projective limit, orthocomplemented subspaces, linear functional, dual spaces 2016 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol36iss6art4 https://www.opuscula.agh.edu.pl/vol36/6/art/opuscula_math_3645.pdf Minimal realizations of generalized Nevanlinna functions Seppo Hassi, Hendrik Luit Wietsma Minimal realizations of generalized Nevanlinna functions that carry the information on their generalized poles of nonpositive type in an explicit form are established. These realizations are based on a modification of the basic canonical factorization of generalized Nevanlinna functions whereby the non-minimality problems in realizations that are based directly on the canonical factorization are circumvented. generalized Nevanlinna functions, selfadjoint (multi-valued) operators, (minimal) realizations 2016 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol36iss6art5 https://www.opuscula.agh.edu.pl/vol36/6/art/opuscula_math_3646.pdf Dispersion estimates for spherical Schrödinger equations: the effect of boundary conditions Markus Holzleitner, Aleksey Kostenko, Gerald Teschl We investigate the dependence of the \(L^1\to L^{\infty}\) dispersive estimates for one-dimensional radial Schrödinger operators on boundary conditions at \(0\). In contrast to the case of additive perturbations, we show that the change of a boundary condition at zero results in the change of the dispersive decay estimates if the angular momentum is positive, \(l\in (0,1/2)\). However, for nonpositive angular momenta, \(l\in (-1/2,0]\), the standard \(O(|t|^{-1/2})\) decay remains true for all self-adjoint realizations. Schrödinger equation, dispersive estimates, scattering 2016 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol36iss6art6 https://www.opuscula.agh.edu.pl/vol36/6/art/opuscula_math_3647.pdf Elementary operators - still not elementary? Martin Mathieu Properties of elementary operators, that is, finite sums of two-sided multiplications on a Banach algebra, have been studied under a vast variety of aspects by numerous authors. In this paper we review recent advances in a new direction that seems not to have been explored before: the question when an elementary operator is spectrally bounded or spectrally isometric. As with other investigations, a number of subtleties occur which show that elementary operators are still not elementary to handle. spectral isometries, elementary operators, Jordan isomorphisms 2016 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol36iss6art7 https://www.opuscula.agh.edu.pl/vol36/6/art/opuscula_math_3648.pdf 2-hyperreflexivity and hyporeflexivity of power partial isometries Kamila Piwowarczyk, Marek Ptak Power partial isometries are not always hyperreflexive neither reflexive. In the present paper it will be shown that power partial isometries are always hyporeflexive and \(2\)-hyperreflexive. power partial isometry, reflexive subspace, hyperreflexive subspace, hyperreflexive operator, hyporeflexive algebra 2016 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol36iss6art8 https://www.opuscula.agh.edu.pl/vol36/6/art/opuscula_math_3649.pdf On the spectrum of periodic perturbations of certain unbounded Jacobi operators Jaouad Sahbani It is known that a purely off-diagonal Jacobi operator with coefficients \(a_n=n^{\alpha}\), \(\alpha\in(0,1]\), has a purely absolutely continuous spectrum filling the whole real axis. We show that a 2-periodic perturbation of these operators creates a non trivial gap in the spectrum. essential spectrum, spectral gap, periodic perturbation 2016 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol37iss1 https://www.opuscula.agh.edu.pl/om-vol37iss1art1 https://www.opuscula.agh.edu.pl/vol37/1/art/opuscula_math_3701.pdf Limit-point criteria for the matrix Sturm-Liouville operator and its powers Irina N. Braeutigam We consider matrix Sturm-Liouville operators generated by the formal expression \[l[y]=-(P(y^{\prime}-Ry))^{\prime}-R^*P(y^{\prime}-Ry)+Qy,\] in the space \(L^2_n(I)\), \(I:=[0, \infty)\). Let the matrix functions \(P:=P(x)\), \(Q:=Q(x)\) and \(R:=R(x)\) of order \(n\) (\(n \in \mathbb{N}\)) be defined on \(I\), \(P\) is a nondegenerate matrix, \(P\) and \(Q\) are Hermitian matrices for \(x \in I\) and the entries of the matrix functions \(P^{-1}\), \(Q\) and \(R\) are measurable on \(I\) and integrable on each of its closed finite subintervals. The main purpose of this paper is to find conditions on the matrices \(P\), \(Q\) and \(R\) that ensure the realization of the limit-point case for the minimal closed symmetric operator generated by \(l^k[y]\) (\(k \in \mathbb{N}\)). In particular, we obtain limit-point conditions for Sturm-Liouville operators with matrix-valued distributional coefficients. quasi-derivative, quasi-differential operator, matrix Sturm-Liouville operator, deficiency numbers, distributions 2017 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol37iss1art2 https://www.opuscula.agh.edu.pl/vol37/1/art/opuscula_math_3702.pdf The LQ/KYP problem for infinite-dimensional systems Piotr Grabowski Our aim is to present a solution to a general linear-quadratic (LQ) problem as well as to a Kalman-Yacubovich-Popov (KYP) problem for infinite-dimensional systems with bounded operators. The results are then applied, via the reciprocal system approach, to the question of solvability of some Lur'e resolving equations arising in the stability theory of infinite-dimensional systems in factor form with unbounded control and observation operators. To be more precise the Lur'e resolving equations determine a Lyapunov functional candidate for some closed-loop feedback systems on the base of some properties of an uncontrolled (open-loop) system. Our results are illustrated in details by an example of a temperature of a rod stabilization automatic control system. control of infinite-dimensional systems, semigroups, infinite-time LQ-control problem, Lur'e feedback systems 2017 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol37iss1art3 https://www.opuscula.agh.edu.pl/vol37/1/art/opuscula_math_3703.pdf Towards theory of C-symmetries S. Kuzhel, V. Sudilovskaya The concept of \(\mathcal{C}\)-symmetry originally appeared in \(\mathcal{PT}\)-symmetric quantum mechanics is studied within the Krein spaces framework. Krein space, \(J\)-self-adjoint operator, \(J\)-symmetric operator, Friedrichs extension, \(\mathcal{C}\)-symmetry 2017 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol37iss1art4 https://www.opuscula.agh.edu.pl/vol37/1/art/opuscula_math_3704.pdf Seminormal systems of operators in Clifford environments Mircea Martin The primary goal of our article is to implement some standard spin geometry techniques related to the study of Dirac and Laplace operators on Dirac vector bundles into the multidimensional theory of Hilbert space operators. The transition from spin geometry to operator theory relies on the use of Clifford environments, which essentially are Clifford algebra augmentations of unital complex \(C^*\)-algebras that enable one to set up counterparts of the geometric Bochner-Weitzenböck and Bochner-Kodaira-Nakano curvature identities for systems of elements of a \(C^*\)-algebra. The so derived self-commutator identities in conjunction with Bochner's method provide a natural motivation for the definitions of several types of seminormal systems of operators. As part of their study, we single out certain spectral properties, introduce and analyze a singular integral model that involves Riesz transforms, and prove some self-commutator inequalities. multidimensional operator theory, joint seminormality, Riesz transforms, Putnam inequality 2017 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol37iss1art5 https://www.opuscula.agh.edu.pl/vol37/1/art/opuscula_math_3705.pdf Eigenvalue asymptotics for the Sturm-Liouville operator with potential having a strong local negative singularity Medet Nursultanov, Grigori Rozenblum We find asymptotic formulas for the eigenvalues of the Sturm-Liouville operator on the finite interval, with potential having a strong negative singularity at one endpoint. This is the case of limit circle in H. Weyl sense. We establish that, unlike the case of an infinite interval, the asymptotics for positive eigenvalues does not depend on the potential and it is the same as in the regular case. The asymptotics of the negative eigenvalues may depend on the potential quite strongly, however there are always asymptotically fewer negative eigenvalues than positive ones. By unknown reasons this type of problems had not been studied previously. Sturm-Liouville operator, singular potential, asymptotics of eigenvalues 2017 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol37iss1art6 https://www.opuscula.agh.edu.pl/vol37/1/art/opuscula_math_3706.pdf The basis property of eigenfunctions in the problem of a nonhomogeneous damped string Łukasz Rzepnicki The equation which describes the small vibrations of a nonhomogeneous damped string can be rewritten as an abstract Cauchy problem for the densely defined closed operator \(i A\). We prove that the set of root vectors of the operator \(A\) forms a basis of subspaces in a certain Hilbert space \(H\). Furthermore, we give the rate of convergence for the decomposition with respect to this basis. In the second main result we show that with additional assumptions the set of root vectors of the operator \(A\) is a Riesz basis for \(H\). nonhomogeneous damped string, Hilbert space, Riesz basis, modulus of continuity, basis with parentheses, basis of subspaces, string equation 2017 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol37iss1art7 https://www.opuscula.agh.edu.pl/vol37/1/art/opuscula_math_3707.pdf The inverse scattering transform in the form of a Riemann-Hilbert problem for the Dullin-Gottwald-Holm equation Dmitry Shepelsky, Lech Zielinski The Cauchy problem for the Dullin-Gottwald-Holm (DGH) equation \[u_t-\alpha^2 u_{xxt}+2\omega u_x +3uu_x+\gamma u_{xxx}=\alpha^2 (2u_x u_{xx} + uu_{xxx})\] with zero boundary conditions (as \(|x|\to\infty\)) is treated by the Riemann-Hilbert approach to the inverse scattering transform method. The approach allows us to give a representation of the solution to the Cauchy problem, which can be efficiently used for further studying the properties of the solution, particularly, in studying its long-time behavior. Using the proposed formalism, smooth solitons as well as non-smooth cuspon solutions are presented. Dullin-Gottwald-Holm equation, Camassa-Holm equation, inverse scattering transform, Riemann-Hilbert problem 2017 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol37iss1art8 https://www.opuscula.agh.edu.pl/vol37/1/art/opuscula_math_3708.pdf Hankel and Toeplitz operators: continuous and discrete representations Dmitri R. Yafaev We find a relation guaranteeing that Hankel operators realized in the space of sequences \(\mathcal{l}^2 (\mathbb{Z}_{+})\) and in the space of functions \(L^2 (\mathbb{R}_{+})\) are unitarily equivalent. This allows us to obtain exhaustive spectral results for two classes of unbounded Hankel operators in the space \(\mathcal{l}^2 (\mathbb{Z}_{+})\) generalizing in different directions the classical Hilbert matrix. We also discuss a link between representations of Toeplitz operators in the spaces \(\mathcal{l}^2 (\mathbb{Z}_{+})\) and \(L^2 (\mathbb{R}_{+})\). unbounded Hankel and Toeplitz operators, various representations, moment problems, generalized Hilbert matrices 2017 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol37iss2 https://www.opuscula.agh.edu.pl/om-vol37iss2art1 https://www.opuscula.agh.edu.pl/vol37/2/art/opuscula_math_3709.pdf Non-factorizable C-valued functions induced by finite connected graphs Ilwoo Cho In this paper, we study factorizability of \(\mathbb{C}\)-valued formal series at fixed vertices, called the graph zeta functions, induced by the reduced length on the graph groupoids of given finite connected directed graphs. The construction of such functions is motivated by that of Redei zeta functions. In particular, we are interested in (i) "non-factorizability" of such functions, and (ii) certain factorizable functions induced by non-factorizable functions. By constructing factorizable functions from our non-factorizable functions, we study relations between graph zeta functions and well-known number-theoretic objects, the Riemann zeta function and the Euler totient function. directed graphs, graph groupoids, Redei zeta functions, graph zeta functions, non-factorizable graphs, gluing on graphs 2017 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol37iss2art2 https://www.opuscula.agh.edu.pl/vol37/2/art/opuscula_math_3710.pdf Fractional boundary value problems on the half line Assia Frioui, Assia Guezane-Lakoud, Rabah Khaldi In this paper, we focus on the solvability of a fractional boundary value problem at resonance on an unbounded interval. By constructing suitable operators, we establish an existence theorem upon the coincidence degree theory of Mawhin. The obtained results are illustrated by an example. boundary value problem at resonance, existence of solution, unbounded interval, coincidence degree of Mawhin, fractional differential equation 2017 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol37iss2art3 https://www.opuscula.agh.edu.pl/vol37/2/art/opuscula_math_3711.pdf Existence of three solutions for impulsive nonlinear fractional boundary value problems Shapour Heidarkhani, Massimiliano Ferrara, Giuseppe Caristi, Amjad Salari In this work we present new criteria on the existence of three solutions for a class of impulsive nonlinear fractional boundary-value problems depending on two parameters. We use variational methods for smooth functionals defined on reflexive Banach spaces in order to achieve our results. fractional differential equation, impulsive condition, classical solution, variational methods, critical point theory 2017 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol37iss2art4 https://www.opuscula.agh.edu.pl/vol37/2/art/opuscula_math_3712.pdf Compact generalized weighted composition operators on the Bergman space Qinghua Hu, Xiangling Zhu We characterize the compactness of the generalized weighted composition operators acting on the Bergman space. Bergman space, generalized weighted composition operator, compactness 2017 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol37iss2art5 https://www.opuscula.agh.edu.pl/vol37/2/art/opuscula_math_3713.pdf Control system defined by some integral operator Marek Majewski In the paper we consider a nonlinear control system governed by the Volterra integral operator. Using a version of the global implicit function theorem we prove that the control system under consideration is well-posed and robust, i.e. for any admissible control \(u\) there exists a uniquely defined trajectory \(x_{u}\) which continuously depends on control \(u\) and the operator \(u\mapsto x_{u}\) is continuously differentiable. The novelty of this paper is, among others, the application of the Bielecki norm in the space of solutions which allows us to weaken standard assumptions. Volterra equation, implicit function theorem, sensitivity 2017 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol37iss2art6 https://www.opuscula.agh.edu.pl/vol37/2/art/opuscula_math_3714.pdf The interaction between PDE and graphs in multiscale modeling Fernando A. Morales, Sebastián Naranjo Álvarez In this article an upscaling model is presented for complex networks with highly clustered regions exchanging/trading quantities of interest at both, microscale and macroscale level. Such an intricate system is approximated by a partitioned open map in \(\mathbb{R}^{2}\) or \(\mathbb{R}^{3}\). The behavior of the quantities is modeled as flowing in the map constructed and thus it is subject to be described using partial differential equations. We follow this approach using the Darcy Porous Media, saturated fluid flow model in mixed variational formulation. coupled PDE systems, mixed formulations, porous media, analytic graph theory, complex networks 2017 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol37iss3 https://www.opuscula.agh.edu.pl/om-vol37iss3art1 https://www.opuscula.agh.edu.pl/vol37/3/art/opuscula_math_3715.pdf Existence of three solutions for impulsive multi-point boundary value problems Martin Bohner, Shapour Heidarkhani, Amjad Salari, Giuseppe Caristi This paper is devoted to the study of the existence of at least three classical solutions for a second-order multi-point boundary value problem with impulsive effects. We use variational methods for smooth functionals defined on reflexive Banach spaces in order to achieve our results. Also by presenting an example, we ensure the applicability of our results. multi-point boundary value problem, impulsive condition, classical solution, variational method, three critical points theorem 2017 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol37iss3art2 https://www.opuscula.agh.edu.pl/vol37/3/art/opuscula_math_3716.pdf On the uniform perfectness of equivariant diffeomorphism groups for principal G manifolds Kazuhiko Fukui We proved in [K. Abe, K. Fukui, On commutators of equivariant diffeomorphisms, Proc. Japan Acad. 54 (1978), 52-54] that the identity component \(\text{Diff}\,^r_{G,c}(M)_0\) of the group of equivariant \(C^r\)-diffeomorphisms of a principal \(G\) bundle \(M\) over a manifold \(B\) is perfect for a compact connected Lie group \(G\) and \(1 \leq r \leq \infty\) (\(r \neq \dim B + 1\)). In this paper, we study the uniform perfectness of the group of equivariant \(C^r\)-diffeomorphisms for a principal \(G\) bundle \(M\) over a manifold \(B\) by relating it to the uniform perfectness of the group of \(C^r\)-diffeomorphisms of \(B\) and show that under a certain condition, \(\text{Diff}\,^r_{G,c}(M)_0\) is uniformly perfect if \(B\) belongs to a certain wide class of manifolds. We characterize the uniform perfectness of the group of equivariant \(C^r\)-diffeomorphisms for principal \(G\) bundles over closed manifolds of dimension less than or equal to 3, and in particular we prove the uniform perfectness of the group for the 3-dimensional case and \(r\neq 4\). uniform perfectness, principal \(G\) manifold, equivariant diffeomorphism 2017 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol37iss3art3 https://www.opuscula.agh.edu.pl/vol37/3/art/opuscula_math_3717.pdf General solutions of second-order linear difference equations of Euler type Akane Hongyo, Naoto Yamaoka The purpose of this paper is to give general solutions of linear difference equations which are related to the Euler-Cauchy differential equation \(y^{\prime\prime}+(\lambda/t^2)y=0\) or more general linear differential equations. We also show that the asymptotic behavior of solutions of the linear difference equations are similar to solutions of the linear differential equations. Euler-Cauchy equations, oscillation, conditionally oscillatory 2017 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol37iss3art4 https://www.opuscula.agh.edu.pl/vol37/3/art/opuscula_math_3718.pdf Sufficient conditions for optimality for a mathematical model of drug treatment with pharmacodynamics Maciej Leszczyński, Elżbieta Ratajczyk, Urszula Ledzewicz, Heinz Schättler We consider an optimal control problem for a general mathematical model of drug treatment with a single agent. The control represents the concentration of the agent and its effect (pharmacodynamics) is modelled by a Hill function (i.e., Michaelis-Menten type kinetics). The aim is to minimize a cost functional consisting of a weighted average related to the state of the system (both at the end and during a fixed therapy horizon) and to the total amount of drugs given. The latter is an indirect measure for the side effects of treatment. It is shown that optimal controls are continuous functions of time that change between full or no dose segments with connecting pieces that take values in the interior of the control set. Sufficient conditions for the strong local optimality of an extremal controlled trajectory in terms of the existence of a solution to a piecewise defined Riccati differential equation are given. optimal control, sufficient conditions for optimality, method of characteristics, pharmacodynamic model 2017 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol37iss3art5 https://www.opuscula.agh.edu.pl/vol37/3/art/opuscula_math_3719.pdf Positive solutions of a singular fractional boundary value problem with a fractional boundary condition Jeffrey W. Lyons, Jeffrey T. Neugebauer For \(\alpha\in(1,2]\), the singular fractional boundary value problem \[D^{\alpha}_{0^+}x+f\left(t,x,D^{\mu}_{0^+}x\right)=0,\quad 0\lt t\lt 1,\] satisfying the boundary conditions \(x(0)=D^{\beta}_{0^+}x(1)=0\), where \(\beta\in(0,\alpha-1]\), \(\mu\in(0,\alpha-1]\), and \(D^{\alpha}_{0^+}\), \(D^{\beta}_{0^+}\) and \(D^{\mu}_{0^+}\) are Riemann-Liouville derivatives of order \(\alpha\), \(\beta\) and \(\mu\) respectively, is considered. Here \(f\) satisfies a local Carathéodory condition, and \(f(t,x,y)\) may be singular at the value 0 in its space variable \(x\). Using regularization and sequential techniques and Krasnosel'skii's fixed point theorem, it is shown this boundary value problem has a positive solution. An example is given. fractional differential equation, singular problem, fixed point 2017 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol37iss3art6 https://www.opuscula.agh.edu.pl/vol37/3/art/opuscula_math_3720.pdf On strongly spanning k-edge-colorable subgraphs Vahan V. Mkrtchyan, Gagik N. Vardanyan A subgraph \(H\) of a multigraph \(G\) is called strongly spanning, if any vertex of \(G\) is not isolated in \(H\). \(H\) is called maximum \(k\)-edge-colorable, if \(H\) is proper \(k\)-edge-colorable and has the largest size. We introduce a graph-parameter \(sp(G)\), that coincides with the smallest \(k\) for which a multigraph \(G\) has a maximum \(k\)-edge-colorable subgraph that is strongly spanning. Our first result offers some alternative definitions of \(sp(G)\). Next, we show that \(\Delta(G)\) is an upper bound for \(sp(G)\), and then we characterize the class of multigraphs \(G\) that satisfy \(sp(G)=\Delta(G)\). Finally, we prove some bounds for \(sp(G)\) that involve well-known graph-theoretic parameters. \(k\)-edge-colorable subgraph, maximum \(k\)-edge-colorable subgraph, strongly spanning \(k\)-edge-colorable subgraph, \([1,k]\)-factor 2017 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol37iss3art7 https://www.opuscula.agh.edu.pl/vol37/3/art/opuscula_math_3721.pdf On the inverse signed total domination number in graphs D. A. Mojdeh, B. Samadi In this paper, we study the inverse signed total domination number in graphs and present new sharp lower and upper bounds on this parameter. For example by making use of the classic theorem of Turán (1941), we present a sharp upper bound on \(K_{r+1}\)-free graphs for \(r\geq 2\). Also, we bound this parameter for a tree from below in terms of its order and the number of leaves and characterize all trees attaining this bound. inverse signed total dominating function, inverse signed total domination number, \(k\)-tuple total domination number 2017 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol37iss3art8 https://www.opuscula.agh.edu.pl/vol37/3/art/opuscula_math_3722.pdf On criteria for algebraic independence of collections of functions satisfying algebraic difference relations Hiroshi Ogawara This paper gives conditions for algebraic independence of a collection of functions satisfying a certain kind of algebraic difference relations. As applications, we show algebraic independence of two collections of special functions: (1) Vignéras' multiple gamma functions and derivatives of the gamma function, (2) the logarithmic function, \(q\)-exponential functions and \(q\)-polylogarithm functions. In a similar way, we give a generalization of Ostrowski's theorem. difference algebra, systems of algebraic difference equations, algebraic independence, Vignéras' multiple gamma functions, \(q\)-polylogarithm functions 2017 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol37iss4 https://www.opuscula.agh.edu.pl/om-vol37iss4art1 https://www.opuscula.agh.edu.pl/vol37/4/art/opuscula_math_3723.pdf On Fibonacci numbers in edge coloured trees Urszula Bednarz, Dorota Bród, Anetta Szynal-Liana, Iwona Włoch, Małgorzata Wołowiec-Musiał In this paper we show the applications of the Fibonacci numbers in edge coloured trees. We determine the second smallest number of all \((A,2B)\)-edge colourings in trees. We characterize the minimum tree achieving this second smallest value. edge colouring, tree, tripod, Fibonacci numbers 2017 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol37iss4art2 https://www.opuscula.agh.edu.pl/vol37/4/art/opuscula_math_3724.pdf Colourings of (k-r,k)-trees M. Borowiecki, H. P. Patil Trees are generalized to a special kind of higher dimensional complexes known as \((j,k)\)-trees ([L. W. Beineke, R. E. Pippert, On the structure of \((m,n)\)-trees, Proc. 8th S-E Conf. Combinatorics, Graph Theory and Computing, 1977, 75-80]), and which are a natural extension of \(k\)-trees for \(j=k-1\). The aim of this paper is to study\((k-r,k)\)-trees ([H. P. Patil, Studies on \(k\)-trees and some related topics, PhD Thesis, University of Warsaw, Poland, 1984]), which are a generalization of \(k\)-trees (or usual trees when \(k=1\)). We obtain the chromatic polynomial of \((k-r,k)\)-trees and show that any two \((k-r,k)\)-trees of the same order are chromatically equivalent. However, if \(r\neq 1\) in any \((k-r,k)\)-tree \(G\), then it is shown that there exists another chromatically equivalent graph \(H\), which is not a \((k-r,k)\)-tree. Further, the vertex-partition number and generalized total colourings of \((k-r,k)\)-trees are obtained. We formulate a conjecture about the chromatic index of \((k-r,k)\)-trees, and verify this conjecture in a number of cases. Finally, we obtain a result of [M. Borowiecki, W. Chojnacki, Chromatic index of \(k\)-trees, Discuss. Math. 9 (1988), 55-58] as a corollary in which \(k\)-trees of Class 2 are characterized. chromatic polynomial, partition number, colouring, tree 2017 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol37iss4art3 https://www.opuscula.agh.edu.pl/vol37/4/art/opuscula_math_3725.pdf Spanning trees with a bounded number of leaves Junqing Cai, Evelyne Flandrin, Hao Li, Qiang Sun In 1998, H. Broersma and H. Tuinstra proved that: Given a connected graph \(G\) with \(n\geq 3\) vertices, if \(d(u)+d(v)\geq n-k+1\) for all non-adjacent vertices \(u\) and \(v\) of \(G\) (\(k\geq 1\)), then \(G\) has a spanning tree with at most \(k\) leaves. In this paper, we generalize this result by using implicit degree sum condition of \(t\) (\(2\leq t\leq k\)) independent vertices and we prove what follows: Let \(G\) be a connected graph on \(n\geq 3\) vertices and \(k\geq 2\) be an integer. If the implicit degree sum of any \(t\) independent vertices is at least \(\frac{t(n-k)}{2}+1\) for (\(k\geq t\geq 2\)), then \(G\) has a spanning tree with at most \(k\) leaves. spanning tree, implicit degree, leaves 2017 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol37iss4art4 https://www.opuscula.agh.edu.pl/vol37/4/art/opuscula_math_3726.pdf The metric dimension of circulant graphs and their Cartesian products Kevin Chau, Shonda Gosselin Let \(G=(V,E)\) be a connected graph (or hypergraph) and let \(d(x,y)\) denote the distance between vertices \(x,y\in V(G)\). A subset \(W\subseteq V(G)\) is called a resolving set for \(G\) if for every pair of distinct vertices \(x,y\in V(G)\), there is \(w\in W\) such that \(d(x,w)\neq d(y,w)\). The minimum cardinality of a resolving set for \(G\) is called the metric dimension of \(G\), denoted by \(\beta(G)\). The circulant graph \(C_n(1,2,\ldots,t)\) has vertex set \(\{v_0,v_1,\ldots,v_{n-1}\}\) and edges \(v_iv_{i+j}\) where \(0\leq i\leq n-1\) and \(1\leq j\leq t\) and the indices are taken modulo \(n\) (\(2\leq t\leq\left\lfloor\frac{n}{2}\right\rfloor\)). In this paper we determine the exact metric dimension of the circulant graphs \(C_n(1,2,\ldots,t)\), extending previous results due to Borchert and Gosselin (2013), Grigorious et al. (2014), and Vetrík (2016). In particular, we show that \(\beta(C_n(1,2,\ldots,t))=\beta(C_{n+2t}(1,2,\ldots,t))\) for large enough \(n\), which implies that the metric dimension of these circulants is completely determined by the congruence class of \(n\) modulo \(2t\). We determine the exact value of \(\beta(C_n(1,2,\ldots,t))\) for \(n\equiv 2\bmod 2t\) and \(n\equiv (t+1)\bmod 2t\) and we give better bounds on the metric dimension of these circulants for \(n\equiv 0\bmod 2t\) and \(n\equiv 1 \bmod 2t\). In addition, we bound the metric dimension of Cartesian products of circulant graphs. metric dimension, circulant graph, cartesian product 2017 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol37iss4art5 https://www.opuscula.agh.edu.pl/vol37/4/art/opuscula_math_3727.pdf Acyclic sum-list-colouring of grids and other classes of graphs Ewa Drgas-Burchardt, Agata Drzystek In this paper we consider list colouring of a graph \(G\) in which the sizes of lists assigned to different vertices can be different. We colour \(G\) from the lists in such a way that each colour class induces an acyclic graph. The aim is to find the smallest possible sum of all the list sizes, such that, according to the rules, \(G\) is colourable for any particular assignment of the lists of these sizes. This invariant is called the \(D_1\)-sum-choice-number of \(G\). In the paper we investigate the \(D_1\)-sum-choice-number of graphs with small degrees. Especially, we give the exact value of the \(D_1\)-sum-choice-number for each grid \(P_n\square P_m\), when at least one of the numbers \(n\), \(m\) is less than five, and for each generalized Petersen graph. Moreover, we present some results that estimate the \(D_1\)-sum-choice-number of an arbitrary graph in terms of the decycling number, other graph invariants and special subgraphs. sum-list colouring, acyclic colouring, grids, generalized Petersen graphs 2017 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol37iss4art6 https://www.opuscula.agh.edu.pl/vol37/4/art/opuscula_math_3728.pdf A note on incomplete regular tournaments with handicap two of order n≡8(mod 16) Dalibor Froncek A \(d\)-handicap distance antimagic labeling of a graph \(G=(V,E)\) with \(n\) vertices is a bijection \(f:V\to \{1,2,\ldots ,n\}\) with the property that \(f(x_i)=i\) and the sequence of weights \(w(x_1),w(x_2),\ldots,w(x_n)\) (where \(w(x_i)=\sum_{x_i x_j\in E}f(x_j)\)) forms an increasing arithmetic progression with common difference \(d\). A graph \(G\) is a \(d\)-handicap distance antimagic graph if it allows a \(d\)-handicap distance antimagic labeling. We construct a class of \(k\)-regular \(2\)-handicap distance antimagic graphs for every order \(n\equiv8\pmod{16}\), \(n\geq56\) and \(6\leq k\leq n-50\). incomplete tournaments, handicap tournaments, distance magic labeling, handicap labeling 2017 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol37iss4art7 https://www.opuscula.agh.edu.pl/vol37/4/art/opuscula_math_3729.pdf Anti-Ramsey numbers for disjoint copies of graphs Izolda Gorgol, Agnieszka Görlich A subgraph of an edge-colored graph is called rainbow if all of its edges have different colors. For a graph \(G\) and a positive integer \(n\), the anti-Ramsey number \(ar(n,G)\) is the maximum number of colors in an edge-coloring of \(K_n\) with no rainbow copy of \(H\). Anti-Ramsey numbers were introduced by Erdȍs, Simonovits and Sós and studied in numerous papers. Let \(G\) be a graph with anti-Ramsey number \(ar(n,G)\). In this paper we show the lower bound for \(ar(n,pG)\), where \(pG\) denotes \(p\) vertex-disjoint copies of \(G\). Moreover, we prove that in some special cases this bound is sharp. anti-Ramsey number, rainbow number, disjoint copies 2017 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol37iss4art8 https://www.opuscula.agh.edu.pl/vol37/4/art/opuscula_math_3730.pdf A general 2-part Erdȍs-Ko-Rado theorem Gyula O. H. Katona A two-part extension of the famous Erdȍs-Ko-Rado Theorem is proved. The underlying set is partitioned into \(X_1\) and \(X_2\). Some positive integers \(k_i\), \(\ell_i\) (\(1\leq i\leq m\)) are given. We prove that if \(\mathcal{F}\)) is an intersecting family containing members \(F\) such that \(|F\cap X_1|=k_i\), \(|F\cap X_2|=\ell_i\) holds for one of the values \(i\) (\(1\leq i\leq m\)) then \(|\mathcal{F}|\) cannot exceed the size of the largest subfamily containing one element. extremal set theory, two-part problem, intersecting family 2017 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol37iss4art9 https://www.opuscula.agh.edu.pl/vol37/4/art/opuscula_math_3731.pdf Toward Wojda's conjecture on digraph packing Jerzy Konarski, Andrzej Żak Given a positive integer \(m\leq n/2\), Wojda conjectured in 1985 that if \(D_1\) and \(D_2\) are digraphs of order \(n\) such that \(|A(D_1)|\leq n-m\) and \(|A(D_2)|\leq 2n-\lfloor n/m\rfloor-1\) then \(D_1\) and \(D_2\) pack. The cases when \(m=1\) or \(m = n/2\) follow from known results. Here we prove the conjecture for \(m\geq\sqrt{8n}+418275\). packing, digraph, size 2017 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol37iss4art10 https://www.opuscula.agh.edu.pl/vol37/4/art/opuscula_math_3732.pdf A hierarchy of maximal intersecting triple systems Joanna Polcyn, Andrzej Ruciński We reach beyond the celebrated theorems of Erdȍs-Ko-Rado and Hilton-Milner, and a recent theorem of Han-Kohayakawa, and determine all maximal intersecting triples systems. It turns out that for each \(n\geq 7\) there are exactly 15 pairwise non-isomorphic such systems (and 13 for \(n=6\)). We present our result in terms of a hierarchy of Turán numbers \(\operatorname{ex}^{(s)}(n; M_2^{3})\), \(s\geq 1\), where \(M_2^{3}\) is a pair of disjoint triples. Moreover, owing to our unified approach, we provide short proofs of the above mentioned results (for triple systems only). The triangle \(C_3\) is defined as \(C_3=\{\{x_1,y_3,x_2\},\{x_1,y_2,x_3\},\{x_2,y_1,x_3\}\}\). Along the way we show that the largest intersecting triple system \(H\) on \(n\geq 6\) vertices, which is not a star and is triangle-free, consists of \(\max\{10,n\}\) triples. This facilitates our main proof's philosophy which is to assume that \(H\) contains a copy of the triangle and analyze how the remaining edges of \(H\) intersect that copy. maximal intersecting family, 3-uniform hypergraph, triple system 2017 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol37iss4art11 https://www.opuscula.agh.edu.pl/vol37/4/art/opuscula_math_3733.pdf On the chromatic number of (P_{5},windmill)-free graphs Ingo Schiermeyer In this paper we study the chromatic number of \((P_5, windmill)\)-free graphs. For integers \(r,p\geq 2\) the windmill graph \(W_{r+1}^p=K_1 \vee pK_r\) is the graph obtained by joining a single vertex (the center) to the vertices of \(p\) disjoint copies of a complete graph \(K_r\). Our main result is that every \((P_5, windmill)\)-free graph \(G\) admits a polynomial \(\chi\)-binding function. Moreover, we will present polynomial \(\chi\)-binding functions for several other subclasses of \(P_5\)-free graphs. vertex colouring, perfect graphs, \(\chi\)-binding function, forbidden induced subgraph 2017 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol37iss4art12 https://www.opuscula.agh.edu.pl/vol37/4/art/opuscula_math_3734.pdf Fan's condition on induced subgraphs for circumference and pancyclicity Wojciech Wideł Let \(\mathcal{H}\) be a family of simple graphs and \(k\) be a positive integer. We say that a graph \(G\) of order \(n\geq k\) satisfies Fan's condition with respect to \(\mathcal{H}\) with constant \(k\), if for every induced subgraph \(H\) of \(G\) isomorphic to any of the graphs from \(\mathcal{H}\) the following holds: \[\forall u,v\in V(H)\colon d_H(u,v)=2\,\Rightarrow \max\{d_G(u),d_G(v)\}\geq k/2.\] If \(G\) satisfies the above condition, we write \(G\in\mathcal{F}(\mathcal{H},k)\). In this paper we show that if \(G\) is \(2\)-connected and \(G\in\mathcal{F}(\{K_{1,3},P_4\},k)\), then \(G\) contains a cycle of length at least \(k\), and that if \(G\in\mathcal{F}(\{K_{1,3},P_4\},n)\), then \(G\) is pancyclic with some exceptions. As corollaries we obtain the previous results by Fan, Benhocine and Wojda, and Ning. Fan's condition, circumference, hamiltonian cycle, pancyclicity 2017 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol37iss5 https://www.opuscula.agh.edu.pl/om-vol37iss5art1 https://www.opuscula.agh.edu.pl/vol37/5/art/opuscula_math_3735.pdf Block colourings of 6-cycle systems Paola Bonacini, Mario Gionfriddo, Lucia Marino Let \(\Sigma=(X,\mathcal{B})\) be a \(6\)-cycle system of order \(v\), so \(v\equiv 1,9\mod 12\). A \(c\)-colouring of type \(s\) is a map \(\phi\colon\mathcal {B}\rightarrow \mathcal{C}\), with \(C\) set of colours, such that exactly \(c\) colours are used and for every vertex \(x\) all the blocks containing \(x\) are coloured exactly with \(s\) colours. Let \(\frac{v-1}{2}=qs+r\), with \(q, r\geq 0\). \(\phi\) is equitable if for every vertex \(x\) the set of the \(\frac{v-1}{2}\) blocks containing \(x\) is partitioned in \(r\) colour classes of cardinality \(q+1\) and \(s-r\) colour classes of cardinality \(q\). In this paper we study bicolourings and tricolourings, for which, respectively, \(s=2\) and \(s=3\), distinguishing the cases \(v=12k+1\) and \(v=12k+9\). In particular, we settle completely the case of \(s=2\), while for \(s=3\) we determine upper and lower bounds for \(c\). 6-cycles, block-colourings, G-decompositions 2017 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol37iss5art2 https://www.opuscula.agh.edu.pl/vol37/5/art/opuscula_math_3736.pdf Semicircular elements induced by p-adic number fields Ilwoo Cho, Palle E. T. Jorgensen In this paper, we study semicircular-like elements, and semicircular elements induced by \(p\)-adic analysis, for each prime \(p\). Starting from a \(p\)-adic number field \(\mathbb{Q}_{p}\), we construct a Banach \(*\)-algebra \(\mathfrak{LS}_{p}\), for a fixed prime \(p\), and show the generating elements \(Q_{p,j}\) of \(\mathfrak{LS}_{p}\) form weighted-semicircular elements, and the corresponding scalar-multiples \(\Theta_{p,j}\) of \(Q_{p,j}\) become semicircular elements, for all \(j\in\mathbb{Z}\). The main result of this paper is the very construction of suitable linear functionals \(\tau_{p,j}^{0}\) on \(\mathfrak{LS}_{p}\), making \(Q_{p,j}\) be weighted-semicircular, for all \(j\in\mathbb{Z}\). free probability, primes, \(p\)-adic number fields \(\mathbb{Q}_{p}\), Hilbert-space representations, \(C^{*}\)-algebras, wighted-semicircular elements, semicircular elements 2017 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol37iss5art3 https://www.opuscula.agh.edu.pl/vol37/5/art/opuscula_math_3737.pdf Existence of minimal and maximal solutions to RL fractional integro-differential initial value problems Z. Denton, J. D. Ramírez In this work we investigate integro-differential initial value problems with Riemann Liouville fractional derivatives where the forcing function is a sum of an increasing function and a decreasing function. We will apply the method of lower and upper solutions and develop two monotone iterative techniques by constructing two sequences that converge uniformly and monotonically to minimal and maximal solutions. In the first theorem we will construct two natural sequences and in the second theorem we will construct two intertwined sequences. Finally, we illustrate our results with an example. Riemann Liouville derivative, integro-differential equation, monotone method 2017 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol37iss5art4 https://www.opuscula.agh.edu.pl/vol37/5/art/opuscula_math_3738.pdf On 3-total edge product cordial connected graphs Jaroslav Ivančo A \(k\)-total edge product cordial labeling is a variant of the well-known cordial labeling. In this paper we characterize connected graphs of order at least 15 admitting a 3-total edge product cordial labeling. 3-total edge product cordial labelings, 3-TEPC graphs 2017 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol37iss5art5 https://www.opuscula.agh.edu.pl/vol37/5/art/opuscula_math_3739.pdf On nonexistence of global in time solution for a mixed problem for a nonlinear evolution equation with memory generalizing the Voigt-Kelvin rheological model Petro Pukach, Volodymyr Il'kiv, Zinovii Nytrebych, Myroslava Vovk The paper deals with investigating of the first mixed problem for a fifth-order nonlinear evolutional equation which generalizes well known equation of the vibrations theory. We obtain sufficient conditions of nonexistence of a global solution in time variable. boundary value problem, beam vibrations, nonlinear evolution equation, Voigt-Kelvin model, memory, blowup 2017 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol37iss5art6 https://www.opuscula.agh.edu.pl/vol37/5/art/opuscula_math_3740.pdf Multiplicity results for perturbed fourth-order Kirchhoff-type problems Mohamad Reza Heidari Tavani, Ghasem Alizadeh Afrouzi, Shapour Heidarkhani In this paper, we investigate the existence of three generalized solutions for fourth-order Kirchhoff-type problems with a perturbed nonlinear term depending on two real parameters. Our approach is based on variational methods. multiplicity results, multiple solutions, fourth-order Kirchhoff-type equation, variational methods, critical point theory 2017 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol37iss6 https://www.opuscula.agh.edu.pl/om-vol37iss6art1 https://www.opuscula.agh.edu.pl/vol37/6/art/opuscula_math_3741.pdf On the Steklov problem involving the p(x)-Laplacian with indefinite weight Khaled Ben Ali, Abdeljabbar Ghanmi, Khaled Kefi Under suitable assumptions, we study the existence of a weak nontrivial solution for the following Steklov problem involving the \(p(x)\)-Laplacian \[\begin{cases}\Delta_{p(x)}u=a(x)|u|^{p(x)-2}u \quad \text{in }\Omega, \\ |\nabla u|^{p(x)-2}\frac{\partial u}{\partial \nu}=\lambda V(x)|u|^{q(x)-2}u \quad \text{on }\partial \Omega.\end{cases}\] Our approach is based on min-max method and Ekeland's variational principle. \(p(x)\)-Laplace operator, Steklov problem, variable exponent Sobolev spaces, variational methods, Ekeland's variational principle 2017 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol37iss6art2 https://www.opuscula.agh.edu.pl/vol37/6/art/opuscula_math_3742.pdf The second Cushing-Henson conjecture for the Beverton-Holt q-difference equation Martin Bohner, Sabrina H. Streipert In this paper, we study the second Cushing-Henson conjecture for the Beverton-Holt difference equation with periodic inherent growth rate and periodic carrying capacity in the quantum calculus setting. We give a short summary of recent results regarding the Beverton-Holt difference and \(q\)-difference equation and introduce the theory of quantum calculus briefly. Next, we analyze the second Cushing-Henson conjecture. We extend recent studies in [The Beverton-Holt q-difference equation with periodic growth rate, Difference Equations, Discrete Dynamical Systems, and Applications, Springer-Verlag, Berlin, Heidelberg, New York, 2015, pp. 3-14] and state a modified formulation of the second Cushing-Henson conjecture for the Beverton-Holt \(q\)-difference equation as a generalization of existing formulations. Beverton-Holt equation, Cushing-Henson conjectures, \(q\)-difference equation, periodic solution 2017 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol37iss6art3 https://www.opuscula.agh.edu.pl/vol37/6/art/opuscula_math_3743.pdf A direct approach to linear-quadratic stochastic control Tyrone E. Duncan, Bozenna Pasik-Duncan A direct approach is used to solve some linear-quadratic stochastic control problems for Brownian motion and other noise processes. This direct method does not require solving Hamilton-Jacobi-Bellman partial differential equations or backward stochastic differential equations with a stochastic maximum principle or the use of a dynamic programming principle. The appropriate Riccati equation is obtained as part of the optimization problem. The noise processes can be fairly general including the family of fractional Brownian motions. linear-quadratic Gaussian control, Riccati equation for optimization, stochastic control 2017 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol37iss6art4 https://www.opuscula.agh.edu.pl/vol37/6/art/opuscula_math_3744.pdf Ideals with linear quotients in Segre products Gioia Failla We establish that the Segre product between a polynomial ring on a field \(K\) in \(m\) variables and the second squarefree Veronese subalgebra of a polynomial ring on \(K\) in \(n\) variables has the intersection degree equal to three. We describe a class of monomial ideals of the Segre product with linear quotients. monomial algebras, graded ideals, linear resolutions 2017 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol37iss6art5 https://www.opuscula.agh.edu.pl/vol37/6/art/opuscula_math_3745.pdf Oscillatory and asymptotic behavior of a third-order nonlinear neutral differential equation John R. Graef, Ercan Tunҫ, Said R. Grace This paper discusses oscillatory and asymptotic properties of solutions of a class of third-order nonlinear neutral differential equations. Some new sufficient conditions for a solution of the equation to be either oscillatory or to converges to zero are presented. The results obtained can easily be extended to more general neutral differential equations as well as to neutral dynamic equations on time scales. Two examples are provided to illustrate the results. third order, neutral differential equations, asymptotic behavior, nonoscillatory, oscillatory solution 2017 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol37iss6art6 https://www.opuscula.agh.edu.pl/vol37/6/art/opuscula_math_3746.pdf On the hyper-order of transcendental meromorphic solutions of certain higher order linear differential equations Karima Hamani, Benharrat Belaïdi In this paper, we investigate the growth of meromorphic solutions of the linear differential equation \[f^{(k)}+h_{k-1}(z)e^{P_{k-1}(z)}f^{(k-1)}+\ldots +h_{0}(z)e^{P_{0}(z)}f=0,\] where \(k\geq 2\) is an integer, \(P_{j}(z)\) (\(j=0,1,\ldots ,k-1\)) are nonconstant polynomials and \(h_{j}(z)\) are meromorphic functions. Under some conditions, we determine the hyper-order of these solutions. We also consider nonhomogeneous linear differential equations. linear differential equation, transcendental meromorphic function, order of growth, hyper-order 2017 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol37iss6art7 https://www.opuscula.agh.edu.pl/vol37/6/art/opuscula_math_3747.pdf On the structure of compact graphs Reza Nikandish, Farzad Shaveisi A simple graph \(G\) is called a compact graph if \(G\) contains no isolated vertices and for each pair \(x\), \(y\) of non-adjacent vertices of \(G\), there is a vertex \(z\) with \(N(x)\cup N(y)\subseteq N(z)\), where \(N(v)\) is the neighborhood of \(v\), for every vertex \(v\) of \(G\). In this paper, compact graphs with sufficient number of edges are studied. Also, it is proved that every regular compact graph is strongly regular. Some results about cycles in compact graphs are proved, too. Among other results, it is proved that if the ascending chain condition holds for the set of neighbors of a compact graph \(G\), then the descending chain condition holds for the set of neighbors of \(G\). compact graph, vertex degree, cycle, neighborhood 2017 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol37iss6art8 https://www.opuscula.agh.edu.pl/vol37/6/art/opuscula_math_3748.pdf Oscillation of solutions to non-linear difference equations with several advanced arguments Sandra Pinelas, Julio G. Dix This work concerns the oscillation and asymptotic properties of solutions to the non-linear difference equation with advanced arguments \[x_{n+1}- x_n =\sum_{i=1}^m f_{i,n}( x_{n+h_{i,n}}).