Opuscula MathematicaA 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.
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AGH University of Science and Technology PressenOpuscula Mathematica1232-92742300-6919Opuscula Mathematicahttps://www.opuscula.agh.edu.pl/img/opuscula00_0.jpg
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Exponential stability results for variable delay difference equationsSufficient conditions that guarantee exponential decay to zero of the variable delay difference equation \[x(n+1)=a(n)x(n)+b(n)x(n-g(n))\] are obtained. These sufficient conditions are deduced via inequalities by employing Lyapunov functionals. In addition, a criterion for the instability of the zero solution is established. The results in the paper generalizes some results in the literature.
https://www.opuscula.agh.edu.pl/vol41/1/art/opuscula_math_4107.pdf
Exponential stability results for variable delay difference equationsErnest Yanksonexponential stability, Lyapunov functional, instabilitydoi:10.7494/OpMath.2021.41.1.145Opuscula Math. 41, no. 1 (2021), 145-155, https://doi.org/10.7494/OpMath.2021.41.1.145Opuscula Mathematica20212021https://doi.org/10.7494/OpMath.2021.41.1.145https://www.opuscula.agh.edu.pl/vol41/1/art/opuscula_math_4107.pdf411145155 Title: Exponential stability results for variable delay difference equations.

Author(s): Ernest Yankson.

Abstract: Sufficient conditions that guarantee exponential decay to zero of the variable delay difference equation \[x(n+1)=a(n)x(n)+b(n)x(n-g(n))\] are obtained. These sufficient conditions are deduced via inequalities by employing Lyapunov functionals. In addition, a criterion for the instability of the zero solution is established. The results in the paper generalizes some results in the literature. Keywords: exponential stability, Lyapunov functional, instability. Mathematics Subject Classification: 34D20, 34D40, 34K20. Journal: Opuscula Mathematica. Citation: Opuscula Math. 41, no. 1 (2021), 145-155, https://doi.org/10.7494/OpMath.2021.41.1.145.

]]>Reiterated periodic homogenization of integral functionals with convex and nonstandard growth integrandsMultiscale periodic homogenization is extended to an Orlicz-Sobolev setting. It is shown by the reiteraded periodic two-scale convergence method that the sequence of minimizers of a class of highly oscillatory minimizations problems involving convex functionals, converges to the minimizers of a homogenized problem with a suitable convex function.
https://www.opuscula.agh.edu.pl/vol41/1/art/opuscula_math_4106.pdf
Reiterated periodic homogenization of integral functionals with convex and nonstandard growth integrandsJoel Fotso TachagoHubert NnangElvira Zappaleconvex function, reiterated two-scale convergence, relaxation, Orlicz-Sobolev spacesdoi:10.7494/OpMath.2021.41.1.113Opuscula Math. 41, no. 1 (2021), 113-143, https://doi.org/10.7494/OpMath.2021.41.1.113Opuscula Mathematica20212021https://doi.org/10.7494/OpMath.2021.41.1.113https://www.opuscula.agh.edu.pl/vol41/1/art/opuscula_math_4106.pdf411113143 Title: Reiterated periodic homogenization of integral functionals with convex and nonstandard growth integrands.

Author(s): Joel Fotso Tachago, Hubert Nnang, Elvira Zappale.

Abstract: Multiscale periodic homogenization is extended to an Orlicz-Sobolev setting. It is shown by the reiteraded periodic two-scale convergence method that the sequence of minimizers of a class of highly oscillatory minimizations problems involving convex functionals, converges to the minimizers of a homogenized problem with a suitable convex function. Keywords: convex function, reiterated two-scale convergence, relaxation, Orlicz-Sobolev spaces. Mathematics Subject Classification: 35B27, 35B40, 35J25, 46J10, 49J45. Journal: Opuscula Mathematica. Citation: Opuscula Math. 41, no. 1 (2021), 113-143, https://doi.org/10.7494/OpMath.2021.41.1.113.

