A Class Of Generalized Differential Variational-Hemivariational Inequalities

Differential variational inequalities which are formulated by a time-dependent variational inequality and a differential equation can be employed as a powerful mathematical model for solving various problems from engineering.The main goal of the dissertation is to introduce a class of generalized differential variational-hemivariational inequalities in Banach spaces,and study deeply on existence,uniqueness,the dependence of the solution on initial data,the convergence of the solution,and applications.The dissertation includes six chapters.Chapter 1 mainly describes the research background and the domestic as well as foreign research situation of differential variational/hemivariational inequalities.Also,we provide the main work of this paper.Chapter 2 is devoted to recall some necessary preliminaries which contains operator semigroup theory,set-valued analysis,nonsmooth analysis theory and so on.In Chapter 3,we introduce a class of generalized differential variational-hemivariational inequalities in Banach spaces.By using KKM theorem,Minty principle and the properties of Clarke subdifferential,the solvability of the problem is proved.In Chapter 4,we further explore the uniqueness,the continuous dependence of the solution on the initial value,and the convergence of the solution for the problem in which our method is based on penalty method,fixed point principle,and monotone operator theory.To illustrate the applicability,Chapter 5 is concerned to study a partial differential system with coupling effect given in,which is formulated by a parabolic equation with homogeneous Dirichlet boundary condition and a time-dependent elliptic equation with mixed boundary conditions.In Chapter 6,we conclude the main work of the dissertation and then prospect some further study work in the near future.