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Opuscula Mathematica
Journal Prestige (SJR): 0.378  Citation Impact (citeScore): 1 Number of Followers: 0  Open Access journalISSN (Print) 1232-9274 - ISSN (Online) 2300-6919 Published by AGH University of Science and Technology Press  [6 journals]
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- The crossing numbers of join products of four graphs of order five with
paths and cycles
Abstract:
Author(s): Michal Staš, Mária Timková.
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. In the paper, we extend known results concerning crossing numbers of join products of four small graphs with paths and cycles. The crossing numbers of the join products \(G^\ast + P_n\) and \(G^\ast + C_n\) for the disconnected graph \(G^\ast\) consisting of the complete tripartite graph \(K_{1,1,2}\) and one isolated vertex are given, where \(P_n\) and \(C_n\) are the path and the cycle on \(n\) vertices, respectively. In the paper also the crossing numbers of \(H^\ast + P_n\) and \(H^\ast + C_n\) are determined, where \(H^\ast\) is isomorphic to the complete tripartite graph \(K_{1,1,3}\). Finally, by adding new edges to the graphs \(G^\ast\) and \(H^\ast\), we are able to obtain crossing numbers of join products of two other graphs \(G_1\) and \(H_1\) with paths and cycles.
Keywords: graph, crossing number, join product, path, cycle, separating cycle.
Mathematics Subject Classification: 05C10, 05C38.
Journal: Opuscula Mathematica.
Citation: Opuscula Math. 43, no. 6 (2023), 865-883, https://doi.org/10.7494/OpMath.2023.43.6.865.
PubDate: Sat, 22 Jul 2023 09:00:08 +020
- Every graph is local antimagic total and its applications
Abstract:
Author(s): Gee-Choon Lau, Karl Schaffer, Wai Chee Shiu.
Abstract: Let \(G = (V,E)\) be a simple graph of order \(p\) and size \(q\). A graph \(G\) is called local antimagic (total) if \(G\) admits a local antimagic (total) labeling. A bijection \(g : E \to \{1,2,\ldots,q\}\) is called a local antimagic labeling of \(G\) if for any two adjacent vertices \(u\) and \(v\), we have \(g^+(u) \neq g^+(v)\), where \(g^+(u) = \sum_{e\in E(u)} g(e)\), and \(E(u)\) is the set of edges incident to \(u\). Similarly, a bijection \(f:V(G)\cup E(G)\to \{1,2,\ldots,p+q\}\) is called a local antimagic total labeling of \(G\) if for any two adjacent vertices \(u\) and \(v\), we have \(w_f(u)\neq w_f(v)\), where \(w_f(u) = f(u) + \sum_{e\in E(u)} f(e)\). Thus, any local antimagic (total) labeling induces a proper vertex coloring of \(G\) if vertex \(v\) is assigned the color \(g^+(v)\) (respectively, \(w_f(u)\)). The local antimagic (total) chromatic number, denoted \(\chi_{la}(G)\) (respectively \(\chi_{lat}(G)\)), is the minimum number of induced colors taken over local antimagic (total) labeling of \(G\). We provide a short proof that every graph \(G\) is local antimagic total. The proof provides sharp upper bound to \(\chi_{lat}(G)\). We then determined the exact \(\chi_{lat}(G)\), where \(G\) is a complete bipartite graph, a path, or the Cartesian product of two cycles. Consequently, the \(\chi_{la}(G\vee K_1)\) is also obtained. Moreover, we determined the \(\chi_{la}(G\vee K_1)\) and hence the \(\chi_{lat}(G)\) for a class of 2-regular graphs \(G\) (possibly with a path). The work of this paper also provides many open problems on \(\chi_{lat}(G)\). We also conjecture that each graph \(G\) of order at least 3 has \(\chi_{lat}(G)\leq \chi_{la}(G)\).
Keywords: local antimagic (total) chromatic number, Cartesian product, join product.
Mathematics Subject Classification: 05C78, 05C15.
Journal: Opuscula Mathematica.
Citation: Opuscula Math. 43, no. 6 (2023), 841-864, https://doi.org/10.7494/OpMath.2023.43.6.841.
PubDate: Sat, 22 Jul 2023 09:00:07 +020
- On the path partition of graphs
Abstract:
Author(s): Mekkia Kouider, Mohamed Zamime.
Abstract: Let \(G\) be a graph of order \(n\). The maximum and minimum degree of \(G\) are denoted by \(\Delta\) and \(\delta\), respectively. The path partition number \(\mu(G)\) of a graph \(G\) is the minimum number of paths needed to partition the vertices of \(G\). Magnant, Wang and Yuan conjectured that \[\mu(G)\leq\max \left\{\frac{n}{\delta+1},\frac{(\Delta-\delta)n}{\Delta+\delta}\right\}.\] In this work, we give a positive answer to this conjecture, for \(\Delta\geq 2\delta\).
Keywords: path, partition.
Mathematics Subject Classification: 05C20.
Journal: Opuscula Mathematica.
Citation: Opuscula Math. 43, no. 6 (2023), 829-839, https://doi.org/10.7494/OpMath.2023.43.6.829.
PubDate: Sat, 22 Jul 2023 09:00:06 +020
- On minimum intersections of certain secondary dominating sets in graphs
Abstract:
Author(s): Anna Kosiorowska, Adrian Michalski, Iwona Włoch.
