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.Thu, 08 Sep 2022 10:30:00 +0200Opuscula Mathematicahttps://www.opuscula.agh.edu.pl/img/opuscula00_0.jpg
https://www.opuscula.agh.edu.pl
Notes on aplications of the dual fountain theorem to local and nonlocal elliptic equations with variable exponent
https://www.opuscula.agh.edu.pl/vol42/5/art/opuscula_math_4235.pdf
https://www.opuscula.agh.edu.pl/vol42/5/art/opuscula_math_4235.pdfThu, 08 Sep 2022 10:00:06 +0200 Author(s): Robert Stegliński.

Abstract: 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. Keywords: dual fountain theorem, \(p(\cdot)\)-Laplacian, fractional \(p(\cdot)\)-Laplacian, infinitely many solutions. Mathematics Subject Classification: 35J60, 35D30, 35J20. Journal: Opuscula Mathematica. Citation: Opuscula Math. 42, no. 5 (2022), 751-761, https://doi.org/10.7494/OpMath.2022.42.5.751.

]]>Positive stationary solutions of convection-diffusion equations for superlinear sources
https://www.opuscula.agh.edu.pl/vol42/5/art/opuscula_math_4234.pdf
https://www.opuscula.agh.edu.pl/vol42/5/art/opuscula_math_4234.pdfThu, 08 Sep 2022 10:00:05 +0200 Author(s): Aleksandra Orpel.

Abstract: 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. Keywords: semipositone problems, positive stationary solutions, minimal solutions with finite energy, sub and supersolutions methods. Mathematics Subject Classification: 35B09, 35B40, 35J15, 35J61. Journal: Opuscula Mathematica. Citation: Opuscula Math. 42, no. 5 (2022), 727-749, https://doi.org/10.7494/OpMath.2022.42.5.727.

]]>On the numerical solution of one inverse problem for a linearized two-dimensional system of Navier-Stokes equations
https://www.opuscula.agh.edu.pl/vol42/5/art/opuscula_math_4233.pdf
https://www.opuscula.agh.edu.pl/vol42/5/art/opuscula_math_4233.pdfThu, 08 Sep 2022 10:00:04 +0200 Author(s): Muvasharkhan Jenaliyev, Murat Ramazanov, Madi Yergaliyev.

Abstract: 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. Keywords: Navier-Stokes equations, inverse problem, numerical solution. Mathematics Subject Classification: 35Q30, 35R30, 65N21. Journal: Opuscula Mathematica. Citation: Opuscula Math. 42, no. 5 (2022), 709-725, https://doi.org/10.7494/OpMath.2022.42.5.709.

]]>Nonlinear Choquard equations on hyperbolic space
https://www.opuscula.agh.edu.pl/vol42/5/art/opuscula_math_4232.pdf
https://www.opuscula.agh.edu.pl/vol42/5/art/opuscula_math_4232.pdfThu, 08 Sep 2022 10:00:03 +0200 Author(s): Haiyang He.

Abstract: 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. Keywords: nonlinear Choquard equation, hyperbolic space, existence solutions, Hardy-Littlewood-Sobolev inequality. Mathematics Subject Classification: 35A01, 35J60. Journal: Opuscula Mathematica. Citation: Opuscula Math. 42, no. 5 (2022), 691-708, https://doi.org/10.7494/OpMath.2022.42.5.691.

]]>Stability switches in a linear differential equation with two delays
https://www.opuscula.agh.edu.pl/vol42/5/art/opuscula_math_4231.pdf
https://www.opuscula.agh.edu.pl/vol42/5/art/opuscula_math_4231.pdfThu, 08 Sep 2022 10:00:02 +0200 Author(s): Yuki Hata, Hideaki Matsunaga.

Abstract: 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. Keywords: delay differential equations, stability switches, two delays. Mathematics Subject Classification: 34K06, 34K20, 34K25. Journal: Opuscula Mathematica. Citation: Opuscula Math. 42, no. 5 (2022), 673-690, https://doi.org/10.7494/OpMath.2022.42.5.673.

]]>Properties of even order linear functional differential equations with deviating arguments of mixed type
https://www.opuscula.agh.edu.pl/vol42/5/art/opuscula_math_4230.pdf
https://www.opuscula.agh.edu.pl/vol42/5/art/opuscula_math_4230.pdfThu, 08 Sep 2022 10:00:01 +0200 Author(s): Jozef Dzurina.

