Existence results for general inequality problems with constraints

Abstract

This paper is concerned with existence results for inequality problems of type $F^0\left(u;v\right)+{\Psi }^{\prime }\left(u;v\right)\ge 0$, for all $v\in X$, where $X$ is a Banach space, $F:X\to ℝ$ is locally Lipschitz, and $\Psi :X\to (-\infty +\infty ]$ is proper, convex, and lower semicontinuous. Here $F^0$ stands for the generalized directional derivative of $F$ and ${\Psi }^{\prime }$ denotes the directional derivative of $\Psi$. The applications we consider focus on the variational-hemivariational inequalities involving the $p$-Laplacian operator.