Opuscula Mathematica
https://www.opuscula.agh.edu.pl
enA list of articles of the latest volume. The journal Opuscula Mathematica publishes original research articles that are of significant importance in all areas of Discrete Mathematics, Functional Analysis, Differential Equations, Mathematical Physics, Nonlinear Analysis, Numerical Analysis, Probability Theory and Statistics, Theory of Optimal Control and Optimization, Financial Mathematics and Mathematical Economic Theory, Operations Research, and other areas of Applied Mathematics.Sat, 23 Feb 2019 21:30:00 +0100Opuscula Mathematicahttps://www.opuscula.agh.edu.pl/img/opuscula00_0.jpg
https://www.opuscula.agh.edu.pl
General multiplicative Zagreb indices of graphs with given clique number
https://www.opuscula.agh.edu.pl/vol39/3/art/opuscula_math_3927.pdf
https://www.opuscula.agh.edu.pl/vol39/3/art/opuscula_math_3927.pdfSat, 23 Feb 2019 21:00:08 +0100 Author(s): Tomáš Vetrík, Selvaraj Balachandran.

Abstract: 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. Keywords: clique number, multiplicative Zagreb index, chromatic number. Mathematics Subject Classification: 05C35, 05C07. Journal: Opuscula Mathematica. Citation: Opuscula Math. 39, no. 3 (2019), 433-446, https://doi.org/10.7494/OpMath.2019.39.3.433.

]]>The complexity of open k-monopolies in graphs for negative k
https://www.opuscula.agh.edu.pl/vol39/3/art/opuscula_math_3926.pdf
https://www.opuscula.agh.edu.pl/vol39/3/art/opuscula_math_3926.pdfSat, 23 Feb 2019 21:00:07 +0100 Author(s): Iztok Peterin.

Abstract: 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. Keywords: open \(k\)-monopolies, complexity, total domination. Mathematics Subject Classification: 05C85, 05C69, 05C07. Journal: Opuscula Mathematica. Citation: Opuscula Math. 39, no. 3 (2019), 425-431, https://doi.org/10.7494/OpMath.2019.39.3.425.

]]>Metric dimension of Andrásfai graphs
https://www.opuscula.agh.edu.pl/vol39/3/art/opuscula_math_3925.pdf
https://www.opuscula.agh.edu.pl/vol39/3/art/opuscula_math_3925.pdfSat, 23 Feb 2019 21:00:06 +0100 Author(s): S. Batool Pejman, Shiroyeh Payrovi, Ali Behtoei.

Abstract: 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 \(dim_M(And(k)\square C_n)\). Keywords: resolving set, metric dimension, Andrásfai graph, Cayley graph, Cartesian product. Mathematics Subject Classification: 05C12, 05C25. Journal: Opuscula Mathematica. Citation: Opuscula Math. 39, no. 3 (2019), 415-423, https://doi.org/10.7494/OpMath.2019.39.3.415.

]]>Remarks on global solutions to the initial-boundary value problem for quasilinear degenerate parabolic equations with a nonlinear source term
https://www.opuscula.agh.edu.pl/vol39/3/art/opuscula_math_3924.pdf
https://www.opuscula.agh.edu.pl/vol39/3/art/opuscula_math_3924.pdfSat, 23 Feb 2019 21:00:05 +0100 Author(s): Mitsuhiro Nakao.

Abstract: 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. Keywords: degenerate quasilinear parabolic equation, nonlinear source term, Moser's method. Mathematics Subject Classification: 35B40, 35D35, 58J35, 58K30. Journal: Opuscula Mathematica. Citation: Opuscula Math. 39, no. 3 (2019), 395-414, https://doi.org/10.7494/OpMath.2019.39.3.395.

]]>Decomposing complete 3-uniform hypergraph K_{n}^{(3)} into 7-cycles
https://www.opuscula.agh.edu.pl/vol39/3/art/opuscula_math_3923.pdf
https://www.opuscula.agh.edu.pl/vol39/3/art/opuscula_math_3923.pdfSat, 23 Feb 2019 21:00:04 +0100 Author(s): Meihua, Meiling Guan, Jirimutu.

