Existence theorems of the variational‐hemivariational inequalities

This paper is devoted to the existence of solutions for the variational‐hemivariational inequalities in reflexive Banach spaces. Using the notion of the stable $\varphi$ ‐quasimonotonicity and the properties of Clarke’s generalized directional derivative and Clarke’s generalized gradient, some existence results of solutions are proved 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 results presented in this paper generalize and improve some known results.