Existence And Uniqueness Of Mild Solutions Of Some Fractional Integro-differential Equations

In this paper, we study the existence and uniqueness of mild solutions of some fractional integro-differential equations by using the theory of operator and fixed point theorem mainly.Chapter 2 focus on a Cauchy problem for fractional impulsive integro-differential equations involving nonlocal initial conditions, where the linear part is a generator of a solution operator on a complex Banach space. A suitable mild solution for the Cauchy problem is introduced by applying the Laplace transform of integro differential equations. The existence and uniqueness of mild solutions for the Cauchy problem, under various criterions, are proved. In the last part of this chapter, we construct an example to illustrate the feasibility of our results.In Chapter 3, we study the existence of mild solution for a Cauchy problem of fractionalpartial neutral differential equation with nonlocal initial condition. A new existence theorem which allows us to relax the compactness and Lipschitz continuity on nonlocal item, ensuring the existence of mild solution for the Cauchy problem, is presented. In the last part of Chapter 3, we give an example to illustrate the feasibility of abstract results.In Chapter 4, we establish a sufficient condition for the controllability for Caputo fractional derivative of order q (0< q< 1) neutral differential systems with state-dependent delay by using the fractional power of operators, Sadovskii's fixed point theorem and Leray-Schauder Alternative's fixed point theorem.In Chapter 5, we study the controllability for a class of partial neutral functional integeo-differential systems with delay. By using the resolvent operator, Sadovskii's fixed point theorem and Leray-Schauder Alternative's fixed point theorem, two sufficient conditions for the controllability of partial neutral functional integeo-differential systems with state-dependent delay in a Banach space are established. The results extend and improve the related reports in the literatures.