]]>On the crossing numbers of join products of W_{4}+P_{n} and W_{4}+C_{n}The crossing number \(\mathrm{cr}(G)\) of a graph \(G\) is the minimum number of edge crossings over all drawings of \(G\) in the plane. The main aim of the paper is to give the crossing number of the join product \(W_4+P_n\) and \(W_4+C_n\) for the wheel \(W_4\) on five vertices, where \(P_n\) and \(C_n\) are the path and the cycle on \(n\) vertices, respectively. Yue et al. conjectured that the crossing number of \(W_m+C_n\) is equal to \(Z(m+1)Z(n)+(Z(m)-1) \big \lfloor \frac{n}{2} \big \rfloor + n+ \big\lceil\frac{m}{2}\big\rceil +2\), for all \(m,n \geq 3\), and where the Zarankiewicz's number \(Z(n)=\big \lfloor \frac{n}{2} \big \rfloor \big \lfloor \frac{n-1}{2} \big \rfloor\) is defined for \(n\geq 1\). Recently, this conjecture was proved for \(W_3+C_n\) by Klešč. We establish the validity of this conjecture for \(W_4+C_n\) and we also offer a new conjecture for the crossing number of the join product \(W_m+P_n\) for \(m\geq 3\) and \(n\geq 2\).
https://www.opuscula.agh.edu.pl/vol41/1/art/opuscula_math_4105.pdf
On the crossing numbers of join products of W_{4}+P_{n} and W_{4}+C_{n}Michal StašJuraj Valiskagraph, crossing number, join product, cyclic permutation, path, cycledoi:10.7494/OpMath.2021.41.1.95Opuscula Math. 41, no. 1 (2021), 95-112, https://doi.org/10.7494/OpMath.2021.41.1.95Opuscula Mathematica20212021https://doi.org/10.7494/OpMath.2021.41.1.95https://www.opuscula.agh.edu.pl/vol41/1/art/opuscula_math_4105.pdf41195112 Title: On the crossing numbers of join products of W_{4}+P_{n} and W_{4}+C_{n}.

Author(s): Michal Staš, Juraj Valiska.

Abstract: The crossing number \(\mathrm{cr}(G)\) of a graph \(G\) is the minimum number of edge crossings over all drawings of \(G\) in the plane. The main aim of the paper is to give the crossing number of the join product \(W_4+P_n\) and \(W_4+C_n\) for the wheel \(W_4\) on five vertices, where \(P_n\) and \(C_n\) are the path and the cycle on \(n\) vertices, respectively. Yue et al. conjectured that the crossing number of \(W_m+C_n\) is equal to \(Z(m+1)Z(n)+(Z(m)-1) \big \lfloor \frac{n}{2} \big \rfloor + n+ \big\lceil\frac{m}{2}\big\rceil +2\), for all \(m,n \geq 3\), and where the Zarankiewicz's number \(Z(n)=\big \lfloor \frac{n}{2} \big \rfloor \big \lfloor \frac{n-1}{2} \big \rfloor\) is defined for \(n\geq 1\). Recently, this conjecture was proved for \(W_3+C_n\) by Klešč. We establish the validity of this conjecture for \(W_4+C_n\) and we also offer a new conjecture for the crossing number of the join product \(W_m+P_n\) for \(m\geq 3\) and \(n\geq 2\). Keywords: graph, crossing number, join product, cyclic permutation, path, cycle. Mathematics Subject Classification: 05C10, 05C38. Journal: Opuscula Mathematica. Citation: Opuscula Math. 41, no. 1 (2021), 95-112, https://doi.org/10.7494/OpMath.2021.41.1.95.

]]>Remarks on the existence of nonoscillatory solutions of half-linear ordinary differential equations, IWe consider the half-linear differential equation of the form \[(p(t)|x'|^{\alpha}\mathrm{sgn} x')' + q(t)|x|^{\alpha}\mathrm{sgn} x = 0, \quad t\geq t_{0},\] under the assumption \(\int_{t_{0}}^{\infty}p(s)^{-1/\alpha}ds =\infty\). It is shown that if a certain condition is satisfied, then the above equation has a pair of nonoscillatory solutions with specific asymptotic behavior as \(t \to \infty\).
https://www.opuscula.agh.edu.pl/vol41/1/art/opuscula_math_4104.pdf
Remarks on the existence of nonoscillatory solutions of half-linear ordinary differential equations, IManabu Naitoasymptotic behavior, nonoscillatory solution, half-linear differential equation, Hardy-type inequalitydoi:10.7494/OpMath.2021.41.1.71Opuscula Math. 41, no. 1 (2021), 71-94, https://doi.org/10.7494/OpMath.2021.41.1.71Opuscula Mathematica20212021https://doi.org/10.7494/OpMath.2021.41.1.71https://www.opuscula.agh.edu.pl/vol41/1/art/opuscula_math_4104.pdf4117194 Title: Remarks on the existence of nonoscillatory solutions of half-linear ordinary differential equations, I.