Abstract: 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. Keywords: uniform hypergraph, 7-cycle, cycle decomposition. Mathematics Subject Classification: 05C65, 05C85. Journal: Opuscula Mathematica. Citation: Opuscula Math. 39, no. 3 (2019), 383-393, https://doi.org/10.7494/OpMath.2019.39.3.383.

]]>On the zeros of the Macdonald functions
https://www.opuscula.agh.edu.pl/vol39/3/art/opuscula_math_3922.pdf
https://www.opuscula.agh.edu.pl/vol39/3/art/opuscula_math_3922.pdfSat, 23 Feb 2019 21:00:03 +0100 Author(s): Yuji Hamana, Hiroyuki Matsumoto, Tomoyuki Shirai.

Abstract: 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. Keywords: zeros, Macdonald functions, Bessel functions. Mathematics Subject Classification: 33C10, 30C15, 32A60, 33F05. Journal: Opuscula Mathematica. Citation: Opuscula Math. 39, no. 3 (2019), 361-382, https://doi.org/10.7494/OpMath.2019.39.3.361.

]]>A partial refining of the Erdős-Kelly regulation
https://www.opuscula.agh.edu.pl/vol39/3/art/opuscula_math_3921.pdf
https://www.opuscula.agh.edu.pl/vol39/3/art/opuscula_math_3921.pdfSat, 23 Feb 2019 21:00:02 +0100 Author(s): Joanna Górska, Zdzisław Skupień.

Abstract: 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. Keywords: inducing \(\Delta\)-regulation, cost of regulation. Mathematics Subject Classification: 05C35, 05C75. Journal: Opuscula Mathematica. Citation: Opuscula Math. 39, no. 3 (2019), 355-360, https://doi.org/10.7494/OpMath.2019.39.3.355.

]]>Oscillations of equations caused by several deviating arguments
https://www.opuscula.agh.edu.pl/vol39/3/art/opuscula_math_3920.pdf
https://www.opuscula.agh.edu.pl/vol39/3/art/opuscula_math_3920.pdfSat, 23 Feb 2019 21:00:01 +0100 Author(s): George E. Chatzarakis.

Abstract: 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. Keywords: differential equation, non-monotone argument, oscillatory solution, nonoscillatory solution. Mathematics Subject Classification: 34K11, 34K06. Journal: Opuscula Mathematica. Citation: Opuscula Math. 39, no. 3 (2019), 321-353, https://doi.org/10.7494/OpMath.2019.39.3.321.

]]>Global well-posedness of a class of fourth-order strongly damped nonlinear wave equations
https://www.opuscula.agh.edu.pl/vol39/2/art/opuscula_math_3919.pdf
https://www.opuscula.agh.edu.pl/vol39/2/art/opuscula_math_3919.pdfFri, 07 Dec 2018 12:00:11 +0100 Author(s): Yang Yanbing, Md Salik Ahmed, Qin Lanlan, Xu Runzhang.

Abstract: 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. Keywords: fourth-order nonlinear wave equation, strong damping, blow up, global existence. Mathematics Subject Classification: 35B44, 35L35, 35L05. Journal: Opuscula Mathematica. Citation: Opuscula Math. 39, no. 2 (2019), 297-313, https://doi.org/10.7494/OpMath.2019.39.2.297.

]]>Extremal length and Dirichlet problem on Klein surfaces
https://www.opuscula.agh.edu.pl/vol39/2/art/opuscula_math_3918.pdf
https://www.opuscula.agh.edu.pl/vol39/2/art/opuscula_math_3918.pdfFri, 07 Dec 2018 12:00:10 +0100 Author(s): Monica Roşiu.

Abstract: 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. Keywords: Klein surface, extremal length, extremal distance. Mathematics Subject Classification: 30F50, 35J05, 31A15. Journal: Opuscula Mathematica. Citation: Opuscula Math. 39, no. 2 (2019), 281-296, https://doi.org/10.7494/OpMath.2019.39.2.281.

]]>Isotropic and anisotropic double-phase problems: old and new
https://www.opuscula.agh.edu.pl/vol39/2/art/opuscula_math_3917.pdf
https://www.opuscula.agh.edu.pl/vol39/2/art/opuscula_math_3917.pdfFri, 07 Dec 2018 12:00:09 +0100 Author(s): Vicenţiu D. Rădulescu.

