Existence results for general inequality problems with constraints

Abstract

This paper is concerned with existence results for inequality problems of type F0(u;v)+Ψ′(u;v)≥0, for all v∈X, where X is a Banach space, F:X→ℝ is locally Lipschitz, and Ψ:X→(−∞+∞] is proper, convex, and lower semicontinuous. Here F0 stands for the generalized directional derivative of F and Ψ′ denotes the directional derivative of Ψ. The applications we consider focus on the variational-hemivariational inequalities involving the p-Laplacian operator.