On the well-posedness of variational-hemivariational inequalities and associated fixed point problems

Abstract

. 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 λ > 0, we use the resolvent of this operator to construct an auxiliary fixed point problem, denoted P λ . Next, we perform a parallel study of problems P and P λ 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 λ is proved by using the Banach contraction principle in the study of Problem P λ . 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 P λ allow us to deduce a convergence criterion in the study of Problem P .