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.Fri, 27 Oct 2023 09:10:00 +0200Opuscula Mathematicahttps://www.opuscula.agh.edu.pl/img/opuscula00_0.jpg
https://www.opuscula.agh.edu.pl
The extensive 1-median problem with radius on networks
https://www.opuscula.agh.edu.pl/vol44/1/art/opuscula_math_4407.pdf
https://www.opuscula.agh.edu.pl/vol44/1/art/opuscula_math_4407.pdfFri, 27 Oct 2023 09:00:07 +0200 Author(s): Tran Hoai Ngoc Nhan, Nguyen Thanh Hung, Kien Trung Nguyen.

Abstract: The median location problem concerns finding locations of one or several new facilities that minimize the overall weighted distances from the existing to the new facilities. We address the problem of locating one new facility with a radius \(r\) on networks. Furthermore, the radius \(r\) is flexible and the objective function is the conic combination of the traditional 1-median function and the value \(r\). We call this problem an extensive 1-median problem with radius on networks. To solve the problem, we first induce the so-called finite dominating set, that contains all points on the underlying network and radius values which are candidate for the optimal solution of the problem. This helps to develop a combinatorial algorithm that solves the problem on a general network \(G=(V,E)\) in \(O(|E||V|^3)\) time. We also consider the underlying problem with improved algorithm on trees. Based the convexity of the objective function with variable radius, we develop a linear time algorithm to find an extensive 1-median with radius on the underlying tree. Keywords: extensive facility, median problem, tree, convex. Mathematics Subject Classification: 90B10, 90C35. Journal: Opuscula Mathematica. Citation: Opuscula Math. 44, no. 1 (2024), 135-149, https://doi.org/10.7494/OpMath.2024.44.1.135.

]]>Singular quasilinear convective systems involving variable exponents
https://www.opuscula.agh.edu.pl/vol44/1/art/opuscula_math_4406.pdf
https://www.opuscula.agh.edu.pl/vol44/1/art/opuscula_math_4406.pdfFri, 27 Oct 2023 09:00:06 +0200 Author(s): Abdelkrim Moussaoui, Dany Nabab, Jean Vélin.

Abstract: The paper deals with the existence of solutions for quasilinear elliptic systems involving singular and convection terms with variable exponents. The approach combines the sub-supersolutions method and Schauder's fixed point theorem. Keywords: \(p(x)\)-Laplacian, variable exponents, fixed point, singular system, gradient estimate, regularity. Mathematics Subject Classification: 35J75, 35J48, 35J92. Journal: Opuscula Mathematica. Citation: Opuscula Math. 44, no. 1 (2024), 105-134, https://doi.org/10.7494/OpMath.2024.44.1.105.

]]>Conditional mean embedding and optimal feature selection via positive definite kernels
https://www.opuscula.agh.edu.pl/vol44/1/art/opuscula_math_4405.pdf
https://www.opuscula.agh.edu.pl/vol44/1/art/opuscula_math_4405.pdfFri, 27 Oct 2023 09:00:05 +0200 Author(s): Palle E.T. Jorgensen, Myung-Sin Song, James Tian.

Abstract: Motivated by applications, we consider new operator-theoretic approaches to conditional mean embedding (CME). Our present results combine a spectral analysis-based optimization scheme with the use of kernels, stochastic processes, and constructive learning algorithms. For initially given non-linear data, we consider optimization-based feature selections. This entails the use of convex sets of kernels in a construction o foptimal feature selection via regression algorithms from learning models. Thus, with initial inputs of training data (for a suitable learning algorithm), each choice of a kernel \(K\) in turn yields a variety of Hilbert spaces and realizations of features. A novel aspect of our work is the inclusion of a secondary optimization process over a specified convex set of positive definite kernels, resulting in the determination of "optimal" feature representations. Keywords: positive-definite kernels, reproducing kernel Hilbert space, stochastic processes, frames, machine learning, embedding problems, optimization. Mathematics Subject Classification: 47N10, 47A52, 47B32, 42A82, 42C15, 62H12, 62J07, 65J20, 68T07, 90C20. Journal: Opuscula Mathematica. Citation: Opuscula Math. 44, no. 1 (2024), 79-103, https://doi.org/10.7494/OpMath.2024.44.1.79.

]]>Two-weight norm inequalities for rough fractional integral operators on Morrey spaces
https://www.opuscula.agh.edu.pl/vol44/1/art/opuscula_math_4404.pdf
https://www.opuscula.agh.edu.pl/vol44/1/art/opuscula_math_4404.pdfFri, 27 Oct 2023 09:00:04 +0200 Author(s): Kwok-Pun Ho.

Abstract: We establish the two-weight norm inequalities for the rough fractional integral operators on Morrey spaces. Keywords: two-weight norm inequalities, rough fractional integral operators, Morrey spaces. Mathematics Subject Classification: 42B20, 42B25, 46E30. Journal: Opuscula Mathematica. Citation: Opuscula Math. 44, no. 1 (2024), 67-77, https://doi.org/10.7494/OpMath.2024.44.1.67.