\] We establish sufficient conditions for the existence of positive, and negative solutions. Then we obtain conditions for solutions to be bounded, convergent to positive infinity and to negative infinity and to zero. Also we obtain conditions for all solutions to be oscillatory. advanced difference equation, non-oscillatory solution 2017 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol38iss1 https://www.opuscula.agh.edu.pl/om-vol38iss1art1 https://www.opuscula.agh.edu.pl/vol38/1/art/opuscula_math_3801.pdf Upper bounds for the extended energy of graphs and some extended equienergetic graphs Chandrashekar Adiga, B. R. Rakshith In this paper, we give two upper bounds for the extended energy of a graph one in terms of ordinary energy, maximum degree and minimum degree of a graph, and another bound in terms of forgotten index, inverse degree sum, order of a graph and minimum degree of a graph which improves an upper bound of Das et al. from [On spectral radius and energy of extended adjacency matrix of graphs, Appl. Math. Comput. 296 (2017), 116-123]. We present a pair of extended equienergetic graphs on \(n\) vertices for \(n\equiv 0(\text{mod } 8)\) starting with a pair of extended equienergetic non regular graphs on \(8\) vertices and also we construct a pair of extended equienergetic graphs on \(n\) vertices for all \(n\geq 9\) starting with a pair of equienergetic regular graphs on \(9\) vertices. energy of a graph, extended energy of a graph, extended equienergetic graphs 2018 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol38iss1art2 https://www.opuscula.agh.edu.pl/vol38/1/art/opuscula_math_3802.pdf The spectrum problem for digraphs of order 4 and size 5 Ryan C. Bunge, Steven DeShong, Saad I. El-Zanati, Alexander Fischer, Dan P. Roberts, Lawrence Teng The paw graph consists of a triangle with a pendant edge attached to one of the three vertices. We obtain a multigraph by adding exactly one repeated edge to the paw. Now, let \(D\) be a directed graph obtained by orientating the edges of that multigraph. For 12 of the 18 possibilities for \(D\), we establish necessary and sufficient conditions on \(n\) for the existence of a \((K^{*}_{n},D)\)-design. Partial results are given for the remaining 6 possibilities for \(D\). spectrum problem, digraph decompositions 2018 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol38iss1art3 https://www.opuscula.agh.edu.pl/vol38/1/art/opuscula_math_3803.pdf Existence of positive solutions to a discrete fractional boundary value problem and corresponding Lyapunov-type inequalities Amar Chidouh, Delfim F. M. Torres We prove existence of positive solutions to a boundary value problem depending on discrete fractional operators. Then, corresponding discrete fractional Lyapunov-type inequalities are obtained. fractional difference equations, Lyapunov-type inequalities, fractional boundary value problems, positive solutions 2018 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol38iss1art4 https://www.opuscula.agh.edu.pl/vol38/1/art/opuscula_math_3804.pdf Asymptotic profiles for a class of perturbed Burgers equations in one space dimension F. Dkhil, M. A. Hamza, B. Mannoubi In this article we consider the Burgers equation with some class of perturbations in one space dimension. Using various energy functionals in appropriate weighted Sobolev spaces rewritten in the variables \(\frac{\xi}{\sqrt\tau}\) and \(\log\tau\), we prove that the large time behavior of solutions is given by the self-similar solutions of the associated Burgers equation. Burgers equation, self-similar variables, asymptotic behavior, self-similar solutions 2018 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol38iss1art5 https://www.opuscula.agh.edu.pl/vol38/1/art/opuscula_math_3805.pdf Wiener index of strong product of graphs Iztok Peterin, Petra Žigert Pleteršek The Wiener index of a connected graph \(G\) is the sum of distances between all pairs of vertices of \(G\). The strong product is one of the four most investigated graph products. In this paper the general formula for the Wiener index of the strong product of connected graphs is given. The formula can be simplified if both factors are graphs with the constant eccentricity. Consequently, closed formulas for the Wiener index of the strong product of a connected graph \(G\) of constant eccentricity with a cycle are derived. Wiener index, graph product, strong product 2018 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol38iss1art6 https://www.opuscula.agh.edu.pl/vol38/1/art/opuscula_math_3806.pdf On the stability of some systems of exponential difference equations N. Psarros, G. Papaschinopoulos, C. J. Schinas In this paper we prove the stability of the zero equilibria of two systems of difference equations of exponential type, which are some extensions of an one-dimensional biological model. The stability of these systems is investigated in the special case when one of the eigenvalues is equal to -1 and the other eigenvalue has absolute value less than 1, using centre manifold theory. In addition, we study the existence and uniqueness of positive equilibria, the attractivity and the global asymptotic stability of these equilibria of some related systems of difference equations. difference equations, asymptotic behaviour, global stability, centre manifold, biological dynamics 2018 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol38iss1art7 https://www.opuscula.agh.edu.pl/vol38/1/art/opuscula_math_3807.pdf Study of ODE limit problems for reaction-diffusion equations Jacson Simsen, Mariza Stefanello Simsen, Aleksandra Zimmermann In this work we study ODE limit problems for reaction-diffusion equations for large diffusion and we study the sensitivity of nonlinear ODEs with respect to initial conditions and exponent parameters. Moreover, we prove continuity of the flow and weak upper semicontinuity of a family of global attractors for reaction-diffusion equations with spatially variable exponents when the exponents go to 2 in \(L^{\infty}(\Omega)\) and the diffusion coefficients go to infinity. ODE limit problems, shadow systems, reaction-diffusion equations, parabolic problems, variable exponents, attractors, upper semicontinuity 2018 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol38iss2 https://www.opuscula.agh.edu.pl/om-vol38iss2art1 https://www.opuscula.agh.edu.pl/vol38/2/art/opuscula_math_3808.pdf Adelic analysis and functional analysis on the finite Adele ring Ilwoo Cho In this paper, we study operator theory on the \(*\)-algebra \(\mathcal{M}_{\mathcal{P}}\), consisting of all measurable functions on the finite Adele ring \(A_{\mathbb{Q}}\), in extended free-probabilistic sense. Even though our \(*\)-algebra \(\mathcal{M}_{\mathcal{P}}\) is commutative, our Adelic-analytic data and properties on \(\mathcal{M}_{\mathcal{P}}\) are understood as certain free-probabilistic results under enlarged sense of (noncommutative) free probability theory (well-covering commutative cases). From our free-probabilistic model on \(A_{\mathbb{Q}}\), we construct the suitable Hilbert-space representation, and study a \(C^{*}\)-algebra \(M_{\mathcal{P}}\) generated by \(\mathcal{M}_{\mathcal{P}}\) under representation. In particular, we focus on operator-theoretic properties of certain generating operators on \(M_{\mathcal{P}}\). representations, \(C^{*}\)-algebras, \(p\)-adic number fields, the Adele ring, the finite Adele ring 2018 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol38iss2art2 https://www.opuscula.agh.edu.pl/vol38/2/art/opuscula_math_3809.pdf Existence results for Kirchhoff type systems with singular nonlinearity A. Firouzjai, G. A. Afrouzi, S. Talebi Using the method of sub-super solutions, we study the existence of positive solutions for a class of singular nonlinear semipositone systems involving nonlocal operator. sub-supersolution, infinite semipositone systems, singular weights, Kirchhoff-type 2018 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol38iss2art3 https://www.opuscula.agh.edu.pl/vol38/2/art/opuscula_math_3810.pdf Flat structure and potential vector fields related with algebraic solutions to Painlevé VI equation Mitsuo Kato, Toshiyuki Mano, Jiro Sekiguchi A potential vector field is a solution of an extended WDVV equation which is a generalization of a WDVV equation. It is expected that potential vector fields corresponding to algebraic solutions of Painlevé VI equation can be written by using polynomials or algebraic functions explicitly. The purpose of this paper is to construct potential vector fields corresponding to more than thirty non-equivalent algebraic solutions. flat structure, Painlevé VI equation, algebraic solution, potential vector field 2018 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol38iss2art4 https://www.opuscula.agh.edu.pl/vol38/2/art/opuscula_math_3811.pdf Trace formulas for perturbations of operators with Hilbert-Schmidt resolvents Bishnu Prasad Sedai Trace formulas for self-adjoint perturbations \(V\) of self-adjoint operators \(H\) such that \(V\) is in Schatten class were obtained in the works of L.S. Koplienko, M.G. Krein, and the joint paper of D. Potapov, A. Skripka and F. Sukochev. In this article, we obtain an analogous trace formula under the assumptions that the perturbation \(V\) is bounded and the resolvent of \(H\) belongs to Hilbert-Schmidt class. trace formulas 2018 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol38iss2art5 https://www.opuscula.agh.edu.pl/vol38/2/art/opuscula_math_3812.pdf Stochastic differential equations for random matrices processes in the nonlinear framework Sara Stihi, Hacène Boutabia, Selma Meradji In this paper, we investigate the processes of eigenvalues and eigenvectors of a symmetric matrix valued process \(X_{t}\), where \(X_{t}\) is the solution of a general SDE driven by a \(G\)-Brownian motion matrix. Stochastic differential equations of these processes are given. This extends results obtained by P. Graczyk and J. Malecki in [Multidimensional Yamada-Watanabe theorem and its applications to particle systems, J. Math. Phys. 54 (2013), 021503]. \(G\)-Brownian motion matrix, \(G\)-stochastic differential equations, random matrices, eigenvalues, eigenvectors 2018 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol38iss3 https://www.opuscula.agh.edu.pl/om-vol38iss3art1 https://www.opuscula.agh.edu.pl/vol38/3/art/opuscula_math_3813.pdf Solutions to p(x)-Laplace type equations via nonvariational techniques Mustafa Avci In this article, we consider a class of nonlinear Dirichlet problems driven by a Leray-Lions type operator with variable exponent. The main result establishes an existence property by means of nonvariational arguments, that is, nonlinear monotone operator theory and approximation method. Under some natural conditions, we show that a weak limit of approximate solutions is a solution of the given quasilinear elliptic partial differential equation involving variable exponent. Leray-Lions type operator, nonlinear monotone operator, approximation, variable Lebesgue spaces 2018 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol38iss3art2 https://www.opuscula.agh.edu.pl/vol38/3/art/opuscula_math_3814.pdf Backward stochastic variational inequalities driven by multidimensional fractional Brownian motion Dariusz Borkowski, Katarzyna Jańczak-Borkowska We study the existence and uniqueness of the backward stochastic variational inequalities driven by \(m\)-dimensional fractional Brownian motion with Hurst parameters \(H_k\) (\(k=1,\ldots m\)) greater than \(1/2\). The stochastic integral used throughout the paper is the divergence type integral. backward stochastic differential equation, fractional Brownian motion, backward stochastic variational inequalities, subdifferential operator 2018 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol38iss3art3 https://www.opuscula.agh.edu.pl/vol38/3/art/opuscula_math_3815.pdf Improved iterative oscillation tests for first-order deviating differential equations George E. Chatzarakis, Irena Jadlovská In this paper, improved oscillation conditions are established for the oscillation of all solutions of differential equations with non-monotone deviating arguments and nonnegative coefficients. They lead to a procedure that checks for oscillations by iteratively computing \(\lim \sup\) and \(\lim \inf\) on terms recursively defined on the equation's coefficients and deviating argument. This procedure significantly improves all known oscillation criteria. The results and the improvement achieved over the other known conditions are illustrated by two examples, numerically solved in MATLAB. differential equation, non-monotone argument, oscillatory solution, nonoscillatory solution 2018 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol38iss3art4 https://www.opuscula.agh.edu.pl/vol38/3/art/opuscula_math_3816.pdf Forbidden configurations for hypohamiltonian graphs Igor Fabrici, Tomáš Madaras, Mária Timková A graph \(G\) is called hypohamiltonian if \(G\) is not hamiltonian, but \(G-x\) is hamiltonian for each vertex \(x\) of \(G\). We present a list of 331 forbidden configurations which do not appear in hypohamiltonian graphs. hypohamiltonian graph, forbidden configuration, long cycle 2018 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol38iss3art5 https://www.opuscula.agh.edu.pl/vol38/3/art/opuscula_math_3817.pdf On expansive and anti-expansive tree maps Sergiy Kozerenko With every self-map on the vertex set of a finite tree one can associate the directed graph of a special type which is called the Markov graph. Expansive and anti-expansive tree maps are two extremal classes of maps with respect to the number of loops in their Markov graphs. In this paper we prove that a tree with at least two vertices has a perfect matching if and only if it admits an expansive cyclic permutation of its vertices. Also, we show that for every tree with at least three vertices there exists an expansive map with a weakly connected (strongly connected provided the tree has a perfect matching) Markov graph as well as anti-expansive map with a strongly connected Markov graph. maps on trees, Markov graphs, Sharkovsky's theorem 2018 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol38iss3art6 https://www.opuscula.agh.edu.pl/vol38/3/art/opuscula_math_3818.pdf On the boundedness of equivariant homeomorphism groups Jacek Lech, Ilona Michalik, Tomasz Rybicki Given a principal \(G\)-bundle \(\pi:M\to B\), let \(\mathcal{H}_G(M)\) be the identity component of the group of \(G\)-equivariant homeomorphisms on \(M\). The problem of the uniform perfectness and boundedness of \(\mathcal{H}_G(M)\) is studied. It occurs that these properties depend on the structure of \(\mathcal{H}(B)\), the identity component of the group of homeomorphisms of \(B\), and of \(B\) itself. Most of the obtained results still hold in the \(C^r\) category. principal \(G\)-manifold, equivariant homeomorphism, uniformly perfect, bounded, \(C^r\) equivariant diffeomorphism 2018 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol38iss3art7 https://www.opuscula.agh.edu.pl/vol38/3/art/opuscula_math_3819.pdf On domination multisubdivision number of unicyclic graphs Joanna Raczek The paper continues the interesting study of the domination subdivision number and the domination multisubdivision number. On the basis of the constructive characterization of the trees with the domination subdivision number equal to 3 given in [H. Aram, S.M. Sheikholeslami, O. Favaron, Domination subdivision number of trees, Discrete Math. 309 (2009), 622-628], we constructively characterize all connected unicyclic graphs with the domination multisubdivision number equal to 3. We end with further questions and open problems. domination number, domination subdivision number, domination multisubdivision number, trees, unicyclic graphs 2018 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol38iss3art8 https://www.opuscula.agh.edu.pl/vol38/3/art/opuscula_math_3820.pdf Graphons and renormalization of large Feynman diagrams Ali Shojaei-Fard The article builds a new enrichment of the Connes-Kreimer renormalization Hopf algebra of Feynman diagrams in the language of graph functions. graph functions, Dyson-Schwinger equations, Connes-Kreimer renormalization Hopf algebra 2018 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol38iss4 https://www.opuscula.agh.edu.pl/om-vol38iss4art1 https://www.opuscula.agh.edu.pl/vol38/4/art/opuscula_math_3821.pdf Positive definite functions and dual pairs of locally convex spaces Daniel Alpay, Saak Gabriyelyan Using pairs of locally convex topological vector spaces in duality and topologies defined by directed families of sets bounded with respect to the duality, we prove general factorization theorems and general dilation theorems for operator-valued positive definite functions. positive definite function, locally convex space, dual pair, the (strong) factorization property, dilation theory 2018 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol38iss4art2 https://www.opuscula.agh.edu.pl/vol38/4/art/opuscula_math_3822.pdf On spectra of quadratic operator pencils with rank one gyroscopic linear part Olga Boyko, Olga Martynyuk, Vyacheslav Pivovarchik The spectrum of a selfadjoint quadratic operator pencil of the form \(\lambda^2M-\lambda G-A\) is investigated where \(M\geq 0\), \(G\geq 0\) are bounded operators and \(A\) is selfadjoint bounded below is investigated. It is shown that in the case of rank one operator \(G\) the eigenvalues of such a pencil are of two types. The eigenvalues of one of these types are independent of the operator \(G\). Location of the eigenvalues of both types is described. Examples for the case of the Sturm-Liouville operators \(A\) are given. quadratic operator pencil, gyroscopic force, eigenvalues, algebraic multiplicity 2018 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol38iss4art3 https://www.opuscula.agh.edu.pl/vol38/4/art/opuscula_math_3823.pdf Banach *-algebras generated by semicircular elements induced by certain orthogonal projections Ilwoo Cho, Palle E. T. Jorgensen The main purpose of this paper is to study structure theorems of Banach \(*\)-algebras generated by semicircular elements. In particular, we are interested in the cases where given semicircular elements are induced by orthogonal projections in a \(C^{*}\)-probability space. free probability, orthogonal projections, weighted-semicircular elements, semicircular elements 2018 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol38iss4art4 https://www.opuscula.agh.edu.pl/vol38/4/art/opuscula_math_3824.pdf On the non-existence of zero modes Daniel M. Elton We consider magnetic fields on \(\mathbb{R}^3\) which are parallel to a conformal Killing field. When the latter generates a simple rotation we show that a Weyl-Dirac operator with such a magnetic field cannot have a zero mode. In particular this allows us to expand the class of non zero mode producing magnetic fields to include examples of non-trivial smooth compactly supported fields. Weyl-Dirac operator, zero modes 2018 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol38iss4art5 https://www.opuscula.agh.edu.pl/vol38/4/art/opuscula_math_3825.pdf Linear Sturm-Liouville problems with Riemann-Stieltjes integral boundary conditions Qingkai Kong, Thomas E. St. George We study second-order linear Sturm-Liouville problems involving general homogeneous linear Riemann-Stieltjes integral boundary conditions. Conditions are obtained for the existence of a sequence of positive eigenvalues with consecutive zero counts of the eigenfunctions. Additionally, we find interlacing relationships between the eigenvalues of such Sturm-Liouville problems and those of Sturm-Liouville problems with certain two-point separated boundary conditions. nodal solutions, integral boundary value problems, Sturm-Liouville problems, eigenvalues, matching method 2018 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol38iss4art6 https://www.opuscula.agh.edu.pl/vol38/4/art/opuscula_math_3826.pdf Toeplitz versus Hankel: semibounded operators Dmitri R. Yafaev Our goal is to compare various results for Toeplitz \(T\) and Hankel \(H\) operators. We consider semibounded operators and find necessary and sufficient conditions for their quadratic forms to be closable. This property allows one to define \(T\) and \(H\) as self-adjoint operators under minimal assumptions on their matrix elements. We also describe domains of the closed Toeplitz and Hankel quadratic forms. semibounded Toeplitz, Hankel and Wiener-Hopf operators, closable and closed quadratic forms 2018 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol38iss5 https://www.opuscula.agh.edu.pl/om-vol38iss5art1 https://www.opuscula.agh.edu.pl/vol38/5/art/opuscula_math_3827.pdf The spectral theorem for locally normal operators Aurelian Gheondea We prove the spectral theorem for locally normal operators in terms of a locally spectral measure. In order to do this, we first obtain some characterisations of local projections and we single out and investigate the concept of a locally spectral measure. locally Hilbert space, locally \(C^*\)-algebra, locally normal operator, local projection, locally spectral measure 2018 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol38iss5art2 https://www.opuscula.agh.edu.pl/vol38/5/art/opuscula_math_3828.pdf Spectrum of J-frame operators Juan Giribet, Matthias Langer, Leslie Leben, Alejandra Maestripieri, Francisco Martínez Pería, Carsten Trunk A \(J\)-frame is a frame \(\mathcal{F}\) for a Krein space \((\mathcal{H},[\cdot,\cdot ])\) which is compatible with the indefinite inner product \([\cdot,\cdot ]\) in the sense that it induces an indefinite reconstruction formula that resembles those produced by orthonormal bases in \(\mathcal{H}\). With every \(J\)-frame the so-called \(J\)-frame operator is associated, which is a self-adjoint operator in the Krein space \(\mathcal{H}\). The \(J\)-frame operator plays an essential role in the indefinite reconstruction formula. In this paper we characterize the class of \(J\)-frame operators in a Krein space by a \(2\times 2\) block operator representation. The \(J\)-frame bounds of \(\mathcal{F}\) are then recovered as the suprema and infima of the numerical ranges of some uniformly positive operators which are build from the entries of the \(2\times 2\) block representation. Moreover, this \(2\times 2\) block representation is utilized to obtain enclosures for the spectrum of \(J\)-frame operators, which finally leads to the construction of a square root. This square root allows a complete description of all \(J\)-frames associated with a given \(J\)-frame operator. frame, Krein space, block operator matrix, spectrum 2018 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol38iss5art3 https://www.opuscula.agh.edu.pl/vol38/5/art/opuscula_math_3829.pdf Small-gain theorem for a class of abstract parabolic systems Piotr Grabowski We consider a class of abstract control system of parabolic type with observation which the state, input and output spaces are Hilbert spaces. The state space operator is assumed to generate a linear exponentially stable analytic semigroup. An observation and control action are allowed to be described by unbounded operators. It is assumed that the observation operator is admissible but the control operator may be not. Such a system is controlled in a feedback loop by a controller with static characteristic being a globally Lipschitz map from the space of outputs into the space of controls. Our main interest is to obtain a perturbation theorem of the small-gain-type which guarantees that null equilibrium of the closed-loop system will be globally asymptotically stable in Lyapunov's sense. control of infinite-dimensional systems, semigroups, infinite-time LQ-control problem, Lur'e feedback systems 2018 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol38iss5art4 https://www.opuscula.agh.edu.pl/vol38/5/art/opuscula_math_3830.pdf Krein-von Neumann extension of an even order differential operator on a finite interval Yaroslav I. Granovskyi, Leonid L. Oridoroga We describe the Krein-von Neumann extension of minimal operator associated with the expression \(\mathcal{A}:=(-1)^n\frac{d^{2n}}{dx^{2n}}\) on a finite interval \((a,b)\) in terms of boundary conditions. All non-negative extensions of the operator \(A\) as well as extensions with a finite number of negative squares are described. non-negative extension, Friedrichs' extension, Krein-von Neumann extension, boundary triple, Weyl function 2018 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol38iss5art5 https://www.opuscula.agh.edu.pl/vol38/5/art/opuscula_math_3831.pdf On one condition of absolutely continuous spectrum for self-adjoint operators and its applications Eduard Ianovich In this work the method of analyzing of the absolutely continuous spectrum for self-adjoint operators is considered. For the analysis it is used an approximation of a self-adjoint operator \(A\) by a sequence of operators \(A_n\) with absolutely continuous spectrum on a given interval \([a,b]\) which converges to \(A\) in a strong sense on a dense set. The notion of equi-absolute continuity is also used. It was found a sufficient condition of absolute continuity of the operator \(A\) spectrum on the finite interval \([a,b]\) and the condition for that the corresponding spectral density belongs to the class \(L_p[a,b]\) (\(p\ge 1\)). The application of this method to Jacobi matrices is considered. As one of the results we obtain the following assertion: Under some mild assumptions, suppose that there exist a constant \(C\gt 0\) and a positive function \(g(x)\in L_p[a,b]\) (\(p\ge 1\)) such that for all \(n\) sufficiently large and almost all \(x\in[a,b]\) the estimate \(\frac{1}{g(x)}\le b_n(P_{n+1}^2(x)+P_{n}^2(x))\le C\) holds, where \(P_n(x)\) are 1st type polynomials associated with Jacobi matrix (in the sense of Akhiezer) and \(b_n\) is a second diagonal sequence of Jacobi matrix. Then the spectrum of Jacobi matrix operator is purely absolutely continuous on \([a,b]\) and for the corresponding spectral density \(f(x)\) we have \(f(x)\in L_p[a,b]\). self-adjoint operators, absolutely continuous spectrum, equi-absolute continuity, spectral density, Jacobi matrices 2018 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol38iss5art6 https://www.opuscula.agh.edu.pl/vol38/5/art/opuscula_math_3832.pdf Inverse scattering problems for half-line Schrödinger operators and Banach algebras Yaroslav Mykytyuk, Nataliia Sushchyk The inverse scattering problem for half-line Schrödinger operators with potentials from the Marchenko class is shown to be closely related to some Banach algebra of functions on the line. In particular, it is proved that the topological conditions in the Marchenko theorem can be replaced by the condition that the scattering function should belong to this Banach algebra. inverse scattering, Schrödinger operator, Banach algebra 2018 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol38iss5art7 https://www.opuscula.agh.edu.pl/vol38/5/art/opuscula_math_3833.pdf Eigenvalue asymptotics for potential type operators on Lipschitz surfaces of codimension greater than 1 Grigori Rozenblum, Grigory Tashchiyan For potential type integral operators on a Lipschitz submanifold the asymptotic formula for eigenvalues is proved. The reasoning is based upon the study of the rate of operator convergence as smooth surfaces approximate the Lipschitz one. integral operators, potential theory, eigenvalue asymptotics 2018 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol38iss6 https://www.opuscula.agh.edu.pl/om-vol38iss6art1 https://www.opuscula.agh.edu.pl/vol38/6/art/opuscula_math_3834.pdf Improved bounds for solutions of ϕ-Laplacians Waldo Arriagada, Jorge Huentutripay In this short paper we prove a parametric version of the Harnack inequality for \(\phi\)-Laplacian equations. In this sense, the estimates are optimal and represent an improvement of previous bounds for this kind of operators. Orlicz-Sobolev space, Harnack inequality, \(\phi\)-Laplacian 2018 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol38iss6art2 https://www.opuscula.agh.edu.pl/vol38/6/art/opuscula_math_3835.pdf On signed arc total domination in digraphs Leila Asgharsharghi, Abdollah Khodkar, S. M. Sheikholeslami Let \(D=(V,A)\) be a finite simple digraph and \(N(uv)=\{u^{\prime}v^{\prime}\neq uv \mid u=u^{\prime}\text{ or }v=v^{\prime}\}\) be the open neighbourhood of \(uv\) in \(D\). A function \(f: A\rightarrow \{-1, +1\}\) is said to be a signed arc total dominating function (SATDF) of \(D\) if \(\sum _{e^{\prime}\in N(uv)}f(e^{\prime})\geq 1\) holds for every arc \(uv\in A\). The signed arc total domination number \(\gamma^{\prime}_{st}(D)\) is defined as \(\gamma^{\prime}_{st}(D)= \operatorname{min}\{\sum_{e\in A}f(e)\mid f \text{ is an SATDF of }D\}\). In this paper we initiate the study of the signed arc total domination in digraphs and present some lower bounds for this parameter. signed arc total dominating function, signed arc total domination number, domination in digraphs 2018 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol38iss6art3 https://www.opuscula.agh.edu.pl/vol38/6/art/opuscula_math_3836.pdf On locally irregular decompositions of subcubic graphs Olivier Baudon, Julien Bensmail, Hervé Hocquard, Mohammed Senhaji, Éric Sopena A graph \(G\) is locally irregular if every two adjacent vertices of \(G\) have different degrees. A locally irregular decomposition of \(G\) is a partition \(E_1,\dots,E_k\) of \(E(G)\) such that each \(G[E_i]\) is locally irregular. Not all graphs admit locally irregular decompositions, but for those who are decomposable, in that sense, it was conjectured by Baudon, Bensmail, Przybyło and Woźniak that they decompose into at most 3 locally irregular graphs. Towards that conjecture, it was recently proved by Bensmail, Merker and Thomassen that every decomposable graph decomposes into at most 328 locally irregular graphs. We here focus on locally irregular decompositions of subcubic graphs, which form an important family of graphs in this context, as all non-decomposable graphs are subcubic. As a main result, we prove that decomposable subcubic graphs decompose into at most 5 locally irregular graphs, and only at most 4 when the maximum average degree is less than \(\frac{12}{5}\). We then consider weaker decompositions, where subgraphs can also include regular connected components, and prove the relaxations of the conjecture above for subcubic graphs. locally irregular edge-colouring, irregular chromatic index, subcubic graphs 2018 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol38iss6art4 https://www.opuscula.agh.edu.pl/vol38/6/art/opuscula_math_3837.pdf Zig-zag facial total-coloring of plane graphs Július Czap, Stanislav Jendroľ, Margit Voigt In this paper we introduce the concept of zig-zag facial total-coloring of plane graphs. We obtain lower and upper bounds for the minimum number of colors which is necessary for such a coloring. Moreover, we give several sharpness examples and formulate some open problems. plane graph, facial coloring, total-coloring, zig-zag coloring 2018 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol38iss6art5 https://www.opuscula.agh.edu.pl/vol38/6/art/opuscula_math_3838.pdf Essential norm of an integral-type operator from ω-Bloch spaces to μ-Zygmund spaces on the unit ball Juntao Du, Xiangling Zhu In this paper, we give an estimate for the essential norm of an integral-type operator from \(\omega\)-Bloch spaces to \(\mu\)-Zygmund spaces on the unit ball. essential norm, integral-type operator, \(\omega\)-Bloch space, \(\mu\)-Zygmund space 2018 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol38iss6art6 https://www.opuscula.agh.edu.pl/vol38/6/art/opuscula_math_3839.pdf Hubtic number in graphs Shadi Ibrahim Khalaf, Veena Mathad, Sultan Senan Mahde The maximum order of partition of the vertex set \(V(G)\) into hub sets is called hubtic number of \(G\) and denoted by \(\xi(G)\). In this paper we determine the hubtic number of some standard graphs. Also we obtain bounds for \(\xi(G)\). And we characterize the class of all \((p,q)\) graphs for which \(\xi(G)=p\). hubtic number, hub number, partition 2018 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol38iss6art7 https://www.opuscula.agh.edu.pl/vol38/6/art/opuscula_math_3840.pdf Circulant matrices: norm, powers, and positivity Marko Lindner In their recent paper "The spectral norm of a Horadam circulant matrix", Merikoski, Haukkanen, Mattila and Tossavainen study under which conditions the spectral norm of a general real circulant matrix \({\bf C}\) equals the modulus of its row/column sum. We improve on their sufficient condition until we have a necessary one. Our results connect the above problem to positivity of sufficiently high powers of the matrix \({\bf C^\top C}\). We then generalize the result to complex circulant matrices. spectral norm, circulant matrix, eventually positive semigroups 2018 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol38iss6art8 https://www.opuscula.agh.edu.pl/vol38/6/art/opuscula_math_3841.pdf Minimal unavoidable sets of cycles in plane graphs Tomáš Madaras, Martina Tamášová A set \(S\) of cycles is minimal unavoidable in a graph family \(\cal{G}\) if each graph \(G \in \cal{G}\) contains a cycle from \(S\) and, for each proper subset \(S^{\prime}\subset S\), there exists an infinite subfamily \(\cal{G}^{\prime}\subseteq\cal{G}\) such that no graph from \(\cal{G}^{\prime}\) contains a cycle from \(S^{\prime}\). In this paper, we study minimal unavoidable sets of cycles in plane graphs of minimum degree at least 3 and present several graph constructions which forbid many cycle sets to be unavoidable. We also show the minimality of several small sets consisting of short cycles. plane graph, polyhedral graph, set of cycles 2018 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol38iss6art9 https://www.opuscula.agh.edu.pl/vol38/6/art/opuscula_math_3842.pdf Estimation of the distortion risk premium for heavy-tailed losses under serial dependence Hakim Ouadjed In the actuarial literature, many authors have studied estimation of the reinsurance premium for heavy tailed i.i.d. sequences, especially for the Proportional Hazard (PH) due to Wang. The main aim of this paper is to extend this estimation for heavy tailed dependent sequences satisfying some mixing dependence structure. In this study we prove that the new estimator is asymptotically normal. The behavior of the estimator is examined using simulation for MA(1) process. extreme value theory, mixing processes, tail index estimation 2018 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol38iss6art10 https://www.opuscula.agh.edu.pl/vol38/6/art/opuscula_math_3843.pdf Initial value problem for the time-dependent linear Schrödinger equation with a point singular potential by the unified transform method Yan Rybalko We study an initial value problem for the one-dimensional non-stationary linear Schrödinger equation with a point singular potential. In our approach, the problem is considered as a system of coupled initial-boundary value (IBV) problems on two half-lines, to which we apply the unified approach to IBV problems for linear and integrable nonlinear equations, also known as the Fokas unified transform method. Following the ideas of this method, we obtain the integral representation of the solution of the initial value problem. the Fokas unified transform method, Schrödinger equation, interface problems 2018 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol38iss6art11 https://www.opuscula.agh.edu.pl/vol38/6/art/opuscula_math_3844.pdf Corrigendum to "Acyclic sum-list-colouring of grids and other classes of graphs" [Opuscula Math. 37, no. 4 (2017), 535-556] Ewa Drgas-Burchardt, Agata Drzystek This note provides some minor corrections to the article [Acyclic sum-list-colouring of grids and other classes of graphs, Opuscula Math. 37, no. 4 (2017), 535-556]. sum-list colouring, acyclic colouring, grids, generalized Petersen graphs 2018 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol39iss1 https://www.opuscula.agh.edu.pl/om-vol39iss1art1 https://www.opuscula.agh.edu.pl/vol39/1/art/opuscula_math_3901.pdf Difference equations with impulses Marius Danca, Michal Fečkan, Michal Pospíšil Difference equations with impulses are studied focussing on the existence of periodic or bounded orbits, asymptotic behavior and chaos. So impulses are used to control the dynamics of the autonomous difference equations. A model of supply and demand is also considered when Li-Yorke chaos is shown among others. difference equations, impulses, stability, fixed points, Li-Yorke chaos 2019 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol39iss1art2 https://www.opuscula.agh.edu.pl/vol39/1/art/opuscula_math_3902.pdf Dynamic system with random structure for modeling security and risk management in cyberspace Irada Dzhalladova, Miroslava Růžičková We deal with the investigation of \(L_{2}\)-stability of the trivial solution to the system of difference equations with coefficients depending on a semi-Markov chain. In our considerations, random transformations of solutions are assumed. Necessary and sufficient conditions for \(L_{2}\)-stability of the trivial solution to such systems are obtained. A method is proposed for constructing Lyapunov functions and the conditions for its existence are justified. The dynamic system and methods discussed in the paper are very well suited for use as models for protecting information in cyberspace. semi-Markov chain, random transformation of solutions, the Lyapunov function, \(L_{2}\)-stability, systems of difference equations, jumps of solutions, cybersecurity 2019 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol39iss1art3 https://www.opuscula.agh.edu.pl/vol39/1/art/opuscula_math_3903.pdf Oscillatory behavior of even-order nonlinear differential equations with a sublinear neutral term John R. Graef, Said R. Grace, Ercan Tunç The authors present a new technique for the linearization of even-order nonlinear differential equations with a sublinear neutral term. They establish some new oscillation criteria via comparison with higher-order linear delay differential inequalities as well as with first-order linear delay differential equations whose oscillatory characters are known. Examples are provided to illustrate the theorems. oscillatory behavior, neutral differential equation, even-order 2019 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol39iss1art4 https://www.opuscula.agh.edu.pl/vol39/1/art/opuscula_math_3904.pdf Boundary value problems with solutions in convex sets Gerd Herzog, Peer Chr. Kunstmann By means of the continuation method for contractions we prove the existence of solutions of Dirichlet boundary value problems in convex sets. As an application we prove the existence of concave solutions of certain boundary value problems in ordered Banach spaces. Dirichlet boundary value problems, solutions in convex sets, continuation method, ordered Banach spaces, concave solutions 2019 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol39iss1art5 https://www.opuscula.agh.edu.pl/vol39/1/art/opuscula_math_3905.pdf On the convergence of solutions to second-order neutral difference equations Małgorzata Migda, Janusz Migda, Małgorzata Zdanowicz A second-order nonlinear neutral difference equation with a quasi-difference is studied. Sufficient conditions are established under which for every real constant there exists a solution of the considered equation convergent to this constant. second-order difference equation, asymptotic behavior, quasi-differences, Krasnoselskii's fixed point theorem 2019 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol39iss1art6 https://www.opuscula.agh.edu.pl/vol39/1/art/opuscula_math_3906.pdf The existence of consensus of a leader-following problem with Caputo fractional derivative Ewa Schmeidel In this paper, consensus of a leader-following problem is investigated. The leader-following problem describes a dynamics of the leader and a number of agents. The trajectory of the leader is given. The dynamics of each agent depends on the leader trajectory and others agents' inputs. Here, the leader-following problem is modeled by a system of nonlinear equations with Caputo fractional derivative, which can be rewritten as a system of Volterra equations. The main tools in the investigation are the properties of the resolvent kernel corresponding to the Volterra equations, and Schauder fixed point theorem. By study of the existence of an asymptotically stable solution of a suitable Volterra system, the sufficient conditions for consensus of the leader-following problem are obtained. leader-following problem, Caputo fractional differential equation, consensus, nonlinear system, Schauder fixed point theorem 2019 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol39iss1art7 https://www.opuscula.agh.edu.pl/vol39/1/art/opuscula_math_3907.pdf Oscillation criteria for even order neutral difference equations S. Selvarangam, S. A. Rupadevi, E. Thandapani, S. Pinelas In this paper, we present some new sufficient conditions for oscillation of even order nonlinear neutral difference equation of the form \[\Delta^m(x_n+ax_{n-\tau_1}+bx_{n+\tau_2})+p_nx_{n-\sigma_1}^{\alpha}+q_nx_{n+\sigma_2}^{\beta}=0,\quad n\geq n_0\gt0,\] where \(m\geq 2\) is an even integer, using arithmetic-geometric mean inequality. Examples are provided to illustrate the main results. even order, neutral difference equation, oscillation, asymptotic behavior, mixed type 2019 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol39iss1art8 https://www.opuscula.agh.edu.pl/vol39/1/art/opuscula_math_3908.pdf Pseudo-differential equations and conical potentials: 2-dimensional case Vladimir B. Vasilyev We consider two-dimensional elliptic pseudo-differential equation in a plane sector. Using a special representation for an elliptic symbol and the formula for a general solution we study the Dirichlet problem for such equation. This problem was reduced to a system of linear integral equations and then after some transformations to a system of linear algebraic equations. The unique solvability for the Dirichlet problem was proved in Sobolev-Slobodetskii spaces and a priori estimate for the solution is given. pseudo-differential equation, wave factorization, Dirichlet problem, system of linear integral equations 2019 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol39iss2 https://www.opuscula.agh.edu.pl/om-vol39iss2art1 https://www.opuscula.agh.edu.pl/vol39/2/art/opuscula_math_3909.pdf On unique solvability of a Dirichlet problem with nonlinearity depending on the derivative Michał Bełdziński, Marek Galewski In this work we consider second order Dirichelet boundary value problem with nonlinearity depending on the derivative. Using a global diffeomorphism theorem we propose a new variational approach leading to the existence and uniqueness result for such problems. diffeomorphism, uniqueness, non-potential problems, variational methods, monotone methods, Palais-Smale condition 2019 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol39iss2art2 https://www.opuscula.agh.edu.pl/vol39/2/art/opuscula_math_3910.pdf Some remarks on the coincidence set for the Signorini problem Miguel de Benito Delgado, Jesus Ildefonso Díaz We study some properties of the coincidence set for the boundary Signorini problem, improving some results from previous works by the second author and collaborators. Among other new results, we show here that the convexity assumption on the domain made previously in the literature on the location of the coincidence set can be avoided under suitable alternative conditions on the data. Signorini problem, coincidence set, location estimates, free boundary problem, contact problems 2019 