The Existence Of Solutions For Hemivariational Inequalities And Optimal Control

In the last years the study on hemivariational inequalities has gained a great development in analysis from both the theoretical and applications. It has many applications in the fields of engineering, control, physics, biology, economy and so on. The thesis is divided into five chapters:In chapter 1, we give the background, the present situation and the development trend of research on hemivariational inequalities and the main research of this thesis.In chapter 2, we introduce the necessary preliminaries including basic definitions, prop-erties and some related lemmas for Bochner integral, fractional differential-integral, operator semigroup, multimap, Clarke’s subdifferential and the measure of noncompactness and so on.In chapter 3, the purpose of this chapter is to study optimal control for generalized quasi-variational hemivariational inequalities involving multivalued mapping. Under some suitable conditions, we give existence results of the optimal control. We also consider the convergence behavior of the optimal control when the data for the underlying quasi-variational hemivaria-tional inequalities is contaminated by some noise. In the last section, we give an example to illustrate our main results.In chapter 4, by using the method of the measure of noncompactness, the fixed points the-orem of multivalued maps and the properties of Clarke’s subdifferential, we give the existence result and controllability for the fractional evolution inclusions of Clarke’s subdifferential type in a separable reflexive Banach space.In chapter 5, we make a brief summary for this paper and introduce our future work.