Existence Solutions For Differential Inclusions And Differential Variational Inequalities

Derivative inclusion arise in the mathematical modelling of certain problems in economics,optimal control,stochastic analysis.It have obtained comprehensive applications on the modeling of many phenomena in various fields of engineering,physics and economics,and are widely investigated by many authors.There has been a significant development in derivative inclusion in recent years.In this thesis,we use the theorem of fractional differential,set valued mapping and several fixed point theorems discussed the existence of solution for derivative inclusion and fractional differential variational inequalities.This thesis consists of six chapters:In chapter 1,we present the background,the current situation and the future tendency of the research on fractional differential,derivative inclusion and variational inequalities and the main research of this thesis.In chapter 2,we introduce the preliminaries which will be used in this thesis,including some definitions,properties of fractional calculus,set valued mapping and conclusions of function spaces as well as some fixed point theorems which are helpful in our study.In chapter 3,by using the compression mapping principle in Banach space we studied with the existence of solutions for fractional impulsive differential equations.We got two differential results,The first result using mathematical induction,obtained with the existence of solutions for fractional differential equations.The second results is derived from the compression mapping principle in Banach space for the existence of impulsive fractional differential equation.In chapter 4,we establish sufficient conditions for the solutions of derivative inclusion.The cases of convex-valued right hand sides are considered.by using fixed point method,we get the existence results of derivative inclusion.In chapter 5,we studied the existence of solutions for impulsive derivative inclusion.Based on the results of the fourth chapter,fractional derivative inclusion problem is generalized to fractional impulsive derivative inclusion.we establish sufficient conditions,by using fixed point theorems of set valued mapping,we get the the existence results of impulsive derivative inclusion.In chapter 6,we studied the existence of solutions for fractional impulsive differential variational inequalities in finite dimensional spaces.On the base of previous studies,we establish sufficient conditions,by using a nonlinear alternative for multivalued maps and Filippove implicit function lemma,we get the existence of the solution.In chapter 7,we give a conclusion of our present work and make a plan for further research.