Abstract: 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 \(p(t)\sim a/t^n.\) Keywords: higher order differential equations, mixed argument, monotonic properties, oscillation. Mathematics Subject Classification: 34K11, 34C10. Journal: Opuscula Mathematica. Citation: Opuscula Math. 42, no. 5 (2022), 659-671, https://doi.org/10.7494/OpMath.2022.42.5.659.

]]>The crossing numbers of join products of paths with three graphs of order five
https://www.opuscula.agh.edu.pl/vol42/4/art/opuscula_math_4229.pdf
https://www.opuscula.agh.edu.pl/vol42/4/art/opuscula_math_4229.pdfThu, 30 Jun 2022 10:30:07 +0200 Author(s): Michal Staš, Mária Švecová.

Abstract: 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 \(P_n\). Keywords: graph, crossing number, join product, cyclic permutation, path. Mathematics Subject Classification: 05C10, 05C38. Journal: Opuscula Mathematica. Citation: Opuscula Math. 42, no. 4 (2022), 635-651, https://doi.org/10.7494/OpMath.2022.42.4.635.

]]>Critical cases in neutral functional differential equations, arising from hydraulic engineering
https://www.opuscula.agh.edu.pl/vol42/4/art/opuscula_math_4228.pdf
https://www.opuscula.agh.edu.pl/vol42/4/art/opuscula_math_4228.pdfThu, 30 Jun 2022 10:30:06 +0200 Author(s): Vladimir Răsvan.

Abstract: 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. Keywords: \(1D\) hyperbolic partial differential equations, neutral functional differential equations, difference operator, critical case. Mathematics Subject Classification: 34K20, 34K40, 35B35, 35L50. Journal: Opuscula Mathematica. Citation: Opuscula Math. 42, no. 4 (2022), 605-633, https://doi.org/10.7494/OpMath.2022.42.4.605.

]]>Fractional operators and their commutators on generalized Orlicz spaces
https://www.opuscula.agh.edu.pl/vol42/4/art/opuscula_math_4227.pdf
https://www.opuscula.agh.edu.pl/vol42/4/art/opuscula_math_4227.pdfThu, 30 Jun 2022 10:30:05 +0200 Author(s): Arttu Karppinen.

Abstract: 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. Keywords: maximal operator, commutator, fractional operator, generalized Orlicz, Musielak-Orlicz. Mathematics Subject Classification: 46E30, 42B35, . Journal: Opuscula Mathematica. Citation: Opuscula Math. 42, no. 4 (2022), 583-604, https://doi.org/10.7494/OpMath.2022.42.4.583.

]]>Nordhaus-Gaddum bounds for upper total domination
https://www.opuscula.agh.edu.pl/vol42/4/art/opuscula_math_4226.pdf
https://www.opuscula.agh.edu.pl/vol42/4/art/opuscula_math_4226.pdfThu, 30 Jun 2022 10:30:04 +0200 Author(s): Teresa W. Haynes, Michael A. Henning.

Abstract: 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. Keywords: upper total domination, Nordhaus-Gaddum bounds. Mathematics Subject Classification: 05C69. Journal: Opuscula Mathematica. Citation: Opuscula Math. 42, no. 4 (2022), 573-582, https://doi.org/10.7494/OpMath.2022.42.4.573.

]]>Upper bounds on distance vertex irregularity strength of some families of graphs
https://www.opuscula.agh.edu.pl/vol42/4/art/opuscula_math_4225.pdf
https://www.opuscula.agh.edu.pl/vol42/4/art/opuscula_math_4225.pdfThu, 30 Jun 2022 10:30:03 +0200 Author(s): Sylwia Cichacz, Agnieszka Görlich, Andrea Semaničová-Feňovčíková.

Abstract: 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. Keywords: distance vertex irregularity strength of a graph, hypercube, tree. Mathematics Subject Classification: 05C65, 05C78. Journal: Opuscula Mathematica. Citation: Opuscula Math. 42, no. 4 (2022), 561-571, https://doi.org/10.7494/OpMath.2022.42.4.561.

]]>Oscillation of even order linear functional differential equations with mixed deviating arguments
https://www.opuscula.agh.edu.pl/vol42/4/art/opuscula_math_4224.pdf
https://www.opuscula.agh.edu.pl/vol42/4/art/opuscula_math_4224.pdfThu, 30 Jun 2022 10:30:02 +0200 Author(s): Blanka Baculikova.

