Research On Two Kinds Of Differential Hemivariational Inequalities

The differential hemivariational inequality problem is composed of differential equations and hemivariational inequalities.In this thesis,we study the existence of solutions to two kinds of differential hemivariational inequalities and the upper semicontinuity of hemivariational inequality solution mapping.The thesis consists of the following five chapters.In chapter 1,we mainly introduce the research background of differential hemivariational inequality,the research status at home and abroad,and the main work of this thesis.In chapter 2,we review preliminary knowledge required in this paper,including important definitions and theorems in the theory of function space,operator semigroup,nonsmooth analysis,set-valued analysis and monotone operator.In chapter 3,we introduce a class differential evolution hemivariational inequality system.Which couples an abstract parabolic evolution hemivariational inequality and a nonlinear differential equation in a Banach space.First,by applying surjectivity result for pseudomonotone multivalued mapping and the properties of Clarke’s subgradient,we show the nonempty of the solution set for the parabolic hemivariational inequality.Then,some topological properties of the solution set are established such as boundedness,closedness and convexity.Furthermore,we explore the upper semicontinuity of the solution mapping.Finally,we prove the solution set of the system is nonempty by applying Arzela-Ascoli theorem and BohnenblustKarlin fixed point theorem.In chapter 4,we study a class of differential hyperbolic hemivariational inequality systems coupled by an abstract hyperbolic hemivariational inequality and a second-order nonlinear differential equation.First,the integral operator is introduced to transform the hyperbolic evolution hemivariational inequality problem into a parabolic one.Then,with the help of the surjectivity result of pseudo-monotone operators,the Clarke subdifferential property proves that the solution set of the hyperbolic evolution hemivariational inequality is nonempty,convex and compact,and the solution map is upper semicontinuous.Finally,the existence of solutions to the system is proved by the fixed point theory.In chapter 5,we summarize the work of this paper and put forward some ideas for the future work.