Pub Date : 2023-01-01DOI: 10.7494/opmath.2023.43.6.741
A. Alshehri, Noha Aljaber, H. Altamimi, Rasha Alessa, M. Majdoub
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)quadtext{for }(t,x)in(0,infty)times mathbb{R}^{N},] where (alphainmathbb{R}), (pgt 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_{ttoinfty} 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.
{"title":"Nonexistence of global solutions for a nonlinear parabolic equation with a forcing term","authors":"A. Alshehri, Noha Aljaber, H. Altamimi, Rasha Alessa, M. Majdoub","doi":"10.7494/opmath.2023.43.6.741","DOIUrl":"https://doi.org/10.7494/opmath.2023.43.6.741","url":null,"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)quadtext{for }(t,x)in(0,infty)times mathbb{R}^{N},] where (alphainmathbb{R}), (pgt 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_{ttoinfty} 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.","PeriodicalId":45563,"journal":{"name":"Opuscula Mathematica","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71343440","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-01-01DOI: 10.7494/opmath.2023.43.1.19
T. Godoy
Let (Omega) be a (C^{2}) bounded domain in (mathbb{R}^{n}) such that (partialOmega=Gamma_{1}cupGamma_{2}), where (Gamma_{1}) and (Gamma_{2}) are disjoint closed subsets of (partialOmega), and consider the problem(-Delta u=g(cdot,u)) in (Omega), (u=tau) on (Gamma_{1}), (frac{partial u}{partialnu}=eta) on (Gamma_{2}), where (0leqtauin W^{frac{1}{2},2}(Gamma_{1})), (etain(H_{0,Gamma_{1}}^{1}(Omega))^{prime}), and (g:Omega times(0,infty)rightarrowmathbb{R}) is a nonnegative Carath�odory function. Under suitable assumptions on (g) and (eta) we prove the existence and uniqueness of a positive weak solution of this problem. Our assumptions allow (g) to be singular at (s=0) and also at (xin S) for some suitable subsets (Ssubsetoverline{Omega}). The Dirichlet problem (-Delta u=g(cdot,u)) in (Omega), (u=sigma) on (partialOmega) is also studied in the case when (0leqsigmain W^{frac{1}{2},2}(Omega)).
{"title":"Singular elliptic problems with Dirichlet or mixed Dirichlet-Neumann non-homogeneous boundary conditions","authors":"T. Godoy","doi":"10.7494/opmath.2023.43.1.19","DOIUrl":"https://doi.org/10.7494/opmath.2023.43.1.19","url":null,"abstract":"Let (Omega) be a (C^{2}) bounded domain in (mathbb{R}^{n}) such that (partialOmega=Gamma_{1}cupGamma_{2}), where (Gamma_{1}) and (Gamma_{2}) are disjoint closed subsets of (partialOmega), and consider the problem(-Delta u=g(cdot,u)) in (Omega), (u=tau) on (Gamma_{1}), (frac{partial u}{partialnu}=eta) on (Gamma_{2}), where (0leqtauin W^{frac{1}{2},2}(Gamma_{1})), (etain(H_{0,Gamma_{1}}^{1}(Omega))^{prime}), and (g:Omega times(0,infty)rightarrowmathbb{R}) is a nonnegative Carath�odory function. Under suitable assumptions on (g) and (eta) we prove the existence and uniqueness of a positive weak solution of this problem. Our assumptions allow (g) to be singular at (s=0) and also at (xin S) for some suitable subsets (Ssubsetoverline{Omega}). The Dirichlet problem (-Delta u=g(cdot,u)) in (Omega), (u=sigma) on (partialOmega) is also studied in the case when (0leqsigmain W^{frac{1}{2},2}(Omega)).","PeriodicalId":45563,"journal":{"name":"Opuscula Mathematica","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71342620","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-01-01DOI: 10.7494/opmath.2023.43.2.173
T. Haynes, Jason T. Hedetniemi, S. Hedetniemi, A. A. McRae, Raghuveer Mohan
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.
