Research On A Class Of System Of Variational--Hemivariational Inequalities With Volterra Integral Terms

As a generalization of variational inequalities,hemivariational inequalities and their systems play an important role in the fields of mechanics,physics,engineering science,economics,optimal control,etc.Therefore,using research tools such as KKM theorem,finite element method and surjective lemma,various types of hemivariational inequalities and their systems have been extensively studied,and abundant research results have been obtained.In this thesis,we consider a generalized system of variational-hemivariational in-equalities,which can be used to study a class of thermoviscoelastic frictional contact prob-lems.First of all,in the Bochner-Lebesgue space,a system of variational-hemivariational inequalities with history-dependent operators is considered and its solvability is proved.Secondly,in the continuous function space,we consider a system of variational-hemivaria-tional inequalities concerning both history-dependent operators and Volterra integral terms,and give conditions of solvability for the corresponding problem.Different from the tra-ditional research methods in the literature for solvability of the system of variational-hemivariational inequalities,we directly use a generalized fixed point theorem to study the unique solvability of the system of variational-hemivariational inequalities.The main research results are as follows.Firstly,in chapter three,we consider a system of variational-hemivariational inequali-ties with history-dependent operators.It is mainly composed of two variational-hemivaria-tional inequalities with history-dependent operators.Moreover,the convex energy func-tionals and the Clarke direction derivative of the local Lipschitz functionals have a cross effect between the two inequalities in the system.For the study of the solvability of this system,we mainly complete it in two steps: the first step is to define an operator related to the solution of the original problem by proving the unique solvability of the auxiliary problem? the second step is to verify that the defined operator has a unique fixed point to obtain the uniqueness of solution to the system.Then,based on the theoretical research results obtained in chapter three and the re-search significance in the modeling of contact problem of Volterra integral term,in chapter four we focus a system of variational-hemivariational inequalities with history-dependent operators and Volterra integral terms,in which the Volterra integral terms also have cross-effects in the system.At the same time,we use the generalized fixed point theorem to ob-tain the existence and uniqueness of solution to the system of variational-hemivariational inequalities in the continuous function space.Finally,in order to clarify the generalization of the model studied in this thesis,we consider several degenerate cases for the models studied in chapter three and chapter four respectively,and make comparison among the corresponding solvability conditions when different research methods are used.