A set-valued approach for quasi-variational–hemivariational inequality systems

Abstract In this paper, we investigate a coupled system of quasi-variational–hemivariational inequalities, with variable constraint sets that depend on the solution of the system. First, we consider the case of nonempty bounded closed convex constraint sets and employ a fixed point theorem for set-valued maps to show the non-emptiness and weak compactness of the solution set. If one of the constraint sets is unbounded, we impose an additional coercivity condition to ensure the existence of solutions. As an application, in the last section of the paper, we consider a mathematical model which describes the contact between a deformable elastic body and Winkler-type foundation. We show that the primal-dual variational formulation of the model leads to a coupled system of quasi-variational–hemivariational inequalities and we employ the theoretical results obtained in the previous sections to achieve the weak solvability of the problem.