Abstract
In this paper, we introduce and investigate a new kind of coupled systems, called coupled variational inequalities, which consist of two elliptic mixed variational inequalities on Banach spaces. Under general assumptions, by employing Kakutani-Ky Fan fixed point theorem combined with Minty technique, we prove that the set of solutions for the coupled variational inequality (CVI, for short) under consideration is nonempty and weak compact. Then, two uniqueness theorems are delivered via using the monotonicity arguments, and a stability result for the solutions of CVI is proposed, through the perturbations of duality mappings. Furthermore, an optimal control problem governed by CVI is introduced, and a solvability result for the optimal control problem is established. Finally, to illustrate the applicability of the theoretical results, we study a coupled elliptic mixed boundary value system with nonlocal effect and multivalued boundary conditions, and a feedback control problem involving a least energy condition with respect to the control variable, respectively.
Highlights
Introduction and Mathematical PrerequisitesIn numerous complicated natural phenomenon, physical constitutive laws, chemical processes, and economic models are often leaded to inequalities rather thanDedicated to Professor Franco Giannessi on the occasion of his 85th birthday.Communicated by Massimo Pappalardo.Journal of Optimization Theory and Applications the more commonly seen equations
This subsection is devoted to the investigation of an elliptic mixed boundary value system with distributed control in which the distributed control is described by a least energy equation which explicitly relies on the status variable
We have introduced and studied a new kind of coupled variational inequalities on Banach spaces
Summary
In numerous complicated natural phenomenon, physical constitutive laws, chemical processes, and economic models are often leaded to inequalities rather than. Among the results we mention: Pang-Stewart [38] in 2008 systematically introduced and studied a class of dynamical systems on finite-dimensional spaces, which is formulated as a combination of ordinary differential equations and time-dependent variational inequalities. They represent powerful mathematical tools with applications to various problems involving both dynamics and constraints arising in mechanical impact processes, electrical circuits with ideal diodes, Coulomb friction for contacting bodies, economical dynamics, dynamic traffic networks. Theorem 5 Let Y be a reflexive Banach space and D ⊆ Y be a nonempty, bounded, closed and convex set.
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