A new general class of systems of elliptic quasi-variational–hemivariational inequalities

Abstract

In this paper we study a general class of systems of strongly coupled quasi-variational–hemivariational inequalities with implicit constraints in reflexive Banach spaces. Each inequality contains a convex potential, a locally Lipschitz superpotential, and a implicit obstacle set. The solvability of the system is established based on a fixed point approach, monotonicity arguments, and nonsmooth analysis. The result generalizes several theorems on various particular cases on variational, variational–hemivariational, and quasi-variational inequalities. We illustrate the applicability of the abstract theory by a nonlinear static thermoelastic frictional contact problem with implicit obstacles for which we provide existence and regularity results.