Nonlinear Quasi-hemivariational Inequalities: Existence and Optimal Control

Abstract

In this paper, we investigate a generalized nonlinear quasi-hemivariational inequality (QHI) involving a multivalued map in a Banach space. Under general assumptions, by using a fixed point theorem combined with the theory of nonsmooth analysis and the Minty technique, we prove that the set of solutions for the hemivariational inequality associated to the QHI problem is nonempty, bounded, closed, and convex. Then, we prove the existence of a solution to QHI. Furthermore, an optimal control problem governed by QVI is introduced, and a solvability result for the optimal control problem is established. Finally, an approximation of an elastic contact problem with the constitutive law involving a convex subdifferential inclusion is studied as an illustrative application, in which approximate contact boundary conditions are described by a multivalued version of the normal compliance contact condition with frictionless effect and a frictional contact law with the slip dependent coefficient of friction.