Abstract: 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. Keywords: differential operator with unbalanced growth, double-phase energy, variable exponent. Mathematics Subject Classification: 35J60, 35J65, 58E05. Journal: Opuscula Mathematica. Citation: Opuscula Math. 39, no. 2 (2019), 259-279, https://doi.org/10.7494/OpMath.2019.39.2.259.

]]>Existence and multiplicity results for quasilinear equations in the Heisenberg group
https://www.opuscula.agh.edu.pl/vol39/2/art/opuscula_math_3916.pdf
https://www.opuscula.agh.edu.pl/vol39/2/art/opuscula_math_3916.pdfFri, 07 Dec 2018 12:00:08 +0100 Author(s): Patrizia Pucci.

Abstract: 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 \((u_k)_k\). Keywords: Heisenberg group, entire solutions, critical exponents. Mathematics Subject Classification: 35J62, 35J70, 35B08, 35J20, 35B09. Journal: Opuscula Mathematica. Citation: Opuscula Math. 39, no. 2 (2019), 247-257, https://doi.org/10.7494/OpMath.2019.39.2.247.

]]>On a Robin (p,q)-equation with a logistic reaction
https://www.opuscula.agh.edu.pl/vol39/2/art/opuscula_math_3915.pdf
https://www.opuscula.agh.edu.pl/vol39/2/art/opuscula_math_3915.pdfFri, 07 Dec 2018 12:00:07 +0100 Author(s): Nikolaos S. Papageorgiou, Calogero Vetro, Francesca Vetro.

Abstract: 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. Keywords: positive solutions, superdiffusive reaction, local minimizers, maximum principle, minimal positive solutions, Robin boundary condition, indefinite potential. Mathematics Subject Classification: 35J20, 35J60. Journal: Opuscula Mathematica. Citation: Opuscula Math. 39, no. 2 (2019), 227-245, https://doi.org/10.7494/OpMath.2019.39.2.227.

]]>Controllability of degenerate and singular parabolic problems: the double strong case with Neumann boundary conditions
https://www.opuscula.agh.edu.pl/vol39/2/art/opuscula_math_3914.pdf
https://www.opuscula.agh.edu.pl/vol39/2/art/opuscula_math_3914.pdfFri, 07 Dec 2018 12:00:06 +0100 Author(s): Genni Fragnelli, Dimitri Mugnai.

Abstract: 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. Keywords: strongly singular/degenerate equations, non smooth coefficients, null controllability. Mathematics Subject Classification: 35Q93, 93B05, 34H05. Journal: Opuscula Mathematica. Citation: Opuscula Math. 39, no. 2 (2019), 207-225, https://doi.org/10.7494/OpMath.2019.39.2.207.

]]>Existence results and a priori estimates for solutions of quasilinear problems with gradient terms
https://www.opuscula.agh.edu.pl/vol39/2/art/opuscula_math_3913.pdf
https://www.opuscula.agh.edu.pl/vol39/2/art/opuscula_math_3913.pdfFri, 07 Dec 2018 12:00:05 +0100 Author(s): Roberta Filippucci, Chiara Lini.

Abstract: 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 \(u=0\). Keywords: existence result, quasilinear problems, a priori estimates. Mathematics Subject Classification: 35J92, 35J70. Journal: Opuscula Mathematica. Citation: Opuscula Math. 39, no. 2 (2019), 195-206, https://doi.org/10.7494/OpMath.2019.39.2.195.

]]>Infinitely many solutions for some nonlinear supercritical problems with break of symmetry
https://www.opuscula.agh.edu.pl/vol39/2/art/opuscula_math_3912.pdf
https://www.opuscula.agh.edu.pl/vol39/2/art/opuscula_math_3912.pdfFri, 07 Dec 2018 12:00:04 +0100 Author(s): Anna Maria Candela, Addolorata Salvatore.

Abstract: 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. Keywords: quasilinear elliptic equation, weak Cerami-Palais-Smale condition, Ambrosetti-Rabinowitz condition, break of symmetry, perturbation method, supercritical growth. Mathematics Subject Classification: 35J20, 35J62, 35J66, 58E05. Journal: Opuscula Mathematica. Citation: Opuscula Math. 39, no. 2 (2019), 175-194, https://doi.org/10.7494/OpMath.2019.39.2.175.

