Coupled Variational Inequalities: Existence, Stability and Optimal Control

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.