Abstract: 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. Keywords: higher order differential equations, mixed argument, monotonic properties, oscillation. Mathematics Subject Classification: 34K11, 34C10. Journal: Opuscula Mathematica. Citation: Opuscula Math. 42, no. 4 (2022), 549-560, https://doi.org/10.7494/OpMath.2022.42.4.549.

]]>The strong 3-rainbow index of some certain graphs and its amalgamation
https://www.opuscula.agh.edu.pl/vol42/4/art/opuscula_math_4223.pdf
https://www.opuscula.agh.edu.pl/vol42/4/art/opuscula_math_4223.pdfThu, 30 Jun 2022 10:30:01 +0200 Author(s): Zata Yumni Awanis, A.N.M. Salman.

Abstract: 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. Keywords: amalgamation, rainbow coloring, rainbow Steiner tree, strong \(k\)-rainbow index. Mathematics Subject Classification: 05C05, 05C15, 05C40. Journal: Opuscula Mathematica. Citation: Opuscula Math. 42, no. 4 (2022), 527-547, https://doi.org/10.7494/OpMath.2022.42.4.527.

]]>Existence of positive continuous weak solutions for some semilinear elliptic eigenvalue problems
https://www.opuscula.agh.edu.pl/vol42/3/art/opuscula_math_4222.pdf
https://www.opuscula.agh.edu.pl/vol42/3/art/opuscula_math_4222.pdfFri, 29 Apr 2022 21:30:07 +0200 Author(s): Noureddine Zeddini, Rehab Saeed Sari.

Abstract: 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 \(K(D)\). Keywords: Green function, Kato class, nonlinear elliptic systems, positive solution, maximum principle, Schauder fixed point theorem. Mathematics Subject Classification: 31A35, 31B35, 31A16, 35B09, 35B50, 35J08, 35J57. Journal: Opuscula Mathematica. Citation: Opuscula Math. 42, no. 3 (2022), 489-519, https://doi.org/10.7494/OpMath.2022.42.3.489.

Abstract: 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. Keywords: spectral operators, chains, triangular decomposition, Laurent operators, Jacobi matrices. Mathematics Subject Classification: 47B40, 47B28, 47B36, 47B35, 47B39. Journal: Opuscula Mathematica. Citation: Opuscula Math. 42, no. 3 (2022), 459-487, https://doi.org/10.7494/OpMath.2022.42.3.459.

]]>Distance irregularity strength of graphs with pendant vertices
https://www.opuscula.agh.edu.pl/vol42/3/art/opuscula_math_4220.pdf
https://www.opuscula.agh.edu.pl/vol42/3/art/opuscula_math_4220.pdfFri, 29 Apr 2022 21:30:05 +0200 Author(s): Faisal Susanto, Kristiana Wijaya, Slamin, Andrea Semaničová-Feňovčíková.

Abstract: 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. Keywords: vertex \(k\)-labeling, distance irregular vertex \(k\)-labeling, distance irregularity strength, pendant vertices. Mathematics Subject Classification: 05C78, 05C12. Journal: Opuscula Mathematica. Citation: Opuscula Math. 42, no. 3 (2022), 439-458, https://doi.org/10.7494/OpMath.2022.42.3.439.

]]>On Ambarzumian type theorems for tree domains
https://www.opuscula.agh.edu.pl/vol42/3/art/opuscula_math_4219.pdf
https://www.opuscula.agh.edu.pl/vol42/3/art/opuscula_math_4219.pdfFri, 29 Apr 2022 21:30:04 +0200 Author(s): Vyacheslav Pivovarchik.

Abstract: 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. Keywords: Sturm-Liouville equation, eigenvalue, equilateral tree, star graph, Dirichlet boundary condition, Neumann boundary condition. Mathematics Subject Classification: 34B45, 34B24, 34L20. Journal: Opuscula Mathematica. Citation: Opuscula Math. 42, no. 3 (2022), 427-437, https://doi.org/10.7494/OpMath.2022.42.3.427.

]]>Growth of solutions of a class of linear fractional differential equations with polynomial coefficients
https://www.opuscula.agh.edu.pl/vol42/3/art/opuscula_math_4218.pdf
https://www.opuscula.agh.edu.pl/vol42/3/art/opuscula_math_4218.pdfFri, 29 Apr 2022 21:30:03 +0200 Author(s): Saada Hamouda, Sofiane Mahmoudi.