]]>Nonlinear elliptic equations involving the p-Laplacian with mixed Dirichlet-Neumann boundary conditions
https://www.opuscula.agh.edu.pl/vol39/2/art/opuscula_math_3911.pdf
https://www.opuscula.agh.edu.pl/vol39/2/art/opuscula_math_3911.pdfFri, 07 Dec 2018 12:00:03 +0100 Author(s): Gabriele Bonanno, Giuseppina D'Aguì, Angela Sciammetta.

Abstract: 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. Keywords: mixed problem, critical points. Mathematics Subject Classification: 35J25, 35J20. Journal: Opuscula Mathematica. Citation: Opuscula Math. 39, no. 2 (2019), 159-174, https://doi.org/10.7494/OpMath.2019.39.2.159.

]]>Some remarks on the coincidence set for the Signorini problem
https://www.opuscula.agh.edu.pl/vol39/2/art/opuscula_math_3910.pdf
https://www.opuscula.agh.edu.pl/vol39/2/art/opuscula_math_3910.pdfFri, 07 Dec 2018 12:00:02 +0100 Author(s): Miguel de Benito Delgado, Jesus Ildefonso Díaz.

Abstract: 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. Keywords: Signorini problem, coincidence set, location estimates, free boundary problem, contact problems. Mathematics Subject Classification: 35J86, 35R35, 35R70, 35B60. Journal: Opuscula Mathematica. Citation: Opuscula Math. 39, no. 2 (2019), 145-157, https://doi.org/10.7494/OpMath.2019.39.2.145.

]]>On unique solvability of a Dirichlet problem with nonlinearity depending on
the derivative
https://www.opuscula.agh.edu.pl/vol39/2/art/opuscula_math_3909.pdf
https://www.opuscula.agh.edu.pl/vol39/2/art/opuscula_math_3909.pdfFri, 07 Dec 2018 12:00:01 +0100 Author(s): Michał Bełdziński, Marek Galewski.

Abstract: 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. Keywords: diffeomorphism, uniqueness, non-potential problems, variational methods, monotone methods, Palais-Smale condition. Mathematics Subject Classification: 34A12, 47H30, 47J07. Journal: Opuscula Mathematica. Citation: Opuscula Math. 39, no. 2 (2019), 131-144, https://doi.org/10.7494/OpMath.2019.39.2.131.

]]>Pseudo-differential equations and conical potentials: 2-dimensional case
https://www.opuscula.agh.edu.pl/vol39/1/art/opuscula_math_3908.pdf
https://www.opuscula.agh.edu.pl/vol39/1/art/opuscula_math_3908.pdfTue, 07 Aug 2018 10:00:08 +0200 Author(s): Vladimir B. Vasilyev.

Abstract: 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. Keywords: pseudo-differential equation, wave factorization, Dirichlet problem, system of linear integral equations. Mathematics Subject Classification: 35S15, 45A05. Journal: Opuscula Mathematica. Citation: Opuscula Math. 39, no. 1 (2019), 109-124, https://doi.org/10.7494/OpMath.2019.39.1.109.

]]>Oscillation criteria for even order neutral difference equations
https://www.opuscula.agh.edu.pl/vol39/1/art/opuscula_math_3907.pdf
https://www.opuscula.agh.edu.pl/vol39/1/art/opuscula_math_3907.pdfTue, 07 Aug 2018 10:00:07 +0200 Author(s): S. Selvarangam, S. A. Rupadevi, E. Thandapani, S. Pinelas.

Abstract: 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. Keywords: even order, neutral difference equation, oscillation, asymptotic behavior, mixed type. Mathematics Subject Classification: 39A10. Journal: Opuscula Mathematica. Citation: Opuscula Math. 39, no. 1 (2019), 91-108, https://doi.org/10.7494/OpMath.2019.39.1.91.

]]>The existence of consensus of a leader-following problem with Caputo fractional derivative
https://www.opuscula.agh.edu.pl/vol39/1/art/opuscula_math_3906.pdf
https://www.opuscula.agh.edu.pl/vol39/1/art/opuscula_math_3906.pdfTue, 07 Aug 2018 10:00:06 +0200 Author(s): Ewa Schmeidel.