Abstract: In this paper we consider secondary dominating sets, also named as \((1,k)\)-dominating sets, introduced by Hedetniemi et al. in 2008. In particular, we study intersections of the \((1,1)\)-dominating sets and proper \((1,2)\)-dominating sets. We introduce \((1,\overline{2})\)-intersection index as the minimum possible cardinality of such intersection and determine its value for some classes of graphs.
Keywords: dominating set, 2-dominating set, \((1,2)\)-dominating set, proper \((1,2)\)-dominating set, domination numbers, \((1,\overline{2})\)-intersection index.
Mathematics Subject Classification: 05C69, 05C76.
Journal: Opuscula Mathematica.
Citation: Opuscula Math. 43, no. 6 (2023), 813-827, https://doi.org/10.7494/OpMath.2023.43.6.813.
PubDate: Sat, 22 Jul 2023 09:00:05 +020
- On the concept of generalization of I-density points
Abstract:
Author(s): Jacek Hejduk, Renata Wiertelak.
Abstract: This paper deals with essential generalization of \(\mathcal{I}\)-density points and \(\mathcal{I}\)-density topology. In particular, there is an example showing that this generalization of \(\mathcal{I}\)-density point yields the stronger concept of density point than the notion of \(\mathcal{I}(\mathcal{J})\)-density. Some properties of topologies generated by operators related to this essential generalization of density points are provided.
Keywords: density topology, generalization of density topology.
Mathematics Subject Classification: 54A05, 54A10.
Journal: Opuscula Mathematica.
Citation: Opuscula Math. 43, no. 6 (2023), 803-811, https://doi.org/10.7494/OpMath.2023.43.6.803.
PubDate: Sat, 22 Jul 2023 09:00:04 +020
- Oscillation conditions for difference equations with several variable
delays
Abstract:
Author(s): Bassant M. El-Matary, Hassan A. El-Morshedy, Vasileios Benekas, Ioannis P. Stavroulakis.
Abstract: A technique is developed to establish a new oscillation criterion for a first-order linear difference equation with several delays and non-negative coefficients. Our result improves recent oscillation criteria and covers the cases of monotone and non-monotone delays. Moreover, the paper is concluded with an illustrative example to show the applicability and strength of our result.
Keywords: oscillation, difference equations, non-monotone delays, first order.
Mathematics Subject Classification: 39A10, 39A21.
Journal: Opuscula Mathematica.
Citation: Opuscula Math. 43, no. 6 (2023), 789-801, https://doi.org/10.7494/OpMath.2023.43.6.789.
PubDate: Sat, 22 Jul 2023 09:00:03 +020
- Regularity and existence of solutions to parabolic equations with
nonstandard p(x,t),q(x,t)-growth conditions
Abstract:
Author(s): Hamid El Bahja.
Abstract: We study the Cauchy-Dirichlet problem for a class of nonlinear parabolic equations driven by nonstandard \(p(x,t),q(x,t)\)-growth condition. We prove theorems of existence and uniqueness of weak solutions in suitable Orlicz-Sobolev spaces, derive global and local in time \(L^{\infty}\) bounds for the weak solutions.
Keywords: existence theory, nonlinear parabolic problems, nonstandard growth, regularity theory.
Mathematics Subject Classification: 35K55, 35K65.
Journal: Opuscula Mathematica.
Citation: Opuscula Math. 43, no. 6 (2023), 759-788, https://doi.org/10.7494/OpMath.2023.43.6.759.
PubDate: Sat, 22 Jul 2023 09:00:02 +020
- Nonexistence of global solutions for a nonlinear parabolic equation with a
forcing term
Abstract:
Author(s): Aisha Alshehri, Noha Aljaber, Haya Altamimi, Rasha Alessa, Mohamed Majdoub.
Abstract: The purpose of this work is to analyze the blow-up of solutions of a nonlinear parabolic equation with a forcing term depending on both time and space variables \[u_t-\Delta u= x ^{\alpha} u ^{p}+{\mathtt a}(t)\,{\mathbf w}(x)\quad\text{for }(t,x)\in(0,\infty)\times \mathbb{R}^{N},\] where \(\alpha\in\mathbb{R}\), \(p\gt 1\), and \({\mathtt a}(t)\) as well as \({\mathbf w}(x)\) are suitable given functions. We generalize and somehow improve earlier existing works by considering a wide class of forcing terms that includes the most common investigated example \(t^\sigma\,{\mathbf w}(x)\) as a particular case. Using the test function method and some differential inequalities, we obtain sufficient criteria for the nonexistence of global weak solutions. This criterion mainly depends on the value of the limit \(\lim_{t\to\infty} \frac{1}{t}\,\int_0^t\,{\mathtt a}(s)\,ds\). The main novelty lies in our treatment of the nonstandard condition on the forcing term.
Keywords: nonlinear heat equation, forcing term, blow-up, test-function, differential inequalities.
Mathematics Subject Classification: 35K05, 35A01, 35B44.
Journal: Opuscula Mathematica.
Citation: Opuscula Math. 43, no. 6 (2023), 741-758, https://doi.org/10.7494/OpMath.2023.43.6.741.