Abstract: 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. Keywords: linear fractional differential equations, growth of solutions, Caputo fractional derivative operator. Mathematics Subject Classification: 34M10, 26A33. Journal: Opuscula Mathematica. Citation: Opuscula Math. 42, no. 3 (2022), 415-426, https://doi.org/10.7494/OpMath.2022.42.3.415.

]]>New aspects for the oscillation of first-order difference equations with deviating arguments
https://www.opuscula.agh.edu.pl/vol42/3/art/opuscula_math_4217.pdf
https://www.opuscula.agh.edu.pl/vol42/3/art/opuscula_math_4217.pdfFri, 29 Apr 2022 21:30:02 +0200 Author(s): Emad R. Attia, Bassant M. El-Matary.

Abstract: 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. Keywords: difference equations, oscillation, non-monotone advanced arguments. Mathematics Subject Classification: 39A10, 39A21. Journal: Opuscula Mathematica. Citation: Opuscula Math. 42, no. 3 (2022), 393-413, https://doi.org/10.7494/OpMath.2022.42.3.393.

]]>Monodromy invariant Hermitian forms for second order Fuchsian differential equations with four singularities
https://www.opuscula.agh.edu.pl/vol42/3/art/opuscula_math_4216.pdf
https://www.opuscula.agh.edu.pl/vol42/3/art/opuscula_math_4216.pdfFri, 29 Apr 2022 21:30:01 +0200 Author(s): Shunya Adachi.

Abstract: 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. Keywords: Fuchsian differential equations, monodromy representation, monodromy invariant Hermitian form. Mathematics Subject Classification: 34M35, 34M15. Journal: Opuscula Mathematica. Citation: Opuscula Math. 42, no. 3 (2022), 361-391, https://doi.org/10.7494/OpMath.2022.42.3.361.

]]>Ground states of coupled critical Choquard equations with weighted potentials
https://www.opuscula.agh.edu.pl/vol42/2/art/opuscula_math_4215.pdf
https://www.opuscula.agh.edu.pl/vol42/2/art/opuscula_math_4215.pdfFri, 25 Feb 2022 14:30:09 +0100 Author(s): Gaili Zhu, Chunping Duan, Jianjun Zhang, Huixing Zhang.

Abstract: 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. Keywords: ground states, Choquard equations, Hardy-Littlewood-Sobolev inequality, lower critical exponent. Mathematics Subject Classification: 35B25, 35B33, 35J61. Journal: Opuscula Mathematica. Citation: Opuscula Math. 42, no. 2 (2022), 337-354, https://doi.org/10.7494/OpMath.2022.42.2.337.

]]>On some inverse problem for bi-parabolic equation with observed data in L^{p} spaces
https://www.opuscula.agh.edu.pl/vol42/2/art/opuscula_math_4214.pdf
https://www.opuscula.agh.edu.pl/vol42/2/art/opuscula_math_4214.pdfFri, 25 Feb 2022 14:30:08 +0100 Author(s): Nguyen Huy Tuan.

Abstract: 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. Keywords: bi-parabolic equations, Fourier truncation method, inverse source parabolic, inverse initial problem, Sobolev embeddings, Sobolev embeddings. Mathematics Subject Classification: 35A05, 35A08. Journal: Opuscula Mathematica. Citation: Opuscula Math. 42, no. 2 (2022), 305-335, https://doi.org/10.7494/OpMath.2022.42.2.305.

]]>Entire solutions for some critical equations in the Heisenberg group
https://www.opuscula.agh.edu.pl/vol42/2/art/opuscula_math_4213.pdf
https://www.opuscula.agh.edu.pl/vol42/2/art/opuscula_math_4213.pdfFri, 25 Feb 2022 14:30:07 +0100 Author(s): Patrizia Pucci, Letizia Temperini.

Abstract: 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. Keywords: Heisenberg group, entire solutions, critical exponents. Mathematics Subject Classification: 35J62, 35J70, 35B08, 35J20, 35B09. Journal: Opuscula Mathematica. Citation: Opuscula Math. 42, no. 2 (2022), 279-303, https://doi.org/10.7494/OpMath.2022.42.2.279.

]]>Double phase problems: a survey of some recent results
https://www.opuscula.agh.edu.pl/vol42/2/art/opuscula_math_4212.pdf
https://www.opuscula.agh.edu.pl/vol42/2/art/opuscula_math_4212.pdfFri, 25 Feb 2022 14:30:06 +0100 Author(s): Nikolaos S. Papageorgiou.

