Upper And Lower Solution Method For Impulsive Fractional Differential Equations

In this paper, we investigate the impulsive fractional differential equations. First, we introduce the mild solutions of impulsive fractional differential equations with order 0< a< 1 and 1< a< 2. Then, we deal with Captuo and Riemann-Liouville impulsive fractional differential equations using the method of upper and lower solution via Hausdorff measure of noncompactness. At last, the existence theorem of the mild solutions for impulsive fractional differential equations with nonlocal condition and finite delay is obtained. Our paper is organized as follows:In chapter 1, firstly, we introduce the research background and current situation of frac-tional differential equation; secondly, the main content of the paper is given; then, we show the innovation of the paper; finally, the preliminary knowledge of our paper is introduced.Chapter 2 is the core part of the paper. Considering the different formulas of mild solution for impulsive fractional evolution equation, we point out the correct expression of the mild solution and prove it. Then we further investigate the solution of linear fractional impulsive evolution equations with order 1< a< 2. Finally, we explain the reason that the solutions to an impulsive evolution equation can not be distinct.In chapter 3, there are two subsections. In subsection 1, we investigate the existence of the mild solutions for a class of impulsive fractional partial differential equations with order 1< a< 2 by upper and lower solution method using the theory of Hausdorff measure of noncompactness and properties of solution operators. In subsection 2, we investigate the existence of solutions for a class of impulsive Riemann-Liouville fractional differential equations with order 1<?< 2 by upper and lower solution method via Hausdorff measure of noncompactness.In chapter 4, we investigate the existence theorem of the mild solutions for impulsive fractional differential equations with nonlocal condition and finite delay. Subsection 1 is concerned with the impulsive fractional differential equations with nonlocal conditions of order 1< a< 2. The properties of solution operators and Krasnoselskii's fixed point theorem are used to obtain the mild solutions of the equations which are proved and its existence results. In subsection 2, the existence theorem of the mild solutions for impulsive fractional differential equations with nonlocal condition and finite delay of order 0< a< 1 in a Banach space is obtained by using Monch's fixed point theorem.