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A Class of Elliptic Quasi–Variational–Hemivariational Inequalities with Applications

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arXiv 2023年

作者： Migórski, Stanislaw Yao, Jen-Chih Zeng, Shengda College of Applied Mathematics Chengdu University of Information Technology Sichuan Province Chengdu610225 China Jagiellonian University in Krakow Chair of Optimization and Control ul. Lojasiewicza 6 Krakow30348 Poland Center for General Education China Medical University Taichung Taiwan Guangxi Colleges Universities Key Laboratory of Complex System Optimization and Big Data Processing Yulin Normal University Guangxi Yulin537000 China Jagiellonian University in Krakow Faculty of Mathematics and Computer Science ul. Lojasiewicza 6 Krakow30348 Poland

equalities in reflexive Banach spaces. The inequalities contain a convex potential, a locally Lipschitz superpotential, and a solution-dependent set of constraints. Solution existence and compactness of the solution set to the inequality problem are established based on the Kakutani–Ky Fan–Glicksberg fixed point theorem. Two examples of the interior and boundary semipermeability models illustrate the applicability of our *** Codes 35L15, 35L86, 35L87 © 2023, CC BY.

关键词： Banach spaces

A NEW CLASS OF EVOLUTION MULTIVALUED QUASI-VARIATIONAL INEQUALITIES I: EXISTENCE AND NONSMOOTH OPTIMAL CONTROL

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arXiv 2024年

作者： Zeng, Shengda Radulescu, Vicentiu D. Guangxi Colleges and Universities Key Laboratory of Complex System Optimization and Big Data Processing Yulin Normal University Guangxi Yulin537000 China Jagiellonian University in Krakow Faculty of Mathematics and Computer Science ul. Lojasiewicza 6 Krakow30-348 Poland Faculty of Applied Mathematics AGH University of Kraków Kraków30-059 Poland Department of Mathematics University of Craiova Craiova200585 Romania Simion Stoilow Institute of Mathematics The Romanian Academy Bucharest010702 Romania

In this paper, we consider a new kind of evolution multivalued quasi-variational inequalities with feedback effect and a nonlinear bifunction which contain several (evolution) quasi-variational/hemivariational inequalities as special cases. The main contribution of this paper is twofold. The first goal is to establish a novel framework for proving the existence of solutions and the compactness of solution set to the evolution multivalued quasi-variational inequalities, under quite mild assumptions. Whereas, the second contribution is to introduce and study a nonlinear and nonsmooth optimal control problem governed by an evolution multivalued quasi-variational inequality, and then to obtain the sufficient conditions for guaranteeing the solvability of the nonlinear and nonsmooth optimal control problem under consideration. Such nonlinear and nonsmooth optimal control problem could as a useful model to explore the simultaneous distributed-boundary optimal control problems driven by evolution multivalued quasi-variational inequalities, and optimal parameters identification for evolution multivalued quasi-variational inequalities. Copyright © 2024, The Authors. All rights reserved.

关键词： Optimal control systems

DOUBLE PHASE IMPLICIT OBSTACLE PROBLEMS WITH CONVECTION AND MULTIVALUED MIXED BOUNDARY VALUE CONDITIONS

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arXiv 2021年

作者： Zeng, Shengda Rǎdulescu, Vicenţiu D. Winkert, Patrick Guangxi Colleges Universities Key Laboratory of Complex System Optimization and Big Data Processing Yulin Normal University Guangxi Yulin537000 China Jagiellonian University in Krakow Faculty of Mathematics and Computer Science ul. Lojasiewicza 6 Krakow30-348 Poland Faculty of Applied Mathematics Agh University of Science and Technology Kraków30-059 Poland Department of Mathematics University of Craiova Craiova200585 Romania Technische Universität Berlin Institut für Mathematik Straße des 17. Juni 136 Berlin10623 Germany

In this paper we consider a mixed boundary value problem with a nonhomogeneous, nonlinear differential operator (called double phase operator), a nonlinear convection term (a reaction term depending on the gradient), three multivalued terms and an implicit obstacle constraint. Under very general assumptions on the data, we prove that the solution set of such implicit obstacle problem is nonempty (so there is at least one solution) and weakly compact. The proof of our main result uses the Kakutani-Ky Fan fixed point theorem for multivalued operators along with the theory of nonsmooth analysis and variational methods for pseudomonotone operators. Copyright © 2021, The Authors. All rights reserved.

关键词： Fixed point arithmetic

Fuzzy Stability of the Orthogonal Cauchy Functional equations

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Chinese Quarterly Journal of Mathematics 2018年第3期33卷 302-312页

作者： BI Lv-qing DAI Song-song HU BO School of Electronics and Communication Engineering Guangxi Colleges and Universities KeyLaboratory of Complex System Optimization and Big Data Processing Yulin Normal UniversityYulin537000P.R.China. School of Information Science and Engineering Xiamen UniversityXiamen 361005P.R.China. School of Mechanical and Electrical Engineering Guizhou Normal UniversityGuiyang 550025P.R.China.

