Approximate Controllability Of Sobolev Type Fractional Stochastic Functional Integro-differential Inclusions

Differential inclusion theory is an important branch of the non-linear analysis.It is closely related to other branches of mathematics such as differential equation,optimal control and optimization.Due to the need for application and the develop-ment of mathematics itself.In recent years,more and more scholars begin to study the differential inclusion systems and a lot of results have been obtained.In this pa-per,we study the approximate controllability of Sobolev type fractional stochastic functional integro-differential inclusions.This paper is divided into three chapters,the main contents are as follows.Chapter 1,we introduce the development status of differential inclusion at home and abroad,and give the differential inclusion system of this paper.Chapter 2,we give the knowledge of the signs,theorems,definitions and so on which are used in the latter.Chapter 3,we study the existence of solutions for Sobolev type fractional s-tochastic functional integro-differential inclusions and give the general form of the solution by using fractional calculation,the semigroup theories and so on.The existence of the solution for the system is proved by the Bohenblust-Karlin fixed point theorem.Chapter 4,we show the approximate controllability of Sobolev type fractional stochastic functional integro-differential inclusions.Under the premise of the exis-tence of the solution for the system,and the sufficient conditions for the approximate controllability of the system are obtained.