Coupled Variational Inequalities: Existence, Stability and Optimal Control

Abstract In this paper, we introduce and investigate a new kind of coupled systems, called variational inequalities, which consist two elliptic mixed inequalities on Banach spaces. Under general assumptions, by employing Kakutani-Ky Fan fixed point theorem combined with Minty technique, prove that the set solutions for inequality (CVI, short) under consideration is nonempty weak compact. Then, uniqueness theorems are delivered via using monotonicity arguments, stability result CVI proposed, through perturbations duality mappings. Furthermore, an optimal control problem governed introduced, solvability established. Finally, to illustrate applicability theoretical results, study boundary value system nonlocal effect multivalued conditions, feedback involving least energy condition respect variable, respectively.