Differential Variational Inequalities And Relevant Problems

Differential variational inequalities are defined by an ordinary differential equation parameterized by an algebraic variable that is required to be a solution of a variational inequality containing the state variable of the system, which are an useful tool for representing models involving both dynamics and constraints in the form of inequalities. However, the thesis will research differential variational inequalities in infinite space by using five chapters.In Chapter 1, we focus our attention on the background, the domestic and foreign research situation of differential variational inequalities and then elaborate the main work of this thesis.In Chapter 2, we will recall some important definitions, properties and conclusions concerning Bochner integral, theory for semi-group of operators and set-valued functions etc.In Chapter 3, we introduce differential mixed variational inequalities(3.1) in Banach spaces. By using KKM theorem, theory of semigroups, fixed point theorem of set-valued mappings and Filippov Implicit Function Lemma, we prove the solution set of differential mixed variational inequalities is nonempty and compact.In Chapter 4, we propose the well-posedness and the well-posedness in the generalized sense of differential mixed quasi-variational inequalities(4.1) in Hilbert spaces. Base on these concepts we show the relation between metric characterizations and well-posedness of mixed quasi-variational inequalities(4.1). Then we prove that the solution set of mixed quasi-variational inequalities(4.1) is compact, if mixed quasi-variational inequalities(4.1) is well-posed in the generalized sense.Finally, Chapter 5 concludes our present works and then discusses some further study ideas in the future.