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol39iss2art3 https://www.opuscula.agh.edu.pl/vol39/2/art/opuscula_math_3911.pdf Nonlinear elliptic equations involving the p-Laplacian with mixed Dirichlet-Neumann boundary conditions Gabriele Bonanno, Giuseppina D'Aguì, Angela Sciammetta In this paper, a nonlinear differential problem involving the \(p\)-Laplacian operator with mixed boundary conditions is investigated. In particular, the existence of three non-zero solutions is established by requiring suitable behavior on the nonlinearity. Concrete examples illustrate the abstract results. mixed problem, critical points 2019 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol39iss2art4 https://www.opuscula.agh.edu.pl/vol39/2/art/opuscula_math_3912.pdf Infinitely many solutions for some nonlinear supercritical problems with break of symmetry Anna Maria Candela, Addolorata Salvatore In this paper, we prove the existence of infinitely many weak bounded solutions of the nonlinear elliptic problem \[\begin{cases}-\operatorname{div}(a(x,u,\nabla u))+A_t(x,u,\nabla u) = g(x,u)+h(x)&\text{in }\Omega,\\ u=0 &\text{on }\partial\Omega,\end{cases}\] where \(\Omega \subset \mathbb{R}^N\) is an open bounded domain, \(N\geq 3\), and \(A(x,t,\xi)\), \(g(x,t)\), \(h(x)\) are given functions, with \(A_t = \frac{\partial A}{\partial t}\), \(a = \nabla_{\xi} A\), such that \(A(x,\cdot,\cdot)\) is even and \(g(x,\cdot)\) is odd. To this aim, we use variational arguments and the Rabinowitz's perturbation method which is adapted to our setting and exploits a weak version of the Cerami-Palais-Smale condition. Furthermore, if \(A(x,t,\xi)\) grows fast enough with respect to \(t\), then the nonlinear term related to \(g(x,t)\) may have also a supercritical growth. quasilinear elliptic equation, weak Cerami-Palais-Smale condition, Ambrosetti-Rabinowitz condition, break of symmetry, perturbation method, supercritical growth 2019 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol39iss2art5 https://www.opuscula.agh.edu.pl/vol39/2/art/opuscula_math_3913.pdf Existence results and a priori estimates for solutions of quasilinear problems with gradient terms Roberta Filippucci, Chiara Lini In this paper we establish a priori estimates and then an existence theorem of positive solutions for a Dirichlet problem on a bounded smooth domain in \(\mathbb{R}^N\) with a nonlinearity involving gradient terms. The existence result is proved with no use of a Liouville theorem for the limit problem obtained via the usual blow up method, in particular we refer to the modified version by Ruiz. In particular our existence theorem extends a result by Lorca and Ubilla in two directions, namely by considering a nonlinearity which includes in the gradient term a power of \(u\) and by removing the growth condition for the nonlinearity \(f\) at \(u=0\). existence result, quasilinear problems, a priori estimates 2019 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol39iss2art6 https://www.opuscula.agh.edu.pl/vol39/2/art/opuscula_math_3914.pdf Controllability of degenerate and singular parabolic problems: the double strong case with Neumann boundary conditions Genni Fragnelli, Dimitri Mugnai We prove a null controllability result for a parabolic problem with Neumann boundary conditions. We consider non smooth coefficients in presence of a strongly singular potential and a strongly degenerate coefficient, both vanishing at an interior point. This paper concludes the study of the Neumann case. strongly singular/degenerate equations, non smooth coefficients, null controllability 2019 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol39iss2art7 https://www.opuscula.agh.edu.pl/vol39/2/art/opuscula_math_3915.pdf On a Robin (p,q)-equation with a logistic reaction Nikolaos S. Papageorgiou, Calogero Vetro, Francesca Vetro We consider a nonlinear nonhomogeneous Robin equation driven by the sum of a \(p\)-Laplacian and of a \(q\)-Laplacian (\((p,q)\)-equation) plus an indefinite potential term and a parametric reaction of logistic type (superdiffusive case). We prove a bifurcation-type result describing the changes in the set of positive solutions as the parameter \(\lambda \gt 0\) varies. Also, we show that for every admissible parameter \(\lambda \gt 0\), the problem admits a smallest positive solution. positive solutions, superdiffusive reaction, local minimizers, maximum principle, minimal positive solutions, Robin boundary condition, indefinite potential 2019 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol39iss2art8 https://www.opuscula.agh.edu.pl/vol39/2/art/opuscula_math_3916.pdf Existence and multiplicity results for quasilinear equations in the Heisenberg group Patrizia Pucci In this paper we complete the study started in [Existence of entire solutions for quasilinear equations in the Heisenberg group, Minimax Theory Appl. 4 (2019)] on entire solutions for a quasilinear equation \((\mathcal{E}_{\lambda})\) in \(\mathbb{H}^{n}\), depending on a real parameter \(\lambda\), which involves a general elliptic operator \(\mathbf{A}\) in divergence form and two main nonlinearities. Here, in the so called sublinear case, we prove existence for all \(\lambda\gt 0\) and, for special elliptic operators \(\mathbf{A}\), existence of infinitely many solutions \((u_k)_k\). Heisenberg group, entire solutions, critical exponents 2019 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol39iss2art9 https://www.opuscula.agh.edu.pl/vol39/2/art/opuscula_math_3917.pdf Isotropic and anisotropic double-phase problems: old and new Vicenţiu D. Rădulescu We are concerned with the study of two classes of nonlinear problems driven by differential operators with unbalanced growth, which generalize the \((p,q)\)- and \((p(x),q(x))\)-Laplace operators. The associated energy is a double-phase functional, either isotropic or anisotropic. The content of this paper is in relationship with pioneering contributions due to P. Marcellini and G. Mingione. differential operator with unbalanced growth, double-phase energy, variable exponent 2019 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol39iss2art10 https://www.opuscula.agh.edu.pl/vol39/2/art/opuscula_math_3918.pdf Extremal length and Dirichlet problem on Klein surfaces Monica Roşiu The object of this paper is to extend the method of extremal length to Klein surfaces by solving conformally invariant extremal problems on the complex double. Within this method we define the extremal length, the extremal distance, the conjugate extremal distance, the modulus, the reduced extremal distance on a Klein surface and we study their dependences on arcs. Klein surface, extremal length, extremal distance 2019 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol39iss2art11 https://www.opuscula.agh.edu.pl/vol39/2/art/opuscula_math_3919.pdf Global well-posedness of a class of fourth-order strongly damped nonlinear wave equations Yang Yanbing, Md Salik Ahmed, Qin Lanlan, Xu Runzhang Global well-posedness and finite time blow up issues for some strongly damped nonlinear wave equation are investigated in the present paper. For subcritical initial energy by employing the concavity method we show a finite time blow up result of the solution. And for critical initial energy we present the global existence, asymptotic behavior and finite time blow up of the solution in the framework of the potential well. Further for supercritical initial energy we give a sufficient condition on the initial data such that the solution blows up in finite time. fourth-order nonlinear wave equation, strong damping, blow up, global existence 2019 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol39iss3 https://www.opuscula.agh.edu.pl/om-vol39iss3art1 https://www.opuscula.agh.edu.pl/vol39/3/art/opuscula_math_3920.pdf Oscillations of equations caused by several deviating arguments George E. Chatzarakis Linear delay or advanced differential equations with variable coefficients and several not necessarily monotone arguments are considered, and some new oscillation criteria are given. More precisely, sufficient conditions, involving \(\lim\sup\) and \(\lim\inf\), are obtained, which essentially improve several known criteria existing in the literature. Examples illustrating the results are also given, numerically solved in MATLAB. differential equation, non-monotone argument, oscillatory solution, nonoscillatory solution 2019 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol39iss3art2 https://www.opuscula.agh.edu.pl/vol39/3/art/opuscula_math_3921.pdf A partial refining of the Erdős-Kelly regulation Joanna Górska, Zdzisław Skupień The aim of this note is to advance the refining of the Erdős-Kelly result on graphical inducing regularization. The operation of inducing regulation (on graphs or multigraphs) with prescribed maximum vertex degree is originated by D. König in 1916. As is shown by Chartrand and Lesniak in their textbook Graphs & Digraphs (1996), an iterated construction for graphs can result in a regularization with many new vertices. Erdős and Kelly have presented (1963, 1967) a simple and elegant numerical method of determining for any simple \(n\)-vertex graph \(G\) with maximum vertex degree \(\Delta\), the exact minimum number, say \(\theta =\theta(G)\), of new vertices in a \(\Delta\)-regular graph \(H\) which includes \(G\) as an induced subgraph. The number \(\theta(G)\), which we call the cost of regulation of \(G\), has been upper-bounded by the order of \(G\), the bound being attained for each \(n\ge4\), e.g. then the edge-deleted complete graph \(K_n-e\) has \(\theta=n\). For \(n\ge 4\), we present all factors of \(K_n\) with \(\theta=n\) and next \(\theta=n-1\). Therein in case \(\theta=n-1\) and \(n\) odd only, we show that a specific extra structure, non-matching, is required. inducing \(\Delta\)-regulation, cost of regulation 2019 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol39iss3art3 https://www.opuscula.agh.edu.pl/vol39/3/art/opuscula_math_3922.pdf On the zeros of the Macdonald functions Yuji Hamana, Hiroyuki Matsumoto, Tomoyuki Shirai We are concerned with the zeros of the Macdonald functions or the modified Bessel functions of the second kind with real index. By using the explicit expressions for the algebraic equations satisfied by the zeros, we describe the behavior of the zeros when the index moves. Results by numerical computations are also presented. zeros, Macdonald functions, Bessel functions 2019 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol39iss3art4 https://www.opuscula.agh.edu.pl/vol39/3/art/opuscula_math_3923.pdf Decomposing complete 3-uniform hypergraph K_{n}^{(3)} into 7-cycles Meihua, Meiling Guan, Jirimutu We use the Katona-Kierstead definition of a Hamiltonian cycle in a uniform hypergraph. A decomposition of complete \(k\)-uniform hypergraph \(K^{(k)}_{n}\) into Hamiltonian cycles was studied by Bailey-Stevens and Meszka-Rosa. For \(n\equiv 2,4,5\pmod 6\), we design an algorithm for decomposing the complete 3-uniform hypergraphs into Hamiltonian cycles by using the method of edge-partition. A decomposition of \(K^{(3)}_{n}\) into 5-cycles has been presented for all admissible \(n\leq17\), and for all \(n=4^{m}+1\) when \(m\) is a positive integer. In general, the existence of a decomposition into 5-cycles remains open. In this paper, we show if \(42~|~(n-1)(n-2)\) and if there exist \(\lambda=\frac{(n-1)(n-2)}{42}\) sequences \((k_{i_{0}},k_{i_{1}},\ldots,k_{i_{6}})\) on \(D_{all}(n)\), then \(K^{(3)}_{n}\) can be decomposed into 7-cycles. We use the method of edge-partition and cycle sequence. We find a decomposition of \(K^{(3)}_{37}\) and \(K^{(3)}_{43}\) into 7-cycles. uniform hypergraph, 7-cycle, cycle decomposition 2019 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol39iss3art5 https://www.opuscula.agh.edu.pl/vol39/3/art/opuscula_math_3924.pdf Remarks on global solutions to the initial-boundary value problem for quasilinear degenerate parabolic equations with a nonlinear source term Mitsuhiro Nakao We give an existence theorem of global solution to the initial-boundary value problem for \(u_{t}-\operatorname{div}\{\sigma(|\nabla u|^2)\nabla u\}=f(u)\) under some smallness conditions on the initial data, where \(\sigma (v^2)\) is a positive function of \(v^2\ne 0\) admitting the degeneracy property \(\sigma(0)=0\). We are interested in the case where \(\sigma(v^2)\) has no exponent \(m \geq 0\) such that \(\sigma(v^2) \geq k_0|v|^m , k_0 \gt 0\). A typical example is \(\sigma(v^2)=\operatorname{log}(1+v^2)\). \(f(u)\) is a function like \(f=|u|^{\alpha} u\). A decay estimate for \(\|\nabla u(t)\|_{\infty}\) is also given. degenerate quasilinear parabolic equation, nonlinear source term, Moser's method 2019 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol39iss3art6 https://www.opuscula.agh.edu.pl/vol39/3/art/opuscula_math_3925.pdf Metric dimension of Andrásfai graphs S. Batool Pejman, Shiroyeh Payrovi, Ali Behtoei A set \(W\subseteq V(G)\) is called a resolving set, if for each pair of distinct vertices \(u,v\in V(G)\) there exists \(t\in W\) such that \(d(u,t)\neq d(v,t)\), where \(d(x,y)\) is the distance between vertices \(x\) and \(y\). The cardinality of a minimum resolving set for \(G\) is called the metric dimension of \(G\) and is denoted by \(\dim_M(G)\). This parameter has many applications in different areas. The problem of finding metric dimension is NP-complete for general graphs but it is determined for trees and some other important families of graphs. In this paper, we determine the exact value of the metric dimension of Andrásfai graphs, their complements and \(And(k)\square P_n\). Also, we provide upper and lower bounds for \(dim_M(And(k)\square C_n)\). resolving set, metric dimension, Andrásfai graph, Cayley graph, Cartesian product 2019 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol39iss3art7 https://www.opuscula.agh.edu.pl/vol39/3/art/opuscula_math_3926.pdf The complexity of open k-monopolies in graphs for negative k Iztok Peterin Let \(G\) be a graph with vertex set \(V(G)\), \(\delta(G)\) minimum degree of \(G\) and \(k\in\left\{1-\left\lceil\frac{\delta(G)}{2}\right\rceil,\ldots ,\left\lfloor \frac{\delta(G)}{2}\right\rfloor\right\}\). Given a nonempty set \(M\subseteq V(G)\) a vertex \(v\) of \(G\) is said to be \(k\)-controlled by \(M\) if \(\delta_M(v)\ge\frac{\delta_{V(G)}(v)}{2}+k\) where \(\delta_M(v)\) represents the number of neighbors of \(v\) in \(M\). The set \(M\) is called an open \(k\)-monopoly for \(G\) if it \(k\)-controls every vertex \(v\) of \(G\). In this short note we prove that the problem of computing the minimum cardinality of an open \(k\)-monopoly in a graph for a negative integer \(k\) is NP-complete even restricted to chordal graphs. open \(k\)-monopolies, complexity, total domination 2019 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol39iss3art8 https://www.opuscula.agh.edu.pl/vol39/3/art/opuscula_math_3927.pdf General multiplicative Zagreb indices of graphs with given clique number Tomáš Vetrík, Selvaraj Balachandran We obtain lower and upper bounds on general multiplicative Zagreb indices for graphs of given clique number and order. Bounds on the basic multiplicative Zagreb indices and on the multiplicative sum Zagreb index follow from our results. We also determine graphs with the smallest and the largest indices. clique number, multiplicative Zagreb index, chromatic number 2019 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol39iss4 https://www.opuscula.agh.edu.pl/om-vol39iss4art1 https://www.opuscula.agh.edu.pl/vol39/4/art/opuscula_math_3928.pdf Applications of PDEs inpainting to magnetic particle imaging and corneal topography Andrea Andrisani, Rosa Maria Mininni, Francesca Mazzia, Giuseppina Settanni, Alessandro Iurino, Sabina Tangaro, Andrea Tateo, Roberto Bellotti In this work we propose a novel application of Partial Differential Equations (PDEs) inpainting techniques to two medical contexts. The first one concerning recovering of concentration maps for superparamagnetic nanoparticles, used as tracers in the framework of Magnetic Particle Imaging. The analysis is carried out by two set of simulations, with and without adding a source of noise, to show that the inpainted images preserve the main properties of the original ones. The second medical application is related to recovering data of corneal elevation maps in ophthalmology. A new procedure consisting in applying the PDEs inpainting techniques to the radial curvature image is proposed. The images of the anterior corneal surface are properly recovered to obtain an approximation error of the required precision. We compare inpainting methods based on second, third and fourth-order PDEs with standard approximation and interpolation techniques. PDEs inpainting, medical imaging, magnetic particle imaging, radial curvature image, anterior surface of a cornea 2019 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol39iss4art2 https://www.opuscula.agh.edu.pl/vol39/4/art/opuscula_math_3929.pdf Oscillatory results for second-order noncanonical delay differential equations Jozef Džurina, Irena Jadlovská, Ioannis P. Stavroulakis The main purpose of this paper is to improve recent oscillation results for the second-order half-linear delay differential equation \[\left(r(t)\left(y'(t)\right)^\gamma\right)'+q(t)y^\gamma(\tau(t))= 0, \quad t\geq t_0,\] under the condition \[\int_{t_0}^{\infty}\frac{\text{d} t}{r^{1/\gamma}(t)} \lt \infty.\] Our approach is essentially based on establishing sharper estimates for positive solutions of the studied equation than those used in known works. Two examples illustrating the results are given. linear differential equation, delay, second-order, noncanonical, oscillation 2019 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol39iss4art3 https://www.opuscula.agh.edu.pl/vol39/4/art/opuscula_math_3930.pdf Decomposition of Gaussian processes, and factorization of positive definite kernels Palle Jorgensen, Feng Tian We establish a duality for two factorization questions, one for general positive definite (p.d.) kernels \(K\), and the other for Gaussian processes, say \(V\). The latter notion, for Gaussian processes is stated via Ito-integration. Our approach to factorization for p.d. kernels is intuitively motivated by matrix factorizations, but in infinite dimensions, subtle measure theoretic issues must be addressed. Consider a given p.d. kernel \(K\), presented as a covariance kernel for a Gaussian process \(V\). We then give an explicit duality for these two seemingly different notions of factorization, for p.d. kernel \(K\), vs for Gaussian process \(V\). Our result is in the form of an explicit correspondence. It states that the analytic data which determine the variety of factorizations for \(K\) is the exact same as that which yield factorizations for \(V\). Examples and applications are included: point-processes, sampling schemes, constructive discretization, graph-Laplacians, and boundary-value problems. reproducing kernel Hilbert space, frames, generalized Ito-integration, the measurable category, analysis/synthesis, interpolation, Gaussian free fields, non-uniform sampling, optimization, transform, covariance, feature space 2019 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol39iss4art4 https://www.opuscula.agh.edu.pl/vol39/4/art/opuscula_math_3931.pdf On unitary equivalence of bilateral operator valued weighted shifts Jakub Kośmider We establish a characterization of unitary equivalence of two bilateral operator valued weighted shifts with quasi-invertible weights by an operator of diagonal form. We provide an example of unitary equivalence between shifts with weights defined on \(\mathbb{C}^2\) which cannot be given by any unitary operator of diagonal form. The paper also contains some remarks regarding unitary operators that can give unitary equivalence of bilateral operator valued weighted shifts. unitary equivalence, bilateral weighted shift, quasi-invertible weights, partial isometry 2019 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol39iss4art5 https://www.opuscula.agh.edu.pl/vol39/4/art/opuscula_math_3932.pdf Description of the scattering data for Sturm-Liouville operators on the half-line Yaroslav Mykytyuk, Nataliia Sushchyk We describe the set of the scattering data for self-adjoint Sturm-Liouville operators on the half-line with potentials belonging to \(L_1(\mathbb{R}_+,\rho(x)\,\text{d} x)\), where \(\rho:\mathbb{R}_+\to\mathbb{R}_+\) is a monotonically nondecreasing function from some family \(\mathscr{R}\). In particular, \(\mathscr{R}\) includes the functions \(\rho(x)=(1+x)^{\alpha}\) with \(\alpha\geq 1\). inverse scattering, Schrödinger operator, Banach algebra 2019 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol39iss4art6 https://www.opuscula.agh.edu.pl/vol39/4/art/opuscula_math_3933.pdf The intersection graph of annihilator submodules of a module S.B. Pejman, Sh. Payrovi, S. Babaei Let \(R\) be a commutative ring and \(M\) be a Noetherian \(R\)-module. The intersection graph of annihilator submodules of \(M\), denoted by \(GA(M)\) is an undirected simple graph whose vertices are the classes of elements of \(Z_R(M)\setminus \text{Ann}_R(M)\), for \(a,b \in R\) two distinct classes \([a]\) and \([b]\) are adjacent if and only if \(\text{Ann}_M(a)\cap \text{Ann}_M(b)\neq 0\). In this paper, we study diameter and girth of \(GA(M)\) and characterize all modules that the intersection graph of annihilator submodules are connected. We prove that \(GA(M)\) is complete if and only if \(Z_R(M)\) is an ideal of \(R\). Also, we show that if \(M\) is a finitely generated \(R\)-module with \(r(\text{Ann}_R(M))\neq \text{Ann}_R(M)\) and \(|m-\text{Ass}_R(M)|=1\) and \(GA(M)\) is a star graph, then \(r(\text{Ann}_R(M))\) is not a prime ideal of \(R\) and \(|V(GA(M))|=|\text{Min}\,\text{Ass}_R(M)|+1\). prime submodule, annihilator submodule, intersection annihilator graph 2019 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol39iss5 https://www.opuscula.agh.edu.pl/om-vol39iss5art1 https://www.opuscula.agh.edu.pl/vol39/5/art/opuscula_math_3934.pdf On properties of minimizers of a control problem with time-distributed functional related to parabolic equations I. V. Astashova, A. V. Filinovskiy We consider a control problem given by a mathematical model of the temperature control in industrial hothouses. The model is based on one-dimensional parabolic equations with variable coefficients. The optimal control is defined as a minimizer of a quadratic cost functional. We describe qualitative properties of this minimizer, study the structure of the set of accessible temperature functions, and prove the dense controllability for some set of control functions. parabolic equation, extremal problem, quadratic cost functional, minimizer, exact controllability, dense controllability 2019 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol39iss5art2 https://www.opuscula.agh.edu.pl/vol39/5/art/opuscula_math_3935.pdf On the imaginary part of coupling resonance points Nurulla Azamov, Tom Daniels We prove for rank one perturbations that the imaginary part of a coupling resonance point is inversely proportional by a factor of \(-2\) to the rate of change of the scattering phase, as a function of the coupling variable, evaluated at the real part of the resonance point. This equality is analogous to the Breit-Wigner formula from quantum scattering theory. For more general relatively trace class perturbations, we also give a formula for the spectral shift function in terms of coupling resonance points, non-real and real. scattering matrix, scattering phase, resonance point, Breit-Wigner formula 2019 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol39iss5art3 https://www.opuscula.agh.edu.pl/vol39/5/art/opuscula_math_3936.pdf On 1-rotational decompositions of complete graphs into tripartite graphs Ryan C. Bunge Consider a tripartite graph to be any simple graph that admits a proper vertex coloring in at most 3 colors. Let \(G\) be a tripartite graph with \(n\) edges, one of which is a pendent edge. This paper introduces a labeling on such a graph \(G\) used to achieve 1-rotational \(G\)-decompositions of \(K_{2nt}\) for any positive integer \(t\). It is also shown that if \(G\) with a pendent edge is the result of adding an edge to a path on \(n\) vertices, then \(G\) admits such a labeling. graph decomposition, 1-rotational, vertex labeling 2019 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol39iss5art4 https://www.opuscula.agh.edu.pl/vol39/5/art/opuscula_math_3937.pdf Direct and inverse spectral problems for Dirac systems with nonlocal potentials Kamila Dębowska, Leonid P. Nizhnik The main purposes of this paper are to study the direct and inverse spectral problems of the one-dimensional Dirac operators with nonlocal potentials. Based on informations about the spectrum of the operator, we find the potential and recover the form of the Dirac system. The methods used allow us to reduce the situation to the one-dimensional case. In accordance with the given assumptions and conditions we consider problems in a specific way. We describe the spectrum, the resolvent, the characteristic function etc. Illustrative examples are also given. inverse spectral problem, nonlocal potential, nonlocal boundary conditions, Dirac system 2019 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol39iss5art5 https://www.opuscula.agh.edu.pl/vol39/5/art/opuscula_math_3938.pdf Positive solutions for the one-dimensional p-Laplacian with nonlinear boundary conditions D. D. Hai, X. Wang We prove the existence of positive solutions for the \(p\)-Laplacian problem \[\begin{cases}-(r(t)\phi (u^{\prime }))^{\prime }=\lambda g(t)f(u),& t\in (0,1),\\au(0)-H_{1}(u^{\prime }(0))=0,\\cu(1)+H_{2}(u^{\prime}(1))=0,\end{cases}\] where \(\phi (s)=|s|^{p-2}s\), \(p\gt 1\), \(H_{i}:\mathbb{R}\rightarrow\mathbb{R}\) can be nonlinear, \(i=1,2\), \(f:(0,\infty )\rightarrow \mathbb{R}\) is \(p\)-superlinear or \(p\)-sublinear at \(\infty\) and is allowed be singular \((\pm\infty)\) at \(0\), and \(\lambda\) is a positive parameter. \(p\)-Laplacian, semipositone, nonlinear boundary conditions, positive solutions 2019 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol39iss5art6 https://www.opuscula.agh.edu.pl/vol39/5/art/opuscula_math_3939.pdf On edge product cordial graphs Jaroslav Ivančo An edge product cordial labeling is a variant of the well-known cordial labeling. In this paper we characterize graphs admitting an edge product cordial labeling. Using this characterization we investigate the edge product cordiality of broad classes of graphs, namely, dense graphs, dense bipartite graphs, connected regular graphs, unions of some graphs, direct products of some bipartite graphs, joins of some graphs, maximal \(k\)-degenerate and related graphs, product cordial graphs. edge product cordial labelings, dense graphs, regular graphs 2019 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol39iss5art7 https://www.opuscula.agh.edu.pl/vol39/5/art/opuscula_math_3940.pdf On the Lebesgue and Sobolev spaces on a time-scale Ewa Skrzypek, Katarzyna Szymańska-Dębowska We consider the generalized Lebesgue and Sobolev spaces on a bounded time-scale. We study the standard properties of these spaces and compare them to the classical known results for the Lebesgue and Sobolev spaces on a bounded interval. These results provide the necessary framework for the study of boundary value problems on bounded time-scales. Lebesgue spaces, Sobolev spaces, modular spaces, time-scales, boundary value problems on time-scales 2019 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol39iss5art8 https://www.opuscula.agh.edu.pl/vol39/5/art/opuscula_math_3941.pdf Large and moderate deviation principles for nonparametric recursive kernel distribution estimators defined by stochastic approximation method Yousri Slaoui In this paper we prove large and moderate deviations principles for the recursive kernel estimators of a distribution function defined by the stochastic approximation algorithm. We show that the estimator constructed using the stepsize which minimize the Mean Integrated Squared Error (MISE) of the class of the recursive estimators defined by Mokkadem et al. gives the same pointwise large deviations principle (LDP) and moderate deviations principle (MDP) as the Nadaraya kernel distribution estimator. distribution estimation, stochastic approximation algorithm, large and moderate deviations principles 2019 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol39iss6 https://www.opuscula.agh.edu.pl/om-vol39iss6art1 https://www.opuscula.agh.edu.pl/vol39/6/art/opuscula_math_3942.pdf Oscillatory criteria for second order differential equations with several sublinear neutral terms Blanka Baculikova In this paper, sufficient conditions for oscillation of the second order differential equations with several sublinear neutral terms are established. The results obtained generalize and extend those reported in the literature. Several examples are included to illustrate the importance and novelty of the presented results. second order neutral differential equation, sub-linear neutral term, oscillation 2019 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol39iss6art2 https://www.opuscula.agh.edu.pl/vol39/6/art/opuscula_math_3943.pdf Vertices with the second neighborhood property in Eulerian digraphs Michael Cary The Second Neighborhood Conjecture states that every simple digraph has a vertex whose second out-neighborhood is at least as large as its first out-neighborhood, i.e. a vertex with the Second Neighborhood Property. A cycle intersection graph of an even graph is a new graph whose vertices are the cycles in a cycle decomposition of the original graph and whose edges represent vertex intersections of the cycles. By using a digraph variant of this concept, we prove that Eulerian digraphs which admit a simple cycle intersection graph not only adhere to the Second Neighborhood Conjecture, but that local simplicity can, in some cases, also imply the existence of a Seymour vertex in the original digraph. Eulerian digraph, second neighborhood conjecture, cycle decomposition, cycle intersection graph 2019 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol39iss6art3 https://www.opuscula.agh.edu.pl/vol39/6/art/opuscula_math_3944.pdf Deformation of semicircular and circular laws via p-adic number fields and sampling of primes Ilwoo Cho, Palle E. T. Jorgensen In this paper, we study semicircular elements and circular elements in a certain Banach \(*\)-probability space \((\mathfrak{LS},\tau ^{0})\) induced by analysis on the \(p\)-adic number fields \(\mathbb{Q}_{p}\) over primes \(p\). In particular, by truncating the set \(\mathcal{P}\) of all primes for given suitable real numbers \(t\lt s\) in \(\mathbb{R}\), two different types of truncated linear functionals \(\tau_{t_{1}\lt t_{2}}\), and \(\tau_{t_{1}\lt t_{2}}^{+}\) are constructed on the Banach \(*\)-algebra \(\mathfrak{LS}\). We show how original free distributional data (with respect to \(\tau ^{0}\)) are distorted by the truncations on \(\mathcal{P}\) (with respect to \(\tau_{t\lt s}\), and \(\tau_{t\lt s}^{+}\)). As application, distorted free distributions of the semicircular law, and those of the circular law are characterized up to truncation. free probability, primes, \(p\)-adic number fields, Banach \(*\)-probability spaces, semicircular elements, circular elements, truncated linear functionals 2019 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol39iss6art4 https://www.opuscula.agh.edu.pl/vol39/6/art/opuscula_math_3945.pdf Graphs with equal domination and certified domination numbers Magda Dettlaff, Magdalena Lemańska, Mateusz Miotk, Jerzy Topp, Radosław Ziemann, Paweł Żyliński A set \(D\) of vertices of a graph \(G=(V_G,E_G)\) is a dominating set of \(G\) if every vertex in \(V_G-D\) is adjacent to at least one vertex in \(D\). The domination number (upper domination number, respectively) of \(G\), denoted by \(\gamma(G)\) (\(\Gamma(G)\), respectively), is the cardinality of a smallest (largest minimal, respectively) dominating set of \(G\). A subset \(D\subseteq V_G\) is called a certified dominating set of \(G\) if \(D\) is a dominating set of \(G\) and every vertex in \(D\) has either zero or at least two neighbors in \(V_G-D\). The cardinality of a smallest (largest minimal, respectively) certified dominating set of \(G\) is called the certified (upper certified, respectively) domination number of \(G\) and is denoted by \(\gamma_{\rm cer}(G)\) (\(\Gamma_{\rm cer}(G)\), respectively). In this paper relations between domination, upper domination, certified domination and upper certified domination numbers of a graph are studied. domination, certified domination 2019 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol39iss6art5 https://www.opuscula.agh.edu.pl/vol39/6/art/opuscula_math_3946.pdf Lightweight paths in graphs Jochen Harant, Stanislav Jendrol' Let \(k\) be a positive integer, \(G\) be a graph on \(V(G)\) containing a path on \(k\) vertices, and \(w\) be a weight function assigning each vertex \(v\in V(G)\) a real weight \(w(v)\). Upper bounds on the weight \(w(P)=\sum_{v\in V(P)}w(v)\) of \(P\) are presented, where \(P\) is chosen among all paths of \(G\) on \(k\) vertices with smallest weight. weighted graph, lightweight path 2019 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol39iss6art6 https://www.opuscula.agh.edu.pl/vol39/6/art/opuscula_math_3947.pdf On existence and global attractivity of periodic solutions of nonlinear delay differential equations Chuanxi Qian, Justin Smith Consider the delay differential equation with a forcing term \[\tag{\(\ast\)} x'(t)=-f(t,x(t))+g(t,x(t-\tau ))+r(t), \quad t \geq 0\] where \(f(t,x): [0,\infty) \times [0,\infty) \to \mathbb{R}\), \(g(t,x): [0,\infty) \times [0,\infty) \to [0,\infty)\) are continuous functions and \(\omega\)-periodic in \(t\), \(r(t): [0,\infty) \to\mathbb{R}\) is a continuous function and \(\tau \in (0,\infty)\) is a positive constant. We first obtain a sufficient condition for the existence of a unique nonnegative periodic solution \(\tilde{x}(t)\) of the associated unforced differential equation of Eq. (\(\ast\)) \[\tag{\(\ast\ast\)} x'(t)=-f(t,x(t))+g(t,x(t-\tau)), \quad t \geq 0.\] Then we obtain a sufficient condition so that every nonnegative solution of the forced equation (\(\ast\)) converges to this nonnegative periodic solution \(\tilde{x}(t)\) of the associated unforced equation(\(\ast\ast\)). Applications from mathematical biology and numerical examples are also given. delay differential equation, periodic solution, global attractivity 2019 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol40iss1 https://www.opuscula.agh.edu.pl/om-vol40iss1art1 https://www.opuscula.agh.edu.pl/vol40/1/art/opuscula_math_4001.pdf On some convergence results for fractional periodic Sobolev spaces Vincenzo Ambrosio In this note we extend the well-known limiting formulas due to Bourgain-Brezis-Mironescu and Maz'ya-Shaposhnikova, to the setting of fractional Sobolev spaces on the torus. We also give a \(\Gamma\)-convergence result in the spirit of Ponce. The main theorems are obtained by using the nice structure of Fourier series. fractional periodic Sobolev spaces, Fourier series, \(\Gamma\)-convergence 2020 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol40iss1art2 https://www.opuscula.agh.edu.pl/vol40/1/art/opuscula_math_4002.pdf Some multiplicity results of homoclinic solutions for second order Hamiltonian systems Sara Barile, Addolorata Salvatore We look for homoclinic solutions \(q:\mathbb{R} \rightarrow \mathbb{R}^N\) to the class of second order Hamiltonian systems \[-\ddot{q} + L(t)q = a(t) \nabla G_1(q) - b(t) \nabla G_2(q) + f(t) \quad t \in \mathbb{R}\] where \(L: \mathbb{R}\rightarrow \mathbb{R}^{N \times N}\) and \(a,b: \mathbb{R}\rightarrow \mathbb{R}\) are positive bounded functions, \(G_1, G_2: \mathbb{R}^N \rightarrow \mathbb{R}\) are positive homogeneous functions and \(f:\mathbb{R}\rightarrow\mathbb{R}^N\). Using variational techniques and the Pohozaev fibering method, we prove the existence of infinitely many solutions if \(f\equiv 0\) and the existence of at least three solutions if \(f\) is not trivial but small enough. second order Hamiltonian systems, homoclinic solutions, variational methods, compact embeddings 2020 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol40iss1art3 https://www.opuscula.agh.edu.pl/vol40/1/art/opuscula_math_4003.pdf On solvability of elliptic boundary value problems via global invertibility Michał Bełdziński, Marek Galewski In this work we apply global invertibility result in order to examine the solvability of elliptic equations with both Neumann and Dirichlet boundary conditions. diffeomorphism, Dirichlet conditions, Laplace operator, Neumann conditions, uniqueness 2020 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol40iss1art4 https://www.opuscula.agh.edu.pl/vol40/1/art/opuscula_math_4004.pdf On the regularity of solution to the time-dependent p-Stokes system Luigi C. Berselli, Michael Růžička In this paper we consider the time evolutionary \(p\)-Stokes problem in a smooth and bounded domain. This system models the unsteady motion or certain non-Newtonian incompressible fluids in the regime of slow motions, when the convective term is negligible. We prove results of space/time regularity, showing that first-order time-derivatives and second-order space-derivatives of the velocity and first-order space-derivatives of the pressure belong to rather natural Lebesgue spaces. regularity, evolution problem, \(p\)-Stokes 2020 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol40iss1art5 https://www.opuscula.agh.edu.pl/vol40/1/art/opuscula_math_4005.pdf Nonhomogeneous equations with critical exponential growth and lack of compactness Giovany M. Figueiredo, Vicenţiu D. Rădulescu We study the existence and multiplicity of positive solutions for the following class of quasilinear problems \[-\operatorname{div}(a(|\nabla u|^{p})| \nabla u|^{p-2}\nabla u)+V(\epsilon x)b(|u|^{p})|u|^{p-2}u=f(u) \qquad\text{ in } \mathbb{R}^N,\] where \(\epsilon\) is a positive parameter. We assume that \(V:\mathbb{R}^N \to \mathbb{R}\) is a continuous potential and \(f:\mathbb{R}\to\mathbb{R}\) is a smooth reaction term with critical exponential growth. exponential critical growth, quasilinear equation, Trudinger-Moser inequality, Moser iteration 2020 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol40iss1art6 https://www.opuscula.agh.edu.pl/vol40/1/art/opuscula_math_4006.pdf Fractional p&q-Laplacian problems with potentials vanishing at infinity Teresa Isernia In this paper we prove the existence of a positive and a negative ground state weak solution for the following class of fractional \(p\&q\)-Laplacian problems \[\begin{aligned} (-\Delta)_{p}^{s} u + (-\Delta)_{q}^{s} u + V(x) (|u|^{p-2}u + |u|^{q-2}u)= K(x) f(u) \quad \text{ in } \mathbb{R}^{N},\end{aligned}\] where \(s\in (0, 1)\), \(1\lt p\lt q \lt\frac{N}{s}\), \(V: \mathbb{R}^{N}\to \mathbb{R}\) and \(K: \mathbb{R}^{N}\to \mathbb{R}\) are continuous, positive functions, allowed for vanishing behavior at infinity, \(f\) is a continuous function with quasicritical growth and the leading operator \((-\Delta)^{s}_{t}\), with \(t\in \{p,q\}\), is the fractional \(t\)-Laplacian operator. fractional \(p\&q\)-Laplacian, vanishing potentials, ground state solution 2020 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol40iss1art7 https://www.opuscula.agh.edu.pl/vol40/1/art/opuscula_math_4007.pdf Global existence and blow up of solution for semi-linear hyperbolic equation with the product of logarithmic and power-type nonlinearity Wei Lian, Md Salik Ahmed, Runzhang Xu In this paper we consider the semilinear wave equation with the multiplication of logarithmic and polynomial nonlinearities. We establish the global existence and finite time blow up of solutions at three different energy levels (\(E(0)\lt d\), \(E(0)=d\) and \(E(0)\gt 0\)) using potential well method. The results in this article shed some light on using potential wells to classify the solutions of the semilinear wave equation with the product of polynomial and logarithmic nonlinearity. global existence, blow-up, logarithmic and polynomial nonlinearity, potential well 2020 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol40iss1art8 https://www.opuscula.agh.edu.pl/vol40/1/art/opuscula_math_4008.pdf A multiplicity theorem for parametric superlinear (p,q)-equations Florin-Iulian Onete, Nikolaos S. Papageorgiou, Calogero Vetro We consider a parametric nonlinear Robin problem driven by the sum of a \(p\)-Laplacian and of a \(q\)-Laplacian (\((p,q)\)-equation). The reaction term is \((p-1)\)-superlinear but need not satisfy the Ambrosetti-Rabinowitz condition. Using variational tools, together with truncation and comparison techniques and critical groups, we show that for all small values of the parameter, the problem has at least five nontrivial smooth solutions, all with sign information. superlinear reaction, constant sign and nodal solutions, extremal solutions, nonlinear regularity, nonlinear maximum principle, critical groups 2020 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol40iss1art9 https://www.opuscula.agh.edu.pl/vol40/1/art/opuscula_math_4009.pdf Concentration-compactness results for systems in the Heisenberg group Patrizia Pucci, Letizia Temperini In this paper we complete the study started in [P. Pucci, L. Temperini, Existence for (p,q) critical systems in the Heisenberg group, Adv. Nonlinear Anal. 9 (2020), 895–922] on some variants of the concentration-compactness principle in bounded PS domains \(\Omega\) of the Heisenberg group \(\mathbb{H}^n\). The concentration-compactness principle is a basic tool for treating nonlinear problems with lack of compactness. The results proved here can be exploited when dealing with some kind of elliptic systems involving critical nonlinearities and Hardy terms. Heisenberg group, concentration-compactness, critical exponents, Hardy terms 2020 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol40iss2 https://www.opuscula.agh.edu.pl/om-vol40iss2art1 https://www.opuscula.agh.edu.pl/vol40/2/art/opuscula_math_4010.pdf On the deformed Besov-Hankel spaces Salem Ben Saïd, Mohamed Amine Boubatra, Mohamed Sifi In this paper we introduce function spaces denoted by \(BH_{\kappa,\beta}^{p,r}\) (\(0\lt\beta\lt 1\), \(1\leq p, r \leq +\infty\)) as subspaces of \(L^p\) that we call deformed Besov-Hankel spaces. We provide characterizations of these spaces in terms of Bochner-Riesz means in the case \(1\leq p\leq +\infty\) and in terms of partial Hankel integrals in the case \(1\lt p\lt +\infty\) associated to the deformed Hankel operator by a parameter \(\kappa\gt 0\). For \(p=r=+\infty\), we obtain an approximation result involving partial Hankel integrals. deformed Hankel kernel, Besov spaces, Bochner-Riesz means, partial Hankel integrals 2020 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol40iss2art2 https://www.opuscula.agh.edu.pl/vol40/2/art/opuscula_math_4011.pdf Maximum packings of the λ-fold complete 3-uniform hypergraph with loose 3-cycles Ryan C. Bunge, Dontez Collins, Daryl Conko-Camel, Saad I. El-Zanati, Rachel Liebrecht, Alexander Vasquez It is known that the 3-uniform loose 3-cycle decomposes the complete 3-uniform hypergraph of order \(v\) if and only if \(v \equiv 0, 1,\text{ or }2\ (\operatorname{mod} 9)\). For all positive integers \(\lambda\) and \(v\), we find a maximum packing with loose 3-cycles of the \(\lambda\)-fold complete 3-uniform hypergraph of order \(v\). We show that, if \(v \geq 6\), such a packing has a leave of two or fewer edges. maximum packing, \(\lambda\)-fold complete 3-uniform hypergraph, loose 3-cycle 2020 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol40iss2art3 https://www.opuscula.agh.edu.pl/vol40/2/art/opuscula_math_4012.pdf On the asymptotic behavior of nonoscillatory solutions of certain fractional differential equations with positive and negative terms John R. Graef, Said R. Grace, Ercan Tunç This paper is concerned with the asymptotic behavior of the nonoscillatory solutions of the forced fractional differential equation with positive and negative terms of the form \[^{C}D_{c}^{\alpha}y(t)+f(t,x(t))=e(t)+k(t)x^{\eta}(t)+h(t,x(t)),\] where \(t\geq c \geq 1\), \(\alpha \in (0,1)\), \(\eta \geq 1\) is the ratio of positive odd integers, and \(^{C}D_{c}^{\alpha}y\) denotes the Caputo fractional derivative of \(y\) of order \(\alpha\). The cases \[y(t)=(a(t)(x^{\prime}(t))^{\eta})^{\prime} \quad \text{and} \quad y(t)=a(t)(x^{\prime}(t))^{\eta}\] are considered. The approach taken here can be applied to other related fractional differential equations. Examples are provided to illustrate the relevance of the results obtained. integro-differential equations, fractional differential equations, nonoscillatory solutions, boundedness, Caputo derivative 2020 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol40iss2art4 https://www.opuscula.agh.edu.pl/vol40/2/art/opuscula_math_4013.pdf Asymptotic expansion of large eigenvalues for a class of unbounded Jacobi matrices Ayoub Harrat, El Hassan Zerouali, Lech Zielinski We investigate a class of infinite tridiagonal matrices which define unbounded self-adjoint operators with discrete spectrum. Our purpose is to establish the asymptotic expansion of large eigenvalues and to compute two correction terms explicitly. tridiagonal matrix, band matrix, unbounded self-adjoint operator, discrete spectrum, large eigenvalues, asymptotics 2020 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol40iss2art5 https://www.opuscula.agh.edu.pl/vol40/2/art/opuscula_math_4014.pdf Subdivision of hypergraphs and their colorings Moharram N. Iradmusa In this paper we introduce the subdivision of hypergraphs, study their properties and parameters and investigate their weak and strong chromatic numbers in various cases. hypergraph, uniform hypergraph, subdivision of hypergraph 