Abstract: 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. Keywords: double phase integrand, generalized Orlicz spaces, regularity theory, maximum principle, Nehari manifold. Mathematics Subject Classification: 35J20, 35J60. Journal: Opuscula Mathematica. Citation: Opuscula Math. 42, no. 2 (2022), 257-278, https://doi.org/10.7494/OpMath.2022.42.2.257.

]]>Exponential decay of solutions to a class of fourth-order nonlinear hyperbolic equations modeling the oscillations of suspension bridges
https://www.opuscula.agh.edu.pl/vol42/2/art/opuscula_math_4211.pdf
https://www.opuscula.agh.edu.pl/vol42/2/art/opuscula_math_4211.pdfFri, 25 Feb 2022 14:30:05 +0100 Author(s): Yang Liu, Chao Yang.

Abstract: 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. Keywords: fourth-order nonlinear hyperbolic equations, weak solutions, exponential decay, a family of potential wells. Mathematics Subject Classification: 35L35, 35D30, 35B40. Journal: Opuscula Mathematica. Citation: Opuscula Math. 42, no. 2 (2022), 239-255, https://doi.org/10.7494/OpMath.2022.42.2.239.

]]>Blowup phenomena for some fourth-order strain wave equations at arbitrary positive initial energy level
https://www.opuscula.agh.edu.pl/vol42/2/art/opuscula_math_4210.pdf
https://www.opuscula.agh.edu.pl/vol42/2/art/opuscula_math_4210.pdfFri, 25 Feb 2022 14:30:04 +0100 Author(s): Qiang Lin, Yongbing Luo.

Abstract: 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. Keywords: fourth-order strain wave equation, arbitrary positive initial energy, blowup, blowup time. Mathematics Subject Classification: 35L05, 35A01, 35L55. Journal: Opuscula Mathematica. Citation: Opuscula Math. 42, no. 2 (2022), 219-238, https://doi.org/10.7494/OpMath.2022.42.2.219.

]]>The d-bar formalism for the modified Veselov-Novikov equation on the half-plane
https://www.opuscula.agh.edu.pl/vol42/2/art/opuscula_math_4209.pdf
https://www.opuscula.agh.edu.pl/vol42/2/art/opuscula_math_4209.pdfFri, 25 Feb 2022 14:30:03 +0100 Author(s): Guenbo Hwang, Byungsoo Moon.

Abstract: 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. Keywords: initial-boundary value problem, integrable nonlinear PDE, spectral analysis, \(d\)-bar. Mathematics Subject Classification: 35G31, 35Q53, 37K15. Journal: Opuscula Mathematica. Citation: Opuscula Math. 42, no. 2 (2022), 179-217, https://doi.org/10.7494/OpMath.2022.42.2.179.

]]>Ground states for fractional nonlocal equations with logarithmic nonlinearity
https://www.opuscula.agh.edu.pl/vol42/2/art/opuscula_math_4208.pdf
https://www.opuscula.agh.edu.pl/vol42/2/art/opuscula_math_4208.pdfFri, 25 Feb 2022 14:30:02 +0100 Author(s): Lifeng Guo, Yan Sun, Guannan Shi.

Abstract: 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. Keywords: linking theorem, ground state, logarithmic nonlinearity, variational methods. Mathematics Subject Classification: 35J20, 35B33, 58E05. Journal: Opuscula Mathematica. Citation: Opuscula Math. 42, no. 2 (2022), 157-178, https://doi.org/10.7494/OpMath.2022.42.2.157.

]]>Global existence and blow-up phenomenon for a quasilinear viscoelastic equation with strong damping and source terms
https://www.opuscula.agh.edu.pl/vol42/2/art/opuscula_math_4207.pdf
https://www.opuscula.agh.edu.pl/vol42/2/art/opuscula_math_4207.pdfFri, 25 Feb 2022 14:30:01 +0100 Author(s): Huafei Di, Zefang Song.

Abstract: 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. Keywords: viscoelastic equation, strong damping and source, blow-up, upper and lower bounds, invariant set, potential well. Mathematics Subject Classification: 35L35, 35L75, 35R15. Journal: Opuscula Mathematica. Citation: Opuscula Math. 42, no. 2 (2022), 119-155, https://doi.org/10.7494/OpMath.2022.42.2.119.