Abstract: 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. Keywords: leader-following problem, Caputo fractional differential equation, consensus, nonlinear system, Schauder fixed point theorem. Mathematics Subject Classification: 26A33, 34K20, 45D05. Journal: Opuscula Mathematica. Citation: Opuscula Math. 39, no. 1 (2019), 77-89, https://doi.org/10.7494/OpMath.2019.39.1.77.

]]>On the convergence of solutions to second-order neutral difference equations
https://www.opuscula.agh.edu.pl/vol39/1/art/opuscula_math_3905.pdf
https://www.opuscula.agh.edu.pl/vol39/1/art/opuscula_math_3905.pdfTue, 07 Aug 2018 10:00:05 +0200 Author(s): Małgorzata Migda, Janusz Migda, Małgorzata Zdanowicz.

Abstract: 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. Keywords: second-order difference equation, asymptotic behavior, quasi-differences, Krasnoselskii's fixed point theorem. Mathematics Subject Classification: 39A10, 39A22, 39A30. Journal: Opuscula Mathematica. Citation: Opuscula Math. 39, no. 1 (2019), 61-75, https://doi.org/10.7494/OpMath.2019.39.1.61.

]]>Boundary value problems with solutions in convex sets
https://www.opuscula.agh.edu.pl/vol39/1/art/opuscula_math_3904.pdf
https://www.opuscula.agh.edu.pl/vol39/1/art/opuscula_math_3904.pdfTue, 07 Aug 2018 10:00:04 +0200 Author(s): Gerd Herzog, Peer Chr. Kunstmann.

Abstract: 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. Keywords: Dirichlet boundary value problems, solutions in convex sets, continuation method, ordered Banach spaces, concave solutions. Mathematics Subject Classification: 34B15, 47H10. Journal: Opuscula Mathematica. Citation: Opuscula Math. 39, no. 1 (2019), 49-60, https://doi.org/10.7494/OpMath.2019.39.1.49.

]]>Oscillatory behavior of even-order nonlinear differential equations with a sublinear neutral term
https://www.opuscula.agh.edu.pl/vol39/1/art/opuscula_math_3903.pdf
https://www.opuscula.agh.edu.pl/vol39/1/art/opuscula_math_3903.pdfTue, 07 Aug 2018 10:00:03 +0200 Author(s): John R. Graef, Said R. Grace, Ercan Tunç.

Abstract: 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. Keywords: oscillatory behavior, neutral differential equation, even-order. Mathematics Subject Classification: 34C10, 34K11. Journal: Opuscula Mathematica. Citation: Opuscula Math. 39, no. 1 (2019), 39-47, https://doi.org/10.7494/OpMath.2019.39.1.39.

]]>Dynamic system with random structure for modeling security and risk management in cyberspace
https://www.opuscula.agh.edu.pl/vol39/1/art/opuscula_math_3902.pdf
https://www.opuscula.agh.edu.pl/vol39/1/art/opuscula_math_3902.pdfTue, 07 Aug 2018 10:00:02 +0200 Author(s): Irada Dzhalladova, Miroslava Růžičková.

Abstract: 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. Keywords: semi-Markov chain, random transformation of solutions, the Lyapunov function, \(L_{2}\)-stability, systems of difference equations, jumps of solutions, cybersecurity. Mathematics Subject Classification: 34F05, 60J28. Journal: Opuscula Mathematica. Citation: Opuscula Math. 39, no. 1 (2019), 23-37, https://doi.org/10.7494/OpMath.2019.39.1.23.

]]>Difference equations with impulses
https://www.opuscula.agh.edu.pl/vol39/1/art/opuscula_math_3901.pdf
https://www.opuscula.agh.edu.pl/vol39/1/art/opuscula_math_3901.pdfTue, 07 Aug 2018 10:00:01 +0200 Author(s): Marius Danca, Michal Fečkan, Michal Pospíšil.

Abstract: 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. Keywords: difference equations, impulses, stability, fixed points, Li-Yorke chaos. Mathematics Subject Classification: 37B55, 39A23, 39A33, 39A60. Journal: Opuscula Mathematica. Citation: Opuscula Math. 39, no. 1 (2019), 5-22, https://doi.org/10.7494/OpMath.2019.39.1.5.