2020 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol40iss2art6 https://www.opuscula.agh.edu.pl/vol40/2/art/opuscula_math_4015.pdf Oscillation of time fractional vector diffusion-wave equation with fractional damping R. Ramesh, S. Harikrishnan, J. J. Nieto, P. Prakash In this paper, sufficient conditions for \(H\)-oscillation of solutions of a time fractional vector diffusion-wave equation with forced and fractional damping terms subject to the Neumann boundary condition are established by employing certain fractional differential inequality, where \(H\) is a unit vector in \(\mathbb{R}^n\). The examples are given to illustrate the main results. fractional diffusion-wave equation, \(H\)-oscillation, vector differential equation 2020 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol40iss3 https://www.opuscula.agh.edu.pl/om-vol40iss3art1 https://www.opuscula.agh.edu.pl/vol40/3/art/opuscula_math_4016.pdf A unique weak solution for a kind of coupled system of fractional Schrödinger equations Fatemeh Abdolrazaghi, Abdolrahman Razani In this paper, we prove the existence of a unique weak solution for a class of fractional systems of Schrödinger equations by using the Minty-Browder theorem in the Cartesian space. To this aim, we need to impose some growth conditions to control the source functions with respect to dependent variables. fractional Laplacian, uniqueness, weak solution, nonlinear systems 2020 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol40iss3art2 https://www.opuscula.agh.edu.pl/vol40/3/art/opuscula_math_4017.pdf Stochastic Wiener filter in the white noise space Daniel Alpay, Ariel Pinhas In this paper we introduce a new approach to the study of filtering theory by allowing the system's parameters to have a random character. We use Hida's white noise space theory to give an alternative characterization and a proper generalization to the Wiener filter over a suitable space of stochastic distributions introduced by Kondratiev. The main idea throughout this paper is to use the nuclearity of this space in order to view the random variables as bounded multiplication operators (with respect to the Wick product) between Hilbert spaces of stochastic distributions. This allows us to use operator theory tools and properties of Wiener algebras over Banach spaces to proceed and characterize the Wiener filter equations under the underlying randomness assumptions. Wiener filter, white noise space, Wick product, stochastic distribution 2020 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol40iss3art3 https://www.opuscula.agh.edu.pl/vol40/3/art/opuscula_math_4018.pdf Existence of periodic positive solutions to nonlinear Lotka-Volterra competition systems Mimia Benhadri, Tomás Caraballo, Halim Zeghdoudi We investigate the existence of positive periodic solutions of a nonlinear Lotka-Volterra competition system with deviating arguments. The main tool we use to obtain our result is the Krasnoselskii fixed point theorem. In particular, this paper improves important and interesting work [X.H. Tang, X. Zhou, On positive periodic solution of Lotka-Volterra competition systems with deviating arguments, Proc. Amer. Math. Soc. 134 (2006), 2967-2974]. Moreover, as an application, we also exhibit some special cases of the system, which have been studied extensively in the literature. Krasnoselskii's fixed point theorem, positive periodic solutions, Lotka-Volterra competition systems, variable delays 2020 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol40iss3art4 https://www.opuscula.agh.edu.pl/vol40/3/art/opuscula_math_4019.pdf A note on confidence intervals for deblurred images Michał Biel, Zbigniew Szkutnik We consider pointwise asymptotic confidence intervals for images that are blurred and observed in additive white noise. This amounts to solving a stochastic inverse problem with a convolution operator. Under suitably modified assumptions, we fill some apparent gaps in the proofs published in [N. Bissantz, M. Birke, Asymptotic normality and confidence intervals for inverse regression models with convolution-type operators, J. Multivariate Anal. 100 (2009), 2364-2375]. In particular, this leads to modified bootstrap confidence intervals with much better finite-sample behaviour than the original ones, the validity of which is, in our opinion, questionable. Some simulation results that support our claims and illustrate the behaviour of the confidence intervals are also presented. inverse problems, confidence intervals, convolution, deblurring 2020 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol40iss3art5 https://www.opuscula.agh.edu.pl/vol40/3/art/opuscula_math_4020.pdf A note on bipartite graphs whose [1,k]-domination number equal to their number of vertices Narges Ghareghani, Iztok Peterin, Pouyeh Sharifani A subset \(D\) of the vertex set \(V\) of a graph \(G\) is called an \([1,k]\)-dominating set if every vertex from \(V-D\) is adjacent to at least one vertex and at most \(k\) vertices of \(D\). A \([1,k]\)-dominating set with the minimum number of vertices is called a \(\gamma_{[1,k]}\)-set and the number of its vertices is the \([1,k]\)-domination number \(\gamma_{[1,k]}(G)\) of \(G\). In this short note we show that the decision problem whether \(\gamma_{[1,k]}(G)=n\) is an \(NP\)-hard problem, even for bipartite graphs. Also, a simple construction of a bipartite graph \(G\) of order \(n\) satisfying \(\gamma_{[1,k]}(G)=n\) is given for every integer \(n \geq (k+1)(2k+3)\). domination, \([1,k]\)-domination number, \([1,k]\)-total domination number, bipartite graphs 2020 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol40iss3art6 https://www.opuscula.agh.edu.pl/vol40/3/art/opuscula_math_4021.pdf On the crossing numbers of join products of five graphs of order six with the discrete graph Michal Staš The main purpose of this article is broaden known results concerning crossing numbers for join of graphs of order six. We give the crossing number of the join product \(G^{\ast} + D_n\), where the disconnected graph \(G^{\ast}\) of order six consists of one isolated vertex and of one edge joining two nonadjacent vertices of the \(5\)-cycle. In our proof, the idea of cyclic permutations and their combinatorial properties will be used. Finally, by adding new edges to the graph \(G^{\ast}\), the crossing numbers of \(G_i+D_n\) for four other graphs \(G_i\) of order six will be also established. graph, drawing, crossing number, join product, cyclic permutation 2020 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol40iss4 https://www.opuscula.agh.edu.pl/om-vol40iss4art1 https://www.opuscula.agh.edu.pl/vol40/4/art/opuscula_math_4022.pdf Multiple solutions of boundary value problems on time scales for a φ-Laplacian operator Pablo Amster, Mariel Paula Kuna, Dionicio Pastor Santos We establish the existence and multiplicity of solutions for some boundary value problems on time scales with a \(\varphi\)-Laplacian operator. For this purpose, we employ the concept of lower and upper solutions and the Leray-Schauder degree. The results extend and improve known results for analogous problems with discrete \(p\)-Laplacian as well as those for boundary value problems on time scales. dynamic equations on time scales, nonlinear boundary value problems, upper and lower solutions, Leray-Schauder degree, multiple solutions 2020 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol40iss4art2 https://www.opuscula.agh.edu.pl/vol40/4/art/opuscula_math_4023.pdf An inverse backward problem for degenerate two-dimensional parabolic equation Khalid Atifi, El-Hassan Essoufi, Bouchra Khouiti This paper deals with the determination of an initial condition in the degenerate two-dimensional parabolic equation \[\partial_{t}u-\mathrm{div}\left(a(x,y)I_2\nabla u\right)=f,\quad (x,y)\in\Omega,\; t\in(0,T),\] where \(\Omega\) is an open, bounded subset of \(\mathbb{R}^2\), \(a \in C^1(\bar{\Omega})\) with \(a\geqslant 0\) everywhere, and \(f\in L^{2}(\Omega \times (0,T))\), with initial and boundary conditions \[u(x,y,0)=u_0(x,y), \quad u\mid_{\partial\Omega}=0,\] from final observations. This inverse problem is formulated as a minimization problem using the output least squares approach with the Tikhonov regularization. To show the convergence of the descent method, we prove the Lipschitz continuity of the gradient of the Tikhonov functional. Also we present some numerical experiments to show the performance and stability of the proposed approach. data assimilation, adjoint method, regularization, heat equation, inverse problem, degenerate equations, optimization 2020 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol40iss4art3 https://www.opuscula.agh.edu.pl/vol40/4/art/opuscula_math_4024.pdf Option pricing formulas under a change of numèraire Antonio Attalienti, Michele Bufalo We present some formulations of the Cox-Ross-Rubinstein and Black-Scholes formulas for European options obtained through a suitable change of measure, which corresponds to a change of numèraire for the underlying price process. Among other consequences, a closed formula for the price of an European call option at each node of the multi-period binomial tree is achieved, too. Some of the results contained herein, though comparable with analogous ones appearing elsewhere in the financial literature, provide however a supplementary widening and deepening in view of useful applications in the more challenging framework of incomplete markets. This last issue, having the present paper as a preparatory material, will be treated extensively in a forthcoming paper. Black-Scholes formula, binomial model, martingale measures, numèraire 2020 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol40iss4art4 https://www.opuscula.agh.edu.pl/vol40/4/art/opuscula_math_4025.pdf Facial rainbow edge-coloring of simple 3-connected plane graphs Július Czap A facial rainbow edge-coloring of a plane graph \(G\) is an edge-coloring such that any two edges receive distinct colors if they lie on a common facial path of \(G\). The minimum number of colors used in such a coloring is denoted by \(\text{erb}(G)\). Trivially, \(\text{erb}(G) \geq \text{L}(G)+1\) holds for every plane graph without cut-vertices, where \(\text{L}(G)\) denotes the length of a longest facial path in \(G\). Jendroľ in 2018 proved that every simple \(3\)-connected plane graph admits a facial rainbow edge-coloring with at most \(\text{L}(G)+2\) colors, moreover, this bound is tight for \(\text{L}(G)=3\). He also proved that \(\text{erb}(G) = \text{L}(G)+1\) for \(\text{L}(G)\not\in\{3,4,5\}\). He posed the following conjecture: There is a simple \(3\)-connected plane graph \(G\) with \(\text{L}(G)=4\) and \(\text{erb}(G)=\text{L}(G)+2\). In this note we answer the conjecture in the affirmative. plane graph, facial path, edge-coloring 2020 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol40iss4art5 https://www.opuscula.agh.edu.pl/vol40/4/art/opuscula_math_4026.pdf Properties of solutions to some weighted p-Laplacian equation Prashanta Garain In this paper, we prove some qualitative properties for the positive solutions to some degenerate elliptic equation given by \[-\text{div}\big(w|\nabla u|^{p-2}\nabla u\big)=f(x,u),\quad w\in \mathcal{A}_p,\] on smooth domain and for varying nonlinearity \(f\). \(p\)-Laplacian, degenerate elliptic equations, weighted Sobolev space 2020 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol40iss4art6 https://www.opuscula.agh.edu.pl/vol40/4/art/opuscula_math_4027.pdf Hilbert-Schmidtness of weighted composition operators and their differences on Hardy spaces Ching-on Lo, Anthony Wai-keung Loh Let \(u\) and \(\varphi\) be two analytic functions on the unit disk \(\mathbb{D}\) such that \(\varphi(\mathbb{D}) \subset \mathbb{D}\). A weighted composition operator \(uC_{\varphi}\) induced by \(u\) and \(\varphi\) is defined on \(H^2\), the Hardy space of \(\mathbb{D}\), by \(uC_{\varphi}f := u \cdot f \circ \varphi\) for every \(f\) in \(H^2\). We obtain sufficient conditions for Hilbert-Schmidtness of \(uC_{\varphi}\) on \(H^2\) in terms of function-theoretic properties of \(u\) and \(\varphi\). Moreover, we characterize Hilbert-Schmidt difference of two weighted composition operators on \(H^2\). weighted composition operators, Hardy spaces, compact operators, Hilbert-Schmidt operators 2020 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol40iss4art7 https://www.opuscula.agh.edu.pl/vol40/4/art/opuscula_math_4028.pdf Decompositions of complete 3-uniform hypergraphs into cycles of constant prime length R. Lakshmi, T. Poovaragavan A complete \(3\)-uniform hypergraph of order \(n\) has vertex set \(V\) with \(|V|=n\) and the set of all \(3\)-subsets of \(V\) as its edge set. A \(t\)-cycle in this hypergraph is \(v_1, e_1, v_2, e_2,\dots, v_t, e_t, v_1\) where \(v_1, v_2,\dots, v_t\) are distinct vertices and \(e_1, e_2,\dots, e_t\) are distinct edges such that \(v_i, v_{i+1}\in e_i\) for \(i \in \{1, 2,\dots, t-1\}\) and \(v_t, v_1 \in e_t\). A decomposition of a hypergraph is a partition of its edge set into edge-disjoint subsets. In this paper, we give necessary and sufficient conditions for a decomposition of the complete \(3\)-uniform hypergraph of order \(n\) into \(p\)-cycles, whenever \(p\) is prime. uniform hypergraph, cycle decomposition 2020 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol40iss5 https://www.opuscula.agh.edu.pl/om-vol40iss5art1 https://www.opuscula.agh.edu.pl/vol40/5/art/opuscula_math_4029.pdf Oscillatory criteria via linearization of half-linear second order delay differential equations Blanka Baculíková, Jozef Džurina In the paper, we study oscillation of the half-linear second order delay differential equations of the form \[\left(r(t)(y'(t))^{\alpha}\right)'+p(t)y^{\alpha}(\tau(t))=0.\] We introduce new monotonic properties of its nonoscillatory solutions and use them for linearization of considered equation which leads to new oscillatory criteria. The presented results essentially improve existing ones. second order differential equations, delay, monotonic properties, linearization, oscillation 2020 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol40iss5art2 https://www.opuscula.agh.edu.pl/vol40/5/art/opuscula_math_4030.pdf Existence results for a sublinear second order Dirichlet boundary value problem on the half-line Dahmane Bouafia, Toufik Moussaoui In this paper we study the existence of nontrivial solutions for a boundary value problem on the half-line, where the nonlinear term is sublinear, by using Ekeland's variational principle and critical point theory. Ekeland's variational principle, critical point 2020 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol40iss5art3 https://www.opuscula.agh.edu.pl/vol40/5/art/opuscula_math_4031.pdf On the nonoscillatory behavior of solutions of three classes of fractional difference equations Said Rezk Grace, Jehad Alzabut, Sakthivel Punitha, Velu Muthulakshmi, Hakan Adıgüzel In this paper, we study the nonoscillatory behavior of three classes of fractional difference equations. The investigations are presented in three different folds. Unlike most existing nonoscillation results which have been established by employing Riccati transformation technique, we employ herein an easily verifiable approach based on the fractional Taylor's difference formula, some features of discrete fractional calculus and mathematical inequalities. The theoretical findings are demonstrated by examples. We end the paper by a concluding remark. Caputo difference operator, nonoscillation criteria, fractional difference equation, mathematical inequalities 2020 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol40iss5art4 https://www.opuscula.agh.edu.pl/vol40/5/art/opuscula_math_4032.pdf On some extensions of the A-model Rytis Juršėnas The A-model for finite rank singular perturbations of class \(\mathfrak{H}_{-m-2}\setminus\mathfrak{H}_{-m-1}\), \(m \in \mathbb{N}\), is considered from the perspective of boundary relations. Assuming further that the Hilbert spaces \((\mathfrak{H}_n)_{n\in\mathbb{Z}}\) admit an orthogonal decomposition \(\mathfrak{H}^-_n \oplus \mathfrak{H}^+_n\), with the corresponding projections satisfying \(P^{\pm}_{n+1}\subseteq P^{\pm}_n\), nontrivial extensions in the A-model are constructed for the symmetric restrictions in the subspaces. finite rank higher order singular perturbation, cascade (A) model, peak model, Hilbert space, scale of Hilbert spaces, Pontryagin space, ordinary boundary triple, Krein \(Q\)-function, Weyl function, gamma field, symmetric operator, proper extension, resolvent 2020 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol40iss5art5 https://www.opuscula.agh.edu.pl/vol40/5/art/opuscula_math_4033.pdf Outer independent rainbow dominating functions in graphs Zhila Mansouri, Doost Ali Mojdeh A 2-rainbow dominating function (2-rD function) of a graph \(G=(V,E)\) is a function \(f:V(G)\rightarrow\{\emptyset,\{1\},\{2\},\{1,2\}\}\) having the property that if \(f(x)=\emptyset\), then \(f(N(x))=\{1,2\}\). The 2-rainbow domination number \(\gamma_{r2}(G)\) is the minimum weight of \(\sum_{v\in V(G)}|f(v)|\) taken over all 2-rainbow dominating functions \(f\). An outer-independent 2-rainbow dominating function (OI2-rD function) of a graph \(G\) is a 2-rD function \(f\) for which the set of all \(v\in V(G)\) with \(f(v)=\emptyset\) is independent. The outer independent 2-rainbow domination number \(\gamma_{oir2}(G)\) is the minimum weight of an OI2-rD function of \(G\). In this paper, we study the OI2-rD number of graphs. We give the complexity of the problem OI2-rD of graphs and present lower and upper bounds on \(\gamma_{oir2}(G)\). Moreover, we characterize graphs with some small or large OI2-rD numbers and we also bound this parameter from above for trees in terms of the order, leaves and the number of support vertices and characterize all trees attaining the bound. Finally, we show that any ordered pair \((a,b)\) is realizable as the vertex cover number and OI2-rD numbers of some non-trivial tree if and only if \(a+1\leq b\leq 2a\). outer-independent rainbow domination, \(K_{1,r}\)-free graphs, trees 2020 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol40iss5art6 https://www.opuscula.agh.edu.pl/vol40/5/art/opuscula_math_4034.pdf On 2-rainbow domination number of functigraph and its complement Athena Shaminezhad, Ebrahim Vatandoost Let \(G\) be a graph and \(f:V (G)\rightarrow P(\{1,2\})\) be a function where for every vertex \(v\in V(G)\), with \(f(v)=\emptyset\) we have \(\bigcup_{u\in N_{G}(v)} f(u)=\{1,2\}\). Then \(f\) is a \(2\)-rainbow dominating function or a \(2RDF\) of \(G\). The weight of \(f\) is \(\omega(f)=\sum_{v\in V(G)} |f(v)|\). The minimum weight of all \(2\)-rainbow dominating functions is \(2\)-rainbow domination number of \(G\), denoted by \(\gamma_{r2}(G)\). Let \(G_1\) and \(G_2\) be two copies of a graph G with disjoint vertex sets \(V(G_1)\) and \(V(G_2)\), and let \(\sigma\) be a function from \(V(G_1)\) to \(V(G_2)\). We define the functigraph \(C(G,\sigma)\) to be the graph that has the vertex set \(V(C(G,\sigma)) = V(G_1)\cup V(G_2)\), and the edge set \(E(C(G,\sigma)) = E(G_1)\cup E(G_2 \cup \{uv ; u\in V(G_1), v\in V(G_2), v =\sigma(u)\}\). In this paper, \(2\)-rainbow domination number of the functigraph of \(C(G,\sigma)\) and its complement are investigated. We obtain a general bound for \(\gamma_{r2}(C(G,\sigma))\) and we show that this bound is sharp. 2-rainbow domination number, functigraph, complement, cubic graph 2020 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol40iss5art7 https://www.opuscula.agh.edu.pl/vol40/5/art/opuscula_math_4035.pdf Oscillatory behavior of second-order damped differential equations with a superlinear neutral term Ercan Tunç, Osman Özdemir This article concerns the oscillatory behavior of solutions to second-order damped nonlinear differential equations with a superlinear neutral term. The results are obtained by a Riccati type transformation as well as by an integral criterion. Examples illustrating the results are provided and some suggestions for further research are indicated. oscillation, second-order, neutral differential equation, damping term 2020 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol40iss6 https://www.opuscula.agh.edu.pl/om-vol40iss6art1 https://www.opuscula.agh.edu.pl/vol40/6/art/opuscula_math_4036.pdf General decay rate of a weakly dissipative viscoelastic equation with a general damping Khaleel Anaya, Salim A. Messaoudi In this paper, we consider a weakly dissipative viscoelastic equation with a nonlinear damping. A general decay rate is proved for a wide class of relaxation functions. To support our theoretical findings, some numerical results are provided. general decay, relaxation function, viscoelastic, weakly dissipative equation 2020 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol40iss6art2 https://www.opuscula.agh.edu.pl/vol40/6/art/opuscula_math_4037.pdf Quasilinearization method for finite systems of nonlinear RL fractional differential equations Zachary Denton, Juan Diego Ramírez In this paper the quasilinearization method is extended to finite systems of Riemann-Liouville fractional differential equations of order \(0\lt q\lt 1\). Existence and comparison results of the linear Riemann-Liouville fractional differential systems are recalled and modified where necessary. Using upper and lower solutions, sequences are constructed that are monotonic such that the weighted sequences converge uniformly and quadratically to the unique solution of the system. A numerical example illustrating the main result is given. fractional differential systems, lower and upper solutions, quasilinearization method 2020 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol40iss6art3 https://www.opuscula.agh.edu.pl/vol40/6/art/opuscula_math_4038.pdf A note on attractivity for the intersection of two discontinuity manifolds Fabio V. Difonzo In piecewise smooth dynamical systems, a co-dimension 2 discontinuity manifold can be attractive either through partial sliding or by spiraling. In this work we prove that both attractivity regimes can be analyzed by means of the moments solution, a spiraling bifurcation parameter and a novel attractivity parameter, which changes sign when attractivity switches from sliding to spiraling attractivity or vice-versa. We also study what happens at what we call attractivity transition points, showing that the spiraling bifurcation parameter is always zero at those points. piecewise smooth systems, sliding motion, co-dimension 2, discontinuity manifold, attractivity 2020 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol40iss6art4 https://www.opuscula.agh.edu.pl/vol40/6/art/opuscula_math_4039.pdf On the twisted Dorfman-Courant like brackets Włodzimierz M. Mikulski There are completely described all \(\mathcal{VB}_{m,n}\)-gauge-natural operators \(C\) which, like to the Dorfman-Courant bracket, send closed linear \(3\)-forms \(H\in\Gamma^{l-\rm{clos}}_E(\bigwedge^3T^*E)\) on a smooth (\(\mathcal{C}^{\infty}\)) vector bundle \(E\) into \(\mathbf{R}\)-bilinear operators \[C_H:\Gamma^l_E(TE\oplus T^*E)\times \Gamma^l_E(TE\oplus T^*E)\to \Gamma^l_E(TE\oplus T^*E)\] transforming pairs of linear sections of \(TE\oplus T^*E\to E\) into linear sections of \(TE\oplus T^*E\to E\). Then all such \(C\) which also, like to the twisted Dorfman-Courant bracket, satisfy both some "restricted" condition and the Jacobi identity in Leibniz form are extracted. natural operator, linear vector field, linear form, (twisted) Dorfman-Courant bracket, Jacobi identity in Leibniz form 2020 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol40iss6art5 https://www.opuscula.agh.edu.pl/vol40/6/art/opuscula_math_4040.pdf Existence and decay of finite energy solutions for semilinear dissipative wave equations in time-dependent domains Mitsuhiro Nakao We consider the initial-boundary value problem for semilinear dissipative wave equations in noncylindrical domain \(\bigcup_{0\leq t \lt\infty} \Omega(t)\times\{t\} \subset \mathbb{R}^N\times \mathbb{R}\). We are interested in finite energy solution. We derive an exponential decay of the energy in the case \(\Omega(t)\) is bounded in \(\mathbb{R}^N\) and the estimate \[\int\limits_0^{\infty} E(t)dt \leq C(E(0),\|u(0)\|)\lt \infty\] in the case \(\Omega(t)\) is unbounded. Existence and uniqueness of finite energy solution are also proved. energy decay, global existence, semilinear wave equation, noncylindrical domains 2020 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol40iss6art6 https://www.opuscula.agh.edu.pl/vol40/6/art/opuscula_math_4041.pdf Echoes and glimpses of a distant drum John Wm. Turner To what extent does the spectrum of the Laplacian operator on a domain \(D\) with prescribed boundary conditions determine its shape? This paper first retraces the history of this problem, then Kac's approach in terms of a diffusion process with absorbing boundary conditions. It is shown how the restriction to a polygonal boundary for \(D\) in this method, which required taking the limit of an infinite number of sides to obtain a smooth one, can be avoided by using the Duhamel method. Kac drum problem, inverse methods, diffusion process 2020 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol41iss1 https://www.opuscula.agh.edu.pl/om-vol41iss1art1 https://www.opuscula.agh.edu.pl/vol41/1/art/opuscula_math_4101.pdf Some existence results for a nonlocal non-isotropic problem Rachid Bentifour, Sofiane El-Hadi Miri In this paper we deal with the following problem \[\begin{cases}-\sum\limits_{i=1}^{N}\left[ \left( a+b\int\limits_{\, \Omega }\left\vert \partial _{i}u\right\vert ^{p_{i}}dx\right) \partial _{i}\left( \left\vert \partial _{i}u\right\vert ^{p_{i}-2}\partial _{i}u\right) \right]=\frac{f(x)}{u^{\gamma }}\pm g(x)u^{q-1} & in\ \Omega, \\ u\geq 0 & in\ \Omega, \\ u=0 & on\ \partial \Omega, \end{cases}\] where \(\Omega\) is a bounded regular domain in \(\mathbb{R}^{N}\). We will assume without loss of generality that \(1\leq p_{1}\leq p_{2}\leq \ldots\leq p_{N}\) and that \(f\) and \(g\) are non-negative functions belonging to a suitable Lebesgue space \(L^{m}(\Omega)\), \(1\lt q\lt \overline{p}^{\ast}\), \(a\gt 0\), \(b\gt 0\) and \(0\lt\gamma \lt 1.\) anisotropic operator, integro-differential problem, variational methods 2021 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol41iss1art2 https://www.opuscula.agh.edu.pl/vol41/1/art/opuscula_math_4102.pdf Nonlinear parabolic equation having nonstandard growth condition with respect to the gradient and variable exponent Abderrahim Charkaoui, Houda Fahim, Nour Eddine Alaa We are concerned with the existence of solutions to a class of quasilinear parabolic equations having critical growth nonlinearity with respect to the gradient and variable exponent. Using Schaeffer's fixed point theorem combined with the sub- and supersolution method, we prove the existence results of a weak solutions to the considered problems. variable exponent, quasilinear equation, Schaeffer's fixed point, subsolution, supersolution, weak solution 2021 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol41iss1art3 https://www.opuscula.agh.edu.pl/vol41/1/art/opuscula_math_4103.pdf More on linear and metric tree maps Sergiy Kozerenko We consider linear and metric self-maps on vertex sets of finite combinatorial trees. Linear maps are maps which preserve intervals between pairs of vertices whereas metric maps are maps which do not increase distances between pairs of vertices. We obtain criteria for a given linear or a metric map to be a positive (negative) under some orientation of the edges in a tree, we characterize trees which admit maps with Markov graphs being paths and prove that the converse of any partial functional digraph is isomorphic to a Markov graph for some suitable map on a tree. tree, Markov graph, metric map, non-expanding map, linear map, graph homomorphism 2021 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol41iss1art4 https://www.opuscula.agh.edu.pl/vol41/1/art/opuscula_math_4104.pdf Remarks on the existence of nonoscillatory solutions of half-linear ordinary differential equations, I Manabu Naito We consider the half-linear differential equation of the form \[(p(t)|x'|^{\alpha}\mathrm{sgn} x')' + q(t)|x|^{\alpha}\mathrm{sgn} x = 0, \quad t\geq t_{0},\] under the assumption \(\int_{t_{0}}^{\infty}p(s)^{-1/\alpha}ds =\infty\). It is shown that if a certain condition is satisfied, then the above equation has a pair of nonoscillatory solutions with specific asymptotic behavior as \(t \to \infty\). asymptotic behavior, nonoscillatory solution, half-linear differential equation, Hardy-type inequality 2021 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol41iss1art5 https://www.opuscula.agh.edu.pl/vol41/1/art/opuscula_math_4105.pdf On the crossing numbers of join products of W_{4}+P_{n} and W_{4}+C_{n} Michal Staš, Juraj Valiska The crossing number \(\mathrm{cr}(G)\) of a graph \(G\) is the minimum number of edge crossings over all drawings of \(G\) in the plane. The main aim of the paper is to give the crossing number of the join product \(W_4+P_n\) and \(W_4+C_n\) for the wheel \(W_4\) on five vertices, where \(P_n\) and \(C_n\) are the path and the cycle on \(n\) vertices, respectively. Yue et al. conjectured that the crossing number of \(W_m+C_n\) is equal to \(Z(m+1)Z(n)+(Z(m)-1) \big \lfloor \frac{n}{2} \big \rfloor + n+ \big\lceil\frac{m}{2}\big\rceil +2\), for all \(m,n \geq 3\), and where the Zarankiewicz's number \(Z(n)=\big \lfloor \frac{n}{2} \big \rfloor \big \lfloor \frac{n-1}{2} \big \rfloor\) is defined for \(n\geq 1\). Recently, this conjecture was proved for \(W_3+C_n\) by Klešč. We establish the validity of this conjecture for \(W_4+C_n\) and we also offer a new conjecture for the crossing number of the join product \(W_m+P_n\) for \(m\geq 3\) and \(n\geq 2\). graph, crossing number, join product, cyclic permutation, path, cycle 2021 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol41iss1art6 https://www.opuscula.agh.edu.pl/vol41/1/art/opuscula_math_4106.pdf Reiterated periodic homogenization of integral functionals with convex and nonstandard growth integrands Joel Fotso Tachago, Hubert Nnang, Elvira Zappale Multiscale periodic homogenization is extended to an Orlicz-Sobolev setting. It is shown by the reiteraded periodic two-scale convergence method that the sequence of minimizers of a class of highly oscillatory minimizations problems involving convex functionals, converges to the minimizers of a homogenized problem with a suitable convex function. convex function, reiterated two-scale convergence, relaxation, Orlicz-Sobolev spaces 2021 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol41iss1art7 https://www.opuscula.agh.edu.pl/vol41/1/art/opuscula_math_4107.pdf Exponential stability results for variable delay difference equations Ernest Yankson Sufficient conditions that guarantee exponential decay to zero of the variable delay difference equation \[x(n+1)=a(n)x(n)+b(n)x(n-g(n))\] are obtained. These sufficient conditions are deduced via inequalities by employing Lyapunov functionals. In addition, a criterion for the instability of the zero solution is established. The results in the paper generalizes some results in the literature. exponential stability, Lyapunov functional, instability 2021 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol41iss2 https://www.opuscula.agh.edu.pl/om-vol41iss2art1 https://www.opuscula.agh.edu.pl/vol41/2/art/opuscula_math_4108.pdf The achromatic number of K_{6} □ K_{7} is 18 Mirko Horňák A vertex colouring \(f:V(G)\to C\) of a graph \(G\) is complete if for any two distinct colours \(c_1, c_2 \in C\) there is an edge \(\{v_1,v_2\}\in E(G)\) such that \(f(v_i)=c_i\), \(i=1,2\). The achromatic number of \(G\) is the maximum number \(\text{achr}(G)\) of colours in a proper complete vertex colouring of \(G\). In the paper it is proved that \(\text{achr}(K_6 \square K_7)=18\). This result finalises the determination of \(\text{achr}(K_6 \square K_q)\). complete vertex colouring, achromatic number, Cartesian product 2021 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol41iss2art2 https://www.opuscula.agh.edu.pl/vol41/2/art/opuscula_math_4109.pdf Influence of an L^{p}-perturbation on Hardy-Sobolev inequality with singularity a curve Idowu Esther Ijaodoro, El Hadji Abdoulaye Thiam We consider a bounded domain \(\Omega\) of \(\mathbb{R}^N\), \(N \geq 3\), \(h\) and \(b\) continuous functions on \(\Omega\). Let \(\Gamma\) be a closed curve contained in \(\Omega\). We study existence of positive solutions \(u \in H^1_0(\Omega)\) to the perturbed Hardy-Sobolev equation: \[-\Delta u+hu+bu^{1+\delta}=\rho^{-\sigma}_{\Gamma} u^{2^*_{\sigma}-1} \quad \textrm{ in } \Omega,\] where \(2^*_{\sigma}:=\frac{2(N-\sigma)}{N-2}\) is the critical Hardy-Sobolev exponent, \(\sigma\in [0,2)\), \(0\lt\delta\lt\frac{4}{N-2}\) and \(\rho_{\Gamma}\) is the distance function to \(\Gamma\). We show that the existence of minimizers does not depend on the local geometry of \(\Gamma\) nor on the potential \(h\). For \(N=3\), the existence of ground-state solution may depends on the trace of the regular part of the Green function of \(-\Delta+h\) and or on \(b\). This is due to the perturbative term of order \(1+\delta\). Hardy-Sobolev inequality, positive minimizers, parametrized curve, mass, Green function 2021 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol41iss2art3 https://www.opuscula.agh.edu.pl/vol41/2/art/opuscula_math_4110.pdf On the gauge-natural operators similar to the twisted Dorfman-Courant bracket Włodzimierz M. Mikulski All \(\mathcal{VB}_{m,n}\)-gauge-natural operators \(C\) sending linear \(3\)-forms \(H \in \Gamma^{l}_E(\bigwedge^3T^*E)\) on a smooth (\(\mathcal{C}^\infty\)) vector bundle \(E\) into \(\mathbf{R}\)-bilinear operators \[C_H:\Gamma^l_E(TE \oplus T^*E)\times \Gamma^l_E(TE \oplus T^*E)\to \Gamma^l_E(TE \oplus T^*E)\] transforming pairs of linear sections of \(TE \oplus T^*E \to E\) into linear sections of \( TE \oplus T^*E \to E\) are completely described. The complete descriptions is given of all generalized twisted Dorfman-Courant brackets \(C\) (i.e. \(C\) as above such that \(C_0\) is the Dorfman-Courant bracket) satisfying the Jacobi identity for closed linear \(3\)-forms \(H\). An interesting natural characterization of the (usual) twisted Dorfman-Courant bracket is presented. natural operator, linear vector field, linear form, twisted Dorfman-Courant bracket, the Jacobi identity in Leibniz form 2021 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol41iss2art4 https://www.opuscula.agh.edu.pl/vol41/2/art/opuscula_math_4111.pdf Dimension of the intersection of certain Cantor sets in the plane Steen Pedersen, Vincent T. Shaw In this paper we consider a retained digits Cantor set \(T\) based on digit expansions with Gaussian integer base. Let \(F\) be the set all \(x\) such that the intersection of \(T\) with its translate by \(x\) is non-empty and let \(F_{\beta}\) be the subset of \(F\) consisting of all \(x\) such that the dimension of the intersection of \(T\) with its translate by \(x\) is \(\beta\) times the dimension of \(T\). We find conditions on the retained digits sets under which \(F_{\beta}\) is dense in \(F\) for all \(0\leq\beta\leq 1\). The main novelty in this paper is that multiplication the Gaussian integer base corresponds to an irrational (in fact transcendental) rotation in the complex plane. Cantor set, fractal, self-similar, translation, intersection, dimension, Minkowski dimension 2021 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol41iss2art5 https://www.opuscula.agh.edu.pl/vol41/2/art/opuscula_math_4112.pdf Introduction to dominated edge chromatic number of a graph Mohammad R. Piri, Saeid Alikhani We introduce and study the dominated edge coloring of a graph. A dominated edge coloring of a graph \(G\), is a proper edge coloring of \(G\) such that each color class is dominated by at least one edge of \(G\). The minimum number of colors among all dominated edge coloring is called the dominated edge chromatic number, denoted by \(\chi_{dom}^{\prime}(G)\). We obtain some properties of \(\chi_{dom}^{\prime}(G)\) and compute it for specific graphs. Also examine the effects on \(\chi_{dom}^{\prime}(G)\), when \(G\) is modified by operations on vertex and edge of \(G\). Finally, we consider the \(k\)-subdivision of \(G\) and study the dominated edge chromatic number of these kind of graphs. dominated edge chromatic number, subdivision, operation, corona 2021 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol41iss2art6 https://www.opuscula.agh.edu.pl/vol41/2/art/opuscula_math_4113.pdf Remarks on the outer-independent double Italian domination number Lutz Volkmann Let \(G\) be a graph with vertex set \(V(G)\). If \(u\in V(G)\), then \(N[u]\) is the closed neighborhood of \(u\). An outer-independent double Italian dominating function (OIDIDF) on a graph \(G\) is a function \(f:V(G)\longrightarrow \{0,1,2,3\}\) such that if \(f(v)\in\{0,1\}\) for a vertex \(v\in V(G)\), then \(\sum_{x\in N[v]}f(x)\ge 3\), and the set \(\{u\in V(G):f(u)=0\}\) is independent. The weight of an OIDIDF \(f\) is the sum \(\sum_{v\in V(G)}f(v)\). The outer-independent double Italian domination number \(\gamma_{oidI}(G)\) equals the minimum weight of an OIDIDF on \(G\). In this paper we present Nordhaus-Gaddum type bounds on the outer-independent double Italian domination number which improved corresponding results given in [F. Azvin, N. Jafari Rad, L. Volkmann, Bounds on the outer-independent double Italian domination number, Commun. Comb. Optim. 6 (2021), 123-136]. Furthermore, we determine the outer-independent double Italian domination number of some families of graphs. double Italian domination number, outer-independent double Italian domination number, Nordhaus-Gaddum bound 2021 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol41iss2art7 https://www.opuscula.agh.edu.pl/vol41/2/art/opuscula_math_4114.pdf Uniqueness of series in the Franklin system and the Gevorkyan problems Zygmunt Wronicz In 1870 G. Cantor proved that if \(\lim_{N \rightarrow \infty}\sum_{n=-N}^N c_{n}e^{inx} = 0\), \(\bar{c}_{n}=c_{n}\), then \(c_{n}=0\) for \(n\in\mathbb{Z}\). In 2004 G. Gevorkyan raised the issue that if Cantor's result extends to the Franklin system. He solved this conjecture in 2015. In 2014 Z. Wronicz proved that there exists a Franklin series for which a subsequence of its partial sums converges to zero, where not all coefficients of the series are zero. In the present paper we show that to the uniqueness of the Franklin system \(\lim_{n\rightarrow \infty}\sum_{n=0}^\infty a_{n}f_{n}\) it suffices to prove the convergence its subsequence \(s_{2^{n}}\) to zero by the condition \(a_{n}=o(\sqrt{n})\). It is a solution of the Gevorkyan problem formulated in 2016. Franklin system, orthonormal spline system, uniqueness of series 2021 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol41iss3 https://www.opuscula.agh.edu.pl/om-vol41iss3art1 https://www.opuscula.agh.edu.pl/vol41/3/art/opuscula_math_4115.pdf New characterizations of reproducing kernel Hilbert spaces and applications to metric geometry Daniel Alpay, Palle E.T. Jorgensen We give two new global and algorithmic constructions of the reproducing kernel Hilbert space associated to a positive definite kernel. We further present a general positive definite kernel setting using bilinear forms, and we provide new examples. Our results cover the case of measurable positive definite kernels, and we give applications to both stochastic analysis and metric geometry and provide a number of examples. reproducing kernel, positive definite functions, approximation, algorithms, measures, stochastic processes 2021 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol41iss3art2 https://www.opuscula.agh.edu.pl/vol41/3/art/opuscula_math_4116.pdf Perturbation series for Jacobi matrices and the quantum Rabi model Mirna Charif, Lech Zielinski We investigate eigenvalue perturbations for a class of infinite tridiagonal matrices which define unbounded self-adjoint operators with discrete spectrum. In particular we obtain explicit estimates for the convergence radius of the perturbation series and error estimates for the Quantum Rabi Model including the resonance case. We also give expressions for coefficients near resonance in order to evaluate the quality of the rotating wave approximation due to Jaynes and Cummings. Jacobi matrix, unbounded self-adjoint operators, quasi-degenerate eigenvalue perturbation, perturbation series, quantum Rabi model, rotating wave approximation 2021 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol41iss3art3 https://www.opuscula.agh.edu.pl/vol41/3/art/opuscula_math_4117.pdf Multi-variable quaternionic spectral analysis Ilwoo Cho, Palle E.T. Jorgensen In this paper, we consider finite dimensional vector spaces \(\mathbb{H}^n\) over the ring \(\mathbb{H}\) of all quaternions. In particular, we are interested in certain functions acting on \(\mathbb{H}^n\), and corresponding functional equations. Our main results show that (i) all quaternions of \(\mathbb{H}\) are classified by the spectra of their realizations under representation, (ii) all vectors of \(\mathbb{H}^n\) are classified by a canonical extended setting of (i), and (iii) the usual spectral analysis on the matricial ring \(M_n(\mathbb{C})\) of all \((n \times n)\)-matrices over the complex numbers \(\mathbb{C}\) has close connections with certain "non-linear" functional equations on \(\mathbb{H}^n\) up to the classification of (ii). the quaternions \(\mathbb{H}\), vector spaces \(\mathbb{H}^n\) over \(\mathbb{H}\), \(q\)-spectral forms, \(q\)-spectral functions 2021 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol41iss3art4 https://www.opuscula.agh.edu.pl/vol41/3/art/opuscula_math_4118.pdf Extensions of dissipative operators with closable imaginary part Christoph Fischbacher Given a dissipative operator \(A\) on a complex Hilbert space \(\mathcal{H}\) such that the quadratic form \(f \mapsto \text{Im}\langle f, Af \rangle\) is closable, we give a necessary and sufficient condition for an extension of \(A\) to still be dissipative. As applications, we describe all maximally accretive extensions of strictly positive symmetric operators and all maximally dissipative extensions of a highly singular first-order operator on the interval. extension theory, dissipative operators, ordinary differential operators 2021 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol41iss3art5 https://www.opuscula.agh.edu.pl/vol41/3/art/opuscula_math_4119.pdf Spectrum localization of a perturbed operator in a strip and applications Michael Gil' Let \(A\) and \(\tilde{A}\) be bounded operators in a Hilbert space. We consider the following problem: let the spectrum of \(A\) lie in some strip. In what strip the spectrum of \(\tilde{A}\) lies if \(A\) and \(\tilde{A}\) are "close"? Applications of the obtained results to integral operators and matrices are also discussed. In addition, we apply our perturbation results to approximate the spectral strip of a Hilbert-Schmidt operator by the spectral strips of finite matrices. operator, spectrum, perturbation, approximation, integral operator, matrix 2021 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol41iss3art6 https://www.opuscula.agh.edu.pl/vol41/3/art/opuscula_math_4120.pdf On the S-matrix of Schrödinger operator with nonlocal δ-interaction Anna Główczyk, Sergiusz Kużel Schrödinger operators with nonlocal \(\delta\)-interaction are studied with the use of the Lax-Phillips scattering theory methods. The condition of applicability of the Lax-Phillips approach in terms of non-cyclic functions is established. Two formulas for the \(S\)-matrix are obtained. The first one deals with the Krein-Naimark resolvent formula and the Weyl-Titchmarsh function, whereas the second one is based on modified reflection and transmission coefficients. The \(S\)-matrix \(S(z)\) is analytical in the lower half-plane \(\mathbb{C}_{−}\) when the Schrödinger operator with nonlocal \(\delta\)-interaction is positive self-adjoint. Otherwise, \(S(z)\) is a meromorphic matrix-valued function in \(\mathbb{C}_{−}\) and its properties are closely related to the properties of the corresponding Schrödinger operator. Examples of \(S\)-matrices are given. Lax-Phillips scattering scheme, scattering matrix, \(S\)-matrix, nonlocal \(\delta\)-interaction, non-cyclic function 2021 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol41iss3art7 https://www.opuscula.agh.edu.pl/vol41/3/art/opuscula_math_4121.pdf Quadratic inequalities for functionals in l^{∞} Gerd Herzog, Peer Chr. Kunstmann For a class of operators \(T\) on \(l^{\infty}\) and \(T\)-invariant functionals \(\varphi\) we prove inequalities between \(\varphi(x)\), \(\varphi(x^2)\) and the upper density of the sets \[P_r:=\{n \in \mathbb{N}_0: \varphi((T^{n}x)\cdot x) \gt r\}.