PubDate: Sat, 22 Jul 2023 09:00:01 +020
- Existence and smoothing effects of the initial-boundary value problem for
\partial u/\partial t-\Delta\sigma(u)=0 in time-dependent domains
Abstract:
Author(s): Mitsuhiro Nakao.
Abstract: We show the existence, smoothing effects and decay properties of solutions to the initial-boundary value problem for a generalized porous medium type parabolic equations of the form \[u_t-\Delta \sigma(u) =0 \quad \text{in } Q(0, T)\] with the initial and boundary conditions \[u(0)=u_0 \quad \text{and} \quad u(t) _{\partial \Omega(t)}=0,\] where \(\Omega(t)\) is a bounded domain in \(R^N\) for each \(t \geq 0\) and \[Q(0,T)=\bigcup_{0 \lt t \lt T} \Omega(t) \times \{t\}, \quad T>0.\] Our class of \(\sigma(u)\) includes \(\sigma(u)= u ^m u\), \(\sigma(u)=u \log (1+ u ^m)\), \(0\leq m \leq 2\), and \(\sigma(u)= u ^{m}u/\sqrt{1+ u ^2}\), \(1 \leq m \leq 2\), etc. We derive precise estimates for \(\ u(t)\ _{\Omega(t),\infty}\) and \(\ \nabla\sigma(u(t))\ ^2_{\Omega(t),2}\), \(t\gt 0\), depending on \(\ u_0\ _{\Omega(0),r}\) and the movement of \(\partial\Omega(t)\).
Keywords: quasilinear parabolic equation, time-dependent domain, smoothing effects.
Mathematics Subject Classification: 35B40, 35K92.
Journal: Opuscula Mathematica.
Citation: Opuscula Math. 43, no. 5 (2023), 703-734, https://doi.org/10.7494/OpMath.2023.43.5.703.
PubDate: Sat, 24 Jun 2023 14:00:07 +020
- Global solutions for a nonlinear Kirchhoff type equation with viscosity
Abstract:
Author(s): Eugenio Cabanillas Lapa.
Abstract: In this paper we consider the existence and asymptotic behavior of solutions of the following nonlinear Kirchhoff type problem \[u_{tt}- M\left(\,\displaystyle \int_{\Omega} \nabla u ^{2}\, dx\right)\triangle u - \delta\triangle u_{t}= \mu u ^{\rho-2}u\quad \text{in } \Omega \times ]0,\infty[,\] where \[M(s)=\begin{cases}a-bs &\text{for } s \in [0,\frac{a}{b}[,\\ 0, &\text{for } s \in [\frac{a}{b}, +\infty[.\end{cases}\] If the initial energy is appropriately small, we derive the global existence theorem and its exponential decay.
Keywords: global solutions, nonlinear Kirchhoff type problem, exponential decay.
Mathematics Subject Classification: 35L80, 35L70, 35B33, 35J75.
Journal: Opuscula Mathematica.
Citation: Opuscula Math. 43, no. 5 (2023), 689-701, https://doi.org/10.7494/OpMath.2023.43.5.689.
PubDate: Sat, 24 Jun 2023 14:00:06 +020
- Radial solutions for nonlinear elliptic equation with nonlinear nonlocal
boundary conditions
Abstract:
Author(s): Igor Kossowski.
Abstract: In this article, we prove existence of radial solutions for a nonlinear elliptic equation with nonlinear nonlocal boundary conditions. Our method is based on some fixed point theorem in a cone.
Keywords: nonlocal boundary value problem, radial solutions, elliptic equation, the Krasnosielskii fixed point theorem in cone.
Mathematics Subject Classification: 34B10, 34B15, 47H11.
Journal: Opuscula Mathematica.
Citation: Opuscula Math. 43, no. 5 (2023), 675-687, https://doi.org/10.7494/OpMath.2023.43.5.675.
PubDate: Sat, 24 Jun 2023 14:00:05 +020
- The existence of bipartite almost self-complementary 3-uniform hypergraphs
Abstract:
Author(s): L.N. Kamble, C.M. Deshpande, B.P. Athawale.
Abstract: An almost self-complementary 3-uniform hypergraph on \(n\) vertices exists if and only if \(n\) is congruent to 3 modulo 4 A hypergraph \(H\) with vertex set \(V\) and edge set \(E\) is called bipartite if \(V\) can be partitioned into two subsets \(V_1\) and \(V_2\) such that \(e\cap V_1\neq \emptyset\) and \(e\cap V_2\neq \emptyset\) for any \(e\in E\). A bipartite self-complementary 3-uniform hypergraph \(H\) with partition \((V_1, V_2)\) of the vertex set \(V\) such that \( V_1 =m\) and \( V_2 =n\) exists if and only if either (i) \(m=n\) or (ii) \(m\neq n\) and either \(m\) or \(n\) is congruent to 0 modulo 4 or (iii) \(m\neq n\) and both \(m\) and \(n\) are congruent to 1 or 2 modulo 4. In this paper we define a bipartite almost self-complementary 3-uniform hypergraph \(H\) with partition \((V_1, V_2)\) of a vertex set \(V\) such that \( V_1 =m\) and \( V_2 =n\) and find the conditions on \(m\) and \(n\) for a bipartite 3-uniform hypergraph \(H\) to be almost self-complementary. We also prove the existence of bi-regular bipartite almost self-complementary 3-uniform hypergraphs.
