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-hemivariationallike inequalities in reflexive Banach spaces. Using the notion of the stable (φ, η)-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.

The Solvability of Generalized Systems of Time-Dependent Hemivariational Inequalities Enjoying Symmetric Structure in Reflexive Banach Spaces

“…As a helpful technique, the HVIs and their systems have played a crucial role in the study of quite meaningful problems of mechanics and engineering sciences, e.g., obstacle problems, thermoviscoelastic frictional contact problems, unilateral contact problems in nonlinear elasticity, etc. ; please refer to [2][3][4][5][6]. Via the Clarke's generalized directional derivative and the Clarke's generalized gradient (appearing in Section 2), various HVIs and systems of HVIs (SHVIs, for short), e.g., stationary HVIs, evolutionary HVIs, and their systems, etc., have been investigated by numerous authors in the past more than 30 years; please refer to, e.g., [2][3][4][5][6][7][8][9][10][11][12].…”

“…; please refer to [2][3][4][5][6]. Via the Clarke's generalized directional derivative and the Clarke's generalized gradient (appearing in Section 2), various HVIs and systems of HVIs (SHVIs, for short), e.g., stationary HVIs, evolutionary HVIs, and their systems, etc., have been investigated by numerous authors in the past more than 30 years; please refer to, e.g., [2][3][4][5][6][7][8][9][10][11][12].…”