ON THE WELL-POSEDNESS OF VARIATIONAL-HEMIVARIATIONAL INEQUALITIES AND ASSOCIATED FIXED POINT PROBLEMS

We consider an elliptic variational-hemivariational inequality P in a p-uniformly smooth Banach space. We prove that the inequality is governed by a multivalued maximal monotone operator, and, for each lambda > 0, we use the resolvent of this operator to construct an auxiliary fixed point problem, denoted P-A,. Next, we perform a parallel study of problems P and P-A, based on their intrinsic equivalence. In this way, we prove existence, uniqueness, and well-posedness results with respect to specific Tykhonov triples. The existence of a unique common solution to problems P and P-A, is proved by using the Banach contraction principle in the study of Problem PA,. In contrast, the well-posedness of the problems is obtained by using a monotonicity argument in the study of Problem P. Finally, the properties of Problem PA, allow us to deduce a convergence criterion in the study of Problem P.