\] Applications are given to Banach limits and integrals. Banach algebras of bounded functions, operator-invariant functionals, Banach limits 2021 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol41iss4 https://www.opuscula.agh.edu.pl/om-vol41iss4art1 https://www.opuscula.agh.edu.pl/vol41/4/art/opuscula_math_4122.pdf Total connected domination game Csilla Bujtás, Michael A. Henning, Vesna Iršič, Sandi Klavžar The (total) connected domination game on a graph \(G\) is played by two players, Dominator and Staller, according to the standard (total) domination game with the additional requirement that at each stage of the game the selected vertices induce a connected subgraph of \(G\). If Dominator starts the game and both players play optimally, then the number of vertices selected during the game is the (total) connected game domination number (\(\gamma_{\rm tcg}(G)\)) \(\gamma_{\rm cg}(G)\) of \(G\). We show that \(\gamma_{\rm tcg}(G) \in \{\gamma_{\rm cg}(G),\gamma_{\rm cg}(G) + 1,\gamma_{\rm cg}(G) + 2\}\), and consequently define \(G\) as Class \(i\) if \(\gamma_{\rm tcg}(G) = \gamma_{\rm cg} + i\) for \(i \in \{0,1,2\}\). A large family of Class \(0\) graphs is constructed which contains all connected Cartesian product graphs and connected direct product graphs with minumum degree at least \(2\). We show that no tree is Class \(2\) and characterize Class \(1\) trees. We provide an infinite family of Class \(2\) bipartite graphs. connected domination game, total connected domination game, graph product, tree 2021 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol41iss4art2 https://www.opuscula.agh.edu.pl/vol41/4/art/opuscula_math_4123.pdf Remarks on damped Schrödinger equation of Choquard type Lassaad Chergui This paper is devoted to the Schrödinger-Choquard equation with linear damping. Global existence and scattering are proved depending on the size of the damping coefficient. damped Choquard equation, global existence, scattering, invariant sets 2021 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol41iss4art3 https://www.opuscula.agh.edu.pl/vol41/4/art/opuscula_math_4124.pdf Spectrum of discrete 2n-th order difference operator with periodic boundary conditions and its applications Abdelrachid El Amrouss, Omar Hammouti Let \(n\in\mathbb{N}^{*}\), and \(N\geq n\) be an integer. We study the spectrum of discrete linear \(2n\)-th order eigenvalue problems \[\begin{cases}\sum_{k=0}^{n}(-1)^{k}\Delta^{2k}u(t-k) = \lambda u(t) ,\quad & t\in[1, N]_{\mathbb{Z}}, \\ \Delta^{i}u(-(n-1))=\Delta^{i}u(N-(n-1)),\quad & i\in[0, 2n-1]_{\mathbb{Z}},\end{cases}\] where \(\lambda\) is a parameter. As an application of this spectrum result, we show the existence of a solution of discrete nonlinear \(2n\)-th order problems by applying the variational methods and critical point theory. discrete boundary value problems, 2n-th order, variational methods, critical point theory 2021 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol41iss4art4 https://www.opuscula.agh.edu.pl/vol41/4/art/opuscula_math_4125.pdf Asymptotic expansions for the first hitting times of Bessel processes Yuji Hamana, Ryo Kaikura, Kosuke Shinozaki We study a precise asymptotic behavior of the tail probability of the first hitting time of the Bessel process. We deduce the order of the third term and decide the explicit form of its coefficient. Bessel process, hitting time, tail probability, modified Bessel function, asymptotic expansion, Laplace transform 2021 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol41iss4art5 https://www.opuscula.agh.edu.pl/vol41/4/art/opuscula_math_4126.pdf Reaction-diffusion coupled inclusions with variable exponents and large diffusion Jacson Simsen, Mariza Stefanello Simsen, Petra Wittbold This work concerns the study of asymptotic behavior of coupled systems of \(p(x)\)-Laplacian differential inclusions. We obtain that the generalized semiflow generated by the coupled system has a global attractor, we prove continuity of the solutions with respect to initial conditions and a triple of parameters and we prove upper semicontinuity of a family of global attractors for reaction-diffusion systems with spatially variable exponents when the exponents go to constants greater than 2 in the topology of \(L^{\infty}(\Omega)\) and the diffusion coefficients go to infinity. reaction-diffusion coupled systems, variable exponents, attractors, upper semicontinuity, large diffusion 2021 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol41iss4art6 https://www.opuscula.agh.edu.pl/vol41/4/art/opuscula_math_4127.pdf Region of existence of multiple solutions for a class of Robin type four-point BVPs Amit K. Verma, Nazia Urus, Ravi P. Agarwal This article aims to prove the existence of a solution and compute the region of existence for a class of four-point nonlinear boundary value problems (NLBVPs) defined as \[\begin{gathered} -u''(x)=\psi(x,u,u'), \quad x\in (0,1),\\ u'(0)=\lambda_{1}u(\xi), \quad u'(1)=\lambda_{2} u(\eta),\end{gathered}\] where \(I=[0,1]\), \(0\lt\xi\leq\eta\lt 1\) and \(\lambda_1,\lambda_2\gt 0\). The nonlinear source term \(\psi\in C(I\times\mathbb{R}^2,\mathbb{R})\) is one sided Lipschitz in \(u\) with Lipschitz constant \(L_1\) and Lipschitz in \(u'\) such that \(|\psi(x,u,u')-\psi(x,u,v')|\leq L_2(x)|u'-v'|\). We develop monotone iterative technique (MI-technique) in both well ordered and reverse ordered cases. We prove maximum, anti-maximum principle under certain assumptions and use it to show the monotonic behaviour of the sequences of upper-lower solutions. The sufficient conditions are derived for the existence of solution and verified for two examples. The above NLBVPs is linearised using Newton's quasilinearization method which involves a parameter \(k\) equivalent to \(\max_u\frac{\partial \psi}{\partial u}\). We compute the range of \(k\) for which iterative sequences are convergent. Green's function, monotone iterative technique, maximum principle, multi-point problem 2021 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol41iss4art7 https://www.opuscula.agh.edu.pl/vol41/4/art/opuscula_math_4128.pdf A note on possible density and diameter of counterexamples to the Seymour's second neighborhood conjecture Oleksiy Zelenskiy, Valentyna Darmosiuk, Illia Nalivayko Seymour's second neighborhood conjecture states that every simple digraph without loops or 2-cycles contains a vertex whose second neighborhood is at least as large as its first. In this paper we show, that from falsity of Seymour's second neighborhood conjecture it follows that there exist strongly-connected counterexamples with both low and high density (dense and sparse graph). Moreover, we show that if there is a counterexample to conjecture, then it is possible to construct counterexample with any diameter \(k\geq 3\). graph theory, Seymour's second neighborhood conjecture, density of graph, diameter of graph 2021 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol41iss5 https://www.opuscula.agh.edu.pl/om-vol41iss5art1 https://www.opuscula.agh.edu.pl/vol41/5/art/opuscula_math_4129.pdf Oscillation criteria for linear difference equations with several variable delays Vasileios Benekas, Ábel Garab, Ardak Kashkynbayev, Ioannis P. Stavroulakis We obtain new sufficient criteria for the oscillation of all solutions of linear delay difference equations with several (variable) finite delays. Our results relax numerous well-known limes inferior-type oscillation criteria from the literature by letting the limes inferior be replaced by the limes superior under some additional assumptions related to slow variation. On the other hand, our findings generalize an oscillation criterion recently given for the case of a constant, single delay. oscillation, difference equations, several delays, non-monotone argument, slowly varying function 2021 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol41iss5art2 https://www.opuscula.agh.edu.pl/vol41/5/art/opuscula_math_4130.pdf Extreme points of the Besicovitch-Orlicz space of almost periodic functions equipped with Orlicz norm Fatiha Boulahia, Slimane Hassaine In the present paper, we give criteria for the existence of extreme points of the Besicovitch-Orlicz space of almost periodic functions equipped with Orlicz norm. Some properties of the set of attainable points of the Amemiya norm in this space are also discussed. extreme points, strict convexity, almost periodic functions, Besicovitch-Orlicz spaces of almost periodic functions 2021 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol41iss5art3 https://www.opuscula.agh.edu.pl/vol41/5/art/opuscula_math_4131.pdf Closed range weighted composition operators between L^{p}-spaces Ching-on Lo, Anthony Wai-keung Loh We characterize the closedness of ranges of weighted composition operators between \(L^p\)-spaces, where \(1 \leq p \leq \infty\). When the \(L^p\)-spaces are weighted sequence spaces, several corollaries about this class of operators are also deduced. weighted composition operator, Lebesgue space, closed range 2021 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol41iss5art4 https://www.opuscula.agh.edu.pl/vol41/5/art/opuscula_math_4132.pdf Coboundaries of commuting Borel automorphisms Shrey Sanadhya We show that if \(S\), \(T\) are two commuting automorphisms of a standard Borel space such that they generate a free Borel \(\mathbb{Z}^2\)-action then \(S\) and \(T\) do not have same sets of real valued bounded coboundaries. We also prove a weaker form of Rokhlin Lemma for Borel \(\mathbb{Z}^d\)-actions. coboundries, Rokhlin Lemma, Borel \(\mathbb{Z}^d\)-action 2021 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol41iss5art5 https://www.opuscula.agh.edu.pl/vol41/5/art/opuscula_math_4133.pdf On classical symmetries of ordinary differential equations related to stationary integrable partial differential equations Ivan Tsyfra We study the relationship between the solutions of stationary integrable partial and ordinary differential equations and coefficients of the second-order ordinary differential equations invariant with respect to one-parameter Lie group. The classical symmetry method is applied. We prove that if the coefficients of ordinary differential equation satisfy the stationary integrable partial differential equation with two independent variables then the ordinary differential equation is integrable by quadratures. If special solutions of integrable partial differential equations are chosen then the coefficients satisfy the stationary KdV equations. It was shown that the Ermakov equation belong to a class of these equations. In the framework of the approach we obtained the similar results for generalized Riccati equations. By using operator of invariant differentiation we describe a class of higher order ordinary differential equations for which the group-theoretical method enables us to reduce the order of ordinary differential equation. ordinary differential equation, partial differential equation, integrability, symmetry, quadrature, Lie transformation group 2021 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol41iss5art6 https://www.opuscula.agh.edu.pl/vol41/5/art/opuscula_math_4134.pdf Certain properties of continuous fractional wavelet transform on Hardy space and Morrey space Amit K. Verma, Bivek Gupta In this paper we define a new class of continuous fractional wavelet transform (CFrWT) and study its properties in Hardy space and Morrey space. The theory developed generalize and complement some of already existing results. fractional Fourier transform, continuous fractional wavelet transform, Hardy space, Morrey space 2021 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol41iss5art7 https://www.opuscula.agh.edu.pl/vol41/5/art/opuscula_math_4135.pdf Nonparametric bootstrap confidence bands for unfolding sphere size distributions Jakub Wojdyła The stereological inverse problem of unfolding the distribution of spheres radii from measured planar sections radii, known as the Wicksell's corpuscle problem, is considered. The construction of uniform confidence bands based on the smoothed bootstrap in the Wicksell's problem is presented. Theoretical results on the consistency of the proposed bootstrap procedure are given, where the consistency of the bands means that the coverage probability converges to the nominal level. The finite-sample performance of the proposed method is studied via Monte Carlo simulations and compared with the asymptotic (non-bootstrap) solution described in literature. bootstrap, confidence bands, inverse problem, nonparametric density estimation, Wicksell's problem 2021 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol41iss6 https://www.opuscula.agh.edu.pl/om-vol41iss6art1 https://www.opuscula.agh.edu.pl/vol41/6/art/opuscula_math_4136.pdf Generalized powers and measures Zbigniew Burdak, Marek Kosiek, Patryk Pagacz, Krzysztof Rudol, Marek Słociński Using the winding of measures on torus in "rational directions" special classes of unitary operators and pairs of isometries are defined. This provides nontrivial examples of generalized powers. Operators related to winding Szegö-singular measures are shown to have specific properties of their invariant subspaces. representing measures, Szegö-singular measure, compatible pair of isometries, spectral measure 2021 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol41iss6art2 https://www.opuscula.agh.edu.pl/vol41/6/art/opuscula_math_4137.pdf Spectral properties of certain operators on the free Hilbert space \mathfrak{F}[H_{1},...,H_{N}] and the semicircular law Ilwoo Cho In this paper, we fix \(N\)-many \(l^2\)-Hilbert spaces \(H_k\) whose dimensions are \(n_{k} \in \mathbb{N}^{\infty}=\mathbb{N} \cup \{\infty\}\), for \(k=1,\ldots,N\), for \(N \in \mathbb{N}\setminus\{1\}\). And then, construct a Hilbert space \(\mathfrak{F}=\mathfrak{F}[H_{1},\ldots,H_{N}]\) induced by \(H_{1},\ldots,H_{N}\), and study certain types of operators on \(\mathfrak{F}\). In particular, we are interested in so-called jump-shift operators. The main results (i) characterize the spectral properties of these operators, and (ii) show how such operators affect the semicircular law induced by \(\bigcup^N_{k=1} \mathcal{B}_{k}\), where \(\mathcal{B}_{k}\) are the orthonormal bases of \(H_{k}\), for \(k=1,\ldots,N\). separable Hilbert spaces, free Hilbert spaces, jump operators, shift operators, jump-shift operators, semicircular elements 2021 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol41iss6art3 https://www.opuscula.agh.edu.pl/vol41/6/art/opuscula_math_4138.pdf The Krein-von Neumann extension of a regular even order quasi-differential operator Minsung Cho, Seth Hoisington, Roger Nichols, Brian Udall We characterize by boundary conditions the Krein-von Neumann extension of a strictly positive minimal operator corresponding to a regular even order quasi-differential expression of Shin-Zettl type. The characterization is stated in terms of a specially chosen basis for the kernel of the maximal operator and employs a description of the Friedrichs extension due to Möller and Zettl. Krein-von Neumann extension, regular quasi-differential operator 2021 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol41iss6art4 https://www.opuscula.agh.edu.pl/vol41/6/art/opuscula_math_4139.pdf Corona theorem for strictly pseudoconvex domains Sebastian Gwizdek Nearly 60 years have passed since Lennart Carleson gave his proof of Corona Theorem for unit disc in the complex plane. It was only recently that M. Kosiek and K. Rudol obtained the first positive result for Corona Theorem in multidimensional case. Using duality methods for uniform algebras the authors proved "abstract" Corona Theorem which allowed to solve Corona Problem for a wide class of regular domains. In this paper we expand Corona Theorem to strictly pseudoconvex domains with smooth boundaries. Corona theorem, Banach algebra, uniform algebra, Arens product, Gleason part, band of measures, representing measure 2021 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol41iss6art5 https://www.opuscula.agh.edu.pl/vol41/6/art/opuscula_math_4140.pdf Spontaneous decay of level from spectral theory point of view Eduard Ianovich In quantum field theory it is believed that the spontaneous decay of excited atomic or molecular level is due to the interaction with continuum of field modes. Besides, the atom makes a transition from upper level to lower one so that the probability to find the atom in the excited state tends to zero. In this paper it will be shown that the mathematical model in single-photon approximation may predict another behavior of this probability generally. Namely, the probability to find the atom in the excited state may tend to a nonzero constant so that the atom is not in the pure state finally. This effect is due to that the spectrum of the complete Hamiltonian is not purely absolutely continuous and has a discrete level outside the continuous part. Namely, we state that in the corresponding invariant subspace, determining the time evolution, the spectrum of the complete Hamiltonian when the field is considered in three dimensions may be not purely absolutely continuous and may have an eigenvalue. The appearance of eigenvalue has a threshold character. If the field is considered in two dimensions the spectrum always has an eigenvalue and the decay is absent. spectral theory, quantum field theory, self-adjoint operators, absolutely continuous spectrum, spontaneous decay 2021 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol41iss6art6 https://www.opuscula.agh.edu.pl/vol41/6/art/opuscula_math_4141.pdf Discrete spectra for some complex infinite band matrices Maria Malejki Under suitable assumptions the eigenvalues for an unbounded discrete operator \(A\) in \(l_2\), given by an infinite complex band-type matrix, are approximated by the eigenvalues of its orthogonal truncations. Let \[\Lambda (A)=\{\lambda \in {\rm Lim}_{n\to \infty} \lambda _n : \lambda _n \text{ is an eigenvalue of } A_n \text{ for } n \geq 1 \},\] where \({\rm Lim}_{n\to \infty} \lambda_n\) is the set of all limit points of the sequence \((\lambda_n)\) and \(A_n\) is a finite dimensional orthogonal truncation of \(A\). The aim of this article is to provide the conditions that are sufficient for the relations \(\sigma(A) \subset \Lambda(A)\) or \(\Lambda (A) \subset \sigma (A)\) to be satisfied for the band operator \(A\). unbounded operator, band-type matrix, complex tridiagonal matrix, discrete spectrum, eigenvalue, limit points of eigenvalues 2021 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol41iss6art7 https://www.opuscula.agh.edu.pl/vol41/6/art/opuscula_math_4142.pdf μ-Hankel operators on Hilbert spaces Adolf Mirotin, Ekaterina Kuzmenkova A class of operators is introduced (\(\mu\)-Hankel operators, \(\mu\) is a complex parameter), which generalizes the class of Hankel operators. Criteria for boundedness, compactness, nuclearity, and finite dimensionality are obtained for operators of this class, and for the case \(|\mu| = 1\) their description in the Hardy space is given. Integral representations of \(\mu\)-Hankel operators on the unit disk and on the Semi-Axis are also considered. Hankel operator, \(\mu\)-Hankel operator, Hardy space, integral representation, nuclear operator, integral operator 2021 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol42iss1 https://www.opuscula.agh.edu.pl/om-vol42iss1art1 https://www.opuscula.agh.edu.pl/vol42/1/art/opuscula_math_4201.pdf Uniqueness of solution of a nonlinear evolution dam problem in a heterogeneous porous medium Messaouda Ben Attia, Elmehdi Zaouche, Mahmoud Bousselsal By choosing convenient test functions and using the method of doubling variables, we prove the uniqueness of the solution to a nonlinear evolution dam problem in an arbitrary heterogeneous porous medium of \(\mathbb{R}^n\) (\(n\in \{2,3\}\)) with an impermeable horizontal bottom. test function, method of doubling variables, nonlinear evolution dam problem, heterogeneous porous medium, uniqueness 2022 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol42iss1art2 https://www.opuscula.agh.edu.pl/vol42/1/art/opuscula_math_4202.pdf γ-paired dominating graphs of cycles Pannawat Eakawinrujee, Nantapath Trakultraipruk A paired dominating set of a graph \(G\) is a dominating set whose induced subgraph contains a perfect matching. The paired domination number, denoted by \(\gamma_{pr}(G)\), is the minimum cardinality of a paired dominating set of \(G\). A \(\gamma_{pr}(G)\)-set is a paired dominating set of cardinality \(\gamma_{pr}(G)\). The \(\gamma\)-paired dominating graph of \(G\), denoted by \(PD_{\gamma}(G)\), as the graph whose vertices are \(\gamma_{pr}(G)\)-sets. Two \(\gamma_{pr}(G)\)-sets \(D_1\) and \(D_2\) are adjacent in \(PD_{\gamma}(G)\) if there exists a vertex \(u\in D_1\) and a vertex \(v\notin D_1\) such that \(D_2=(D_1\setminus \{u\})\cup \{v\}\). In this paper, we present the \(\gamma\)-paired dominating graphs of cycles. paired dominating graph, paired dominating set, paired domination number 2022 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol42iss1art3 https://www.opuscula.agh.edu.pl/vol42/1/art/opuscula_math_4203.pdf Kneser-type oscillation criteria for second-order half-linear advanced difference equations N. Indrajith, John R. Graef, E. Thandapani The authors present Kneser-type oscillation criteria for a class of advanced type second-order difference equations. The results obtained are new and they improve and complement known results in the literature. Two examples are provided to illustrate the importance of the main results. second-order difference equations, advanced argument, half-linear, oscillation 2022 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol42iss1art4 https://www.opuscula.agh.edu.pl/vol42/1/art/opuscula_math_4204.pdf Edge homogeneous colorings Tomáš Madaras, Alfréd Onderko, Thomas Schweser We explore four kinds of edge colorings defined by the requirement of equal number of colors appearing, in particular ways, around each vertex or each edge. We obtain the characterization of graphs colorable in such a way that the ends of each edge see (not regarding the edge color itself) \(q\) colors (resp. one end sees \(q\) colors and the color sets for both ends are the same), and a sufficient condition for 2-coloring a graph in a way that the ends of each edge see (with the omission of that edge color) altogether \(q\) colors. The relations of these colorings to \(M_q\)-colorings and role colorings are also discussed; we prove an interpolation theorem for the numbers of colors in edge coloring where all edges around each vertex have \(q\) colors. homogeneous coloring, \(M_q\)-coloring, line graph, role coloring 2022 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol42iss1art5 https://www.opuscula.agh.edu.pl/vol42/1/art/opuscula_math_4205.pdf Solution of the boundary value problem of heat conduction in a cone Murat Ramazanov, Muvasharkhan Jenaliyev, Nurtay Gulmanov In the paper we consider the boundary value problem of heat conduction in a non-cylindrical domain, which is an inverted cone, i.e. in the domain degenerating into a point at the initial moment of time. In this case, the boundary conditions contain a derivative with respect to the time variable; in practice, problems of this kind arise in the presence of the condition of the concentrated heat capacity. We prove a theorem on the solvability of a boundary value problem in weighted spaces of essentially bounded functions. The issues of solvability of the singular Volterra integral equation of the second kind, to which the original problem is reduced, are studied. We use the Carleman-Vekua method of equivalent regularization to solve the obtained singular Volterra integral equation. noncylindrical domain, cone, boundary value problem of heat conduction, singular Volterra integral equation, Carleman-Vekua regularization method 2022 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol42iss1art6 https://www.opuscula.agh.edu.pl/vol42/1/art/opuscula_math_4206.pdf All metric bases and fault-tolerant metric dimension for square of grid Laxman Saha, Mithun Basak, Kalishankar Tiwary For a simple connected graph \(G=(V,E)\) and an ordered subset \(W = \{w_1,w_2,\ldots, w_k\}\) of \(V\), the code of a vertex \(v\in V\), denoted by \(\mathrm{code}(v)\), with respect to \(W\) is a \(k\)-tuple \((d(v,w_1),\ldots, d(v, w_k))\), where \(d(v, w_t)\) represents the distance between \(v\) and \(w_t\). The set \(W\) is called a resolving set of \(G\) if \(\mathrm{code}(u)\neq \mathrm{code}(v)\) for every pair of distinct vertices \(u\) and \(v\). A metric basis of \(G\) is a resolving set with the minimum cardinality. The metric dimension of \(G\) is the cardinality of a metric basis and is denoted by \(\beta(G)\). A set \(F\subset V\) is called fault-tolerant resolving set of \(G\) if \(F\setminus{\{v\}}\) is a resolving set of \(G\) for every \(v\in F\). The fault-tolerant metric dimension of \(G\) is the cardinality of a minimal fault-tolerant resolving set. In this article, a complete characterization of metric bases for \(G_{mn}^2\) has been given. In addition, we prove that the fault-tolerant metric dimension of \(G_{mn}^2\) is 4 if \(m+n\) is even. We also show that the fault-tolerant metric dimension of \(G_{mn}^2\) is at least 5 and at most 6 when \(m+n\) is odd. code, resolving set, metric dimension, fault-tolerant resolving set, fault-tolerant metric dimension 2022 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol42iss2 https://www.opuscula.agh.edu.pl/om-vol42iss2art1 https://www.opuscula.agh.edu.pl/vol42/2/art/opuscula_math_4207.pdf Global existence and blow-up phenomenon for a quasilinear viscoelastic equation with strong damping and source terms Huafei Di, Zefang Song Considered herein is the global existence and non-global existence of the initial-boundary value problem for a quasilinear viscoelastic equation with strong damping and source terms. Firstly, we introduce a family of potential wells and give the invariance of some sets, which are essential to derive the main results. Secondly, we establish the existence of global weak solutions under the low initial energy and critical initial energy by the combination of the Galerkin approximation and improved potential well method involving with \(t\). Thirdly, we obtain the finite time blow-up result for certain solutions with the non-positive initial energy and positive initial energy, and then give the upper bound for the blow-up time \(T^\ast\). Especially, the threshold result between global existence and non-global existence is given under some certain conditions. Finally, a lower bound for the life span \(T^\ast\) is derived by the means of integro-differential inequality techniques. viscoelastic equation, strong damping and source, blow-up, upper and lower bounds, invariant set, potential well 2022 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol42iss2art2 https://www.opuscula.agh.edu.pl/vol42/2/art/opuscula_math_4208.pdf Ground states for fractional nonlocal equations with logarithmic nonlinearity Lifeng Guo, Yan Sun, Guannan Shi In this paper, we study on the fractional nonlocal equation with the logarithmic nonlinearity formed by \[\begin{cases}\mathcal{L}_{K}u(x)+u\log|u|+|u|^{q-2}u=0, & x\in\Omega,\\ u=0, & x\in\mathbb{R}^{n}\setminus\Omega,\end{cases}\] where \(2\lt q\lt 2^{*}_s\), \(L_{K}\) is a non-local operator, \(\Omega\) is an open bounded set of \(\mathbb{R}^{n}\) with Lipschitz boundary. By using the fractional logarithmic Sobolev inequality and the linking theorem, we present the existence theorem of the ground state solutions for this nonlocal problem. linking theorem, ground state, logarithmic nonlinearity, variational methods 2022 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol42iss2art3 https://www.opuscula.agh.edu.pl/vol42/2/art/opuscula_math_4209.pdf The d-bar formalism for the modified Veselov-Novikov equation on the half-plane Guenbo Hwang, Byungsoo Moon We study the modified Veselov-Novikov equation (mVN) posed on the half-plane via the Fokas method, considered as an extension of the inverse scattering transform for boundary value problems. The mVN equation is one of the most natural \((2+1)\)-dimensional generalization of the \((1+1)\)-dimensional modified Korteweg-de Vries equation in the sense as to how the Novikov-Veselov equation is related to the Korteweg-de Vries equation. In this paper, by means of the Fokas method, we present the so-called global relation for the mVN equation, which is an algebraic equation coupled with the spectral functions, and the \(d\)-bar formalism, also known as Pompieu's formula. In addition, we characterize the \(d\)-bar derivatives and the relevant jumps across certain domains of the complex plane in terms of the spectral functions. initial-boundary value problem, integrable nonlinear PDE, spectral analysis, \(d\)-bar 2022 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol42iss2art4 https://www.opuscula.agh.edu.pl/vol42/2/art/opuscula_math_4210.pdf Blowup phenomena for some fourth-order strain wave equations at arbitrary positive initial energy level Qiang Lin, Yongbing Luo In this paper, we study a series of fourth-order strain wave equations involving dissipative structure, which appears in elasto-plastic-microstructure models. By some differential inequalities, we derive the finite time blow up results and the estimates of the upper bound blowup time with arbitrary positive initial energy. We also discuss the influence mechanism of the linear weak damping and strong damping on blowup time, respectively. fourth-order strain wave equation, arbitrary positive initial energy, blowup, blowup time 2022 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol42iss2art5 https://www.opuscula.agh.edu.pl/vol42/2/art/opuscula_math_4211.pdf Exponential decay of solutions to a class of fourth-order nonlinear hyperbolic equations modeling the oscillations of suspension bridges Yang Liu, Chao Yang This paper is concerned with a class of fourth-order nonlinear hyperbolic equations subject to free boundary conditions that can be used to describe the nonlinear dynamics of suspension bridges. fourth-order nonlinear hyperbolic equations, weak solutions, exponential decay, a family of potential wells 2022 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol42iss2art6 https://www.opuscula.agh.edu.pl/vol42/2/art/opuscula_math_4212.pdf Double phase problems: a survey of some recent results Nikolaos S. Papageorgiou We review some recent results on double phase problems. We focus on the relevant function space framework, which is provided by the generalized Orlicz spaces. We also describe the basic tools and methods used to deal with double phase problems, given that there is no global regularity theory for these problems. double phase integrand, generalized Orlicz spaces, regularity theory, maximum principle, Nehari manifold 2022 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol42iss2art7 https://www.opuscula.agh.edu.pl/vol42/2/art/opuscula_math_4213.pdf Entire solutions for some critical equations in the Heisenberg group Patrizia Pucci, Letizia Temperini We complete the study started in the paper [P. Pucci, L.Temperini, On the concentration-compactness principle for Folland-Stein spaces and for fractional horizontal Sobolev spaces, Math. Eng. 5 (2023), Paper no. 007], giving some applications of its abstract results to get existence of solutions of certain critical equations in the entire Heinseberg group. In particular, different conditions for existence are given for critical horizontal \(p\)-Laplacian equations. Heisenberg group, entire solutions, critical exponents 2022 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol42iss2art8 https://www.opuscula.agh.edu.pl/vol42/2/art/opuscula_math_4214.pdf On some inverse problem for bi-parabolic equation with observed data in L^{p} spaces Nguyen Huy Tuan The bi-parabolic equation has many practical significance in the field of heat transfer. The objective of the paper is to provide a regularized problem for bi-parabolic equation when the observed data are obtained in \(L^p\). We are interested in looking at three types of inverse problems. Regularization results in the \(L^2\) space appears in many related papers, but the survey results are rare in \(L^p\), \(p \neq 2\). The first problem related to the inverse source problem when the source function has split form. For this problem, we introduce the error between the Fourier regularized solution and the exact solution in \(L^p\) spaces. For the inverse initial problem for both linear and nonlinear cases, we applied the Fourier series truncation method. Under the terminal input data observed in \(L^p\), we obtain the approximated solution also in the space \(L^p\). Under some reasonable smoothness assumptions of the exact solution, the error between the the regularized solution and the exact solution are derived in the space \(L^p\). This paper seems to generalize to previous results for bi-parabolic equation on this direction. bi-parabolic equations, Fourier truncation method, inverse source parabolic, inverse initial problem, Sobolev embeddings, Sobolev embeddings 2022 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol42iss2art9 https://www.opuscula.agh.edu.pl/vol42/2/art/opuscula_math_4215.pdf Ground states of coupled critical Choquard equations with weighted potentials Gaili Zhu, Chunping Duan, Jianjun Zhang, Huixing Zhang In this paper, we are concerned with the following coupled Choquard type system with weighted potentials \[\begin{cases} -\Delta u+V_{1}(x)u=\mu_{1}(I_{\alpha}\!\ast\![Q(x)|u|^{\frac{N+\alpha}{N}}])Q(x)|u|^{\frac{\alpha}{N}-1}u+\beta(I_{\alpha}\!\ast\![Q(x)|v|^{\frac{N+\alpha}{N}}])Q(x)|u|^{\frac{\alpha}{N}-1}u,\\ -\Delta v+V_{2}(x)v=\mu_{2}(I_{\alpha}\!\ast\![Q(x)|v|^{\frac{N+\alpha}{N}}])Q(x)|v|^{\frac{\alpha}{N}-1}v+\beta(I_{\alpha}\!\ast\![Q(x)|u|^{\frac{N+\alpha}{N}}])Q(x)|v|^{\frac{\alpha}{N}-1}v,\\ u,v\in H^{1}(\mathbb{R}^{N}),\end{cases}\] where \(N\geq3\), \(\mu_{1},\mu_{2},\beta\gt 0\) and \(V_{1}(x)\), \(V_{2}(x)\) are nonnegative functions. Via the variational approach, one positive ground state solution of this system is obtained under some certain assumptions on \(V_{1}(x)\), \(V_{2}(x)\) and \(Q(x)\). Moreover, by using Hardy's inequality and one Pohozǎev identity, a non-existence result of non-trivial solutions is also considered. ground states, Choquard equations, Hardy-Littlewood-Sobolev inequality, lower critical exponent 2022 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol42iss3 https://www.opuscula.agh.edu.pl/om-vol42iss3art1 https://www.opuscula.agh.edu.pl/vol42/3/art/opuscula_math_4216.pdf Monodromy invariant Hermitian forms for second order Fuchsian differential equations with four singularities Shunya Adachi We study the monodromy invariant Hermitian forms for second order Fuchsian differential equations with four singularities. The moduli space of our monodromy representations can be realized by certain affine cubic surface. In this paper we characterize the irreducible monodromies having the non-degenerate invariant Hermitian forms in terms of that cubic surface. The explicit forms of invariant Hermitian forms are also given. Our result may bring a new insight into the study of the Painlevé differential equations. Fuchsian differential equations, monodromy representation, monodromy invariant Hermitian form 2022 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol42iss3art2 https://www.opuscula.agh.edu.pl/vol42/3/art/opuscula_math_4217.pdf New aspects for the oscillation of first-order difference equations with deviating arguments Emad R. Attia, Bassant M. El-Matary We study the oscillation of first-order linear difference equations with non-monotone deviating arguments. Iterative oscillation criteria are obtained which essentially improve, extend, and simplify some known conditions. These results will be applied to some numerical examples. difference equations, oscillation, non-monotone advanced arguments 2022 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol42iss3art3 https://www.opuscula.agh.edu.pl/vol42/3/art/opuscula_math_4218.pdf Growth of solutions of a class of linear fractional differential equations with polynomial coefficients Saada Hamouda, Sofiane Mahmoudi This paper is devoted to the study of the growth of solutions of certain class of linear fractional differential equations with polynomial coefficients involving the Caputo fractional derivatives by using the generalized Wiman-Valiron theorem in the fractional calculus. linear fractional differential equations, growth of solutions, Caputo fractional derivative operator 2022 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol42iss3art4 https://www.opuscula.agh.edu.pl/vol42/3/art/opuscula_math_4219.pdf On Ambarzumian type theorems for tree domains Vyacheslav Pivovarchik It is known that the spectrum of the spectral Sturm-Liouville problem on an equilateral tree with (generalized) Neumann's conditions at all vertices uniquely determines the potentials on the edges in the unperturbed case, i.e. case of the zero potentials on the edges (Ambarzumian's theorem). This case is exceptional, and in general case (when the Dirichlet conditions are imposed at some of the pendant vertices) even two spectra of spectral problems do not determine uniquely the potentials on the edges. We consider the spectral Sturm-Liouville problem on an equilateral tree rooted at its pendant vertex with (generalized) Neumann conditions at all vertices except of the root and the Dirichlet condition at the root. In this case Ambarzumian's theorem can't be applied. We show that if the spectrum of this problem is unperturbed, the spectrum of the Neumann-Dirichlet problem on the root edge is also unperturbed and the spectra of the problems on the complimentary subtrees with (generalized) Neumann conditions at all vertices except the subtrees' roots and the Dirichlet condition at the subtrees' roots are unperturbed then the potential on each edge of the tree is 0 almost everywhere. Sturm-Liouville equation, eigenvalue, equilateral tree, star graph, Dirichlet boundary condition, Neumann boundary condition 2022 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol42iss3art5 https://www.opuscula.agh.edu.pl/vol42/3/art/opuscula_math_4220.pdf Distance irregularity strength of graphs with pendant vertices Faisal Susanto, Kristiana Wijaya, Slamin, Andrea Semaničová-Feňovčíková A vertex \(k\)-labeling \(\phi:V(G)\rightarrow\{1,2,\dots,k\}\) on a simple graph \(G\) is said to be a distance irregular vertex \(k\)-labeling of \(G\) if the weights of all vertices of \(G\) are pairwise distinct, where the weight of a vertex is the sum of labels of all vertices adjacent to that vertex in \(G\). The least integer \(k\) for which \(G\) has a distance irregular vertex \(k\)-labeling is called the distance irregularity strength of \(G\) and denoted by \(\mathrm{dis}(G)\). In this paper, we introduce a new lower bound of distance irregularity strength of graphs and provide its sharpness for some graphs with pendant vertices. Moreover, some properties on distance irregularity strength for trees are also discussed in this paper. vertex \(k\)-labeling, distance irregular vertex \(k\)-labeling, distance irregularity strength, pendant vertices 2022 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol42iss3art6 https://www.opuscula.agh.edu.pl/vol42/3/art/opuscula_math_4221.pdf Spectral resolutions for non-self-adjoint block convolution operators Ewelina Zalot The paper concerns the spectral theory for a class of non-self-adjoint block convolution operators. We mainly discuss the spectral representations of such operators. It is considered the general case of operators defined on Banach spaces. The main results are applied to periodic Jacobi matrices. spectral operators, chains, triangular decomposition, Laurent operators, Jacobi matrices 2022 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol42iss3art7 https://www.opuscula.agh.edu.pl/vol42/3/art/opuscula_math_4222.pdf Existence of positive continuous weak solutions for some semilinear elliptic eigenvalue problems Noureddine Zeddini, Rehab Saeed Sari Let \(D\) be a bounded \(C^{1,1}\)-domain in \(\mathbb{R}^d\), \(d\geq 2\). The aim of this article is twofold. The first goal is to give a new characterization of the Kato class of functions \(K(D)\) that was defined by N. Zeddini for \(d=2\) and by H. Mâagli and M. Zribi for \(d\geq 3\) and adapted to study some nonlinear elliptic problems in \(D\). The second goal is to prove the existence of positive continuous weak solutions, having the global behavior of the associated homogeneous problem, for sufficiently small values of the nonnegative constants \(\lambda\) and \(\mu\) to the following system \(\Delta u=\lambda f(x,u,v)\), \(\Delta v=\mu g(x,u,v)\) in \(D\), \(u=\phi_1\) and \(v=\phi_2\) on \(\partial D\), where \(\phi_1\) and \(\phi_2\) are nontrivial nonnegative continuous functions on \(\partial D\). The functions \(f\) and \(g\) are nonnegative and belong to a class of functions containing in particular all functions of the type \(f(x,u,v)=p(x) u^{\alpha}h_1(v)\) and \(g(x,u,v)=q(x)h_2(u)v^{\beta}\) with \(\alpha\geq 1\), \(\beta \geq 1\), \(h_1\), \(h_2\) are continuous on \([0,\infty)\) and \(p\), \(q\) are nonnegative functions in \(K(D)\). Green function, Kato class, nonlinear elliptic systems, positive solution, maximum principle, Schauder fixed point theorem 2022 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol42iss4 https://www.opuscula.agh.edu.pl/om-vol42iss4art1 https://www.opuscula.agh.edu.pl/vol42/4/art/opuscula_math_4223.pdf The strong 3-rainbow index of some certain graphs and its amalgamation Zata Yumni Awanis, A.N.M. Salman We introduce a strong \(k\)-rainbow index of graphs as modification of well-known \(k\)-rainbow index of graphs. A tree in an edge-colored connected graph \(G\), where adjacent edge may be colored the same, is a rainbow tree if all of its edges have distinct colors. Let \(k\) be an integer with \(2\leq k\leq n\). The strong \(k\)-rainbow index of \(G\), denoted by \(srx_k(G)\), is the minimum number of colors needed in an edge-coloring of \(G\) so that every \(k\) vertices of \(G\) is connected by a rainbow tree with minimum size. We focus on \(k=3\). We determine the strong \(3\)-rainbow index of some certain graphs. We also provide a sharp upper bound for the strong \(3\)-rainbow index of amalgamation of graphs. Additionally, we determine the exact values of the strong \(3\)-rainbow index of amalgamation of some graphs. amalgamation, rainbow coloring, rainbow Steiner tree, strong \(k\)-rainbow index 2022 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol42iss4art2 https://www.opuscula.agh.edu.pl/vol42/4/art/opuscula_math_4224.pdf Oscillation of even order linear functional differential equations with mixed deviating arguments Blanka Baculikova In the paper, we study oscillation and asymptotic properties for even order linear functional differential equations \[y^{(n)}(t)=p(t)y(\tau(t))\] with mixed deviating arguments, i.e. when both delayed and advanced parts of \(\tau(t)\) are significant. The presented results essentially improve existing ones. higher order differential equations, mixed argument, monotonic properties, oscillation 2022 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol42iss4art3 https://www.opuscula.agh.edu.pl/vol42/4/art/opuscula_math_4225.pdf Upper bounds on distance vertex irregularity strength of some families of graphs Sylwia Cichacz, Agnieszka Görlich, Andrea Semaničová-Feňovčíková For a graph \(G\) its distance vertex irregularity strength is the smallest integer \(k\) for which one can find a labeling \(f: V(G)\to \{1, 2, \dots, k\}\) such that \[ \sum_{x\in N(v)}f(x)\neq \sum_{x\in N(u)}f(x)\] for all vertices \(u,v\) of \(G\), where \(N(v)\) is the open neighborhood of \(v\). In this paper we present some upper bounds on distance vertex irregularity strength of general graphs. Moreover, we give upper bounds on distance vertex irregularity strength of hypercubes and trees. distance vertex irregularity strength of a graph, hypercube, tree 2022 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol42iss4art4 https://www.opuscula.agh.edu.pl/vol42/4/art/opuscula_math_4226.pdf Nordhaus-Gaddum bounds for upper total domination Teresa W. Haynes, Michael A. Henning A set \(S\) of vertices in an isolate-free graph \(G\) is a total dominating set if every vertex in \(G\) is adjacent to a vertex in \(S\). A total dominating set of \(G\) is minimal if it contains no total dominating set of \(G\) as a proper subset. The upper total domination number \(\Gamma_t(G)\) of \(G\) is the maximum cardinality of a minimal total dominating set in \(G\). We establish Nordhaus-Gaddum bounds involving the upper total domination numbers of a graph \(G\) and its complement \(\overline{G}\). We prove that if \(G\) is a graph of order \(n\) such that both \(G\) and \(\overline{G}\) are isolate-free, then \(\Gamma_t(G) + \Gamma_t(\overline{G}) \leq n + 2\) and \(\Gamma_t(G)\Gamma_t(\overline{G}) \leq \frac{1}{4}(n+2)^2\), and these bounds are tight. upper total domination, Nordhaus-Gaddum bounds 2022 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol42iss4art5 https://www.opuscula.agh.edu.pl/vol42/4/art/opuscula_math_4227.pdf Fractional operators and their commutators on generalized Orlicz spaces Arttu Karppinen In this paper we examine boundedness of fractional maximal operator. The main focus is on commutators and maximal commutators on generalized Orlicz spaces (also known as Musielak-Orlicz spaces) for fractional maximal functions and Riesz potentials. We prove their boundedness between generalized Orlicz spaces and give a characterization for functions of bounded mean oscillation. maximal operator, commutator, fractional operator, generalized Orlicz, Musielak-Orlicz 2022 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol42iss4art6 https://www.opuscula.agh.edu.pl/vol42/4/art/opuscula_math_4228.pdf Critical cases in neutral functional differential equations, arising from hydraulic engineering Vladimir Răsvan This paper starts from several applications described by initial/boundary value problems for \(1D\) (time and one space variable) hyperbolic partial differential equations whose basic properties and stability of equilibria are studied throughout the same properties for certain associated neutral functional differential equations. It is a common fact that asymptotic stability for neutral functional differential equations is normally obtained under the assumption of asymptotic stability of the difference operator associated to the aforementioned neutral functional differential equations. However the physically meaningful applications presented in the paper have the associated difference operator(s) in critical cases (their stability is, generally speaking, non-asymptotic). Consequently the stability of the considered application models is either non-asymptotic or fragile (in a sense introduced in the paper). The models represent an overview gathered from various fields, processed here in order to emphasize the associated neutral functional differential equations which, consequently, are a challenge to the usual approaches. In the concluding part there are suggested possible ways to overcome these difficulties. \(1D\) hyperbolic partial differential equations, neutral functional differential equations, difference operator, critical case 2022 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol42iss4art7 https://www.opuscula.agh.edu.pl/vol42/4/art/opuscula_math_4229.pdf The crossing numbers of join products of paths with three graphs of order five Michal Staš, Mária Švecová The main aim of this paper is to give the crossing number of the join product \(G^\ast+P_n\) for the disconnected graph \(G^\ast\) of order five consisting of the complete graph \(K_4\) and one isolated vertex, where \(P_n\) is the path on \(n\) vertices. The proofs are done with the help of a lot of well-known exact values for the crossing numbers of the join products of subgraphs of the graph \(G^\ast\) with the paths. Finally, by adding new edges to the graph \(G^\ast\), we are able to obtain the crossing numbers of the join products of two other graphs with the path \(P_n\). graph, crossing number, join product, cyclic permutation, path 2022 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol42iss5 https://www.opuscula.agh.edu.pl/om-vol42iss5art1 https://www.opuscula.agh.edu.pl/vol42/5/art/opuscula_math_4230.pdf Properties of even order linear functional differential equations with deviating arguments of mixed type Jozef Dzurina This paper is concerned with oscillatory behavior of linear functional differential equations of the type \[y^{(n)}(t)=p(t)y(\tau(t))\] with mixed deviating arguments which means that its both delayed and advanced parts are unbounded subset of \((0,\infty)\). Our attention is oriented to the Euler type of equation, i.e. when \(p(t)\sim a/t^n.