Author(s): Manabu Naito.

Abstract: We consider the half-linear differential equation of the form \[(p(t)|x'|^{\alpha}\mathrm{sgn} x')' + q(t)|x|^{\alpha}\mathrm{sgn} x = 0, \quad t\geq t_{0},\] under the assumption \(\int_{t_{0}}^{\infty}p(s)^{-1/\alpha}ds =\infty\). It is shown that if a certain condition is satisfied, then the above equation has a pair of nonoscillatory solutions with specific asymptotic behavior as \(t \to \infty\). Keywords: asymptotic behavior, nonoscillatory solution, half-linear differential equation, Hardy-type inequality. Mathematics Subject Classification: 34C11, 34C10, 26D10. Journal: Opuscula Mathematica. Citation: Opuscula Math. 41, no. 1 (2021), 71-94, https://doi.org/10.7494/OpMath.2021.41.1.71.

]]>More on linear and metric tree mapsWe consider linear and metric self-maps on vertex sets of finite combinatorial trees. Linear maps are maps which preserve intervals between pairs of vertices whereas metric maps are maps which do not increase distances between pairs of vertices. We obtain criteria for a given linear or a metric map to be a positive (negative) under some orientation of the edges in a tree, we characterize trees which admit maps with Markov graphs being paths and prove that the converse of any partial functional digraph is isomorphic to a Markov graph for some suitable map on a tree.
https://www.opuscula.agh.edu.pl/vol41/1/art/opuscula_math_4103.pdf
More on linear and metric tree mapsSergiy Kozerenkotree, Markov graph, metric map, non-expanding map, linear map, graph homomorphismdoi:10.7494/OpMath.2021.41.1.55Opuscula Math. 41, no. 1 (2021), 55-70, https://doi.org/10.7494/OpMath.2021.41.1.55Opuscula Mathematica20212021https://doi.org/10.7494/OpMath.2021.41.1.55https://www.opuscula.agh.edu.pl/vol41/1/art/opuscula_math_4103.pdf4115570 Title: More on linear and metric tree maps.

Author(s): Sergiy Kozerenko.

Abstract: We consider linear and metric self-maps on vertex sets of finite combinatorial trees. Linear maps are maps which preserve intervals between pairs of vertices whereas metric maps are maps which do not increase distances between pairs of vertices. We obtain criteria for a given linear or a metric map to be a positive (negative) under some orientation of the edges in a tree, we characterize trees which admit maps with Markov graphs being paths and prove that the converse of any partial functional digraph is isomorphic to a Markov graph for some suitable map on a tree. Keywords: tree, Markov graph, metric map, non-expanding map, linear map, graph homomorphism. Mathematics Subject Classification: 05C05, 05C12, 05C20, 54E40. Journal: Opuscula Mathematica. Citation: Opuscula Math. 41, no. 1 (2021), 55-70, https://doi.org/10.7494/OpMath.2021.41.1.55.

]]>Nonlinear parabolic equation having nonstandard growth condition with respect to the gradient and variable exponentWe are concerned with the existence of solutions to a class of quasilinear parabolic equations having critical growth nonlinearity with respect to the gradient and variable exponent. Using Schaeffer's fixed point theorem combined with the sub- and supersolution method, we prove the existence results of a weak solutions to the considered problems.
https://www.opuscula.agh.edu.pl/vol41/1/art/opuscula_math_4102.pdf
Nonlinear parabolic equation having nonstandard growth condition with respect to the gradient and variable exponentAbderrahim CharkaouiHouda FahimNour Eddine Alaavariable exponent, quasilinear equation, Schaeffer's fixed point, subsolution, supersolution, weak solutiondoi:10.7494/OpMath.2021.41.1.25Opuscula Math. 41, no. 1 (2021), 25-53, https://doi.org/10.7494/OpMath.2021.41.1.25Opuscula Mathematica20212021https://doi.org/10.7494/OpMath.2021.41.1.25https://www.opuscula.agh.edu.pl/vol41/1/art/opuscula_math_4102.pdf4112553 Title: Nonlinear parabolic equation having nonstandard growth condition with respect to the gradient and variable exponent.

