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