On the existence of solutions and optimal control for set-valued quasi-variational-hemivariational inequalities with applications

In this paper, we study the existence and optimal control of quasi-hemivariational inequalities by a method different from the one based on Minty's technique. We use an approach similar to the Galerkin method based on a minimax inequality formulation associated with the Brezis pseudomonotonicity notion of multi-valued operators, an implicit Browder-Tikhonov regularization method and a fixed point theorem. This leads us to avoid any kind of monotonicity-type conditions used in recent papers to obtain the convexity of the solution set of the variational selections. We provide applications to the optimal control of implicit obstacle problems of fractional Laplacian type involving a generalized gradient operator, and to the optimal control of contact problems for elastic locking materials. Our approach improves some recent results in the literature.