Abstract: We are concerned with the existence of solutions to a class of quasilinear parabolic equations having critical growth nonlinearity with respect to the gradient and variable exponent. Using Schaeffer's fixed point theorem combined with the sub- and supersolution method, we prove the existence results of a weak solutions to the considered problems. Keywords: variable exponent, quasilinear equation, Schaeffer's fixed point, subsolution, supersolution, weak solution. Mathematics Subject Classification: 35D30, 35K59, 35A01, 35K93, 35A16, 47H10. Journal: Opuscula Mathematica. Citation: Opuscula Math. 41, no. 1 (2021), 25-53, https://doi.org/10.7494/OpMath.2021.41.1.25.

]]>Some existence results for a nonlocal non-isotropic problemIn this paper we deal with the following problem \[\begin{cases}-\sum\limits_{i=1}^{N}\left[ \left( a+b\int\limits_{\, \Omega }\left\vert \partial _{i}u\right\vert ^{p_{i}}dx\right) \partial _{i}\left( \left\vert \partial _{i}u\right\vert ^{p_{i}-2}\partial _{i}u\right) \right]=\frac{f(x)}{u^{\gamma }}\pm g(x)u^{q-1} & in\ \Omega, \\ u\geq 0 & in\ \Omega, \\ u=0 & on\ \partial \Omega, \end{cases}\] where \(\Omega\) is a bounded regular domain in \(\mathbb{R}^{N}\). We will assume without loss of generality that \(1\leq p_{1}\leq p_{2}\leq \ldots\leq p_{N}\) and that \(f\) and \(g\) are non-negative functions belonging to a suitable Lebesgue space \(L^{m}(\Omega)\), \(1\lt q\lt \overline{p}^{\ast}\), \(a\gt 0\), \(b\gt 0\) and \(0\lt\gamma \lt 1.\)
https://www.opuscula.agh.edu.pl/vol41/1/art/opuscula_math_4101.pdf
Some existence results for a nonlocal non-isotropic problemRachid BentifourSofiane El-Hadi Mirianisotropic operator, integro-differential problem, variational methodsdoi:10.7494/OpMath.2021.41.1.5Opuscula Math. 41, no. 1 (2021), 5-23, https://doi.org/10.7494/OpMath.2021.41.1.5Opuscula Mathematica20212021https://doi.org/10.7494/OpMath.2021.41.1.5https://www.opuscula.agh.edu.pl/vol41/1/art/opuscula_math_4101.pdf411523 Title: Some existence results for a nonlocal non-isotropic problem.

Abstract: In this paper we deal with the following problem \[\begin{cases}-\sum\limits_{i=1}^{N}\left[ \left( a+b\int\limits_{\, \Omega }\left\vert \partial _{i}u\right\vert ^{p_{i}}dx\right) \partial _{i}\left( \left\vert \partial _{i}u\right\vert ^{p_{i}-2}\partial _{i}u\right) \right]=\frac{f(x)}{u^{\gamma }}\pm g(x)u^{q-1} & in\ \Omega, \\ u\geq 0 & in\ \Omega, \\ u=0 & on\ \partial \Omega, \end{cases}\] where \(\Omega\) is a bounded regular domain in \(\mathbb{R}^{N}\). We will assume without loss of generality that \(1\leq p_{1}\leq p_{2}\leq \ldots\leq p_{N}\) and that \(f\) and \(g\) are non-negative functions belonging to a suitable Lebesgue space \(L^{m}(\Omega)\), \(1\lt q\lt \overline{p}^{\ast}\), \(a\gt 0\), \(b\gt 0\) and \(0\lt\gamma \lt 1.\) Keywords: anisotropic operator, integro-differential problem, variational methods. Mathematics Subject Classification: 35A15, 35B09, 35E15, 35J20. Journal: Opuscula Mathematica. Citation: Opuscula Math. 41, no. 1 (2021), 5-23, https://doi.org/10.7494/OpMath.2021.41.1.5.