Keywords: almost self-complementary 3-uniform hypergraph, bipartite hypergraph, bipartite self-complementary 3-uniform hypergraph, bipartite almost self-complementary 3-uniform hypergraph.
Mathematics Subject Classification: 05C65.
Journal: Opuscula Mathematica.
Citation: Opuscula Math. 43, no. 5 (2023), 663-673, https://doi.org/10.7494/OpMath.2023.43.5.663.
PubDate: Sat, 24 Jun 2023 14:00:04 +020
- One boundary value problem including a spectral parameter in all boundary
conditions
Abstract:
Author(s): Ayşe Kabataş.
Abstract: In this paper, asymptotic formulae for solutions and Green's function of a boundary value problem are investigated when the equation and the boundary conditions contain a spectral parameter.
Keywords: boundary value problems, asymptotics, Green's functions.
Mathematics Subject Classification: 34B08, 34B27, 34L10.
Journal: Opuscula Mathematica.
Citation: Opuscula Math. 43, no. 5 (2023), 651-661, https://doi.org/10.7494/OpMath.2023.43.5.651.
PubDate: Sat, 24 Jun 2023 14:00:03 +020
- Volterra integral operators on a family of Dirichlet-Morrey spaces
Abstract:
Author(s): Lian Hu, Xiaosong Liu.
Abstract: A family of Dirichlet-Morrey spaces \(\mathcal{D}_{\lambda,K}\) of functions analytic in the open unit disk \(\mathbb{D}\) are defined in this paper. We completely characterize the boundedness of the Volterra integral operators \(T_g\), \(I_g\) and the multiplication operator \(M_g\) on the space \(\mathcal{D}_{\lambda,K}\). In addition, the compactness and essential norm of the operators \(T_g\) and \(I_g\) on \(\mathcal{D}_{\lambda,K}\) are also investigated.
Keywords: Dirichlet-Morrey type space, Carleson measure, Volterra integral operators, bounded operator, essential norm.
Mathematics Subject Classification: 30H99, 47B38.
Journal: Opuscula Mathematica.
Citation: Opuscula Math. 43, no. 5 (2023), 633-649, https://doi.org/10.7494/OpMath.2023.43.5.633.
PubDate: Sat, 24 Jun 2023 14:00:02 +020
- A viability result for Carathéodory non-convex differential inclusion
in Banach spaces
Abstract:
Author(s): Nabil Charradi, Saïd Sajid.
Abstract: This paper deals with the existence of solutions to the following differential inclusion: \(\dot{x}(t)\in F(t,x(t))\) a.e. on \([0, T[\) and \(x(t)\in K\), for all \(t \in [0, T]\), where \(F: [0, T]\times K \rightarrow 2^E\) is a Carathéodory multifunction and \(K\) is a closed subset of a separable Banach space \(E\).
Keywords: viability, measurable multifunction, selection, Carathéodory multifunction.
Mathematics Subject Classification: 34A60, 28B20.
Journal: Opuscula Mathematica.
Citation: Opuscula Math. 43, no. 5 (2023), 621-632, https://doi.org/10.7494/OpMath.2023.43.5.621.
PubDate: Sat, 24 Jun 2023 14:00:01 +020
- Solutions for a nonhomogeneous p&q-Laplacian problem via variational
methods and sub-supersolution technique
Abstract:
Author(s): Leandro S. Tavares, J. Vanterler C. Sousa.
Abstract: In this paper it is obtained, through variational methods and sub-supersolution arguments, existence and multiplicity of solutions for a nonhomogeneous problem which arise in several branches of science such as chemical reactions, biophysics and plasma physics. Under a general hypothesis it is proved an existence result and multiple solutions are obtained by considering an additional natural condition.
Keywords: \(p\&q\)-Laplacian operator, nonhomogeneous operator, sub-supersolutions, existence, multiplicity.
Mathematics Subject Classification: 35A15, 35J60.
Journal: Opuscula Mathematica.
Citation: Opuscula Math. 43, no. 4 (2023), 603-613, https://doi.org/10.7494/OpMath.2023.43.4.603.
PubDate: Tue, 13 Jun 2023 18:00:07 +020
- Bernstein operational matrix of differentiation and collocation approach
for a class of three-point singular BVPs: error estimate and convergence
analysis
Abstract:
Author(s): Nikhil Sriwastav, Amit K. Barnwal, Abdul-Majid Wazwaz, Mehakpreet Singh.
Abstract: Singular boundary value problems (BVPs) have widespread applications in the field of engineering, chemical science, astrophysics and mathematical biology. Finding an approximate solution to a problem with both singularity and non-linearity is highly challenging. The goal of the current study is to establish a numerical approach for dealing with problems involving three-point boundary conditions. The Bernstein polynomials and collocation nodes of a domain are used for developing the proposed numerical approach. The straightforward mathematical formulation and easy to code, makes the proposed numerical method accessible and adaptable for the researchers working in the field of engineering and sciences. The priori error estimate and convergence analysis are carried out to affirm the viability of the proposed method. Various examples are considered and worked out in order to illustrate its applicability and effectiveness. The results demonstrate excellent accuracy and efficiency compared to the other existing methods.