\) higher order differential equations, mixed argument, monotonic properties, oscillation 2022 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol42iss5art2 https://www.opuscula.agh.edu.pl/vol42/5/art/opuscula_math_4231.pdf Stability switches in a linear differential equation with two delays Yuki Hata, Hideaki Matsunaga This paper is devoted to the study of the effect of delays on the asymptotic stability of a linear differential equation with two delays \[x'(t)=-ax(t)-bx(t-\tau)-cx(t-2\tau),\quad t\geq 0,\] where \(a\), \(b\), and \(c\) are real numbers and \(\tau\gt 0\). We establish some explicit conditions for the zero solution of the equation to be asymptotically stable. As a corollary, it is shown that the zero solution becomes unstable eventually after undergoing stability switches finite times when \(\tau\) increases only if \(c-a\lt 0\) and \(\sqrt{-8c(c-a)}\lt |b| \lt a+c\). The explicit stability dependence on the changing \(\tau\) is also described. delay differential equations, stability switches, two delays 2022 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol42iss5art3 https://www.opuscula.agh.edu.pl/vol42/5/art/opuscula_math_4232.pdf Nonlinear Choquard equations on hyperbolic space Haiyang He In this paper, our purpose is to prove the existence results for the following nonlinear Choquard equation \[-\Delta_{\mathbb{B}^{N}}u=\int_{\mathbb{B}^N}\dfrac{|u(y)|^{p}}{|2\sinh\frac{\rho(T_y(x))}{2}|^\mu} dV_y \cdot |u|^{p-2}u +\lambda u\] on the hyperbolic space \(\mathbb{B}^N\), where \(\Delta_{\mathbb{B}^{N}}\) denotes the Laplace-Beltrami operator on \(\mathbb{B}^N\), \[\sinh\frac{\rho(T_y(x))}{2}=\dfrac{|T_y(x)|}{\sqrt{1-|T_y(x)|^2}}=\dfrac{|x-y|}{\sqrt{(1-|x|^2)(1-|y|^2)}},\] \(\lambda\) is a real parameter, \(0\lt \mu\lt N\), \(1\lt p\leq 2_\mu^*\), \(N\geq 3\) and \(2_\mu^*:=\frac{2N-\mu}{N-2}\) is the critical exponent in the sense of the Hardy-Littlewood-Sobolev inequality. nonlinear Choquard equation, hyperbolic space, existence solutions, Hardy-Littlewood-Sobolev inequality 2022 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol42iss5art4 https://www.opuscula.agh.edu.pl/vol42/5/art/opuscula_math_4233.pdf On the numerical solution of one inverse problem for a linearized two-dimensional system of Navier-Stokes equations Muvasharkhan Jenaliyev, Murat Ramazanov, Madi Yergaliyev The paper studies the numerical solution of the inverse problem for a linearized two-dimensional system of Navier-Stokes equations in a circular cylinder with a final overdetermination condition. For a biharmonic operator in a circle, a generalized spectral problem has been posed. For the latter, a system of eigenfunctions and eigenvalues is constructed, which is used in the work for the numerical solution of the inverse problem in a circular cylinder with specific numerical data. Graphs illustrating the results of calculations are presented. Navier-Stokes equations, inverse problem, numerical solution 2022 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol42iss5art5 https://www.opuscula.agh.edu.pl/vol42/5/art/opuscula_math_4234.pdf Positive stationary solutions of convection-diffusion equations for superlinear sources Aleksandra Orpel We investigate the existence and multiplicity of positive stationary solutions for acertain class of convection-diffusion equations in exterior domains. This problem leads to the following elliptic equation \[\Delta u(x)+f(x,u(x))+g(x)x\cdot \nabla u(x)=0,\] for \(x\in \Omega_{R}=\{ x \in \mathbb{R}^n, \|x\|\gt R \}\), \(n\gt 2\). The goal of this paper is to show that our problem possesses an uncountable number of nondecreasing sequences of minimal solutions with finite energy in a neighborhood of infinity. We also prove that each of these sequences generates another solution of the problem. The case when \(f(x,\cdot)\) may be negative at the origin, so-called semipositone problem, is also considered. Our results are based on a certain iteration schema in which we apply the sub and supersolution method developed by Noussair and Swanson. The approach allows us to consider superlinear problems with convection terms containing functional coefficient \(g\) without radial symmetry. semipositone problems, positive stationary solutions, minimal solutions with finite energy, sub and supersolutions methods 2022 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol42iss5art6 https://www.opuscula.agh.edu.pl/vol42/5/art/opuscula_math_4235.pdf Notes on aplications of the dual fountain theorem to local and nonlocal elliptic equations with variable exponent Robert Stegliński Using the Dual Fountain Theorem we obtain some existence of infinitely many solutions for local and nonlocal elliptic equations with variable exponent. Our results correct some of the errors that have appeared recently in the literature. dual fountain theorem, \(p(\cdot)\)-Laplacian, fractional \(p(\cdot)\)-Laplacian, infinitely many solutions 2022 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol42iss6 https://www.opuscula.agh.edu.pl/om-vol42iss6art1 https://www.opuscula.agh.edu.pl/vol42/6/art/opuscula_math_4236.pdf New oscillation conditions for first-order linear retarded difference equations with non-monotone arguments Emad R. Attia, Bassant M. El-Matary, George E. Chatzarakis In this paper, we study the oscillatory behavior of the solutions of a first-order difference equation with non-monotone retarded argument and nonnegative coefficients, based on an iterative procedure. We establish some oscillation criteria, involving \(\lim \sup\), which achieve a marked improvement on several known conditions in the literature. Two examples, numerically solved in MAPLE software, are also given to illustrate the applicability and strength of the obtained conditions. oscillation, difference equations, non-monotone argument 2022 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol42iss6art2 https://www.opuscula.agh.edu.pl/vol42/6/art/opuscula_math_4237.pdf Nonnegative solutions for a class of semipositone nonlinear elliptic equations in bounded domains of R^{n} Imed Bachar, Habib Mâagli, Hassan Eltayeb In this paper, we obtain sufficient conditions for the existence of a unique nonnegative continuous solution of semipositone semilinear elliptic problem in bounded domains of \(\mathbb{R}^n\) (\(n\geq 2\)). The global behavior of this solution is also given. nonnegative solution, semipositone, Kato class, fixed point theorem 2022 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol42iss6art3 https://www.opuscula.agh.edu.pl/vol42/6/art/opuscula_math_4238.pdf Strong consistency of the local linear relative regression estimator for censored data Feriel Bouhadjera, Elias Ould Saïd In this paper, we combine the local linear approach to the relative error regression estimation method to build a new estimator of the regression operator when the response variable is subject to random right censoring. We establish the uniform almost sure consistency with rate over a compact set of the proposed estimator. Numerical studies, firstly on simulated data, then on a real data set concerning the death times of kidney transplant patients, were conducted. These practical studies clearly show the superiority of the new estimator compared to competitive estimators. censored data, local linear approach, relative error, regression function, uniform almost sure convergence 2022 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol42iss6art4 https://www.opuscula.agh.edu.pl/vol42/6/art/opuscula_math_4239.pdf Stochastic model of drug concentration level during IV-administration Irada Dzhalladova, Miroslava Růžičková A stochastic model describing the concentration of the drug in the body during its IV-administration is discussed. The paper compares a deterministic model created with certain simplifications with the stochastic model. Fluctuating and irregular patterns of plasma concentrations of some drugs observed during intravenous infusion are explained. An illustrative example is given with certain values of drug infusion rate and drug elimination rate. IV-administration, deterministic model, stochastic differential equation, mean value, delay differential equation 2022 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol42iss6art5 https://www.opuscula.agh.edu.pl/vol42/6/art/opuscula_math_4240.pdf On oscillatory behaviour of third-order half-linear dynamic equations on time scales Said R. Grace, Gokula Nanda Chhatria In this work, we study the oscillation and asymptotic behaviour of third-order nonlinear dynamic equations on time scales. The findings are obtained using an integral criterion as well as a comparison theorem with the oscillatory properties of a first-order dynamic equation. As a consequence, we give conditions which guarantee that all solutions to the aforementioned problem are only oscillatory, different from any other result in the literature. We propose novel oscillation criteria that improve, extend, and simplify existing ones in the literature. The results are associated with a numerical example. We point out that the results are new even for the case \(\mathbb{T}=\mathbb{R}\) or \(\mathbb{T}=\mathbb{Z}\). oscillation, asymptotic behaviour, dynamic equation on time scales, comparison method, Riccati technique 2022 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol42iss6art6 https://www.opuscula.agh.edu.pl/vol42/6/art/opuscula_math_4241.pdf Forced oscillation and asymptotic behavior of solutions of linear differential equations of second order Yutaka Shoukaku The paper deals with the second order nonhomogeneous linear differential equation \[(p(t) y'(t))' + q(t) y(t) = f(t),\] which is oscillatory under the assumption that \(p(t)\) and \(q(t)\) are positive, continuously differentiable and monotone functions on \([0,\infty)\). Throughout this paper we shall use pairs of quadratic forms, which obtained by different methods than Kusano and Yoshida. This form will lead to a property of qualitative behavior, including amplitudes and slopes, of oscillatory solutions of the above equation. In addition, we will discuss the existence of three types (moderately bounded, small, large) of oscillatory solutions, which are based on results due to Kusano and Yoshida. forced oscillation, asymptotic behavior, second order, differential equation 2022 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol43iss1 https://www.opuscula.agh.edu.pl/om-vol43iss1art1 https://www.opuscula.agh.edu.pl/vol43/1/art/opuscula_math_4301.pdf On the blowing up solutions of the 4-d general q-Kuramoto-Sivashinsky equation with exponentially "dominated" nonlinearity and singular weight Sami Baraket, Safia Mahdaoui, Taieb Ouni Let \(\Omega\) be a bounded domain in \(\mathbb{R}^4\) with smooth boundary and let \(x^{1}, x^{2}, \ldots, x^{m}\) be \(m\)-points in \(\Omega\). We are concerned with the problem \[\Delta^{2} u - H(x,u,D^{k}u) = \rho^{4}\prod_{i=1}^{n}|x-p_{i}|^{4\alpha_{i}}f(x)g(u),\] where the principal term is the bi-Laplacian operator, \(H(x,u,D^{k}u)\) is a functional which grows with respect to \(Du\) at most like \(|Du|^{q}\), \(1\leq q\leq 4\), \(f:\Omega\to [0,+\infty[\) is a smooth function satisfying \(f(p_{i}) \gt 0\) for any \(i = 1,\ldots, n\), \(\alpha_{i}\) are positives numbers and \(g :\mathbb R\to [0,+\infty[\) satisfy \(|g(u)|\leq ce^{u}\). In this paper, we give sufficient conditions for existence of a family of positive weak solutions \((u_\rho)_{\rho\gt 0}\) in \(\Omega\) under Navier boundary conditions \(u=\Delta u =0\) on \(\partial\Omega\). The solutions we constructed are singular as the parameters \( ho\) tends to 0, when the set of concentration \(S=\{x^{1},\ldots,x^{m}\}\subset\Omega\) and the set \(\Lambda :=\{p_{1},\ldots, p_{n}\}\subset\Omega\) are not necessarily disjoint. The proof is mainly based on nonlinear domain decomposition method. singular limits, Green's function, nonlinearity, gradient, nonlinear domain decomposition method 2023 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol43iss1art2 https://www.opuscula.agh.edu.pl/vol43/1/art/opuscula_math_4302.pdf Singular elliptic problems with Dirichlet or mixed Dirichlet-Neumann non-homogeneous boundary conditions Tomas Godoy Let \(\Omega\) be a \(C^{2}\) bounded domain in \(\mathbb{R}^{n}\) such that \(\partial\Omega=\Gamma_{1}\cup\Gamma_{2}\), where \(\Gamma_{1}\) and \(\Gamma_{2}\) are disjoint closed subsets of \(\partial\Omega\), and consider the problem\(-\Delta u=g(\cdot,u)\) in \(\Omega\), \(u=\tau\) on \(\Gamma_{1}\), \(\frac{\partial u}{\partial\nu}=\eta\) on \(\Gamma_{2}\), where \(0\leq\tau\in W^{\frac{1}{2},2}(\Gamma_{1})\), \(\eta\in(H_{0,\Gamma_{1}}^{1}(\Omega))^{\prime}\), and \(g:\Omega \times(0,\infty)\rightarrow\mathbb{R}\) is a nonnegative Carathéodory function. Under suitable assumptions on \(g\) and \(\eta\) we prove the existence and uniqueness of a positive weak solution of this problem. Our assumptions allow \(g\) to be singular at \(s=0\) and also at \(x\in S\) for some suitable subsets \(S\subset\overline{\Omega}\). The Dirichlet problem \(-\Delta u=g(\cdot,u)\) in \(\Omega\), \(u=\sigma\) on \(\partial\Omega\) is also studied in the case when \(0\leq\sigma\in W^{\frac{1}{2},2}(\Omega)\). singular elliptic problems, mixed boundary conditions, weak solutions 2023 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol43iss1art3 https://www.opuscula.agh.edu.pl/vol43/1/art/opuscula_math_4303.pdf Existence of positive radial solutions to a p-Laplacian Kirchhoff type problem on the exterior of a ball John R. Graef, Doudja Hebboul, Toufik Moussaoui In this paper the authors study the existence of positive radial solutions to the Kirchhoff type problem involving the \(p\)-Laplacian \[-\Big(a+b\int_{\Omega_e}|\nabla u|^p dx\Big)\Delta_p u=\lambda f\left(|x|,u\right),\ x\in \Omega_e,\quad u=0\ \text{on} \ \partial\Omega_e,\] where \(\lambda \gt 0\) is a parameter, \(\Omega_e = \lbrace x\in\mathbb{R}^N : |x|\gt r_0\rbrace\), \(r_0\gt 0\), \(N \gt p \gt 1\), \(\Delta_p\) is the \(p\)-Laplacian operator, and \(f\in C(\left[ r_0, +\infty\right)\times\left[0,+\infty\right),\mathbb{R})\) is a non-decreasing function with respect to its second variable. By using the Mountain Pass Theorem, they prove the existence of positive radial solutions for small values of \(\lambda\). Kirchhoff problem, \(p\)-Laplacian, positive radial solution, variational methods 2023 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol43iss1art4 https://www.opuscula.agh.edu.pl/vol43/1/art/opuscula_math_4304.pdf Nonoscillation of damped linear differential equations with a proportional derivative controller and its application to Whittaker-Hill-type and Mathieu-type equations Kazuki Ishibashi The proportional derivative (PD) controller of a differential operator is commonly referred to as the conformable derivative. In this paper, we derive a nonoscillation theorem for damped linear differential equations with a differential operator using the conformable derivative of control theory. The proof of the nonoscillation theorem utilizes the Riccati inequality corresponding to the equation considered. The provided nonoscillation theorem gives the nonoscillatory condition for a damped Euler-type differential equation with a PD controller. Moreover, the nonoscillation of the equation with a PD controller that can generalize Whittaker-Hill-type equations is also considered in this paper. The Whittaker-Hill-type equation considered in this study also includes the Mathieu-type equation. As a subtopic of this work, we consider the nonoscillation of Mathieu-type equations with a PD controller while making full use of numerical simulations. nonoscillation, proportional derivative controller, Riccati technique, Mathieu equation, Whittaker-Hill equation 2023 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol43iss1art5 https://www.opuscula.agh.edu.pl/vol43/1/art/opuscula_math_4305.pdf New results on imbalance graphic graphs Sergiy Kozerenko, Andrii Serdiuk An edge imbalance provides a local measure of how irregular a given graph is. In this paper, we study graphs with graphic imbalance sequences. We give a new proof of imbalance graphicness for trees and use the new idea to prove that the same holds for unicyclic graphs. We then show that antiregular graphs are imbalance graphic and consider the join operation on graphs as well as the double graph operation. Our main results are concerning imbalance graphicness of three classes of block graphs: block graphs having all cut vertices in a single block; block graphs in which the subgraph induced by the cut vertices is either a star or a path. In the end, we discuss open questions and conjectures regarding imbalance graphic graphs. edge imbalance, irregularity of a graph, imbalance sequence, graphic sequence 2023 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol43iss1art6 https://www.opuscula.agh.edu.pl/vol43/1/art/opuscula_math_4306.pdf A note on Hausdorff convergence of pseudospectra Marko Lindner, Dennis Schmeckpeper For a bounded linear operator on a Banach space, we study approximation of the spectrum and pseudospectra in the Hausdorff distance. We give sufficient and necessary conditions in terms of pointwise convergence of appropriate spectral quantities. resolvent, spectrum, pseudospectrum, Hausdorff convergence 2023 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol43iss1art7 https://www.opuscula.agh.edu.pl/vol43/1/art/opuscula_math_4307.pdf On incidence coloring of graph fractional powers Mahsa Mozafari-Nia, Moharram N. Iradmusa For any \(n\in \mathbb{N}\), the \(n\)-subdivision of a graph \(G\) is a simple graph \(G^\frac{1}{n}\) which is constructed by replacing each edge of \(G\) with a path of length \(n\). The \(m\)-th power of \(G\) is a graph, denoted by \(G^m\), with the same vertices of \(G\), where two vertices of \(G^m\) are adjacent if and only if their distance in \(G\) is at most \(m\). In [M.N. Iradmusa, On colorings of graph fractional powers, Discrete Math. 310 (2010), no. 10-11, 1551-1556] the \(m\)-th power of the \(n\)-subdivision of \(G\), denoted by \(G^{\frac{m}{n}}\) is introduced as a fractional power of \(G\). The incidence chromatic number of \(G\), denoted by \(\chi_i(G)\), is the minimum integer \(k\) such that \(G\) has an incidence \(k\)-coloring. In this paper, we investigate the incidence chromatic number of some fractional powers of graphs and prove the correctness of the incidence coloring conjecture for some powers of graphs. incidence coloring, incidence chromatic number, subdivision of graph, power of graph 2023 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol43iss2 https://www.opuscula.agh.edu.pl/om-vol43iss2art1 https://www.opuscula.agh.edu.pl/vol43/2/art/opuscula_math_4308.pdf Global attractivity of a higher order nonlinear difference equation with unimodal terms Abdulaziz Almaslokh, Chuanxi Qian In the present paper, we study the asymptotic behavior of the following higher order nonlinear difference equation with unimodal terms \[x(n+1)= ax(n)+ bx(n)g(x(n)) + cx(n-k)g(x(n-k)), \quad n=0, 1, \ldots,\] where \(a\), \(b\) and \(c\) are constants with \(0\lt a\lt 1\), \(0\leq b\lt 1\), \(0\leq c \lt 1\) and \(a+b+c=1\), \(g\in C[[0, \infty), [0, \infty)]\) is decreasing, and \(k\) is a positive integer. We obtain some new sufficient conditions for the global attractivity of positive solutions of the equation. Applications to some population models are also given. higher order difference equation, positive equilibrium, unimodal term, global attractivity, population model 2023 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol43iss2art2 https://www.opuscula.agh.edu.pl/vol43/2/art/opuscula_math_4309.pdf Square-root boundaries for Bessel processes and the hitting times of radial Ornstein-Uhlenbeck processes Yuji Hamana This article deals with the first hitting times of a Bessel process to a square-root boundary. We obtain the explicit form of the distribution function of the hitting time by means of zeros of the confluent hypergeometric function with respect to the first parameter. In deducing the distribution function, the time that a radial Ornstein-Uhlenbeck process reaches a certain point is very useful and plays an important role. We also give its distribution function in the case that the starting point is closer to the origin than the arrival site. Bessel process, confluent hypergeometric function, first hitting time, radial Ornstein-Uhlenbeck process, square-root boundary 2023 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol43iss2art3 https://www.opuscula.agh.edu.pl/vol43/2/art/opuscula_math_4310.pdf Self-coalition graphs Teresa W. Haynes, Jason T. Hedetniemi, Stephen T. Hedetniemi, Alice A. McRae, Raghuveer Mohan A coalition in a graph \(G = (V, E)\) consists of two disjoint sets \(V_1\) and \(V_2\) of vertices, such that neither \(V_1\) nor \(V_2\) is a dominating set, but the union \(V_1 \cup V_2\) is a dominating set of \(G\). A coalition partition in a graph \(G\) of order \(n = |V|\) is a vertex partition \(\pi = \{V_1, V_2, \ldots, V_k\}\) such that every set \(V_i\) either is a dominating set consisting of a single vertex of degree \(n-1\), or is not a dominating set but forms a coalition with another set \(V_j\) which is not a dominating set. Associated with every coalition partition \(\pi\) of a graph \(G\) is a graph called the coalition graph of \(G\) with respect to \(\pi\), denoted \(CG(G,\pi)\), the vertices of which correspond one-to-one with the sets \(V_1, V_2, \ldots, V_k\) of \(\pi\) and two vertices are adjacent in \(CG(G,\pi)\) if and only if their corresponding sets in \(\pi\) form a coalition. The singleton partition \(\pi_1\) of the vertex set of \(G\) is a partition of order \(|V|\), that is, each vertex of \(G\) is in a singleton set of the partition. A graph \(G\) is called a self-coalition graph if \(G\) is isomorphic to its coalition graph \(CG(G,\pi_1)\), where \(\pi_1\) is the singleton partition of \(G\). In this paper, we characterize self-coalition graphs. coalitions in graphs, coalition partitions, coalition graphs, domination 2023 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol43iss2art4 https://www.opuscula.agh.edu.pl/vol43/2/art/opuscula_math_4311.pdf Lower density operators. Φ_{f} versus Φ_{d} Gertruda Ivanova, Elżbieta Wagner-Bojakowska, Władysław Wilczyński Using the new method of the construction of lower density operator introduced in the earlier paper of the first two authors, we study how much the new operator can be different from the classical one. The aim of this paper is to show that if \(f\) is a good adjusted measure-preserving bijection then the lower density operator generated by \(f\) can be really different from the classical density operator. lower density operator, measure-preserving bijection 2023 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol43iss2art5 https://www.opuscula.agh.edu.pl/vol43/2/art/opuscula_math_4312.pdf Asymptotic analysis of the steady advection-diffusion problem in axial domains Fernando A. Morales We present the asymptotic analysis of the steady advection-diffusion equation in a thin tube. The problem is modeled in a mixed-type variational formulation, in order to separate the phenomenon in the axial direction and a transverse one. Such formulation makes visible the natural separation of scales within the problem and permits a successful asymptotic analysis, delivering a limiting form, free from the initial geometric singularity and suitable for approximating the original one. Furthermore, it is shown that the limiting problem can be simplified to a significantly simpler structure. asymptotic analysis, mixed-type variational formulations, advection-diffusion 2023 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol43iss2art6 https://www.opuscula.agh.edu.pl/vol43/2/art/opuscula_math_4313.pdf Existence and asymptotic behavior of nonoscillatory solutions of half-linear ordinary differential equations Manabu Naito We consider the half-linear differential equation \[(|x'|^{\alpha}\mathrm{sgn}\,x')' + q(t)|x|^{\alpha}\mathrm{sgn}\,x = 0, \quad t \geq t_{0},\] under the condition \[\lim_{t\to\infty}t^{\alpha}\int_{t}^{\infty}q(s)ds = \frac{\alpha^{\alpha}}{(\alpha+1)^{\alpha+1}}.\] It is shown that if certain additional conditions are satisfied, then the above equation has a pair of nonoscillatory solutions with specific asymptotic behavior as \(t\to\infty\). asymptotic behavior, nonoscillatory solution, half-linear differential equation 2023 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol43iss2art7 https://www.opuscula.agh.edu.pl/vol43/2/art/opuscula_math_4314.pdf Discrete spectrum of zero order pseudodifferential operators Grigori Rozenblum We study the rate of convergence of eigenvalues to the endpoints of essential spectrum for zero order pseudodifferential operators on a compact manifold. pseudodifferential operators, eigenvalue asymptotics 2023 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol43iss3 https://www.opuscula.agh.edu.pl/om-vol43iss3art1 https://www.opuscula.agh.edu.pl/vol43/3/art/opuscula_math_4315.pdf Operators induced by certain hypercomplex systems Daniel Alpay, Ilwoo Cho In this paper, we consider a family \(\{ \mathbb{H}_{t}\}_{t\in\mathbb{R}}\) of rings of hypercomplex numbers, indexed by the real numbers, which contain both the quaternions and the split-quaternions. We consider natural Hilbert-space representations \(\{(\mathbb{C}^{2},\pi_{t})\}_{t\in\mathbb{R}}\) of the hypercomplex system \(\{ \mathbb{H}_{t}\}_{t\in\mathbb{R}}\), and study the realizations \(\pi_{t}(h)\) of hypercomplex numbers \(h \in \mathbb{H}_{t}\), as \((2\times 2)\)-matrices acting on \(\mathbb{C}^{2}\), for an arbitrarily fixed scale \(t\in\mathbb{R}\). Algebraic, operator-theoretic, spectral-analytic, and free-probabilistic properties of them are considered. scaled hypercomplex ring, scaled hypercomplex monoids, representations, scaled-spectral forms, scaled-spectralization 2023 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol43iss3art2 https://www.opuscula.agh.edu.pl/vol43/3/art/opuscula_math_4316.pdf On efficiency and duality for a class of nonconvex nondifferentiable multiobjective fractional variational control problems Tadeusz Antczak, Manuel Arana-Jimenéz, Savin Treanţă In this paper, we consider the class of nondifferentiable multiobjective fractional variational control problems involving the nondifferentiable terms in the numerators and in the denominators. Under univexity and generalized univexity hypotheses, we prove optimality conditions and various duality results for such nondifferentiable multiobjective fractional variational control problems. The results established in the paper generalize many similar results established earlier in the literature for such nondifferentiable multiobjective fractional variational control problems. nondifferentiable multiobjective fractional variational control problem, efficient solution, optimality conditions, (generalized) univexity, Mond-Weir duality, Wolfe duality 2023 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol43iss3art3 https://www.opuscula.agh.edu.pl/vol43/3/art/opuscula_math_4317.pdf Axiomatic characterizations of Ptolemaic and chordal graphs Manoj Changat, Lekshmi Kamal K. Sheela, Prasanth G. Narasimha-Shenoi The interval function and the induced path function are two well studied class of set functions of a connected graph having interesting properties and applications to convexity, metric graph theory. Both these functions can be framed as special instances of a general set function termed as a transit function defined on the Cartesian product of a non-empty set \(V\) to the power set of \(V\) satisfying the expansive, symmetric and idempotent axioms. In this paper, we propose a set of independent first order betweenness axioms on an arbitrary transit function and provide characterization of the interval function of Ptolemaic graphs and the induced path function of chordal graphs in terms of an arbitrary transit function. This in turn gives new characterizations of the Ptolemaic and chordal graphs. interval function, betweenness axioms, Ptolemaic graphs, transit function, induced path transit function 2023 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol43iss3art4 https://www.opuscula.agh.edu.pl/vol43/3/art/opuscula_math_4318.pdf Existence and asymptotic stability for generalized elasticity equation with variable exponent Mohamed Dilmi, Sadok Otmani In this paper we propose a new mathematical model describing the deformations of an isotropic nonlinear elastic body with variable exponent in dynamic regime. We assume that the stress tensor \(\sigma^{p(\cdot)}\) has the form \[\sigma^{p(\cdot)}(u)=(2\mu +|d(u)|^{p(\cdot)-2})d(u)+\lambda Tr(d(u)) I_{3},\] where \(u\) is the displacement field, \(\mu\), \(\lambda\) are the given coefficients \(d(\cdot)\) and \(I_{3}\) are the deformation tensor and the unit tensor, respectively. By using the Faedo-Galerkin techniques and a compactness result we prove the existence of the weak solutions, then we study the asymptotic behaviour stability of the solutions. asymptotic stability, variable exponent Lebesgue and Sobolev spaces, generalized elasticity equation 2023 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol43iss3art5 https://www.opuscula.agh.edu.pl/vol43/3/art/opuscula_math_4319.pdf On local antimagic total labeling of complete graphs amalgamation Gee-Choon Lau, Wai Chee Shiu Let \(G = (V,E)\) be a connected simple graph of order \(p\) and size \(q\). A graph \(G\) is called local antimagic (total) if \(G\) admits a local antimagic (total) labeling. A bijection \(g : E \to \{1,2,\ldots,q\}\) is called a local antimagic labeling of $ if for any two adjacent vertices \(u\) and \(v\), we have \(g^+(u) \ne g^+(v)\), where \(g^+(u) = \sum_{e\in E(u)} g(e)\), and \(E(u)\) is the set of edges incident to \(u\). Similarly, a bijection \(f:V(G)\cup E(G)\to \{1,2,\ldots,p+q\}\) is called a local antimagic total labeling of \(G\) if for any two adjacent vertices \(u\) and \(v\), we have \(w_f(u)\ne w_f(v)\), where \(w_f(u) = f(u) + \sum_{e\in E(u)} f(e)\). Thus, any local antimagic (total) labeling induces a proper vertex coloring of \(G\) if vertex \(v\) is assigned the color \(g^+(v)\) (respectively, \(w_f(u)\)). The local antimagic (total) chromatic number, denoted \(\chi_{la}(G)\) (respectively \(\chi_{lat}(G)\)), is the minimum number of induced colors taken over local antimagic (total) labeling of \(G\). In this paper, we determined \(\chi_{lat}(G)\) where \(G\) is the amalgamation ofcomplete graphs. Consequently, we also obtained the local antimagic (total) chromatic number of the disjoint union of complete graphs, and the join of \(K_1\) and amalgamation of complete graphs under various conditions. An application of local antimagic total chromatic number is also given. local antimagic (total) chromatic number, amalgamation, complete graphs 2023 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol43iss3art6 https://www.opuscula.agh.edu.pl/vol43/3/art/opuscula_math_4320.pdf New oscillation constraints for even-order delay differential equations Osama Moaaz, Mona Anis, Ahmed A. El-Deeb, Ahmed M. Elshenhab The purpose of this paper is to study the oscillatory properties of solutions to a class of delay differential equations of even order. We focus on criteria that exclude decreasing positive solutions. As in this paper, this type of solution emerges when considering the noncanonical case of even equations. By finding a better estimate of the ratio between the Kneser solution with and without delay, we obtain new constraints that ensure that all solutions to the considered equation oscillate. The new findings improve some previous findings in the literature. delay differential equations, even-order, Kneser solutions, oscillation 2023 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol43iss4 https://www.opuscula.agh.edu.pl/om-vol43iss4art1 https://www.opuscula.agh.edu.pl/vol43/4/art/opuscula_math_4321.pdf The heat equation on time scales Tom Cuchta, Rui A.C. Ferreira We present the use of a Fourier transform on time scales to solve a dynamic heat IVP. This is done by inverting a certain exponential function via contour integral. We include some specific examples and directions for further study. heat equation, time scales, Fourier transform 2023 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol43iss4art2 https://www.opuscula.agh.edu.pl/vol43/4/art/opuscula_math_4322.pdf Generalized derivations and generalized exponential monomials on hypergroups Żywilla Fechner, Eszter Gselmann, László Székelyhidi In one of our former papers "Endomorphisms of the measure algebra of commutative hypergroups" we considered exponential monomials on hypergroups and higher order derivations of the corresponding measure algebra. Continuing with this, we are now looking for the connection between the generalized exponential polynomials of a commutative hypergroup and the higher order derivations of the corresponding measure algebra. moment function, moment sequence, exponential monomial, exponential polynomial, derivation, higher order derivation, hypergroup 2023 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol43iss4art3 https://www.opuscula.agh.edu.pl/vol43/4/art/opuscula_math_4323.pdf Periodic, nonperiodic, and chaotic solutions for a class of difference equations with negative feedback Benjamin B. Kennedy We study the scalar difference equation \[x(k+1) = x(k) + \frac{f(x(k-N))}{N},\] where \(f\) is nonincreasing with negative feedback. This equation is a discretization of the well-studied differential delay equation \[x'(t) = f(x(t-1)).\] We examine explicit families of such equations for which we can find, for infinitely many values of $ and appropriate parameter values, various dynamical behaviors including periodic solutions with large numbers of sign changes per minimal period, solutions that do not converge to periodic solutions, and chaos. We contrast these behaviors with the dynamics of the limiting differential equation. Our primary tool is the analysis of return maps for the difference equations that are conjugate to continuous self-maps of the circle. difference equation, negative feedback, circle map 2023 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol43iss4art4 https://www.opuscula.agh.edu.pl/vol43/4/art/opuscula_math_4324.pdf On the existence of optimal solutions to the Lagrange problem governed by a nonlinear Goursat-Darboux problem of fractional order Marek Majewski In the paper, the Lagrange problem given by a fractional boundary problem with partial derivatives is considered. The main result is the existence of optimal solutions based on the convexity assumption of a certain set. The proof is based on the lower closure theorem and the appropriate implicit measurable function theorem. fractional partial derivative, fractional boundary problem, existence of optimal solutions, Lagrange problem, lower closure theorem 2023 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol43iss4art5 https://www.opuscula.agh.edu.pl/vol43/4/art/opuscula_math_4325.pdf The first eigencurve for a Neumann boundary problem involving p-Laplacian with essentially bounded weights Ahmed Sanhaji, Ahmed Dakkak, Mimoun Moussaoui This article is intended to prove the existence and uniqueness of the first eigencurve, for a homogeneous Neumann problem with singular weights associated with the equation \[-\Delta_{p} u=\alpha m_{1}|u|^{p-2}u+\beta m_{2}|u|^{p-2}u\] in a bounded domain \(\Omega \subset \mathbb{R}^{N}\). We then establish many properties of this eigencurve, particularly the continuity, variational characterization, asymptotic behavior, concavity and the differentiability. \(p\)-Laplacian, first eigencurve, singular weight, Neumann boundary conditions 2023 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol43iss4art6 https://www.opuscula.agh.edu.pl/vol43/4/art/opuscula_math_4326.pdf Bernstein operational matrix of differentiation and collocation approach for a class of three-point singular BVPs: error estimate and convergence analysis Nikhil Sriwastav, Amit K. Barnwal, Abdul-Majid Wazwaz, Mehakpreet Singh Singular boundary value problems (BVPs) have widespread applications in the field of engineering, chemical science, astrophysics and mathematical biology. Finding an approximate solution to a problem with both singularity and non-linearity is highly challenging. The goal of the current study is to establish a numerical approach for dealing with problems involving three-point boundary conditions. The Bernstein polynomials and collocation nodes of a domain are used for developing the proposed numerical approach. The straightforward mathematical formulation and easy to code, makes the proposed numerical method accessible and adaptable for the researchers working in the field of engineering and sciences. The priori error estimate and convergence analysis are carried out to affirm the viability of the proposed method. Various examples are considered and worked out in order to illustrate its applicability and effectiveness. The results demonstrate excellent accuracy and efficiency compared to the other existing methods. Bernstein polynomials, collocation method, three-point singular BVPs, convergence analysis, error estimate 2023 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol43iss4art7 https://www.opuscula.agh.edu.pl/vol43/4/art/opuscula_math_4327.pdf Solutions for a nonhomogeneous p&q-Laplacian problem via variational methods and sub-supersolution technique Leandro S. Tavares, J. Vanterler C. Sousa In this paper it is obtained, through variational methods and sub-supersolution arguments, existence and multiplicity of solutions for a nonhomogeneous problem which arise in several branches of science such as chemical reactions, biophysics and plasma physics. Under a general hypothesis it is proved an existence result and multiple solutions are obtained by considering an additional natural condition. \(p\&q\)-Laplacian operator, nonhomogeneous operator, sub-supersolutions, existence, multiplicity 2023 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol43iss5 https://www.opuscula.agh.edu.pl/om-vol43iss5art1 https://www.opuscula.agh.edu.pl/vol43/5/art/opuscula_math_4328.pdf A viability result for Carathéodory non-convex differential inclusion in Banach spaces Nabil Charradi, Saïd Sajid This paper deals with the existence of solutions to the following differential inclusion: \(\dot{x}(t)\in F(t,x(t))\) a.e. on \([0, T[\) and \(x(t)\in K\), for all \(t \in [0, T]\), where \(F: [0, T]\times K \rightarrow 2^E\) is a Carathéodory multifunction and \(K\) is a closed subset of a separable Banach space \(E\). viability, measurable multifunction, selection, Carathéodory multifunction 2023 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol43iss5art2 https://www.opuscula.agh.edu.pl/vol43/5/art/opuscula_math_4329.pdf Volterra integral operators on a family of Dirichlet-Morrey spaces Lian Hu, Xiaosong Liu A family of Dirichlet-Morrey spaces \(\mathcal{D}_{\lambda,K}\) of functions analytic in the open unit disk \(\mathbb{D}\) are defined in this paper. We completely characterize the boundedness of the Volterra integral operators \(T_g\), \(I_g\) and the multiplication operator \(M_g\) on the space \(\mathcal{D}_{\lambda,K}\). In addition, the compactness and essential norm of the operators \(T_g\) and \(I_g\) on \(\mathcal{D}_{\lambda,K}\) are also investigated. Dirichlet-Morrey type space, Carleson measure, Volterra integral operators, bounded operator, essential norm 2023 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol43iss5art3 https://www.opuscula.agh.edu.pl/vol43/5/art/opuscula_math_4330.pdf One boundary value problem including a spectral parameter in all boundary conditions Ayşe Kabataş In this paper, asymptotic formulae for solutions and Green's function of a boundary value problem are investigated when the equation and the boundary conditions contain a spectral parameter. boundary value problems, asymptotics, Green's functions 2023 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol43iss5art4 https://www.opuscula.agh.edu.pl/vol43/5/art/opuscula_math_4331.pdf The existence of bipartite almost self-complementary 3-uniform hypergraphs L.N. Kamble, C.M. Deshpande, B.P. Athawale An almost self-complementary 3-uniform hypergraph on \(n\) vertices exists if and only if \(n\) is congruent to 3 modulo 4 A hypergraph \(H\) with vertex set \(V\) and edge set \(E\) is called bipartite if \(V\) can be partitioned into two subsets \(V_1\) and \(V_2\) such that \(e\cap V_1\neq \emptyset\) and \(e\cap V_2\neq \emptyset\) for any \(e\in E\). A bipartite self-complementary 3-uniform hypergraph \(H\) with partition \((V_1, V_2)\) of the vertex set \(V\) such that \(|V_1|=m\) and \(|V_2|=n\) exists if and only if either (i) \(m=n\) or (ii) \(m\neq n\) and either \(m\) or \(n\) is congruent to 0 modulo 4 or (iii) \(m\neq n\) and both \(m\) and \(n\) are congruent to 1 or 2 modulo 4. In this paper we define a bipartite almost self-complementary 3-uniform hypergraph \(H\) with partition \((V_1, V_2)\) of a vertex set \(V\) such that \(|V_1|=m\) and \(|V_2|=n\) and find the conditions on \(m\) and \(n\) for a bipartite 3-uniform hypergraph \(H\) to be almost self-complementary. We also prove the existence of bi-regular bipartite almost self-complementary 3-uniform hypergraphs. almost self-complementary 3-uniform hypergraph, bipartite hypergraph, bipartite self-complementary 3-uniform hypergraph, bipartite almost self-complementary 3-uniform hypergraph 2023 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol43iss5art5 https://www.opuscula.agh.edu.pl/vol43/5/art/opuscula_math_4332.pdf Radial solutions for nonlinear elliptic equation with nonlinear nonlocal boundary conditions Igor Kossowski In this article, we prove existence of radial solutions for a nonlinear elliptic equation with nonlinear nonlocal boundary conditions. Our method is based on some fixed point theorem in a cone. nonlocal boundary value problem, radial solutions, elliptic equation, the Krasnosielskii fixed point theorem in cone 2023 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol43iss5art6 https://www.opuscula.agh.edu.pl/vol43/5/art/opuscula_math_4333.pdf Global solutions for a nonlinear Kirchhoff type equation with viscosity Eugenio Cabanillas Lapa In this paper we consider the existence and asymptotic behavior of solutions of the following nonlinear Kirchhoff type problem \[u_{tt}- M\left(\,\displaystyle \int_{\Omega}|\nabla u|^{2}\, dx\right)\triangle u - \delta\triangle u_{t}= \mu|u|^{\rho-2}u\quad \text{in } \Omega \times ]0,\infty[,\] where \[M(s)=\begin{cases}a-bs &\text{for } s \in [0,\frac{a}{b}[,\\ 0, &\text{for } s \in [\frac{a}{b}, +\infty[.\end{cases}\] If the initial energy is appropriately small, we derive the global existence theorem and its exponential decay. global solutions, nonlinear Kirchhoff type problem, exponential decay 2023 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol43iss5art7 https://www.opuscula.agh.edu.pl/vol43/5/art/opuscula_math_4334.pdf Existence and smoothing effects of the initial-boundary value problem for \partial u/\partial t-\Delta\sigma(u)=0 in time-dependent domains Mitsuhiro Nakao We show the existence, smoothing effects and decay properties of solutions to the initial-boundary value problem for a generalized porous medium type parabolic equations of the form \[u_t-\Delta \sigma(u) =0 \quad \text{in } Q(0, T)\] with the initial and boundary conditions \[u(0)=u_0 \quad \text{and} \quad u(t)|_{\partial \Omega(t)}=0,\] where \(\Omega(t)\) is a bounded domain in \(R^N\) for each \(t \geq 0\) and \[Q(0,T)=\bigcup_{0 \lt t \lt T} \Omega(t) \times \{t\}, \quad T>0.