{"title":"Self-coalition graphs","authors":"T. Haynes, Jason T. Hedetniemi, S. Hedetniemi, A. A. McRae, Raghuveer Mohan","doi":"10.7494/opmath.2023.43.2.173","DOIUrl":"https://doi.org/10.7494/opmath.2023.43.2.173","url":null,"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.","PeriodicalId":45563,"journal":{"name":"Opuscula Mathematica","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71342789","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-01-01DOI: 10.7494/opmath.2023.43.3.335
T. Antczak, Manuel Arana-Jimen�z, Savin Trean��
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.
{"title":"On efficiency and duality for a class of nonconvex nondifferentiable multiobjective fractional variational control problems","authors":"T. Antczak, Manuel Arana-Jimen�z, Savin Trean��","doi":"10.7494/opmath.2023.43.3.335","DOIUrl":"https://doi.org/10.7494/opmath.2023.43.3.335","url":null,"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.","PeriodicalId":45563,"journal":{"name":"Opuscula Mathematica","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71342840","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-01-01DOI: 10.7494/opmath.2023.43.4.575
Nikhil Sriwastav, A. Barnwal, A. Wazwaz, Mehakpreet Singh
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.
{"title":"Bernstein operational matrix of differentiation and collocation approach for a class of three-point singular BVPs: error estimate and convergence analysis","authors":"Nikhil Sriwastav, A. Barnwal, A. Wazwaz, Mehakpreet Singh","doi":"10.7494/opmath.2023.43.4.575","DOIUrl":"https://doi.org/10.7494/opmath.2023.43.4.575","url":null,"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.","PeriodicalId":45563,"journal":{"name":"Opuscula Mathematica","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71343017","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-01-01DOI: 10.7494/opmath.2023.43.5.675
Igor Kossowski
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.
{"title":"Radial solutions for nonlinear elliptic equation with nonlinear nonlocal boundary conditions","authors":"Igor Kossowski","doi":"10.7494/opmath.2023.43.5.675","DOIUrl":"https://doi.org/10.7494/opmath.2023.43.5.675","url":null,"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.","PeriodicalId":45563,"journal":{"name":"Opuscula Mathematica","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71343231","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-01-01DOI: 10.7494/opmath.2023.43.5.703
M. Nakao
{"title":"Existence and smoothing effects of the initial-boundary value problem for partial u/partial t-Deltasigma(u)=0 in time-dependent domains","authors":"M. Nakao","doi":"10.7494/opmath.2023.43.5.703","DOIUrl":"https://doi.org/10.7494/opmath.2023.43.5.703","url":null,"abstract":"","PeriodicalId":45563,"journal":{"name":"Opuscula Mathematica","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71343286","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-01-01DOI: 10.7494/opmath.2023.43.4.603
L. S. Tavares, J. V. C. Sousa
{"title":"Solutions for a nonhomogeneous p&q-Laplacian problem via variational methods and sub-supersolution technique","authors":"L. S. Tavares, J. V. C. Sousa","doi":"10.7494/opmath.2023.43.4.603","DOIUrl":"https://doi.org/10.7494/opmath.2023.43.4.603","url":null,"abstract":"","PeriodicalId":45563,"journal":{"name":"Opuscula Mathematica","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71343055","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-01-01DOI: 10.7494/opmath.2023.43.4.507
Benjamin B. Kennedy
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.
{"title":"Periodic, nonperiodic, and chaotic solutions for a class of difference equations with negative feedback","authors":"Benjamin B. Kennedy","doi":"10.7494/opmath.2023.43.4.507","DOIUrl":"https://doi.org/10.7494/opmath.2023.43.4.507","url":null,"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.","PeriodicalId":45563,"journal":{"name":"Opuscula Mathematica","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71343386","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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