Keywords: Bernstein polynomials, collocation method, three-point singular BVPs, convergence analysis, error estimate.
Mathematics Subject Classification: 34B05, 34B15, 34B16, 34B18, 34B27, 34B60.
Journal: Opuscula Mathematica.
Citation: Opuscula Math. 43, no. 4 (2023), 575-601, https://doi.org/10.7494/OpMath.2023.43.4.575.
PubDate: Tue, 13 Jun 2023 18:00:06 +020
- The first eigencurve for a Neumann boundary problem involving p-Laplacian
with essentially bounded weights
Abstract:
Author(s): Ahmed Sanhaji, Ahmed Dakkak, Mimoun Moussaoui.
Abstract: This article is intended to prove the existence and uniqueness of the first eigencurve, for a homogeneous Neumann problem with singular weights associated with the equation \[-\Delta_{p} u=\alpha m_{1} u ^{p-2}u+\beta m_{2} u ^{p-2}u\] in a bounded domain \(\Omega \subset \mathbb{R}^{N}\). We then establish many properties of this eigencurve, particularly the continuity, variational characterization, asymptotic behavior, concavity and the differentiability.
Keywords: \(p\)-Laplacian, first eigencurve, singular weight, Neumann boundary conditions.
Mathematics Subject Classification: 35J30, 35J60, 35J66.
Journal: Opuscula Mathematica.
Citation: Opuscula Math. 43, no. 4 (2023), 559-574, https://doi.org/10.7494/OpMath.2023.43.4.559.
PubDate: Tue, 13 Jun 2023 18:00:05 +020
- On the existence of optimal solutions to the Lagrange problem governed by
a nonlinear Goursat-Darboux problem of fractional order
Abstract:
Author(s): Marek Majewski.
Abstract: In the paper, the Lagrange problem given by a fractional boundary problem with partial derivatives is considered. The main result is the existence of optimal solutions based on the convexity assumption of a certain set. The proof is based on the lower closure theorem and the appropriate implicit measurable function theorem.
Keywords: fractional partial derivative, fractional boundary problem, existence of optimal solutions, Lagrange problem, lower closure theorem.
Mathematics Subject Classification: 35R11, 49J20, 49K20.
Journal: Opuscula Mathematica.
Citation: Opuscula Math. 43, no. 4 (2023), 547-558, https://doi.org/10.7494/OpMath.2023.43.4.547.
PubDate: Tue, 13 Jun 2023 18:00:04 +020
- Periodic, nonperiodic, and chaotic solutions for a class of difference
equations with negative feedback
Abstract:
Author(s): Benjamin B. Kennedy.
Abstract: We study the scalar difference equation \[x(k+1) = x(k) + \frac{f(x(k-N))}{N},\] where \(f\) is nonincreasing with negative feedback. This equation is a discretization of the well-studied differential delay equation \[x'(t) = f(x(t-1)).\] We examine explicit families of such equations for which we can find, for infinitely many values of $ and appropriate parameter values, various dynamical behaviors including periodic solutions with large numbers of sign changes per minimal period, solutions that do not converge to periodic solutions, and chaos. We contrast these behaviors with the dynamics of the limiting differential equation. Our primary tool is the analysis of return maps for the difference equations that are conjugate to continuous self-maps of the circle.
Keywords: difference equation, negative feedback, circle map.
Mathematics Subject Classification: 39A12, 39A23, 39A33.
Journal: Opuscula Mathematica.
Citation: Opuscula Math. 43, no. 4 (2023), 507-546, https://doi.org/10.7494/OpMath.2023.43.4.507.
PubDate: Tue, 13 Jun 2023 18:00:03 +020
- Generalized derivations and generalized exponential monomials on
hypergroups
Abstract:
Author(s): Żywilla Fechner, Eszter Gselmann, László Székelyhidi.
Abstract: In one of our former papers "Endomorphisms of the measure algebra of commutative hypergroups" we considered exponential monomials on hypergroups andhigher order derivations of the corresponding measure algebra. Continuing with this, we are now looking for the connection between the generalized exponential polynomialsof a commutative hypergroup and the higher order derivations of the corresponding measure algebra.
Keywords: moment function, moment sequence, exponential monomial, exponential polynomial, derivation, higher order derivation, hypergroup.
Mathematics Subject Classification: 39B52, 39B72, 43A45, 43A70.
Journal: Opuscula Mathematica.
Citation: Opuscula Math. 43, no. 4 (2023), 493-505, https://doi.org/10.7494/OpMath.2023.43.4.493.
PubDate: Tue, 13 Jun 2023 18:00:02 +020
- The heat equation on time scales
Abstract:
Author(s): Tom Cuchta, Rui A.C. Ferreira.
Abstract: We present the use of a Fourier transform on time scales to solve a dynamic heat IVP. This is done by inverting a certain exponential function via contour integral. We include some specific examples and directions for further study.
Keywords: heat equation, time scales, Fourier transform.
Mathematics Subject Classification: 39A14, 34N05, 42A38.
Journal: Opuscula Mathematica.
Citation: Opuscula Math. 43, no. 4 (2023), 475-491, https://doi.org/10.7494/OpMath.2023.43.4.475.