\] Our class of \(\sigma(u)\) includes \(\sigma(u)=|u|^m u\), \(\sigma(u)=u \log (1+ |u|^m)\), \(0\leq m \leq 2\), and \(\sigma(u)=|u|^{m}u/\sqrt{1+|u|^2}\), \(1 \leq m \leq 2\), etc. We derive precise estimates for \(\|u(t)\|_{\Omega(t),\infty}\) and \(\|\nabla\sigma(u(t))\|^2_{\Omega(t),2}\), \(t\gt 0\), depending on \(\|u_0\|_{\Omega(0),r}\) and the movement of \(\partial\Omega(t)\). quasilinear parabolic equation, time-dependent domain, smoothing effects 2023 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol43iss6 https://www.opuscula.agh.edu.pl/om-vol43iss6art1 https://www.opuscula.agh.edu.pl/vol43/6/art/opuscula_math_4335.pdf Nonexistence of global solutions for a nonlinear parabolic equation with a forcing term Aisha Alshehri, Noha Aljaber, Haya Altamimi, Rasha Alessa, Mohamed Majdoub The purpose of this work is to analyze the blow-up of solutions of a nonlinear parabolic equation with a forcing term depending on both time and space variables \[u_t-\Delta u=|x|^{\alpha} |u|^{p}+{\mathtt a}(t)\,{\mathbf w}(x)\quad\text{for }(t,x)\in(0,\infty)\times \mathbb{R}^{N},\] where \(\alpha\in\mathbb{R}\), \(p\gt 1\), and \({\mathtt a}(t)\) as well as \({\mathbf w}(x)\) are suitable given functions. We generalize and somehow improve earlier existing works by considering a wide class of forcing terms that includes the most common investigated example \(t^\sigma\,{\mathbf w}(x)\) as a particular case. Using the test function method and some differential inequalities, we obtain sufficient criteria for the nonexistence of global weak solutions. This criterion mainly depends on the value of the limit \(\lim_{t\to\infty} \frac{1}{t}\,\int_0^t\,{\mathtt a}(s)\,ds\). The main novelty lies in our treatment of the nonstandard condition on the forcing term. nonlinear heat equation, forcing term, blow-up, test-function, differential inequalities 2023 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol43iss6art2 https://www.opuscula.agh.edu.pl/vol43/6/art/opuscula_math_4336.pdf Regularity and existence of solutions to parabolic equations with nonstandard p(x,t),q(x,t)-growth conditions Hamid El Bahja We study the Cauchy-Dirichlet problem for a class of nonlinear parabolic equations driven by nonstandard \(p(x,t),q(x,t)\)-growth condition. We prove theorems of existence and uniqueness of weak solutions in suitable Orlicz-Sobolev spaces, derive global and local in time \(L^{\infty}\) bounds for the weak solutions. existence theory, nonlinear parabolic problems, nonstandard growth, regularity theory 2023 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol43iss6art3 https://www.opuscula.agh.edu.pl/vol43/6/art/opuscula_math_4337.pdf Oscillation conditions for difference equations with several variable delays Bassant M. El-Matary, Hassan A. El-Morshedy, Vasileios Benekas, Ioannis P. Stavroulakis A technique is developed to establish a new oscillation criterion for a first-order linear difference equation with several delays and non-negative coefficients. Our result improves recent oscillation criteria and covers the cases of monotone and non-monotone delays. Moreover, the paper is concluded with an illustrative example to show the applicability and strength of our result. oscillation, difference equations, non-monotone delays, first order 2023 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol43iss6art4 https://www.opuscula.agh.edu.pl/vol43/6/art/opuscula_math_4338.pdf On the concept of generalization of I-density points Jacek Hejduk, Renata Wiertelak This paper deals with essential generalization of \(\mathcal{I}\)-density points and \(\mathcal{I}\)-density topology. In particular, there is an example showing that this generalization of \(\mathcal{I}\)-density point yields the stronger concept of density point than the notion of \(\mathcal{I}(\mathcal{J})\)-density. Some properties of topologies generated by operators related to this essential generalization of density points are provided. density topology, generalization of density topology 2023 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol43iss6art5 https://www.opuscula.agh.edu.pl/vol43/6/art/opuscula_math_4339.pdf On minimum intersections of certain secondary dominating sets in graphs Anna Kosiorowska, Adrian Michalski, Iwona Włoch In this paper we consider secondary dominating sets, also named as \((1,k)\)-dominating sets, introduced by Hedetniemi et al. in 2008. In particular, we study intersections of the \((1,1)\)-dominating sets and proper \((1,2)\)-dominating sets. We introduce \((1,\overline{2})\)-intersection index as the minimum possible cardinality of such intersection and determine its value for some classes of graphs. dominating set, 2-dominating set, \((1,2)\)-dominating set, proper \((1,2)\)-dominating set, domination numbers, \((1,\overline{2})\)-intersection index 2023 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol43iss6art6 https://www.opuscula.agh.edu.pl/vol43/6/art/opuscula_math_4340.pdf On the path partition of graphs Mekkia Kouider, Mohamed Zamime Let \(G\) be a graph of order \(n\). The maximum and minimum degree of \(G\) are denoted by \(\Delta\) and \(\delta\), respectively. The path partition number \(\mu(G)\) of a graph \(G\) is the minimum number of paths needed to partition the vertices of \(G\). Magnant, Wang and Yuan conjectured that \[\mu(G)\leq\max \left\{\frac{n}{\delta+1},\frac{(\Delta-\delta)n}{\Delta+\delta}\right\}.\] In this work, we give a positive answer to this conjecture, for \(\Delta\geq 2\delta\). path, partition 2023 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol43iss6art7 https://www.opuscula.agh.edu.pl/vol43/6/art/opuscula_math_4341.pdf Every graph is local antimagic total and its applications Gee-Choon Lau, Karl Schaffer, Wai Chee Shiu Let \(G = (V,E)\) be a simple graph of order \(p\) and size \(q\). A graph \(G\) is called local antimagic (total) if \(G\) admits a local antimagic (total) labeling. A bijection \(g : E \to \{1,2,\ldots,q\}\) is called a local antimagic labeling of \(G\) if for any two adjacent vertices \(u\) and \(v\), we have \(g^+(u) \neq g^+(v)\), where \(g^+(u) = \sum_{e\in E(u)} g(e)\), and \(E(u)\) is the set of edges incident to \(u\). Similarly, a bijection \(f:V(G)\cup E(G)\to \{1,2,\ldots,p+q\}\) is called a local antimagic total labeling of \(G\) if for any two adjacent vertices \(u\) and \(v\), we have \(w_f(u)\neq w_f(v)\), where \(w_f(u) = f(u) + \sum_{e\in E(u)} f(e)\). Thus, any local antimagic (total) labeling induces a proper vertex coloring of \(G\) if vertex \(v\) is assigned the color \(g^+(v)\) (respectively, \(w_f(u)\)). The local antimagic (total) chromatic number, denoted \(\chi_{la}(G)\) (respectively \(\chi_{lat}(G)\)), is the minimum number of induced colors taken over local antimagic (total) labeling of \(G\). We provide a short proof that every graph \(G\) is local antimagic total. The proof provides sharp upper bound to \(\chi_{lat}(G)\). We then determined the exact \(\chi_{lat}(G)\), where \(G\) is a complete bipartite graph, a path, or the Cartesian product of two cycles. Consequently, the \(\chi_{la}(G\vee K_1)\) is also obtained. Moreover, we determined the \(\chi_{la}(G\vee K_1)\) and hence the \(\chi_{lat}(G)\) for a class of 2-regular graphs \(G\) (possibly with a path). The work of this paper also provides many open problems on \(\chi_{lat}(G)\). We also conjecture that each graph \(G\) of order at least 3 has \(\chi_{lat}(G)\leq \chi_{la}(G)\). local antimagic (total) chromatic number, Cartesian product, join product 2023 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol43iss6art8 https://www.opuscula.agh.edu.pl/vol43/6/art/opuscula_math_4342.pdf The crossing numbers of join products of four graphs of order five with paths and cycles Michal Staš, Mária Timková The crossing number \(\mathrm{cr}(G)\) of a graph \(G\) is the minimum number of edge crossings over all drawings of \(G\) in the plane. In the paper, we extend known results concerning crossing numbers of join products of four small graphs with paths and cycles. The crossing numbers of the join products \(G^\ast + P_n\) and \(G^\ast + C_n\) for the disconnected graph \(G^\ast\) consisting of the complete tripartite graph \(K_{1,1,2}\) and one isolated vertex are given, where \(P_n\) and \(C_n\) are the path and the cycle on \(n\) vertices, respectively. In the paper also the crossing numbers of \(H^\ast + P_n\) and \(H^\ast + C_n\) are determined, where \(H^\ast\) is isomorphic to the complete tripartite graph \(K_{1,1,3}\). Finally, by adding new edges to the graphs \(G^\ast\) and \(H^\ast\), we are able to obtain crossing numbers of join products of two other graphs \(G_1\) and \(H_1\) with paths and cycles. graph, crossing number, join product, path, cycle, separating cycle 2023 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol44iss1 https://www.opuscula.agh.edu.pl/om-vol44iss1art1 https://www.opuscula.agh.edu.pl/vol44/1/art/opuscula_math_4401.pdf Uniqueness for a class p-Laplacian problems when a parameter is large B. Alreshidi, D.D. Hai We prove uniqueness of positive solutions for the problem \[-\Delta_{p}u=\lambda f(u)\text{ in }\Omega,\ u=0\text{ on }\partial \Omega,\] where \(1\lt p\lt 2\) and \(p\) is close to 2, \(\Omega\) is bounded domain in \(\mathbb{R}^{n}\) with smooth boundary \(\partial \Omega\), \(f:[0,\infty)\rightarrow [0,\infty )\) with \(f(z)\sim z^{\beta }\) at \(\infty\) for some \(\beta \in (0,1)\), and \(\lambda\) is a large parameter. The monotonicity assumption on \(f\) is not required even for \(u\) large. singular \(p\)-Laplacian, uniqueness, positive solutions 2024 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol44iss1art2 https://www.opuscula.agh.edu.pl/vol44/1/art/opuscula_math_4402.pdf Local existence for a viscoelastic Kirchhoff type equation with the dispersive term, internal damping, and logarithmic nonlinearity Sebastião Cordeiro, Carlos Raposo, Jorge Ferreira, Daniel Rocha, Mohammad Shahrouzi This paper concerns a viscoelastic Kirchhoff-type equation with the dispersive term, internal damping, and logarithmic nonlinearity. We prove the local existence of a weak solution via a modified lemma of contraction of the Banach fixed-point theorem. Although the uniqueness of a weak solution is still an open problem, we proved uniqueness locally for specifically suitable exponents. Furthermore, we established a result for local existence without guaranteeing uniqueness, stating a contraction lemma. viscoelastic equation, dispersive term, logarithmic nonlinearity, local existence 2024 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol44iss1art3 https://www.opuscula.agh.edu.pl/vol44/1/art/opuscula_math_4403.pdf Local irregularity conjecture for 2-multigraphs versus cacti Igor Grzelec, Mariusz Woźniak A multigraph is locally irregular if the degrees of the end-vertices of every multiedge are distinct. The locally irregular coloring is an edge coloring of a multigraph \(G\) such that every color induces a locally irregular submultigraph of \(G\). A locally irregular colorable multigraph \(G\) is any multigraph which admits a locally irregular coloring. We denote by \(\textrm{lir}(G)\) the locally irregular chromatic index of a multigraph \(G\), which is the smallest number of colors required in the locally irregular coloring of the locally irregular colorable multigraph \(G\). In case of graphs the definitions are similar. The Local Irregularity Conjecture for 2-multigraphs claims that for every connected graph \(G\), which is not isomorphic to \(K_2\), multigraph \(^2G\) obtained from \(G\) by doubling each edge satisfies \(\textrm{lir}(^2G)\leq 2\). We show this conjecture for cacti. This class of graphs is important for the Local Irregularity Conjecture for 2-multigraphs and the Local Irregularity Conjecture which claims that every locally irregular colorable graph \(G\) satisfies \(\textrm{lir}(G)\leq 3\). At the beginning it has been observed that all not locally irregular colorable graphs are cacti. Recently it has been proved that there is only one cactus which requires 4 colors for a locally irregular coloring and therefore the Local Irregularity Conjecture was disproved. locally irregular coloring, decomposable, cactus graphs, 2-multigraphs 2024 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol44iss1art4 https://www.opuscula.agh.edu.pl/vol44/1/art/opuscula_math_4404.pdf Two-weight norm inequalities for rough fractional integral operators on Morrey spaces Kwok-Pun Ho We establish the two-weight norm inequalities for the rough fractional integral operators on Morrey spaces. two-weight norm inequalities, rough fractional integral operators, Morrey spaces 2024 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol44iss1art5 https://www.opuscula.agh.edu.pl/vol44/1/art/opuscula_math_4405.pdf Conditional mean embedding and optimal feature selection via positive definite kernels Palle E.T. Jorgensen, Myung-Sin Song, James Tian Motivated by applications, we consider new operator-theoretic approaches to conditional mean embedding (CME). Our present results combine a spectral analysis-based optimization scheme with the use of kernels, stochastic processes, and constructive learning algorithms. For initially given non-linear data, we consider optimization-based feature selections. This entails the use of convex sets of kernels in a construction o foptimal feature selection via regression algorithms from learning models. Thus, with initial inputs of training data (for a suitable learning algorithm), each choice of a kernel \(K\) in turn yields a variety of Hilbert spaces and realizations of features. A novel aspect of our work is the inclusion of a secondary optimization process over a specified convex set of positive definite kernels, resulting in the determination of "optimal" feature representations. positive-definite kernels, reproducing kernel Hilbert space, stochastic processes, frames, machine learning, embedding problems, optimization 2024 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol44iss1art6 https://www.opuscula.agh.edu.pl/vol44/1/art/opuscula_math_4406.pdf Singular quasilinear convective systems involving variable exponents Abdelkrim Moussaoui, Dany Nabab, Jean Vélin The paper deals with the existence of solutions for quasilinear elliptic systems involving singular and convection terms with variable exponents. The approach combines the sub-supersolutions method and Schauder's fixed point theorem. \(p(x)\)-Laplacian, variable exponents, fixed point, singular system, gradient estimate, regularity 2024 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol44iss1art7 https://www.opuscula.agh.edu.pl/vol44/1/art/opuscula_math_4407.pdf The extensive 1-median problem with radius on networks Tran Hoai Ngoc Nhan, Nguyen Thanh Hung, Kien Trung Nguyen The median location problem concerns finding locations of one or several new facilities that minimize the overall weighted distances from the existing to the new facilities. We address the problem of locating one new facility with a radius \(r\) on networks. Furthermore, the radius \(r\) is flexible and the objective function is the conic combination of the traditional 1-median function and the value \(r\). We call this problem an extensive 1-median problem with radius on networks. To solve the problem, we first induce the so-called finite dominating set, that contains all points on the underlying network and radius values which are candidate for the optimal solution of the problem. This helps to develop a combinatorial algorithm that solves the problem on a general network \(G=(V,E)\) in \(O(|E||V|^3)\) time. We also consider the underlying problem with improved algorithm on trees. Based the convexity of the objective function with variable radius, we develop a linear time algorithm to find an extensive 1-median with radius on the underlying tree. extensive facility, median problem, tree, convex 2024 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol44iss2 https://www.opuscula.agh.edu.pl/om-vol44iss2art1 https://www.opuscula.agh.edu.pl/vol44/2/art/opuscula_math_4408.pdf On the structure of the diffusion distance induced by the fractional dyadic Laplacian María Florencia Acosta, Hugo Aimar, Ivana Gómez, Federico Morana In this note we explore the structure of the diffusion metric of Coifman-Lafon determined by fractional dyadic Laplacians. The main result is that, for each \(t\gt 0\), the diffusion metric is a function of the dyadic distance, given in \(\mathbb{R}^+\) by \(\delta(x,y) = \inf\{|I|\colon I \text{ is a dyadic interval containing } x \text{ and } y\}\). Even if these functions of \(\delta\) are not equivalent to \(\delta\), the families of balls are the same, to wit, the dyadic intervals. diffusion metrics, dyadic diffusion 2024 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol44iss2art2 https://www.opuscula.agh.edu.pl/vol44/2/art/opuscula_math_4409.pdf Green's functions and existence of solutions of nonlinear fractional implicit difference equations with Dirichlet boundary conditions Alberto Cabada, Nikolay D. Dimitrov, Jagan Mohan Jonnalagadda This article is devoted to deduce the expression of the Green's function related to a general constant coefficients fractional difference equation coupled to Dirichlet conditions. In this case, due to the points where some of the fractional operators are applied, we are in presence of an implicit fractional difference equation. So, due to such a property, it is more complicated to calculate and manage the expression of the Green's function than in the explicit case studied in a previous work of the authors. Contrary to the explicit case, where it is shown that the Green's function is constructed as finite sums, the Green's function constructed here is an infinite series. This fact makes necessary to impose more restrictive assumptions on the parameters that appear in the equation. The expression of the Green's function will be deduced from the Laplace transform on the time scales of the integers. We point out that, despite the implicit character of the considered equation, we can have an explicit expression of the solution by means of the expression of the Green's function. These two facts are not incompatible. Even more, this method allows us to have an explicit expression of the solution of an implicit problem. Finally, we prove two existence results for nonlinear problems, via suitable fixed point theorems. fractional difference, Dirichlet conditions, Green's function, existence of solutions 2024 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol44iss2art3 https://www.opuscula.agh.edu.pl/vol44/2/art/opuscula_math_4410.pdf Parabolic turbulence k-epsilon model with applications in fluid flows through permeable media Hermenegildo Borges de Oliveira In this work, we study a one-equation turbulence \(k\)-epsilon model that governs fluid flows through permeable media. The model problem under consideration here is derived from the incompressible Navier-Stokes equations by the application of a time-averaging operator used in the \(k\)-epsilon modeling and a volume-averaging operator that is characteristic of modeling unsteady porous media flows. For the associated initial- and boundary-value problem, we prove the existence of suitable weak solutions (average velocity field and turbulent kinetic energy) in the space dimensions of physics interest. turbulence, \(k\)-epsilon modelling, permeable media, existence 2024 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol44iss2art4 https://www.opuscula.agh.edu.pl/vol44/2/art/opuscula_math_4411.pdf An inequality for imaginary parts of eigenvalues of non-compact operators with Hilbert-Schmidt Hermitian components Michael Gil' Let \(A\) be a bounded linear operator in a complex separable Hilbert space, \(A^*\) be its adjoint one and \(A_I:=(A-A^*)/(2i)\). Assuming that \(A_I\) is a Hilbert-Schmidt operator, we investigate perturbations of the imaginary parts of the eigenvalues of \(A\). Our results are formulated in terms of the "extended" eigenvalue sets in the sense introduced by T. Kato. Besides, we refine the classical Weyl inequality \(\sum_{k=1}^\infty (\operatorname{Im} \lambda_k(A))^2 \leq N_2^2(A_I)\), where \(\lambda_k(A)\) \((k=1,2, \ldots )\) are the eigenvalues of \(A\) and \(N_2(\cdot)\) is the Hilbert-Schmidt norm. In addition, we discuss applications of our results to the Jacobi operators. Hilbert space, linear operators, eigenvalues 2024 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol44iss2art5 https://www.opuscula.agh.edu.pl/vol44/2/art/opuscula_math_4412.pdf Anisotropic p-Laplace Equations on long cylindrical domain Purbita Jana The main aim of this article is to study the Poisson type problem for anisotropic \(p\)-Laplace type equation on long cylindrical domains. The rate of convergence is shown to be exponential, thereby improving earlier known results for similar type of operators. The Poincaré inequality for a pseudo \(p\)-Laplace operator on an infinite strip-like domain is also studied and the best constant, like in many other situations in literature for other operators, is shown to be the same with the best Poincaré constant of an analogous problem set on a lower dimension. pseudo \(p\)-Laplace equation, cylindrical domains, asymptotic analysis 2024 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol44iss2art6 https://www.opuscula.agh.edu.pl/vol44/2/art/opuscula_math_4413.pdf Positive solutions to a third order nonlocal boundary value problem with a parameter Gabriela Szajnowska, Mirosława Zima We present some sufficient conditions for the existence of positive solutions to a third order differential equation subject to nonlocal boundary conditions. Our approach is based on the Krasnosel'skiĭ-Guo fixed point theorem in cones and the properties of the Green's function corresponding to the BVP under study. The main results are illustrated by suitable examples. boundary value problem, nonlocal boundary conditions, positive solution, cone 2024 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol44iss2art7 https://www.opuscula.agh.edu.pl/vol44/2/art/opuscula_math_4414.pdf Weak signed Roman k-domination in digraphs Lutz Volkmann Let \(k\geq 1\) be an integer, and let \(D\) be a finite and simple digraph with vertex set \(V(D)\). A weak signed Roman \(k\)-dominating function (WSRkDF) on a digraph \(D\) is a function \(f \colon V(D)\rightarrow \{-1,1,2\}\) satisfying the condition that \(\sum_{x \in N^-[v]}f(x)\geq k\) for each \(v\in V(D)\), where \(N^-[v]\) consists of \(v\) and all vertices of \(D\) from which arcs go into \(v\). The weight of a WSRkDF \(f\) is \(w(f)=\sum_{v\in V(D)}f(v)\). The weak signed Roman \(k\)-domination number \(\gamma_{wsR}^k(D)\) is the minimum weight of a WSRkDF on \(D\). In this paper we initiate the study of the weak signed Roman \(k\)-domination number of digraphs, and we present different bounds on \(\gamma_{wsR}^k(D)\). In addition, we determine the weak signed Roman \(k\)-domination number of some classes of digraphs. Some of our results are extensions of well-known properties of the weak signed Roman domination number \(\gamma_{wsR}(D)=\gamma_{wsR}^1(D)\) and the signed Roman \(k\)-domination number \(\gamma_{sR}^k(D).\) digraph, weak signed Roman \(k\)-dominating function, weak signed Roman \(k\)-domination number, signed Roman \(k\)-dominating function, signed Roman \(k\)-domination number 2024 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol44iss3 https://www.opuscula.agh.edu.pl/om-vol44iss3art1 https://www.opuscula.agh.edu.pl/vol44/3/art/opuscula_math_4415.pdf Jan Stochel, a stellar mathematician Sameer Chavan, Raúl Curto, Zenon Jan Jabłoński, Il Bong Jung, Mihai Putinar The occasion for this survey article was the 70th birthday of Jan Stochel, professor at Jagiellonian University, former head of the Chair of Functional Analysis and a prominent member of the Kraków school of operator theory. In the course of his mathematical career, he has dealt, among other things, with various aspects of functional analysis, single and multivariable operator theory, the theory of moments, the theory of orthogonal polynomials, the theory of reproducing kernel Hilbert spaces, and mathematical aspects of quantum mechanics. unbounded subnormal operator, moment problem, composition operator, Cauchy dual 2024 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol44iss3art2 https://www.opuscula.agh.edu.pl/vol44/3/art/opuscula_math_4416.pdf Finitely additive functions in measure theory and applications Daniel Alpay, Palle Jorgensen In this paper, we consider, and make precise, a certain extension of the Radon-Nikodym derivative operator, to functions which are additive, but not necessarily sigma-additive, on a subset of a given sigma-algebra. We give applications to probability theory; in particular, to the study of \(\mu\)-Brownian motion, to stochastic calculus via generalized Itô-integrals, and their adjoints (in the form of generalized stochastic derivatives), to systems of transition probability operators indexed by families of measures \(\mu\), and to adjoints of composition operators. Hilbert space, reproducing kernels, probability space, Gaussian fields, transforms, covariance, Itô integration, Itô calculus, generalized Brownian motion 2024 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol44iss3art3 https://www.opuscula.agh.edu.pl/vol44/3/art/opuscula_math_4417.pdf Shifted model spaces and their orthogonal decompositions M. Cristina Câmara, Kamila Kliś-Garlicka, Marek Ptak We generalize certain well known orthogonal decompositions of model spaces and obtain similar decompositions for the wider class of shifted model spaces, allowing us to establish conditions for near invariance of the latter with respect to certain operators which include, as a particular case, the backward shift \(S^*\). In doing so, we illustrate the usefulness of obtaining appropriate decompositions and, in connection with this, we prove some results on model spaces which are of independent interest. We show moreover how the invariance properties of the kernel of an operator \(T\), with respect to another operator, follow from certain commutation relations between the two operators involved. model space, Toeplitz operator, Toeplitz kernel, truncated Toeplitz operator, nearly invariant, shift invariant 2024 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol44iss3art4 https://www.opuscula.agh.edu.pl/vol44/3/art/opuscula_math_4418.pdf A note on the general moment problem Hamza El Azhar, Abdelouahab Hanine, El Hassan Zerouali In this note we show that given an indeterminate Hamburger moment sequence, it is possible to perturb the first moment in such way that the obtained sequence remains an indeterminate Hamburger moment sequence. As a consequence we prove that every sequence of real numbers is a moment sequence for a signed discrete measure supported in \(\mathbb{R}_+\). general moment problem, charge sequences, atomic measure 2024 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol44iss3art5 https://www.opuscula.agh.edu.pl/vol44/3/art/opuscula_math_4419.pdf Cesàro summability of Taylor series in higher order weighted Dirichlet-type spaces Soumitra Ghara, Rajeev Gupta, Md. Ramiz Reza For a positive integer \(m\) and a finite non-negative Borel measure \(\mu\) on the unit circle, we study the Hadamard multipliers of higher order weighted Dirichlet-type spaces \(\mathcal H_{\mu, m}\). We show that if \(\alpha\gt\frac{1}{2}\), then for any \(f\) in \(\mathcal H_{\mu, m}\) the sequence of generalized Cesàro sums \(\{\sigma_n^{\alpha}[f]\}\) converges to \(f\). We further show that if \(\alpha=\frac{1}{2}\) then for the Dirac delta measure supported at any point on the unit circle, the previous statement breaks down for every positive integer \(m\). weighted Dirichlet-type integrals, Cesàro mean, summability, Hadamard multiplication 2024 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol44iss3art6 https://www.opuscula.agh.edu.pl/vol44/3/art/opuscula_math_4420.pdf On the Möbius invariant principal functions of Pincus Sagar Ghosh, Gadadhar Misra In this semi-expository short note, we prove that the only homogeneous pure hyponormal operator \(T\) with \(\operatorname{rank} (T^*T-TT^*) =1\), modulo unitary equivalence, is the unilateral shift. hyponormal operator, multiplicity, trace formula, homogeneous operators, principal function 2024 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol44iss3art7 https://www.opuscula.agh.edu.pl/vol44/3/art/opuscula_math_4421.pdf Positive solutions for nonparametric anisotropic singular solutions Nikolaos S. Papageorgiou, Vicenţiu D. Rădulescu, Xueying Sun We consider an elliptic equation driven by a nonlinear, nonhomogeneous differential operator with nonstandard growth. The reaction has the combined effects of a singular term and of a "superlinear" perturbation. There is no parameter in the problem. Using variational tools and truncation and comparison techniques, we show the existence of at least two positive smooth solutions. variable Lebesgue and Sobolev spaces, anisotropic regularity, anisotropic maximum principle, truncations and comparisons, Hardy inequality 2024 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol44iss3art8 https://www.opuscula.agh.edu.pl/vol44/3/art/opuscula_math_4422.pdf Reduction of positive self-adjoint extensions Zsigmond Tarcsay, Zoltán Sebestyén We revise Krein's extension theory of semi-bounded Hermitian operators by reducing the problem to finding all positive and contractive extensions of the "resolvent operator" \((I+T)^{-1}\) of \(T\). Our treatment is somewhat simpler and more natural than Krein's original method which was based on the Krein transform \((I-T)(I+T)^{-1}\). Apart from being positive and symmetric, we do not impose any further constraints on the operator \(T\): neither its closedness nor the density of its domain is assumed. Moreover, our arguments remain valid in both real or complex Hilbert spaces. positive selfadjoint contractive extension, nonnegative selfadjoint extension, Friedrichs and Krein-von Neumann extension 2024 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol44iss4 https://www.opuscula.agh.edu.pl/om-vol44iss4art1 https://www.opuscula.agh.edu.pl/vol44/4/art/opuscula_math_4423.pdf Study of fractional semipositone problems on R^{N} Nirjan Biswas Let \(s\in (0,1)\) and \(N\gt 2s\). In this paper, we consider the following class of nonlocal semipositone problems: \[(-\Delta)^s u= g(x)f_a(u)\text{ in }\mathbb{R}^N,\quad u \gt 0\text{ in }\mathbb{R}^N,\] where the weight \(g \in L^1(\mathbb{R}^N) \cap L^{\infty}(\mathbb{R}^N)\) is positive, \(a\gt 0\) is a parameter, and \(f_a \in \mathcal{C}(\mathbb{R})\) is strictly negative on \((-\infty,0]\). For \(f_a\) having subcritical growth and weaker Ambrosetti-Rabinowitz type nonlinearity, we prove that the above problem admits a mountain pass solution \(u_a\), provided \(a\) is near zero. To obtain the positivity of \(u_a\), we establish a Brezis-Kato type uniform estimate of \((u_a)\) in \(L^r(\mathbb{R}^N)\) for every \(r \in [\frac{2N}{N-2s}, \infty]\). semipositone problems, fractional operator, uniform regularity estimates, positive solutions 2024 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol44iss4art2 https://www.opuscula.agh.edu.pl/vol44/4/art/opuscula_math_4424.pdf Degenerate singular parabolic problems with natural growth Mounim El Ouardy, Youssef El Hadfi, Abdelaaziz Sbai In this paper, we study the existence and regularity results for nonlinear singular parabolic problems with a natural growth gradient term \[\begin{cases}\frac{\partial u}{\partial t}-\operatorname{div}((a(x,t)+u^{q})|\nabla u|^{p-2}\nabla u)+d(x,t)\frac{|\nabla u|^{p}}{u^{\gamma}}=f & \text{ in } Q,\\ u(x,t)=0 & \text{ on } \Gamma, \\ u(x,t=0)=u_{0}(x) & \text{ in } \Omega, \end{cases}\] where \(\Omega\) is a bounded open subset of \(\mathbb{R}^{N}\), \(N\gt 2\), \(Q\) is the cylinder \(\Omega \times (0,T)\), \(T\gt 0\), \(\Gamma\) the lateral surface \(\partial \Omega \times (0,T)\), \(2\leq p\lt N\), \(a(x,t)\) and \(b(x,t)\) are positive measurable bounded functions, \(q\geq 0\), \(0\leq\gamma\lt 1\), and \(f\) non-negative function belongs to the Lebesgue space \(L^{m}(Q)\) with \(m\gt 1\), and \(u_{0}\in L^{\infty}(\Omega)\) such that \[\forall\omega\subset\subset\Omega\, \exists D_{\omega}\gt 0:\, u_{0}\geq D_{\omega}\text{ in }\omega.\] More precisely, we study the interaction between the term \(u^{q}\) (\(q>0\)) and the singular lower order term \(d(x,t)|\nabla u|^{p}u^{-\gamma}\) (\(0\lt\gamma\lt 1\)) in order to get a solution to the above problem. The regularizing effect of the term \(u^q\) on the regularity of the solution and its gradient is also analyzed. degenerate parabolic equations, singular parabolic equations, natural growth term 2024 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol44iss4art3 https://www.opuscula.agh.edu.pl/vol44/4/art/opuscula_math_4425.pdf Asymptotic analysis for confluent hypergeometric function in two variables given by double integral Yoshishige Haraoka We study an integrable connection with irregular singularities along a normally crossing divisor. The connection is obtained from an integrable connection of regular singular type by a confluence, and has irregular singularities along \(x=\infty\) and \(y=\infty\). Solutions are expressed by a double integral of Euler type with resonances among the exponents in the integrand. We specify twisted cycles that give main asymptotic behaviors in sectorial domains around \((\infty,\infty)\). Then we obtain linear relations among the twisted cycles, and get an explicit expression of the Stokes multiplier. The methods to derive the asymptotic behaviors for double integrals and to get linear relations among twisted cycles in resonant case, which we developed in this paper, seem to be new. strong asymptotic expansion, Stokes phenomenon, middle convolution, twisted homology 2024 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol44iss4art4 https://www.opuscula.agh.edu.pl/vol44/4/art/opuscula_math_4426.pdf Graphs whose vertex set can be partitioned into a total dominating set and an independent dominating set Teresa W. Haynes, Michael A. Henning A graph \(G\) whose vertex set can be partitioned into a total dominating set and an independent dominating set is called a TI-graph. We give constructions that yield infinite families of graphs that are TI-graphs, as well as constructions that yield infinite families of graphs that are not TI-graphs. We study regular graphs that are TI-graphs. Among other results, we prove that all toroidal graphs are TI-graphs. total domination, vertex partitions, independent domination 2024 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol44iss4art5 https://www.opuscula.agh.edu.pl/vol44/4/art/opuscula_math_4427.pdf Geometric properties of the lattice of polynomials with integer coefficients Artur Lipnicki, Marek J. Śmietański This paper is related to the classic but still being examined issue of approximation of functions by polynomials with integer coefficients. Let \(r\), \(n\) be positive integers with \(n \ge 6r\). Let \(\boldsymbol{P}_n \cap \boldsymbol{M}_r\) be the space of polynomials of degree at most \(n\) on \([0,1]\) with integer coefficients such that \(P^{(k)}(0)/k!\) and \(P^{(k)}(1)/k!\) are integers for \(k=0,\dots,r-1\) and let \(\boldsymbol{P}_n^\mathbb{Z} \cap \boldsymbol{M}_r\) be the additive group of polynomials with integer coefficients. We explore the problem of estimating the minimal distance of elements of \(\boldsymbol{P}_n^\mathbb{Z} \cap \boldsymbol{M}_r\) from \(\boldsymbol{P}_n \cap \boldsymbol{M}_r\) in \(L_2(0,1)\). We give rather precise quantitative estimations for successive minima of \(\boldsymbol{P}_n^\mathbb{Z}\) in certain specific cases. At the end, we study properties of the shortest polynomials in some hyperplane in \(\boldsymbol{P}_n \cap \boldsymbol{M}_r\). approximation by polynomials with integer coefficients, lattice, covering radius, roots of polynomial 2024 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol44iss4art6 https://www.opuscula.agh.edu.pl/vol44/4/art/opuscula_math_4428.pdf On the solvability of some parabolic equations involving nonlinear boundary conditions with L^{1} data Laila Taourirte, Abderrahim Charkaoui, Nour Eddine Alaa We analyze the existence of solutions for a class of quasilinear parabolic equations with critical growth nonlinearities, nonlinear boundary conditions, and \(L^1\) data. We formulate our problems in an abstract form, then using some techniques of functional analysis, such as Leray-Schauder's topological degree associated with the truncation method and very interesting compactness results, we establish the existence of weak solutions to the proposed models. quasilinear parabolic equation, nonlinear boundary conditions, weak solutions, Leray-Schauder topological degree, \(L^1\)-data 2024 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol44iss5 https://www.opuscula.agh.edu.pl/om-vol44iss5art1 https://www.opuscula.agh.edu.pl/vol44/5/art/opuscula_math_4429.pdf Analysis of a multiphase free boundary problem Ahlem Abdelouahab, Sabri Bensid In this paper, we investigate a free boundary problem relevant in several applications, such as tumor growth models. Our problem is expressed as an elliptic equation involving discontinuous nonlinearities in a specified domain with a moving boundary. We establish the existence and uniqueness of solutions and provide a qualitative analysis of the free boundaries generated by the nonlinear term (inner boundaries). Furthermore, we analyze the dynamics of the outer region boundary. The final result demonstrates that under certain conditions, our problem is solvable in the neighborhood of a radial solution. discontinuous nonlinearity, free boundary, perturbation, tumor growth 2024 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol44iss5art2 https://www.opuscula.agh.edu.pl/vol44/5/art/opuscula_math_4430.pdf Unitarily equivalent bilateral weighted shifts with operator weights Michał Buchała The aim of this paper is to study unitarily equivalent bilateral weighted shifts with operator weights. Our purpose is to establish a general characterization of unitary equivalence of such shifts under the assumption that the weights are quasi-invertible. Under further assumptions on weights it was proved that unitary equivalence of bilateral weigthed shifts with operator weights defined on \(\mathbb{C}^{2}\) can always be given by a unitary operator with at most two non-zero diagonals. The paper contains also examples of unitarily equivalent shifts with weights defined on \(\mathbb{C}^{k}\) such that every unitary operator, which intertwines them has at least \(k\) non-zero diagonals. weighted shifts, operator weights, unitary equivalence 2024 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol44iss5art3 https://www.opuscula.agh.edu.pl/vol44/5/art/opuscula_math_4431.pdf Seven largest trees pack Maciej Cisiński, Andrzej Żak The Tree Packing Conjecture (TPC) by Gyárfás states that any set of trees \(T_2,\dots,T_{n-1}, T_n\) such that \(T_i\) has \(i\) vertices pack into \(K_n\). The conjecture is true for bounded degree trees, but in general, it is widely open. Bollobás proposed a weakening of TPC which states that \(k\) largest trees pack. The latter is true if none tree is a star, but in general, it is known only for \(k=5\). In this paper we prove, among other results, that seven largest trees pack. tree, packing, tree packing conjecture 2024 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol44iss5art4 https://www.opuscula.agh.edu.pl/vol44/5/art/opuscula_math_4432.pdf Recovering the shape of an equilateral quantum tree with the Dirichlet conditions at the pendant vertices Anastasia Dudko, Oleksandr Lesechko, Vyacheslav Pivovarchik We consider two spectral problems on an equilateral rooted tree with the standard (continuity and Kirchhoff's type) conditions at the interior vertices (except of the root if it is interior) and Dirichlet conditions at the pendant vertices (except of the root if it is pendant). For the first (Neumann) problem we impose the standard conditions (if the root is an interior vertex) or Neumann condition (if the root is a pendant vertex) at the root, while for the second (Dirichlet) problem we impose the Dirichlet condition at the root. We show that for caterpillar trees the spectra of the Neumann problem and of the Dirichlet problem uniquely determine the shape of the tree. Also, we present an example of co-spectral snowflake graphs. Sturm-Liouville equation, eigenvalue, spectrum, equilateral tree, caterpillar tree, snowflake graph, root, standard conditions, Dirichlet boundary condition, Neumann boundary condition 2024 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol44iss5art5 https://www.opuscula.agh.edu.pl/vol44/5/art/opuscula_math_4433.pdf Isoperimetric inequalities in nonlocal diffusion problems with integrable kernel Gonzalo Galiano We deduce isoperimetric estimates for solutions of linear stationary and evolution problems. Our main result establishes the comparison in norm between the solution of a problem and its symmetric version when nonlocal diffusion defined through integrable kernels is replacing the usual local diffusion defined by a second order differential operator. Since an appropriate kernel rescaling allows to define a sequence of solutions of the nonlocal diffusion problems converging to their local diffusion counterparts, we also find the corresponding isoperimetric inequalities for the latter, i.e. we prove the classical Talenti's theorem. The novelty of our approach is that we replace the measure geometric tools employed in Talenti's proof, such as the geometric isoperimetric inequality or the coarea formula, by the Riesz's rearrangement inequality. Thus, in addition to providing a proof for the nonlocal diffusion case, our technique also introduces an alternative proof to Talenti's theorem. nonlocal diffusion, Schwarz's symmetrization, Talenti's theorem, Riesz's inequality 2024 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol44iss5art6 https://www.opuscula.agh.edu.pl/vol44/5/art/opuscula_math_4434.pdf Wintner-type nonoscillation theorems for conformable linear Sturm-Liouville differential equations Kazuki Ishibashi In this study, we addressed the nonoscillation of th Sturm-Liouville differential equation with a differential operator, which corresponds to a proportional-derivative controller. The equation is a conformable linear differential equation. A Wintner-type nonoscillation theorem was established to be applied to such equations. Using this theorem, we provided a sharp nonoscillation condition that guarantees that all nontrivial solutions to Euler-type conformable linear equations do not oscillate. The main nonoscillation theorems can be proven by introducing a Riccati inequality, which corresponds to the conformable linear equation of the Sturm-Liouville type. nonoscillation, conformable differential equation, proportional-derivative controller, Riccati technique, Euler equation 2024 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol44iss5art7 https://www.opuscula.agh.edu.pl/vol44/5/art/opuscula_math_4435.pdf Properties of the least action level and the existence of ground state solution to fractional elliptic equation with harmonic potential César E. Torres Ledesma, Hernán C. Gutierrez, Jesús A. Rodríguez, Manuel M. Bonilla In this article we consider the following fractional semilinear elliptic equation \[(-\Delta)^su+|x|^2u =\omega u+|u|^{2\sigma}u \quad \text{ in } \mathbb{R}^N,\] where \(s\in (0,1)\), \(N\gt 2s\), \(\sigma\in (0,\frac{2s}{N-2s})\) and \(\omega\in (0, \lambda_1)\). By using variational methods we show the existence of a symmetric decreasing ground state solution of this equation. Moreover, we study some continuity and differentiability properties of the ground state level. Finally, we consider a bifurcation type result. harmonic potential, fractional Sobolev space, ground state solution, bifurcation result, variational method 2024 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol44iss6 https://www.opuscula.agh.edu.pl/om-vol44iss6art1 https://www.opuscula.agh.edu.pl/vol44/6/art/opuscula_math_4436.pdf Closure results for arbitrarily partitionable graphs Julien Bensmail A well-known result of Bondy and Chvátal establishes that a graph of order \(n\) is Hamiltonian if and only if its \(n\)-closure (obtained through repeatedly adding an edge joining any two non-adjacent vertices with degree sum at least \(n\)) also is. In this work, we investigate such closure results for arbitrarily partitionable graphs, a weakening of Hamiltonian graphs being those graphs that can be partitioned into arbitrarily many connected graphs of arbitrary orders. Among other results, we establish closure results for arbitrary partitions into connected graphs of order at most 3, for arbitrary partitions into connected graphs of order exactly any \(\lambda\), and for the property of being arbitrarily partitionable in full. connected partition, arbitrarily partitionable graph, closure, traceability 2024 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol44iss6art2 https://www.opuscula.agh.edu.pl/vol44/6/art/opuscula_math_4437.pdf On a nonlocal p(x)-Laplacian Dirichlet problem involving several critical Sobolev-Hardy exponents Augusto César dos Reis Costa, Ronaldo Lopes da Silva The aim of this work is to present a result of multiplicity of solutions, in generalized Sobolev spaces, for a non-local elliptic problem with \(p(x)\)-Laplace operator containing \(k\) distinct critical Sobolev-Hardy exponents combined with singularity points \[ \begin{cases} M(\psi(u)) (- \Delta_{p(x)} u + |u|^{p(x)-2} u) = \sum_{i=1}^{k} h_i(x) \dfrac{|u|^{p^*_{s_i}(x)-2} u}{|x|^{s_i(x)}} + f(x,u) &\text{in }\Omega, \\ u=0 &\text{on }\partial \Omega, \end{cases} \] where \(\Omega\subset \mathbb{R}^N\) is a bounded domain with \(0 \in \Omega\) and \(1 \lt p^- \leq p(x) \leq p^+ \lt N\). The real function \(M\) is bounded in \([0, +\infty)\) and the functions \(h_i\) \((i=1, \ldots, k)\) and \(f\) are functions whose properties will be given later. To obtain the result we use the Lions' concentration-compactness principle for critical Sobolev-Hardy exponent in the space \(W^{1,p(x)}_{0}(\Omega)\) due to Yu, Fu and Li, and