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. PubDate: Fri, 29 Apr 2022 21:30:07 +020

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. PubDate: Fri, 29 Apr 2022 21:30:06 +020

Abstract: 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. PubDate: Fri, 29 Apr 2022 21:30:05 +020

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. PubDate: Fri, 29 Apr 2022 21:30:04 +020

Abstract: 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. PubDate: Fri, 29 Apr 2022 21:30:03 +020

Abstract: 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. PubDate: Fri, 29 Apr 2022 21:30:02 +020

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. PubDate: Fri, 29 Apr 2022 21:30:01 +020

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. PubDate: Fri, 25 Feb 2022 14:30:09 +010

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. PubDate: Fri, 25 Feb 2022 14:30:08 +010

: 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. PubDate: Fri, 25 Feb 2022 14:30:07 +010

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. PubDate: Fri, 25 Feb 2022 14:30:06 +010

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. PubDate: Fri, 25 Feb 2022 14:30:05 +010

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. PubDate: Fri, 25 Feb 2022 14:30:04 +010

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. PubDate: Fri, 25 Feb 2022 14:30:03 +010

Abstract: 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. PubDate: Fri, 25 Feb 2022 14:30:02 +010

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. PubDate: Fri, 25 Feb 2022 14:30:01 +010

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. PubDate: Thu, 20 Jan 2022 18:00:06 +010

Abstract: 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. PubDate: Thu, 20 Jan 2022 18:00:05 +010

Abstract: 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. PubDate: Thu, 20 Jan 2022 18:00:04 +010

Abstract: 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. PubDate: Thu, 20 Jan 2022 18:00:03 +010

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. PubDate: Thu, 20 Jan 2022 18:00:02 +010

Abstract: 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. PubDate: Thu, 20 Jan 2022 18:00:01 +010