A Fixed Point Approach of Variational-Hemivariational Inequalities

In this paper we provide a new approach in the study of a variational-hemivariational inequality in Hilbert space, based on the theory of maximal monotone operators and the Banach fixed point theorem.
First, we introduce the inequality problem we are interested in, list the assumptions on the data and show that it is governed by a multivalued maximal monotone operator. Then, we prove that solving the variational-hemivariational inequality is equivalent to finding a fixed point for the resolvent of this operator. Based on this equivalence result, we use the Banach contraction principle to prove the unique solvability of the problem. Moreover, we construct the corresponding Picard, Krasnoselski and Mann iterations and deduce their convergence to the unique solution of the variational-hemivariational inequality.