the Fountain Theorem. generalized Lebesgue-Sobolev spaces, \(p(x)\)-Laplacian nonlocal operator, Sobolev-Hardy critical exponents, concentration-compactness principle for critical Sobolev-Hardy exponent, fountain theorem 2024 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol44iss6art3 https://www.opuscula.agh.edu.pl/vol44/6/art/opuscula_math_4438.pdf Facial graceful coloring of plane graphs Július Czap Let \(G\) be a plane graph. Two edges of \(G\) are facially adjacent if they are consecutive on the boundary walk of a face of \(G\). A facial edge coloring of \(G\) is an edge coloring such that any two facially adjacent edges receive different colors. A facial graceful \(k\)-coloring of \(G\) is a proper vertex coloring \(c:V(G)\rightarrow\{1,2,\dots,k\}\) such that the induced edge coloring \(c^{\prime}:E(G)\rightarrow\{1,2,\dots,k-1\}\) defined by \(c^{\prime(uv)}=|c(u)-c(v)|\) is a facial edge coloring. The minimum integer \(k\) for which \(G\) has a facial graceful \(k\)-coloring is denoted by \(\chi_{fg}(G)\). In this paper we prove that \(\chi_{fg}(G)\leq 14\) for every plane graph \(G\) and \(\chi_{fg}(H)\leq 9\) for every outerplane graph \(H\). Moreover, we give exact bounds for cacti and trees. facial edge coloring, plane graph 2024 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol44iss6art4 https://www.opuscula.agh.edu.pl/vol44/6/art/opuscula_math_4439.pdf Positive solutions of nonpositone sublinear elliptic problems Tomas Godoy Consider the problem \(-\Delta u=\lambda f(\cdot, u) \) in \(\Omega\), \(u=0\) on \(\partial\Omega\), \(u\gt 0\) in \(\Omega\), where \(\Omega\) is a bounded domain in \(\mathbb{R}^{n}\) with \(C^{2}\) boundary when \(n\geq2\), \(\lambda\gt 0\), and where \(f\in C (\overline{\Omega}\times[0,\infty)) \) satisfies \(\lim_{s\rightarrow\infty}s^{-p}f(\cdot, s) =\gamma\) for some \(p\in(0,1)\) and some \(\gamma\in C(\overline{\Omega}) \) such that \(\gamma\neq 0\) a.e. in \(\Omega\) and, for some positive constants \(c\) and \(c^{\prime}\), \(\gamma^{-}\leq cd_{\Omega}^{\beta}\) for some \(\beta\in (\frac{n-1}{n},\infty)\) and \((-\Delta)^{-1}\gamma\geq c^{\prime}d_{\Omega}\), where \(d_{\Omega}(x):=dist ( x,\partial \Omega) \) and \(\gamma^{-}:=-\min(0,\gamma)\). Under these assumptions we show that for \(\lambda\) large enough, the above problem has a positive weak solution \(u\in C^{1}(\overline{\Omega})\) such that, for some constant \(c^{\prime\prime}\gt 0\), \(u\geq c^{\prime\prime}d_{\Omega}\) in \(\Omega\). elliptic sublinear problems, nonpositone problems, positive solutions, Leray-Schauder degree 2024 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol44iss6art5 https://www.opuscula.agh.edu.pl/vol44/6/art/opuscula_math_4440.pdf Symmetric 2×2 matrix functions with order preserving property Viera Štoudková Růžičková It is known that the discrete matrix Riccati equation has the order preserving property under some assumptions. In this paper we formulate and prove the converse statement for the case when the dimensions of the matrices are \(2 \times 2\) and the order preserving property holds for all such symmetric matrices. Riccati matrix equation, order preserving property, symmetric matrix function 2024 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol44iss6art6 https://www.opuscula.agh.edu.pl/vol44/6/art/opuscula_math_4441.pdf On expansive three-isometries Laurian Suciu The sub-Brownian 3-isometries in Hilbert spaces are the natural counterparts of the 2-isometries, because all of them have Brownian-type extensions in the sense of J. Agler and M. Stankus. We show that all powers \(T^n\) for \(n\geq 2\) of every expansive 3-isometry \(T\) are sub-Brownian, even if \(T\) does not have such a property. This fact induces some useful relations between the corresponding covariance operators of \(T\). We analyze two matrix representations of \(T\) in order to get some conditions under which \(T\) is sub-Brownian, or \(T\) admits the Wold-type decomposition in the sense of S. Shimorin. We show that the restriction of \(T\) to its range is sub-Brownian of McCullough's type, and that under some conditions on \(\mathcal{N}(T^*)\), \(T\) itself is sub-Brownian, and it admits the Wold-type decomposition. Wold decomposition, 3-isometry, sub-Brownian 3-isometry 2024 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol44iss6art7 https://www.opuscula.agh.edu.pl/vol44/6/art/opuscula_math_4442.pdf Reverse Lieb-Thirring inequality for the half-line matrix Schrödinger operator Ricardo Weder We prove a reverse Lieb-Thirring inequality with a sharp constant for the matrix Schrödinger equation on the half-line. spectral inequalities, matrix Schrödinger equations, Lieb-Thirring inequalities 2024 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol45iss1 https://www.opuscula.agh.edu.pl/om-vol45iss1art1 https://www.opuscula.agh.edu.pl/vol45/1/art/opuscula_math_4501.pdf Graphs with odd and even distances between non-cut vertices Kateryna Antoshyna, Sergiy Kozerenko We prove that in a connected graph, the distances between non-cut vertices are odd if and only if it is the line graph of a strong unique independence tree. We then show that any such tree can be inductively constructed from stars using a simple operation. Further, we study the connected graphs in which the distances between non-cut vertices are even (shortly, NCE-graphs). Our main results on NCE-graphs are the following: we give a criterion of NCE-graphs, show that any bipartite graph is an induced subgraph of an NCE-graph, characterize NCE-graphs with exactly two leaves, characterize graphs that can be subdivided to NCE-graphs, and provide a characterization for NCE-graphs which are maximal with respect to the edge addition operation. non-cut vertex, graph distance, line graph, block, strong unique independence tree 2025 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol45iss1art2 https://www.opuscula.agh.edu.pl/vol45/1/art/opuscula_math_4502.pdf Monotonic properties of Kneser solutions of second order linear differential equations with delayed argument Blanka Baculíková In this paper new monotonic properties of nonoscillatory solutions for second order linear functional differential equations with delayed argument \[y{''}(t)=p(t)y(\tau(t))\] have been established. New properties are used to introduce criteria for elimination of bounded nonoscillatory solutions for studied equations. second order, differential equations, delayed argument, monotonic properties, oscillation 2025 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol45iss1art3 https://www.opuscula.agh.edu.pl/vol45/1/art/opuscula_math_4503.pdf The metric dimension of circulant graphs Tapendra BC, Shonda Dueck A pair of vertices \(x\) and \(y\) in a graph \(G\) are said to be resolved by a vertex \(w\) if the distance from \(x\) to \(w\) is not equal to the distance from \(y\) to \(w\). We say that \(G\) is resolved by a subset of its vertices \(W\) if every pair of vertices in \(G\) is resolved by some vertex in \(W\). The minimum cardinality of a resolving set for \(G\) is called the metric dimension of \(G\), denoted by \(\dim(G)\). The circulant graph \(C_n(1,2,\ldots,t)\) is the Cayley graph \(Cay(\mathbb{Z}_n:\{\pm 1, \pm 2, \ldots, \pm t\})\). In this note we prove that, for \(n=2kt+2t\), \(\dim(C_n(1,2,\ldots,t))\geq t+2\), confirming Conjecture 4.1.2 in [K. Chau, S. Gosselin, The metric dimension of circulant graphs and their Cartesian products, Opuscula Math. 37 (2017), 509-534]. metric dimension, circulant graph 2025 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol45iss1art4 https://www.opuscula.agh.edu.pl/vol45/1/art/opuscula_math_4504.pdf (1,2)-PDS in graphs with the small number of vertices of large degrees Urszula Bednarz, Mateusz Pirga We define and study a perfect \((1,2)\)-dominating set which is a special case of a \((1,2)\)-dominating set. We discuss the existence of a perfect \((1,2)\)-dominating set in graphs with at most two vertices of maximum degree. In particular, we present a complete solution if the maximum degree equals \(n-1\) or \(n-2\). domination, secondary domination, neighborhoods, maximum degree 2025 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol45iss1art5 https://www.opuscula.agh.edu.pl/vol45/1/art/opuscula_math_4505.pdf Total mutual-visibility in Hamming graphs Csilla Bujtás, Sandi Klavžar, Jing Tian If \(G\) is a graph and \(X \subseteq V(G)\), then \(X\) is a total mutual-visibility set if every pair of vertices \(x\) and \(y\) of \(G\) admits the shortest \(x,y\)-path \(P\) with \(V(P) \cap X \subseteq \{x,y\}\). The cardinality of the largest total mutual-visibility set of \(G\) is the total mutual-visibility number \(\mu_{\rm t}(G)\) of \(G\). In this paper the total mutual-visibility number is studied on Hamming graphs, that is, Cartesian products of complete graphs. Different equivalent formulations for the problem are derived. The values \(\mu_{\rm t}(K_{n_1}\square K_{n_2}\square K_{n_3})\) are determined. It is proved that \(\mu_{\rm t}(K_{n_1} \square \cdots \square K_{n_r}) = O(N^{r-2})\), where \(N = n_1+\cdots + n_r\), and that \(\mu_{\rm t}(K_s^{\square r}) = \Theta(s^{r-2})\) for every \(r \geq 3\), where \(K_s^{\square r}\) denotes the Cartesian product of \(r\) copies of \(K_s\). The main theorems are also reformulated as Turán-type results on hypergraphs. mutual-visibility set, total mutual-visibility set, Hamming graph, Turán-type problem 2025 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol45iss1art6 https://www.opuscula.agh.edu.pl/vol45/1/art/opuscula_math_4506.pdf On uniqueness of packing of three copies of 2-factors Igor Grzelec, Tomáš Madaras, Alfréd Onderko The packing of three copies of a graph \(G\) is the union of three edge-disjoint copies (with the same vertex set) of \(G\). In this paper, we completely solve the problem of the uniqueness of packing of three copies of 2-regular graphs. In particular, we show that \(C_3,C_4,C_5,C_6\) and \(2C_3\) have no packing of three copies, \(C_7,C_8,C_3 \cup C_4, C_4 \cup C_4, C_3 \cup C_5\) and \(3C_3\) have unique packing, and any other collection of cycles has at least two distinct packings. uniquely packable graph, 2-factor, 3-packing 2025 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol45iss1art7 https://www.opuscula.agh.edu.pl/vol45/1/art/opuscula_math_4507.pdf Note on robust coloring of planar graphs František Kardoš, Borut Lužar, Roman Soták We consider the robust chromatic number \(\chi_1(G)\) of planar graphs \(G\) and show that there exists an infinite family of planar graphs \(G\) with \(\chi_1(G) = 3\), thus solving a recent problem of Bacsó et al. from [The robust chromatic number of graphs, Graphs Combin. 40 (2024), #89]. robust coloring, robust chromatic number, Tutte graph, planar graph 2025 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol45iss2 https://www.opuscula.agh.edu.pl/om-vol45iss2art1 https://www.opuscula.agh.edu.pl/vol45/2/art/opuscula_math_4508.pdf Combined effects for a class of fractional variational inequalities Shengbing Deng, Wenshan Luo, César E. Torres Ledesma In this paper, we study the existence of a nonnegative weak solution to the following nonlocal variational inequality: \[\int_{\mathbb{R}^N}(-\Delta)^{\frac{s}{2}} u (-\Delta)^{{\frac{s}{2}}}(v-u)dx+\int_{\mathbb{R}^N}(1+\lambda M(x))u(v-u)dx \geq \int_{\mathbb{R}^N}f(u)(v-u)dx, \] for all \(v \in\mathbb{K}\), where \(s\in (0,1)\) and \(M\) is a continuous steep potential well on \(\mathbb{R}^N\). Using penalization techniques from del Pino and Felmer, as well as from Bensoussan and Lions, we establish the existence of nonnegative weak solutions. These solutions localize near the potential well \(\operatorname{Int}(M^{-1}(0))\). fractional variational inequality, variational methods, critical nonlinearity 2025 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol45iss2art2 https://www.opuscula.agh.edu.pl/vol45/2/art/opuscula_math_4509.pdf Representation of solutions of linear discrete systems with constant coefficients and with delays Josef Diblík The paper surveys the results achieved in representing solutions of linear non-homogeneous discrete systems with constant coefficients and with delays and their fractional variants by using special matrices called discrete delayed-type matrices. These are used to express solutions of initial problems in a closed and often simple form. Then, results are briefly discussed achieved by such representations of solutions in stability, controllability and other fields. In addition, a similar topic is dealt with concerning linear non-homogeneous differential equations with delays and their variants. Moreover, some comments are given to this parallel direction pointing some important moments in the developing this theory. An outline of future perspectives in this direction is discussed as well. discrete linear system, constants coefficients, delay, discrete matrix functions, representation of solutions, commutative matrices, non-commutative matrices, controllability 2025 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol45iss2art3 https://www.opuscula.agh.edu.pl/vol45/2/art/opuscula_math_4510.pdf Augmenting graphs to partition their vertices into a total dominating set and an independent dominating set Teresa W. Haynes, Michael A. Henning A graph \(G\) whose vertex set can be partitioned into a total dominating set and an independent dominating set is called a TI-graph. There exist infinite families of graphs that are not TI-graphs. We define the TI-augmentation number \(\operatorname{ti}(G)\) of a graph \(G\) to be the minimum number of edges that must be added to \(G\) to ensure that the resulting graph is a TI-graph. We show that every tree \(T\) of order \(n \geq 5\) satisfies \(\operatorname{ti}(T) \leq \frac{1}{5}n\). We prove that if \(G\) is a bipartite graph of order \(n\) with minimum degree \(\delta(G) \geq 3\), then \(\operatorname{ti}(G) \leq \frac{1}{4}n\), and if \(G\) is a cubic graph of order \(n\), then \(\operatorname{ti}(G) \leq \frac{1}{3}n\). We conjecture that \(\operatorname{ti}(G) \leq \frac{1}{6}n\) for all graphs \(G\) of order \(n\) with \(\delta(G) \geq 3\), and show that there exist connected graphs \(G\) of sufficiently large order \(n\) with \(\delta(G) \geq 3\) such that \(\operatorname{ti}(T) \geq (\frac{1}{6} - \varepsilon) n\) for any given \(\varepsilon \gt 0\). total domination, independent domination, vertex partitions 2025 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol45iss2art4 https://www.opuscula.agh.edu.pl/vol45/2/art/opuscula_math_4511.pdf Complete characterization of graphs with local total antimagic chromatic number 3 Gee-Choon Lau A total labeling of a graph \(G = (V, E)\) is said to be local total antimagic if it is a bijection \(f: V\cup E \to\{1,\ldots,|V|+|E|\}\) such that adjacent vertices, adjacent edges, and pairs of an incident vertex and edge have distinct induced weights where the induced weight of a vertex \(v\) is \(w_f(v) = \sum f(e)\) with \(e\) ranging over all the edges incident to \(v\), and the induced weight of an edge \(uv\) is \(w_f(uv) = f(u) + f(v)\). The local total antimagic chromatic number of \(G\), denoted by \(\chi_{lt}(G)\), is the minimum number of distinct induced vertex and edge weights over all local total antimagic labelings of \(G\). In this paper, we first obtain general lower and upper bounds for \(\chi_{lt}(G)\) and sufficient conditions to construct a graph \(H\) with \(k\) pendant edges and \(\chi_{lt}(H) \in\{\Delta(H)+1, k+1\}\). We then completely characterize graphs \(G\) with \(\chi_{lt}(G)=3\). Many families of (disconnected) graphs \(H\) with \(k\) pendant edges and \(\chi_{lt}(H) \in\{\Delta(H)+1, k+1\}\) are also obtained. local total antimagic, local total antimagic chromatic number 2025 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol45iss2art5 https://www.opuscula.agh.edu.pl/vol45/2/art/opuscula_math_4512.pdf Spectral analysis of infinite Marchenko-Slavin matrices Sergio Palafox, Luis O. Silva This work tackles the problem of spectral characterization of a class of infinite matrices arising from the modeling of small oscillations in a system of interacting particles. The class of matrices under discussion corresponds to the infinite Marchenko-Slavin class. The spectral functions of these matrices are completely characterized, and an algorithm is provided for the reconstruction of the matrix from its spectral function. The techniques used in this work are based on recent results for the spectral characterization of infinite band symmetric matrices with so-called degenerations. inverse spectral problem, band symmetric matrices, spectral measure 2025 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol45iss2art6 https://www.opuscula.agh.edu.pl/vol45/2/art/opuscula_math_4513.pdf Extended symmetry of the Witten-Dijkgraaf-Verlinde-Verlinde equation of Monge-Ampere type Patryk Sitko, Ivan Tsyfra We construct the Lie algebra of extended symmetry group for the Monge-Ampere type Witten-Dijkgraaf-Verlinde-Verlinde (WDVV) equation. This algebra includes novel generators that are unobtainable within the framework of the classical Lie approach and correspond to non-point group transformation of dependent and independent variables. The expansion of symmetry is achieved by introducing new variables through second-order derivatives of the dependent variable. By integrating the Lie equations, we derive transformations that enable the generation of new solutions to the Witten-Dijkgraaf-Verlinde-Verlinde equation from a known one. These transformations yield formulas for obtaining new solutions in implicit form and Bäcklund-type transformations for the nonlinear associativity equations. We also demonstrate that, in the case under study, introducing a substitution of variables via third-order derivatives, as previously used in the literature, does not yield generators of non-point transformations. Instead, this approach produces only the Lie groups of classical point transformations. Furthermore, we perform a group reduction of partial differential equations in two independent variables to a system of ordinary differential equations. This reduction leads to the explicit solution of the fully nonlinear differential equation. Notably, the symmetry group of non-point transformations expands significantly when this method is applied to the second-order differential equation, resulting in a corresponding infinite-dimensional Lie algebra. Finally, we show that auxiliary variables can be systematically derived within the framework of a generalized approach to symmetry reduction of differential equations. non-point symmetries, Witten-Dijkgraaf-Verlinde-Verlinde equation, symmetry group, transformations, Lie algebra 2025 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol45iss2art7 https://www.opuscula.agh.edu.pl/vol45/2/art/opuscula_math_4514.pdf Local properties of graphs that induce global cycle properties Yanyan Wang, Xiaojing Yang A graph \(G\) is locally Hamiltonian if \(G[N(v)]\) is Hamiltonian for every vertex \(v\in V(G)\). In this note, we prove that every locally Hamiltonian graph with maximum degree at least \(|V(G)| - 7\) is weakly pancyclic. Moreover, we show that any connected graph \(G\) with \(\Delta(G)\leq 7\) and \(\delta(G[N(v)])\geq 3\) for every \(v\in V (G)\), is fully cycle extendable. These findings improve some known results by Tang and Vumar. fully cycle extendability, weakly pancyclicity, locally connected 2025 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol45iss3 https://www.opuscula.agh.edu.pl/om-vol45iss3art1 https://www.opuscula.agh.edu.pl/vol45/3/art/opuscula_math_4515.pdf Magnetic Dirichlet Laplacian in curved waveguides Diana Barseghyan, Swanhild Bernstein, Baruch Schneider, Martha Lina Zimmermann For a two-dimensional curved waveguide, it is well known that the spectrum of the Dirichlet Laplacian is unstable with respect to waveguide deformations. This means that if the waveguide is a straight strip then the spectrum of the Dirichlet Laplacian is purely essential. From the other hand, the perturbation of the straight strip produces eigenvalues below the essential spectrum. In this paper, the Dirichlet-Laplace operator with a magnetic field is considered. We explicitly prove that the spectrum of the magnetic Laplacian is stable under small but non-local deformations of the waveguide. magnetic Schrödinger operators, essential spectrum, discrete spectrum 2025 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol45iss3art2 https://www.opuscula.agh.edu.pl/vol45/3/art/opuscula_math_4516.pdf Calculation of explicit expressions for the Hopf bifurcation limit cycles in delay-differential equations José Enríquez Gabeiras, Juan Francisco Padial Molina This paper introduces a methodology to derive explicit power series approximations for the limit cycle periodic solutions of the Hopf bifurcation in autonomous discrete delay differential equations (DDE). The procedure extends the methodology introduced by Casal and Freedman in 1980, by providing a detailed algorithm that iteratively performs systematic calculations up to any desired order of approximation, ensuring a specific error tolerance for any nonlinear DDE presenting a Hopf bifurcation. The methodology is applied to three relevant delay-differential models to illustrate its features: a recently introduced car-following mobility model that explains oscillations in road traffic, a SIR epidemic model for propagation of diseases with temporary immunity, and a simplified macroeconomic system to model business cycles. delay-differential equations, Hopf bifurcation, Poincaré-Lindstedt method, car-following model, SIR epidemic model, macroeconomic model 2025 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol45iss3art3 https://www.opuscula.agh.edu.pl/vol45/3/art/opuscula_math_4517.pdf Upper distance-two domination Jason T. Hedetniemi, Stephen T. Hedetniemi, Thomas M. Lewis Let \(G = (V, E)\) be a graph with vertex set \(V\) and edge set \(E\). A set \(S \subset V\) is a \(2\)-packing in \(G\) if for any two vertices \(u,v \in S\), the distance between them satisfies \(d(u,v) \gt 2\). The upper \(2\)-packing number \(P_2(G)\) is the maximum cardinality of a \(2\)-packing in \(G\). A set \(S \subset V\) is a dominating set for \(G\) if every vertex in \(V - S\) is adjacent to at least one vertex in \(S\). The domination number \(\gamma(G)\) is the minimum cardinality of a dominating set in \(G\). A set \(S \subset V\) is a distance-\(2\) dominating set if for every vertex \(v \in V - S\) there exists a vertex \(u \in S\) such that \(d(u,v) \leq 2\). The upper distance-\(2\) domination number \(\Gamma_{\leq 2}(G)\) is the maximum cardinality of a minimal distance-\(2\) dominating set in \(G\). In this paper we establish two families of graphs \(G\) for which \(P_2(G) = \gamma(G) = \Gamma_{\leq 2}(G)\), which extend several well-known equalities of the form \(P_2(G) = \gamma(G)\). 2-packing, domination, distance-2 domination 2025 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol45iss3art4 https://www.opuscula.agh.edu.pl/vol45/3/art/opuscula_math_4518.pdf Block Jacobi matrices and Titchmarsh-Weyl function Marcin Moszyński, Grzegorz Świderski We collect some results and notions concerning generalizations for block Jacobi matrices of several concepts, which have been important for spectral studies of the simpler and better known scalar Jacobi case. We focus here on some issues related to the matrix Titchmarsh-Weyl function, but we also consider generalizations of some other tools used by subordinacy theory, including the matrix orthogonal polynomials, the notion of finite cyclicity, a variant of a notion of nonsubordinacy, as well as Jitomirskaya-Last type semi-norms. The article brings together some issues already known, our new concepts, and also improvements and strengthening of some results already existing. We give simpler proofs of some known facts, or we add details usually omitted in the existing literature. The introduction contains a separate part devoted to a brief review of the main spectral analysis methods used so far for block Jacobi operators. block Jacobi matrix, matrix measures, Titchmarsh-Weyl function, Liouville-Ostrogradsky formulae 2025 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol45iss3art5 https://www.opuscula.agh.edu.pl/vol45/3/art/opuscula_math_4519.pdf Integral representation of solutions to Dirac systems Łukasz Rzepnicki We introduce a novel integral form for a fundamental set of solutions to one-dimensional Dirac systems with an integrable potential and spectral parameter \(\mu \in \mathbb{C}\). This method enables the construction of solutions that are analytic in \(\mu\) within the half-plane \(\operatorname{Im} \mu\gt -r\), \(r\geq 0\) and \(|\mu| \to \infty\). Consequently, we derive estimates for the solutions that remain valid not just within a horizontal strip but throughout the entire half-plane. Dirac system, integrable potential, integral equations, fundamental system of solutions 2025 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol45iss4 https://www.opuscula.agh.edu.pl/om-vol45iss4art1 https://www.opuscula.agh.edu.pl/vol45/4/art/opuscula_math_4520.pdf Convex geometries yielded by transit functions Manoj Changat, Lekshmi Kamal K. Sheela, Iztok Peterin, Ameera Vaheeda Shanavas Let \(V\) be a finite nonempty set. A transit function is a map \(R:V\times V\rightarrow 2^V\) such that \(R(u,u)=\{u\}\), \(R(u,v)=R(v,u)\) and \(u\in R(u,v)\) holds for every \(u,v\in V\). A set \(K\subseteq V\) is \(R\)-convex if \(R(u,v)\subset K\) for every \(u,v\in K\) and all \(R\)-convex subsets of \(V\) form a convexity \(\mathcal{C}_R\). We consider the Minkowski-Krein-Milman property that every \(R\)-convex set \(K\) in a convexity \(\mathcal{C}_R\) is the convex hull of the set of extreme points of \(K\) from axiomatic point of view and present a characterization of it. Later we consider several well-known transit functions on graphs and present the use of the mentioned characterizations on them. Minkowski-Krein-Milman property, convexity, convex geometry, transit function 2025 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol45iss4art2 https://www.opuscula.agh.edu.pl/vol45/4/art/opuscula_math_4521.pdf Datko-type theorems concerning asymptotic behaviour of exponential type in mean Pham Viet Hai In this paper, we study the concept of exponential (in)stability in mean for stochastic skew-evolution semiflows, in which the exponential (in)stability in the classical sense is replaced by an average with respect to a probability measure. Our paper consists of three major results. The first is to obtain Datko-type characterizations for the exponential stability in mean of stochastic skew-evolution semiflows. Next, we acquire Datko-type characterizations for the exponential instability in mean by extending the stability techniques. The last is to extend Lyapunov-type equations to the case of exponential (in)stability in mean. exponential stability, exponential instability, stochastic skew-evolution semiflows, Datko's theorem, Banach function spaces 2025 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol45iss4art3 https://www.opuscula.agh.edu.pl/vol45/4/art/opuscula_math_4522.pdf Tail probability of the hitting time of Brownian motion to a sphere with fixed hitting sites Yuji Hamana We consider \(d\)-dimensional Brownian motion \(\{B_\mu(t)\}_{t\geqq0}\) with a drift \(\mu\in\mathbb{R}^d\) and the first hitting time \(\sigma_{r,\mu}^{(d)}\) to the sphere with radius \(r\) centered at the origin. This article deals with asymptotic behavior of the probability that both \(t\lt\sigma_{r,\mu}^{(d)}\lt\infty\) and \(B_\mu(\sigma_{r,\mu}^{(d)})\in A\) occur simultaneously, and we obtain that this probability admits an asymptotic expansion in powers of \(1/t\) if \(d\geqq3\) and in that of \(1/\log t\) if \(d=2\) for large \(t\). Moreover, we investigate the case of Brownian motion with no drift. Brownian motion, hitting times and sites, asymptotic expansion 2025 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol45iss4art4 https://www.opuscula.agh.edu.pl/vol45/4/art/opuscula_math_4523.pdf Asymptotic behavior of the solutions of operators that are sum of pseudo p-Laplace type Purbita Jana The article investigates a Poisson-type problem for operators that are finite sum of pseudo \(p\)-Laplace-type operators within long cylindrical domains. It establishes that the rate of convergence is exponential, which is considered optimal. In addition, the study analyzes the asymptotic behavior of the related energy functional. This research contributes to a deeper understanding of the mathematical properties and asymptotic analysis of solutions in this context. pseudo \(p\)-Laplace equation, cylindrical domains, asymptotic analysis 2025 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol45iss4art5 https://www.opuscula.agh.edu.pl/vol45/4/art/opuscula_math_4524.pdf Multiplicity result for mixed local and nonlocal Kirchhoff problems involving critical growth Vinayak Mani Tripathi In this paper, we study the multiplicity of nonnegative solutions for the following nonlocal elliptic problem \[\begin{cases}M\Big(\ \int_{\mathbb{R}^N}|\nabla u|^2dx+\iint_{\mathbb{R}^{2N}}\frac{|u(x)-u(y)|^2}{|x-y|^{N+2s}}dxdy\Big)\mathcal{L}(u) \\ = \lambda {f(x)}|u|^{p-2}u+|u|^{2^*-2}u &\text{ in }\Omega, \\ u=0 &\text{ on }\mathbb R^N\setminus \Omega, \end{cases}\] where \(\Omega\subset\mathbb{R}^N\) is bounded domain with smooth boundary, \(1\lt p\lt 2\lt 2^*=\frac{2N}{N-2}\), \(N\geq 3\), \(\lambda\gt 0\), \(M\) is a Kirchhoff coefficient and \(\mathcal{L}\) denotes the mixed local and nonlocal operator. The weight function \(f\in L^{\frac{2^*}{2^*-p}}(\Omega)\) is allowed to change sign. By applying variational approach based on constrained minimization argument, we show the existence of at least two nonnegative solutions. mixed local and nonlocal operators, Kirchhoff type problem, critical nonlinearity, Nehari manifold 2025 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol45iss4art6 https://www.opuscula.agh.edu.pl/vol45/4/art/opuscula_math_4525.pdf Novel oscillation criteria for third-order semi-canonical differential equations with an advanced neutral term Kumar S. Vidhyaa, Ethiraju Thandapani, Ercan Tunç The main purpose of this paper is to present new oscillation results for nonlinear semi-canonical third-order differential equations with an advanced neutral term. The main idea is first by reducing the studied semi-canonical equation into standard canonical type equation without assuming any extra conditions. Then, by using the comparison method and integral averaging technique, sufficient conditions are established to ensure the oscillation of the reduced canonical equation, which in turn leads to the oscillation of the original equation. Therefore, the technique used here is very useful since the results already known for the canonical equations can be applied to obtain the oscillation of the semi-canonical equations. Two examples are provided to illustrate the importance of the main results. oscillation, third-order, neutral differential equations, semi-canonical 2025 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol45iss4art7 https://www.opuscula.agh.edu.pl/vol45/4/art/opuscula_math_4526.pdf Nontrivial solutions of discrete Kirchhoff-type problem via bifurcation theory Fumei Ye In this paper, we show that the bifurcation points for a discrete Kirchhoff-type problem with only local conditions, and we investigate the existence of positive and negative solutions for the problem when the nonlinear term \(f\) is asymptotically linear at zero and is asymptotically 3-linear at infinity. By using bifurcation techniques and the idea of taking limits of connected branches, under the assumption that \(f\) has some non-zero zeros, some results are also obtained. discrete Kirchhoff-type problem, nontrivial solution, bifurcation, superior limit 2025 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol45iss5 https://www.opuscula.agh.edu.pl/om-vol45iss5art1 https://www.opuscula.agh.edu.pl/vol45/5/art/opuscula_math_4527.pdf Spreading in claw-free cubic graphs Boštjan Brešar, Jaka Hedžet, Michael A. Henning Let \(p \in \mathbb{N}\) and \(q \in \mathbb{N} \cup \lbrace \infty \rbrace\). We study a dynamic coloring of the vertices of a graph \(G\) that starts with an initial subset \(S\) of blue vertices, with all remaining vertices colored white. If a white vertex \(v\) has at least \(p\) blue neighbors and at least one of these blue neighbors of \(v\) has at most \(q\) white neighbors, then by the spreading color change rule the vertex \(v\) is recolored blue. The initial set \(S\) of blue vertices is a \((p,q)\)-spreading set for \(G\) if by repeatedly applying the spreading color change rule all the vertices of \(G\) are eventually colored blue. The \((p,q)\)-spreading set is a generalization of the well-studied concepts of \(k\)-forcing and \(r\)-percolating sets in graphs. For \(q \geq 2\), a \((1,q)\)-spreading set is exactly a \(q\)-forcing set, and the \((1,1)\)-spreading set is a \(1\)-forcing set (also called a zero forcing set), while for \(q = \infty\), a \((p,\infty)\)-spreading set is exactly a \(p\)-percolating set. The \((p,q)\)-spreading number, \(\sigma_{(p,q)}(G)\), of \(G\) is the minimum cardinality of a \((p,q)\)-spreading set. In this paper, we study \((p,q)\)-spreading in claw-free cubic graphs. While the zero-forcing number of claw-free cubic graphs was studied earlier, for each pair of values \(p\) and \(q\) that are not both \(1\) we either determine the \((p,q)\)-spreading number of a claw-free cubic graph \(G\) or show that \(\sigma_{(p,q)}(G)\) attains one of two possible values. bootstrap percolation, zero forcing set, \(k\)-forcing set, spreading 2025 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol45iss5art2 https://www.opuscula.agh.edu.pl/vol45/5/art/opuscula_math_4528.pdf From set-valued dynamical processes to fractals Grzegorz Guzik, Grzegorz Kleszcz We present a general theory of topological semiattractors and attractors for set-valued semigroups. Our results extend and unify those previously obtained by Lasota and Myjak. In particular, we naturally generalize the concept of semifractals for the systems acting on Hausdorff topological spaces. The main tool in our analysis is the notion of topological (Kuratowski) limits. We especially focus on the forward asymptotic behavior of discrete set-valued processes generated by sequences of iterated function systems. In this context, we establish sufficient conditions for the existence of fractal-type limit sets. topological limit, lower semicontinuous multifunction, iterated function system, set-valued process, attractor 2025 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol45iss5art3 https://www.opuscula.agh.edu.pl/vol45/5/art/opuscula_math_4529.pdf Nontrivial solutions for Neumann fractional p-Laplacian problems Chun Li, Dimitri Mugnai, Tai-Jin Zhao In this paper, we investigate some classes of Neumann fractional \(p\)-Laplacian problems. We prove the existence and multiplicity of nontrivial solutions for several different nonlinearities, by using variational methods and critical point theory based on cohomological linking. fractional \(p\)-Laplacian, Neumann boundary condition, linking over cones 2025 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol45iss5art4 https://www.opuscula.agh.edu.pl/vol45/5/art/opuscula_math_4530.pdf A general elliptic equation with intrinsic operator Dumitru Motreanu Existence and bound of a solution is established for a general elliptic equation with intrinsic operator subject to Dirichlet boundary condition. This provides a sufficient condition to the fundamental question if there is a solution belonging to a prescribed ball in the function space. An application deals with an equation involving a convolution product. nonlinear elliptic equation, intrinsic operator, convolution 2025 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol45iss5art5 https://www.opuscula.agh.edu.pl/vol45/5/art/opuscula_math_4531.pdf A note on nonlocal discrete problems involving sign-changing Kirchhoff functions Biagio Ricceri In this note, we establish a multiplicity theorem for a nonlocal discrete problem of the type \[\begin{cases} -\left(a\sum_{m=1}^{n+1}|x_m-x_{m-1}|^2+b\right)(x_{k+1}-2x_k+x_{k-1})=h_k(x_k), &k=1,\ldots ,n, \\ x_0=x_{n+1}=0 \end{cases}\] assuming \(a\gt 0\) and (for the first time) \(b\gt 0\). minimax theorems, Kirchhoff functions, difference equations, variational methods, multiplicity of solutions 2025 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol45iss5art6 https://www.opuscula.agh.edu.pl/vol45/5/art/opuscula_math_4532.pdf Systems of differential inclusions with competing operators and variable exponents Francesca Vetro, Rakib Efendiev In this paper, we study a system of differential inclusions with Dirichlet boundary condition, involving competing operators and variable exponents. More precisely, we investigate the existence of both generalized solutions and weak solutions to the problem under consideration. In order to archive our results, we make use of approximation through finite dimensional subspaces via a Galerkin basis along with minimization and nonsmooth analysis. systems of differential inclusions, hemivariational inequalities, competing operators, Galerkin basis 2025 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol45iss5art7 https://www.opuscula.agh.edu.pl/vol45/5/art/opuscula_math_4533.pdf Normalized solutions for critical Schrödinger equations involving (2,q)-Laplacian Lulu Wei, Yueqiang Song In this paper, we consider the following critical Schrödinger equation involving \((2,q)\)-Laplacian: \[\begin{cases} -\Delta u-\Delta_{q} u=\lambda u+\mu |u|^{\gamma-2}u+|u|^{2^*-2}u \quad\text{in }\mathbb{R}^N, \\ \int_{\mathbb{R}^N} |u|^{2}dx=a^2,\end{cases}\] where \(\Delta_q u =\operatorname{div} (|\nabla u|^{q-2}\nabla u)\) is the \(q\)-Laplacian operator, \(\mu, a\gt 0,\) \(\lambda\in\mathbb{R}\), \(\gamma\in(2,2^*)\), \(q\in(\frac{2N}{N+2},2)\) and \(N\geq3\). The meaningful and interesting phenomenon is the simultaneous occurrence of \((2,q)\)-Laplacian and critical nonlinearity in the above equation. In order to obtain existence of multiple normalized solutions for such equation, we need to make a detailed estimate. More precisely, for the \(L^2\)-subcritical case, we use the truncation technique, concentration-compactness principle and the genus theory to get the existence of multiple normalized solutions. For the \(L^2\)-supercritical case, we obtain a couple of normalized solution for the above equation by a fiber map and concentration-compactness principle. Schrödinger equation, \((2,q)\)-Laplacian, variational methods, critical growth, normalized solutions 2025 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol45iss6 https://www.opuscula.agh.edu.pl/om-vol45iss6art1 https://www.opuscula.agh.edu.pl/vol45/6/art/opuscula_math_4534.pdf A note on distance labeling of graphs Carl Johan Casselgren, Anders Henricsson We study labelings of connected graphs \(G\) using labels \(1,\ldots,|V(G)|\) encoding the distances between vertices in \(G\). Following Lennerstad and Eriksson [Electron. J. Graph Theory Appl. 6 (2018), 152-165], we say that a graph \(G\) which has a labeling \(c\) satisfying that \(d(u,v) \lt d(x,y) \Rightarrow c(u,v) \leq c(x,y)\), where \(c(u,v) = |c(u) - c(v)|\), is a list graph. We show that these graphs are very restricted in nature and enjoy very strong structural properties. Relaxing this condition, we say that a vertex \(u\) in a graph \(G\) with a labeling \(c\) is list-distance consistent if \(d(u,v) \leq d(u,w)\) holds for all vertices \(v\), \(w\) satisfying that \(c(u,w) = c(u,v)+1\). The maximum \(k\) such that \(G\) has a labeling where \(k\) vertices are list-distance consistent is the list-distance consistency \(\operatorname{ldc}(G)\) of \(G\); if \(\operatorname{ldc}(G) = |V(G)|\), then \(G\) is a local list graph. We prove a structural theorem characterizing local list graphs implying that they are a quite restricted family of graphs; this answers a question of Lennerstad. Furthermore, we investigate the parameter \(\operatorname{ldc}(G)\) for various classes of graphs. In particular, we prove that for all \(k\), \(n\) satisfying \(4 \leq k \leq n\) there is a graph \(G\) with \(n\) vertices and \(\operatorname{ldc}(G)=k\), and demonstrate that there are large classes of graphs \(G\) satisfying \(\operatorname{ldc}(G) = 1\). Indeed, we prove that almost every graph have this property, which implies that graphs \(G\) satisfying \(\operatorname{ldc}(G) \gt 1\) are in a sense quite rare (let alone local list graphs). We also suggest further variations on the topic of list graphs for future research. graph labeling, distance labeling 2025 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol45iss6art2 https://www.opuscula.agh.edu.pl/vol45/6/art/opuscula_math_4535.pdf A comprehensive review on the existence of normalized solutions for four classes of nonlinear elliptic equations Sitong Chen, Xianhua Tang This paper provides a comprehensive review of recent results concerning the existence of normalized solutions for four classes of nonlinear elliptic equations: Schrödinger equations, Schrödinger-Poisson equations, Kirchhoff equations, and Choquard equations. normalized solutions, Schrödinger equations, Schrödinger-Poisson equations, Kirchhoff equations, Choquard equations 2025 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol45iss6art3 https://www.opuscula.agh.edu.pl/vol45/6/art/opuscula_math_4536.pdf Anisotropic singular logistic equations João Pablo Pinheiro Da Silva, Giuseppe Failla, Leszek Gasiński, Nikolaos S. Papageorgiou We consider a parametric Dirichlet problem driven by the anisotropic \((p,q)\)-Laplacian and a reaction with a singular term and a superdiffusive logistic perturbation. We prove an existence and nonexistence theorem which is global with respect to the parameter \(\lambda\gt 0\). anisotropic \((p,q)\)-Laplacian, superdiffusive perturbation, anisotropic regularity, Hardy's inequality, strong comparison 2025 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol45iss6art4 https://www.opuscula.agh.edu.pl/vol45/6/art/opuscula_math_4537.pdf Comparison theorems for oscillation and non-oscillation of perturbed Euler type equations Petr Hasil, Jiřina Šišoláková, Michal Veselý The aim of this paper is to present two comparison theorems. These results enable to describe the oscillation behavior of second order Euler type half-linear differential equations with perturbations in both terms using previously obtained oscillation and non-oscillation criteria. We point out that the comparison theorems are easy to use. This fact is also illustrated by a simple example. In addition, the number of perturbations is arbitrary and the last perturbations can be given by very general continuous functions. Note that the presented results are new even in the case of linear equations. comparison theorem, oscillation theory, non-oscillation, half-linear equation, Riccati equation, Prüfer angle 2025 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol45iss6art5 https://www.opuscula.agh.edu.pl/vol45/6/art/opuscula_math_4538.pdf The automorphism groups of domains and the Greene-Krantz conjecture Steven G. Krantz We consider the subject of the automorphism groups of domains in complex space. In particular, we describe and discuss the noted Greene-Krantz conjecture. automorphism group, Greene-Krantz conjecture, boundary orbit accumulation point 2025 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol45iss6art6 https://www.opuscula.agh.edu.pl/vol45/6/art/opuscula_math_4539.pdf On spectral stability for rank one singular perturbations Mario Alberto Ruiz Caballero, Rafael del Río We study the embedded point spectrum of rank one singular perturbations of an arbitrary self-adjoint operator \(A\) on a Hilbert space \(\mathcal{H}\). These perturbations can be regarded as self-adjoint extensions of a densely defined closed symmetric operator \(B\) with deficiency indices \((1,1)\). Assuming the deficiency vector of \(B\) is cyclic for its self-adjoint extensions, we prove that the spectrum of \(A\) contains a dense \(G_{\delta}\) subset on which no eigenvalues occur for the rank one singular perturbations considered. We show this is equivalent to the existence of a dense \(G_{\delta}\) set of rank one singular perturbations of \(A\) such that their eigenvalues are isolated. The approach presented here unifies points of view taken by different authors. self-adjoint extension, rank one singular perturbation, embedded point spectra, singular continuous spectrum 2025 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol45iss6art7 https://www.opuscula.agh.edu.pl/vol45/6/art/opuscula_math_4540.pdf A linear time algorithm to compute vertices that belong to all, some and no minimum dominating sets in a tree and its consequences Radosław Ziemann, Paweł Żyliński We provide a linear time algorithm for determining the sets of vertices that belong to all, some and no minimum dominating sets of a tree, respectively, thus improving the quadratic time algorithm of Benecke and Mynhardt in 2008 [S. Benecke, C.M. Mynhardt, Trees with domination subdivision number one, Australas. J. Comb. 42 (2008), 201-209]. Some algorithmic consequences are also discussed. tree, domination, independence, vertex cover, dissociation, algorithm, linear time, dynamic programming 2025 Opuscula Mathematica ScholarlyArticle https://www.opuscula.agh.edu.pl/om-vol45iss6art8 https://www.opuscula.agh.edu.pl/vol45/6/art/opuscula_math_4541.pdf Corrigendum to "Nontrivial solutions for Neumann fractional p-Laplacian problems" [Opuscula Math. 45, no. 5 (2025), 623-645] Chun Li, Dimitri Mugnai, Tai-Jin Zhao We correct some misprints in [Nontrivial solutions for Neumann fractional p-Laplacian problems, Opuscula Math. 45, no. 5 (2025), 623-645]. 2025 Opuscula Mathematica ScholarlyArticle