]]>All metric bases and fault-tolerant metric dimension for square of grid
https://www.opuscula.agh.edu.pl/vol42/1/art/opuscula_math_4206.pdf
https://www.opuscula.agh.edu.pl/vol42/1/art/opuscula_math_4206.pdfThu, 20 Jan 2022 18:00:06 +0100 Author(s): Laxman Saha, Mithun Basak, Kalishankar Tiwary.

Abstract: 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. Keywords: code, resolving set, metric dimension, fault-tolerant resolving set, fault-tolerant metric dimension. Mathematics Subject Classification: 05C12, 05C05, 05C90, 05C76. Journal: Opuscula Mathematica. Citation: Opuscula Math. 42, no. 1 (2022), 93-111, https://doi.org/10.7494/OpMath.2022.42.1.93.

]]>Solution of the boundary value problem of heat conduction in a cone
https://www.opuscula.agh.edu.pl/vol42/1/art/opuscula_math_4205.pdf
https://www.opuscula.agh.edu.pl/vol42/1/art/opuscula_math_4205.pdfThu, 20 Jan 2022 18:00:05 +0100 Author(s): Murat Ramazanov, Muvasharkhan Jenaliyev, Nurtay Gulmanov.

Abstract: 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. Keywords: noncylindrical domain, cone, boundary value problem of heat conduction, singular Volterra integral equation, Carleman-Vekua regularization method. Mathematics Subject Classification: 35K05, 45D99. Journal: Opuscula Mathematica. Citation: Opuscula Math. 42, no. 1 (2022), 75-91, https://doi.org/10.7494/OpMath.2022.42.1.75.

]]>Edge homogeneous colorings
https://www.opuscula.agh.edu.pl/vol42/1/art/opuscula_math_4204.pdf
https://www.opuscula.agh.edu.pl/vol42/1/art/opuscula_math_4204.pdfThu, 20 Jan 2022 18:00:04 +0100 Author(s): Tomáš Madaras, Alfréd Onderko, Thomas Schweser.

Abstract: 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. Keywords: homogeneous coloring, \(M_q\)-coloring, line graph, role coloring. Mathematics Subject Classification: 05C15. Journal: Opuscula Mathematica. Citation: Opuscula Math. 42, no. 1 (2022), 65-73, https://doi.org/10.7494/OpMath.2022.42.1.65.

]]>Kneser-type oscillation criteria for second-order half-linear advanced difference equations
https://www.opuscula.agh.edu.pl/vol42/1/art/opuscula_math_4203.pdf
https://www.opuscula.agh.edu.pl/vol42/1/art/opuscula_math_4203.pdfThu, 20 Jan 2022 18:00:03 +0100 Author(s): N. Indrajith, John R. Graef, E. Thandapani.

Abstract: 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. Keywords: second-order difference equations, advanced argument, half-linear, oscillation. Mathematics Subject Classification: 39A10. Journal: Opuscula Mathematica. Citation: Opuscula Math. 42, no. 1 (2022), 55-64, https://doi.org/10.7494/OpMath.2022.42.1.55.

]]>γ-paired dominating graphs of cycles
https://www.opuscula.agh.edu.pl/vol42/1/art/opuscula_math_4202.pdf
https://www.opuscula.agh.edu.pl/vol42/1/art/opuscula_math_4202.pdfThu, 20 Jan 2022 18:00:02 +0100 Author(s): Pannawat Eakawinrujee, Nantapath Trakultraipruk.

Abstract: 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. Keywords: paired dominating graph, paired dominating set, paired domination number. Mathematics Subject Classification: 05C69, 05C38. Journal: Opuscula Mathematica. Citation: Opuscula Math. 42, no. 1 (2022), 31-54, https://doi.org/10.7494/OpMath.2022.42.1.31.

]]>Uniqueness of solution of a nonlinear evolution dam problem in a heterogeneous porous medium
https://www.opuscula.agh.edu.pl/vol42/1/art/opuscula_math_4201.pdf
https://www.opuscula.agh.edu.pl/vol42/1/art/opuscula_math_4201.pdfThu, 20 Jan 2022 18:00:01 +0100 Author(s): Messaouda Ben Attia, Elmehdi Zaouche, Mahmoud Bousselsal.

Abstract: 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. Keywords: test function, method of doubling variables, nonlinear evolution dam problem, heterogeneous porous medium, uniqueness. Mathematics Subject Classification: 35A02, 76S05. Journal: Opuscula Mathematica. Citation: Opuscula Math. 42, no. 1 (2022), 5-29, https://doi.org/10.7494/OpMath.2022.42.1.5.