Evolutionary Variational Inequalities with Volterra Type Operators

Mykola Bokalo,Olha Sus

doi:10.13189/ms.2019.070504

Abstract

In this paper, we consider the initial-value problem for parabolic variational inequalities (subdifferential inclusions) with Volterra type operators. We prove the existence and the uniqueness of the solution. Furthermore, the estimates of the solution are obtained. The results are achieved using the Banach's fixed point theorem (the principle of compression mappings). The motivation for this work comes from the evolutionary variational inequalities arising in the study of frictionless contact problems for linear viscoelastic materials with long-term memory. Also, such kind of problems have their application in constructing different models of the injection molding processes.

Highlights

We consider the problem for parabolic variational inequalities with Volterra type operators
Let us remark that problems for evolutionary variational inequalities without Volterra type operators are widely investigated, in [5], [8], [9], [15], [16], [23], [24], [25], [26], and others
Let us note that problems for parabolic variational inequalities or subdifferential inclusions with Volterra type operators have not been considered in the literature before

Summary

Introduction

We consider the problem for parabolic variational inequalities (subdifferential inclusions) with Volterra type operators. Let us remark that problems for evolutionary variational inequalities without Volterra type operators are widely investigated, in [5], [8], [9], [15], [16], [23], [24], [25], [26], and others. Let us note that problems for parabolic variational inequalities or subdifferential inclusions with Volterra type operators have not been considered in the literature before. It serves us as one of the motivations for the study of such kind of problems.

Preliminaries

Statement of the problem and the main results

Proof of the main results

Findings

Conclusions