]]>Local irregularity conjecture for 2-multigraphs versus cacti
https://www.opuscula.agh.edu.pl/vol44/1/art/opuscula_math_4403.pdf
https://www.opuscula.agh.edu.pl/vol44/1/art/opuscula_math_4403.pdfFri, 27 Oct 2023 09:00:03 +0200 Author(s): Igor Grzelec, Mariusz Woźniak.

Abstract: A multigraph is locally irregular if the degrees of the end-vertices of every multiedge are distinct. The locally irregular coloring is an edge coloring of a multigraph \(G\) such that every color induces a locally irregular submultigraph of \(G\). A locally irregular colorable multigraph \(G\) is any multigraph which admits a locally irregular coloring. We denote by \(\textrm{lir}(G)\) the locally irregular chromatic index of a multigraph \(G\), which is the smallest number of colors required in the locally irregular coloring of the locally irregular colorable multigraph \(G\). In case of graphs the definitions are similar. The Local Irregularity Conjecture for 2-multigraphs claims that for every connected graph \(G\), which is not isomorphic to \(K_2\), multigraph \(^2G\) obtained from \(G\) by doubling each edge satisfies \(\textrm{lir}(^2G)\leq 2\). We show this conjecture for cacti. This class of graphs is important for the Local Irregularity Conjecture for 2-multigraphs and the Local Irregularity Conjecture which claims that every locally irregular colorable graph \(G\) satisfies \(\textrm{lir}(G)\leq 3\). At the beginning it has been observed that all not locally irregular colorable graphs are cacti. Recently it has been proved that there is only one cactus which requires 4 colors for a locally irregular coloring and therefore the Local Irregularity Conjecture was disproved. Keywords: locally irregular coloring, decomposable, cactus graphs, 2-multigraphs. Mathematics Subject Classification: 05C15. Journal: Opuscula Mathematica. Citation: Opuscula Math. 44, no. 1 (2024), 49-65, https://doi.org/10.7494/OpMath.2024.44.1.49.

]]>Local existence for a viscoelastic Kirchhoff type equation with the dispersive term, internal damping, and logarithmic nonlinearity
https://www.opuscula.agh.edu.pl/vol44/1/art/opuscula_math_4402.pdf
https://www.opuscula.agh.edu.pl/vol44/1/art/opuscula_math_4402.pdfFri, 27 Oct 2023 09:00:02 +0200 Author(s): Sebastião Cordeiro, Carlos Raposo, Jorge Ferreira, Daniel Rocha, Mohammad Shahrouzi.

Abstract: This paper concerns a viscoelastic Kirchhoff-type equation with the dispersive term, internal damping, and logarithmic nonlinearity. We prove the local existence of a weak solution via a modified lemma of contraction of the Banach fixed-point theorem. Although the uniqueness of a weak solution is still an open problem, we proved uniqueness locally for specifically suitable exponents. Furthermore, we established a result for local existence without guaranteeing uniqueness, stating a contraction lemma. Keywords: viscoelastic equation, dispersive term, logarithmic nonlinearity, local existence. Mathematics Subject Classification: 35A01, 35L20, 35L70. Journal: Opuscula Mathematica. Citation: Opuscula Math. 44, no. 1 (2024), 19-47, https://doi.org/10.7494/OpMath.2024.44.1.19.

]]>Uniqueness for a class p-Laplacian problems when a parameter is large
https://www.opuscula.agh.edu.pl/vol44/1/art/opuscula_math_4401.pdf
https://www.opuscula.agh.edu.pl/vol44/1/art/opuscula_math_4401.pdfFri, 27 Oct 2023 09:00:01 +0200 Author(s): B. Alreshidi, D.D. Hai.

Abstract: We prove uniqueness of positive solutions for the problem \[-\Delta_{p}u=\lambda f(u)\text{ in }\Omega,\ u=0\text{ on }\partial \Omega,\] where \(1\lt p\lt 2\) and \(p\) is close to 2, \(\Omega\) is bounded domain in \(\mathbb{R}^{n}\) with smooth boundary \(\partial \Omega\), \(f:[0,\infty)\rightarrow [0,\infty )\) with \(f(z)\sim z^{\beta }\) at \(\infty\) for some \(\beta \in (0,1)\), and \(\lambda\) is a large parameter. The monotonicity assumption on \(f\) is not required even for \(u\) large. Keywords: singular \(p\)-Laplacian, uniqueness, positive solutions. Mathematics Subject Classification: 35J92, 35J75. Journal: Opuscula Mathematica. Citation: Opuscula Math. 44, no. 1 (2024), 5-17, https://doi.org/10.7494/OpMath.2024.44.1.5.