PubDate: Tue, 13 Jun 2023 18:00:01 +020
- New oscillation constraints for even-order delay differential equations
Abstract:
Author(s): Osama Moaaz, Mona Anis, Ahmed A. El-Deeb, Ahmed M. Elshenhab.
Abstract: The purpose of this paper is to study the oscillatory properties of solutions to a class of delay differential equations of even order. We focus on criteria that exclude decreasing positive solutions. As in this paper, this type of solution emerges when considering the noncanonical case of even equations. By finding a better estimate of the ratio between the Kneser solution with and without delay, we obtain new constraints that ensure that all solutions to the considered equation oscillate. The new findings improve some previous findings in the literature.
Keywords: delay differential equations, even-order, Kneser solutions, oscillation.
Mathematics Subject Classification: 34C10, 34K11.
Journal: Opuscula Mathematica.
Citation: Opuscula Math. 43, no. 3 (2023), 455-467, https://doi.org/10.7494/OpMath.2023.43.3.455.
PubDate: Wed, 17 May 2023 21:00:06 +020
- On local antimagic total labeling of complete graphs amalgamation
Abstract:
Author(s): Gee-Choon Lau, Wai Chee Shiu.
Abstract: Let \(G = (V,E)\) be a connected simple graph of order \(p\) and size \(q\). A graph \(G\) is called local antimagic (total) if \(G\) admits a local antimagic (total) labeling. A bijection \(g : E \to \{1,2,\ldots,q\}\) is called a local antimagic labeling of $ if for any two adjacent vertices \(u\) and \(v\), we have \(g^+(u) \ne g^+(v)\), where \(g^+(u) = \sum_{e\in E(u)} g(e)\), and \(E(u)\) is the set of edges incident to \(u\). Similarly, a bijection \(f:V(G)\cup E(G)\to \{1,2,\ldots,p+q\}\) is called a local antimagic total labeling of \(G\) if for any two adjacent vertices \(u\) and \(v\), we have \(w_f(u)\ne w_f(v)\), where \(w_f(u) = f(u) + \sum_{e\in E(u)} f(e)\). Thus, any local antimagic (total) labeling induces a proper vertex coloring of \(G\) if vertex \(v\) is assigned the color \(g^+(v)\) (respectively, \(w_f(u)\)). The local antimagic (total) chromatic number, denoted \(\chi_{la}(G)\) (respectively \(\chi_{lat}(G)\)), is the minimum number of induced colors taken over local antimagic (total) labeling of \(G\). In this paper, we determined \(\chi_{lat}(G)\) where \(G\) is the amalgamation ofcomplete graphs. Consequently, we also obtained the local antimagic (total) chromatic number of the disjoint union of complete graphs, and the join of \(K_1\) and amalgamation of complete graphs under various conditions. An application of local antimagic total chromatic number is also given.
Keywords: local antimagic (total) chromatic number, amalgamation, complete graphs.
Mathematics Subject Classification: 05C78, 05C15.
Journal: Opuscula Mathematica.
Citation: Opuscula Math. 43, no. 3 (2023), 429-453, https://doi.org/10.7494/OpMath.2023.43.3.429.
PubDate: Wed, 17 May 2023 21:00:05 +020
- Existence and asymptotic stability for generalized elasticity equation
with variable exponent
Abstract:
Author(s): Mohamed Dilmi, Sadok Otmani.
Abstract: In this paper we propose a new mathematical model describing the deformations of an isotropic nonlinear elastic body with variable exponent in dynamic regime. We assume that the stress tensor \(\sigma^{p(\cdot)}\) has the form \[\sigma^{p(\cdot)}(u)=(2\mu + d(u) ^{p(\cdot)-2})d(u)+\lambda Tr(d(u)) I_{3},\] where \(u\) is the displacement field, \(\mu\), \(\lambda\) are the given coefficients \(d(\cdot)\) and \(I_{3}\) are the deformation tensor and the unit tensor, respectively. By using the Faedo-Galerkin techniques and a compactness result we prove the existence of the weak solutions, then we study the asymptotic behaviour stability of the solutions.
Keywords: asymptotic stability, variable exponent Lebesgue and Sobolev spaces, generalized elasticity equation.
Mathematics Subject Classification: 35B37, 35L55, 35L70, 46E30.
Journal: Opuscula Mathematica.
Citation: Opuscula Math. 43, no. 3 (2023), 409-428, https://doi.org/10.7494/OpMath.2023.43.3.409.
PubDate: Wed, 17 May 2023 21:00:04 +020
- Axiomatic characterizations of Ptolemaic and chordal graphs
Abstract:
Author(s): Manoj Changat, Lekshmi Kamal K. Sheela, Prasanth G. Narasimha-Shenoi.
Abstract: The interval function and the induced path function are two well studied class of set functions of a connected graph having interesting properties and applications to convexity, metric graph theory. Both these functions can be framed as special instances of a general set function termed as a transit function defined on the Cartesian product of a non-empty set \(V\) to the power set of \(V\) satisfying the expansive, symmetric and idempotent axioms. In this paper, we propose a set of independent first order betweenness axioms on an arbitrary transit function and provide characterization of the interval function of Ptolemaic graphs and the induced path function of chordal graphs in terms of an arbitrary transit function. This in turn gives new characterizations of the Ptolemaic and chordal graphs.
Keywords: interval function, betweenness axioms, Ptolemaic graphs, transit function, induced path transit function.
