A Class of Variational-Hemivariational Inequalities in Reflexive Banach Spaces

Abstract

We study a new class of elliptic variational-hemivariational inequalities in reflexive Banach spaces. An inequality in the class is governed by a nonlinear operator, a convex set of constraints and two nondifferentiable functionals, among which at least one is convex. We deliver a result on existence and uniqueness of a solution to the inequality. Next, we show the continuous dependence of the solution on the data of the problem and we introduce a penalty method, for which we state and prove a convergence result. Finally, we consider a mathematical model which describes the equilibrium of an elastic body in unilateral contact with a foundation. The model leads to a variational-hemivariational inequality for the displacement field, that we analyse by using our abstract results.

A class of history-dependent variational-hemivariational inequalities

“…Its proof was obtained in our recent paper [23] and was carried out in several steps, based on a surjectivity result for multivalued pseudomonotone operators and the Banach fixed point argument. We complete the statement of Theorem 1 with the remark that for a locally Lipschitz function j : X → R, the hypothesis (2.4)(c) is equivalent to the condition…”

“…3 we introduce the class of history-variational-hemivariational inequalities to be studied, then we state and prove an abstract existence and uniqueness result, Theorem 5. The proof is based on arguments on elliptic variational-hemivariational inequalities obtained in our previous work [23], combined with a fixed point result obtained in [30]. In Sect.…”

“…So is the case of the elliptic variational-hemivariational inequalitities studied recently in [23]. This situation rises the need to extend the results obtained in [20,22,31] to an abstract class of history-dependent variationalhemivariational inequalities and to provide the history-dependent version of the results obtained in [23]. The aim of this paper is to fill this gap by obtaining such kind of abstract results.…”

“…In contrast, the history-dependent variational inequalities considered in [16,32,34] are formulated in the framework of abstract Hilbert spaces. So is the case of the elliptic variational-hemivariational inequalitities studied recently in [23]. This situation rises the need to extend the results obtained in [20,22,31] to an abstract class of history-dependent variationalhemivariational inequalities and to provide the history-dependent version of the results obtained in [23].…”

Penalty and regularization method for variational‐hemivariational inequalities with application to frictional contact

“…The present paper is a continuation of []. Among other results, in [], the existence and uniqueness of solution to the variational‐hemivariational inequality were proved by applying a penalty method.…”