Existence of solutions for a class of variational-hemivariational-like inequalities in Banach spaces

Abstract

This paper is devoted to study the existence of solutions for a class of variational-hemivariational-like inequalities in reflexive Banach spaces. Using the notion of the stable $(\phi,\eta)$-quasimonotonicity, the properties of Clarke's generalized directional derivative and Clarke's generalized gradient, we establish some existence results of solutions when the constrained set is nonempty, bounded (or unbounded), closed and convex. Moreover, a sufficient condition to the boundedness of the solution set and a necessary and sufficient condition to the existence of solutions are also derived.