Mathematics Subject Classification: 05C12, 05C75.
Journal: Opuscula Mathematica.
Citation: Opuscula Math. 43, no. 3 (2023), 393-407, https://doi.org/10.7494/OpMath.2023.43.3.393.
PubDate: Wed, 17 May 2023 21:00:03 +020
- On efficiency and duality for a class of nonconvex nondifferentiable
multiobjective fractional variational control problems
Abstract:
Author(s): Tadeusz Antczak, Manuel Arana-Jimenéz, Savin Treanţă.
Abstract: In this paper, we consider the class of nondifferentiable multiobjective fractional variational control problems involving the nondifferentiable terms in the numerators and in the denominators. Under univexity and generalized univexity hypotheses, we prove optimality conditions and various duality results for such nondifferentiable multiobjective fractional variational control problems. The results established in the paper generalize many similar results established earlier in the literature for such nondifferentiable multiobjective fractional variational control problems.
Keywords: nondifferentiable multiobjective fractional variational control problem, efficient solution, optimality conditions, (generalized) univexity, Mond-Weir duality, Wolfe duality.
Mathematics Subject Classification: 65K10, 90C32, 90C46, 90C30, 90C26.
Journal: Opuscula Mathematica.
Citation: Opuscula Math. 43, no. 3 (2023), 335-391, https://doi.org/10.7494/OpMath.2023.43.3.335.
PubDate: Wed, 17 May 2023 21:00:02 +020
- Operators induced by certain hypercomplex systems
Abstract:
Author(s): Daniel Alpay, Ilwoo Cho.
Abstract: In this paper, we consider a family \(\{ \mathbb{H}_{t}\}_{t\in\mathbb{R}}\) of rings of hypercomplex numbers, indexed by the real numbers, which contain both the quaternions and the split-quaternions. We consider natural Hilbert-space representations \(\{(\mathbb{C}^{2},\pi_{t})\}_{t\in\mathbb{R}}\) of the hypercomplex system \(\{ \mathbb{H}_{t}\}_{t\in\mathbb{R}}\), and study the realizations \(\pi_{t}(h)\) of hypercomplex numbers \(h \in \mathbb{H}_{t}\), as \((2\times 2)\)-matrices acting on \(\mathbb{C}^{2}\), for an arbitrarily fixed scale \(t\in\mathbb{R}\). Algebraic, operator-theoretic, spectral-analytic, and free-probabilistic properties of them are considered.
Keywords: scaled hypercomplex ring, scaled hypercomplex monoids, representations, scaled-spectral forms, scaled-spectralization.
Mathematics Subject Classification: 20G20, 46S10, 47S10.
Journal: Opuscula Mathematica.
Citation: Opuscula Math. 43, no. 3 (2023), 275-333, https://doi.org/10.7494/OpMath.2023.43.3.275.
PubDate: Wed, 17 May 2023 21:00:01 +020
- Discrete spectrum of zero order pseudodifferential operators
Abstract:
Author(s): Grigori Rozenblum.
Abstract: We study the rate of convergence of eigenvalues to the endpoints of essential spectrum for zero order pseudodifferential operators on a compact manifold.
Keywords: pseudodifferential operators, eigenvalue asymptotics.
Mathematics Subject Classification: 47A75, 58J50.
Journal: Opuscula Mathematica.
Citation: Opuscula Math. 43, no. 2 (2023), 247-268, https://doi.org/10.7494/OpMath.2023.43.2.247.
PubDate: Mon, 27 Mar 2023 19:00:07 +020
- Existence and asymptotic behavior of nonoscillatory solutions of
half-linear ordinary differential equations
Abstract:
Author(s): Manabu Naito.
Abstract: We consider the half-linear differential equation \[( x' ^{\alpha}\mathrm{sgn}\,x')' + q(t) x ^{\alpha}\mathrm{sgn}\,x = 0, \quad t \geq t_{0},\] under the condition\[\lim_{t\to\infty}t^{\alpha}\int_{t}^{\infty}q(s)ds = \frac{\alpha^{\alpha}}{(\alpha+1)^{\alpha+1}}.\] It is shown that if certain additional conditions are 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.
Mathematics Subject Classification: 34C11, 34C10.
Journal: Opuscula Mathematica.
Citation: Opuscula Math. 43, no. 2 (2023), 221-246, https://doi.org/10.7494/OpMath.2023.43.2.221.
PubDate: Mon, 27 Mar 2023 19:00:06 +020
- Asymptotic analysis of the steady advection-diffusion problem in axial
domains
Abstract:
Author(s): Fernando A. Morales.
Abstract: We present the asymptotic analysis of the steady advection-diffusion equation in a thin tube. The problem is modeled in a mixed-type variational formulation, in order to separate the phenomenon in the axial direction and a transverse one. Such formulation makes visible the natural separation of scales within the problem and permits a successful asymptotic analysis, delivering a limiting form, free from the initial geometric singularity and suitable for approximating the original one. Furthermore, it is shown that the limiting problem can be simplified to a significantly simpler structure.
Keywords: asymptotic analysis, mixed-type variational formulations, advection-diffusion.
Mathematics Subject Classification: 80A20, 35F15.
Journal: Opuscula Mathematica.
