Complete Controllability And The Existence Of Mild Solution For Impulsive Fractional Differential Equations

In this paper,we investigate the complete controllability and the existence of mild solution for impulsive Caputo and Riemann-Liouville fractional differential equations.The main mathematical techniques used here include the fractional calculus,properties of solution operators,Krasnoselskii's fixed point theorem and Monch's fixed point theorem via measures of noncompactness.Without assuming that the solution operators are compact,we prove the complete controllability and the existence of solution to such equations.Our paper is organized as follows:In chapter 1,firstly,we introduce the research background and current situa-tion of fractional differential equation;secondly,the main content of the paper is given;finally,the preliminary knowledge of our works is introduced.In chapter 2,we investigate the existence of mild solution for impulsive frac-tional neutral function integro-differential evolution equations with infinite delay of order 0<?<1 in a Banach space.The main mathematical techniques used here include the fractional calculus,properties of solution operators,and Monch's fixed point theorem via measures of noncompactness.Without assuming that the solution operators are compact,we prove the existence of mild solution to such equations.In chapter 3,we mainly study controllability of nonlinear fractional impulsive evolution systems.In this chapter,we first give a mild solution expression for nonlinear fractional impulsive evolution systems.By Krasnoselskii's fixed point theorem in the infinite dimensional spaces,we obtained sufficient conditions for controllability results.In chapter 4,firstly,we investigate the existence of solution for impulsive Riemann-Liouville fractional neutral function differential equation with infinite de-lay of order 1<?<2.The existence theorem is obtained by using properties of solution operators,Monch's fixed point theorem via measure of noncompact-ness.Particularly,we do not assume that the solution operators are compact and the function f satisfies a Lipschitz type condition;secondly,we investigate the existence of positive solution of boundary value problem for impulsive Riemann-Liouville fractional differential equation of order 1<?<2.