In this paper, we establish fuzzy stability of the orthogonal Cauchy functional equations f(x + y) = f(x) + f(y), x ⊥ y and the orthogonal Cauchy functional of P exider type f(x + y) = g(x) + h(y), x ⊥ y in which ⊥... 详细信息

关键词： Fuzzy normed space Hyers-Ulam stability Orthogonally Cauchy functionalequation Orthogonally Cauchy functional Pexider equation Orthogonality space

DOUBLE PHASE OBSTACLE PROBLEMS WITH MULTIVALUED CONVECTION AND MIXED BOUNDARY VALUE CONDITIONS

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arXiv 2021年

作者： Zeng, Shengda Radulescu, Vicent Iu D. Winkert, Patrick Guangxi Colleges Universities Key Laboratory of Complex System Optimization and Big Data Processing Yulin Normal University Guangxi Yulin537000 China Department of Mathematics Nanjing University Jiangsu Nanjing210093 China Jagiellonian University in Krakow Faculty of Mathematics and Computer Science ul. Lojasiewicza 6 Krakow30-348 Poland Faculty of Applied Mathematics AGH University of Science and Technology Kraków30-059 Poland Department of Mathematics University of Craiova Craiova200585 Romania China-Romania Research Center in Applied Mathematics Craiova Romania Technische Universität Berlin Institut für Mathematik Straße des 17. Juni 136 Berlin10623 Germany

In this paper, we consider a mixed boundary value problem with a double phase partial differential operator, an obstacle effect and a multivalued reaction convection term. Under very general assumptions, an existence theorem for the mixed boundary value problem under consideration is proved by using a surjectivity theorem for multivalued pseudomonotone operators together with the approximation method of Moreau-Yosida. Then, we introduce a family of the approximating problems without constraints corresponding to the mixed boundary value problem. Denoting by S the solution set of the mixed boundary value problem and by Sn the solution sets of the approximating problems, we establish the following convergence relation ∅ =6 w- lim sup Sn = s- lim sup Sn ⊂ S, n→∞ n→∞ where w-lim supn→∞ Sn and s-lim supn→∞ Sn stand for the weak and the strong Kuratowski upper limit of Sn, respectively. Copyright © 2021, The Authors. All rights reserved.

关键词： Mathematical operators

Anisotropic and Isotropic Implicit Obstacle Problems with Nonlocal Terms and Multivalued Boundary Conditions

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SSRN

SSRN 2022年

作者： Zeng, Shengda Gasinski, Leszek Radulescu, Vicentiu D. Winkert, Patrick Guangxi Colleges Universities Key Laboratory of Complex System Optimization and Big Data Processing Yulin Normal University Guangxi Yulin537000 China Jagiellonian University in Krakow Faculty of Mathematics and Computer Science ul. Lojasiewicza 6 Krakow30-348 Poland Department of Mathematics Pedagogical University of Cracow Podchorazych 2 Cracow30-084 Poland Faculty of Applied Mathematics AGH University of Science and Technology Kraków30-059 Poland Department of Mathematics University of Craiova Craiova200585 Romania China-Romania Research Center in Applied Mathematics Romania Technische Universität Berlin Institut für Mathematik Straße des 17. Juni 136 Berlin10623 Germany

In this paper we study an anisotropic implicit obstacle problem driven by the $(p(\cdot),q(\cdot))$-Laplacian and an isotropic implicit obstacle problem involving a nonlinear convection term (a reaction term depending on the gradient) which contain several interesting and challenging untreated problems. These two implicit obstacle problems have both highly nonlinear and nonlocal functions and three multivalued terms where two of them are appearing on the boundary and the other one is formulated in the domain. Under very general assumptions on the data, we develop general frameworks to examine the nonemptiness and compactness of the set of weak solutions to the problems under consideration. The proofs of our main results use the theory of nonsmooth analysis, Tychonoff's fixed point theorem for multivalued operators, the theory of pseudomonotone operators and variational approach. © 2022, The Authors. All rights reserved.

关键词： Anisotropy

A class of elliptic mixed boundary value problems with (p,q)-Laplacian: existence, comparison and optimal control

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arXiv 2023年

作者： Zeng, Shengda Migórski, Stanislaw Tarzia, Domingo A. Zou, Lang Nguyen, Van Thien Guangxi Colleges Universities Key Laboratory of Complex System Optimization and Big Data Processing Yulin Normal University Yulin537000 China Department of Mathematics Nanjing University Jiangsu Nanjing210093 China Jagiellonian University in Krakow Faculty of Mathematics and Computer Science ul. Lojasiewicza 6 Krakow30348 Poland College of Applied Mathematics Chengdu University of Information Technology Sichuan Province Chengdu610225 China Jagiellonian University in Krakow Chair of Optimization and Control ul. Lojasiewicza 6 Krakow30348 Poland Depto. Matemática FCE Universidad Austral Paraguay 1950 RosarioS2000FZF Argentina CONICET Argentina School of Mathematics and Computational Science Xiangtan University Hunan Province Xiangtan City210000 China Department of Mathematics FPT University Education zone Hoa Lac High Tech Park Km29 Thang Long Highway Thach That ward Hanoi Viet Nam

The paper deals with two nonlinear elliptic equations with (p, q)-Laplacian and the Dirichlet-Neumann-Dirichlet (DND) boundary conditions, and Dirichlet-Neumann-Neumann (DNN) boundary conditions, respectively. Under mild hypotheses, we prove the unique weak solvability of the elliptic mixed boundary value problems. Then, a comparison and a monotonicity results for the solutions of elliptic mixed boundary value problems are established. We examine a convergence result which shows that the solution of (DND) can be approached by the solution of (DNN). Moreover, two optimal control problems governed by (DND) and (DNN), respectively, are considered, and an existence result for optimal control problems is obtained. Finally, we provide a result on asymptotic behavior of optimal controls and system states, when a parameter tends to *** Codes 35J25, 35J66, 35J92, 49J20, 35Bxx © 2023, CC BY.

关键词： Nonlinear equations