Citation: Opuscula Math. 43, no. 2 (2023), 199-220, https://doi.org/10.7494/OpMath.2023.43.2.199.
PubDate: Mon, 27 Mar 2023 19:00:05 +020
- Lower density operators. Φ_{f} versus Φ_{d}
Abstract:
Author(s): Gertruda Ivanova, Elżbieta Wagner-Bojakowska, Władysław Wilczyński.
Abstract: Using the new method of the construction of lower density operator introduced in the earlier paper of the first two authors, we study how much the new operator can be different from the classical one. The aim of this paper is to show that if \(f\) is a good adjusted measure-preserving bijection then the lower density operator generated by \(f\) can be really different from the classical density operator.
Keywords: lower density operator, measure-preserving bijection.
Mathematics Subject Classification: 54C60, 26E25, 28D05.
Journal: Opuscula Mathematica.
Citation: Opuscula Math. 43, no. 2 (2023), 185-197, https://doi.org/10.7494/OpMath.2023.43.2.185.
PubDate: Mon, 27 Mar 2023 19:00:04 +020
- Self-coalition graphs
Abstract:
Author(s): Teresa W. Haynes, Jason T. Hedetniemi, Stephen T. Hedetniemi, Alice A. McRae, Raghuveer Mohan.
Abstract: A coalition in a graph \(G = (V, E)\) consists of two disjoint sets \(V_1\) and \(V_2\) of vertices, such that neither \(V_1\) nor \(V_2\) is a dominating set, but the union \(V_1 \cup V_2\) is a dominating set of \(G\). A coalition partition in a graph \(G\) of order \(n = V \) is a vertex partition \(\pi = \{V_1, V_2, \ldots, V_k\}\) such that every set \(V_i\) either is a dominating set consisting of a single vertex of degree \(n-1\), or is not a dominating set but forms a coalition with another set \(V_j\) which is not a dominating set. Associated with every coalition partition \(\pi\) of a graph \(G\) is a graph called the coalition graph of \(G\) with respect to \(\pi\), denoted \(CG(G,\pi)\), the vertices of which correspond one-to-one with the sets \(V_1, V_2, \ldots, V_k\) of \(\pi\) and two vertices are adjacent in \(CG(G,\pi)\) if and only if their corresponding sets in \(\pi\) form a coalition. The singleton partition \(\pi_1\) of the vertex set of \(G\) is a partition of order \( V \), that is, each vertex of \(G\) is in a singleton set of the partition. A graph \(G\) is called a self-coalition graph if \(G\) is isomorphic to its coalition graph \(CG(G,\pi_1)\), where \(\pi_1\) is the singleton partition of \(G\). In this paper, we characterize self-coalition graphs.
Keywords: coalitions in graphs, coalition partitions, coalition graphs, domination.
Mathematics Subject Classification: 05C69.
Journal: Opuscula Mathematica.
Citation: Opuscula Math. 43, no. 2 (2023), 173-183, https://doi.org/10.7494/OpMath.2023.43.2.173.
PubDate: Mon, 27 Mar 2023 19:00:03 +020
- Square-root boundaries for Bessel processes and the hitting times of
radial Ornstein-Uhlenbeck processes
Abstract:
Author(s): Yuji Hamana.
Abstract: This article deals with the first hitting times of a Bessel process to a square-root boundary. We obtain the explicit form of the distribution function of the hitting time by means of zeros of the confluent hypergeometric function with respect to the first parameter. In deducing the distribution function, the time that a radial Ornstein-Uhlenbeck process reaches a certain point is very useful and plays an important role. We also give its distribution function in the case that the starting point is closer to the origin than the arrival site.
Keywords: Bessel process, confluent hypergeometric function, first hitting time, radial Ornstein-Uhlenbeck process, square-root boundary.
Mathematics Subject Classification: 60J60, 33C15, 44A10.
Journal: Opuscula Mathematica.
Citation: Opuscula Math. 43, no. 2 (2023), 145-172, https://doi.org/10.7494/OpMath.2023.43.2.145.
PubDate: Mon, 27 Mar 2023 19:00:02 +020
- Global attractivity of a higher order nonlinear difference equation with
unimodal terms
Abstract:
Author(s): Abdulaziz Almaslokh, Chuanxi Qian.
Abstract: In the present paper, we study the asymptotic behavior of the following higher order nonlinear difference equation with unimodal terms \[x(n+1)= ax(n)+ bx(n)g(x(n)) + cx(n-k)g(x(n-k)), \quad n=0, 1, \ldots,\] where \(a\), \(b\) and \(c\) are constants with \(0\lt a\lt 1\), \(0\leq b\lt 1\), \(0\leq c \lt 1\) and \(a+b+c=1\), \(g\in C[[0, \infty), [0, \infty)]\) is decreasing, and \(k\) is a positive integer. We obtain some new sufficient conditions for the global attractivity of positive solutions of the equation. Applications to some population models are also given.
Keywords: higher order difference equation, positive equilibrium, unimodal term, global attractivity, population model.
Mathematics Subject Classification: 39A10, 92D25.
Journal: Opuscula Mathematica.
Citation: Opuscula Math. 43, no. 2 (2023), 131-143, https://doi.org/10.7494/OpMath.2023.43.2.131.
PubDate: Mon, 27 Mar 2023 19:00:01 +020
