A class of differential hemivariational inequalities in Banach spaces

Stanisław Migórski,Shengda Zeng

doi:10.1007/s10898-018-0667-5

Abstract

In this paper we investigate an abstract system which consists of a hemivariational inequality of parabolic type combined with a nonlinear evolution equation in the framework of an evolution triple of spaces which is called a differential hemivariational inequality [(DHVI), for short]. A hybrid iterative system corresponding to (DHVI) is introduced by using a temporally semi-discrete method based on the backward Euler difference scheme, i.e., the Rothe method, and a feedback iterative technique. We apply a surjectivity result for pseudomonotone operators and properties of the Clarke subgradient operator to establish existence and a priori estimates for solutions to an approximate problem. Finally, through a limiting procedure for solutions of the hybrid iterative system, the solvability of (DHVI) is proved without imposing any convexity condition on the nonlinear function umapsto f(t,x,u) and compactness of C_0-semigroup e^{A(t)}.

Highlights

It is well known that the theory of variational inequalities, which was initially developed to deal with equilibrium problems, is closely related to the convexity of the energy functionals involved, and is based on various monotonicity arguments
Many problems are described by nonsmooth superpotentials, it is not surprising that, during the last thirty years, a lot of scholars devoted their work to the development of theory and applications of hemivariational inequalities, for example, in contact mechanics [14,35,36,44,51], well-posedness [28,49], control problems [31], nonconvex and nonsmooth inclusions [42,43], and so forth
For the first time, we apply the Rothe method, see [16,51], to study a system of a hemivariational inequality of parabolic type driven by a nonlinear abstract evolution equation

Summary

Introduction

It is well known that the theory of variational inequalities, which was initially developed to deal with equilibrium problems, is closely related to the convexity of the energy functionals involved, and is based on various monotonicity arguments. In our life, many applied problems in engineering, operations research, economical dynamics, and physical sciences, etc., are more precisely described by partial differential equations Based on this motivation, recently, Liu–Zeng–Motreanu [24,26] in 2016 and Liu et al [23] in 2017 proved the existence of solutions for a class of differential mixed variational inequalities in Banach spaces through applying the theory of semigroups, the Filippov implicit function lemma and fixed point theorems for condensing set-valued operators. We consider the following abstract system consisting of a hemivariational inequality of parabolic type combined with a nonlinear abstract evolution equation. For the first time, we apply the Rothe method, see [16,51], to study a system of a hemivariational inequality of parabolic type driven by a nonlinear abstract evolution equation. Through a limiting procedure for the solutions to Problem 16, the existence of solution to Problem 1 is established

Preliminaries

Main results

H for all v