Solutions Of Fractional Impulsive Differential Equations

Nonlinear functional analysis is an important subject in modern mathematics. It can be used to solve kinds of nonlinear problems by abstract theories including topological degree, partial order and cone, monotone operators, variational methods and so on. Many outstanding mathematicians have maken significant contributions in these fields, such as E. Rothe, M. A. Krasnosel’skii, P. Rabinowitz, H. Amann, A. Ambrosetti, etc., as well as some ones in our own country, such as Professor Kung-Ching Chang, Dajun Guo, Jingxian Sun, Yiming Long and so on([1-12]).As a promotion of classic differential equations with integer order, fractional dif-ferential equation is a relatively new subject of research, which can be used to describe some real phenomena better. It is paid more and more attention and is studied by more researchers([13-15,26-37]). Impulsive differential equations can be used to ex-plain the evolution processes, which experience a change of state at certain moments of time. It is widely used in the real world([17-20,38-44]). Therefore, we have reason to believe that it is meaningful to study the solutions of fractional impulsive differential equations. It can help us better understand the world, and will be useful in human activities([45-49]).In this thesis, we will study the solutions of a couple of boundary value problems for fractional impulsive differential equations, by means of several fixed point theorems. Both cases, lower order (0<α≤1) and arbitrary order (n-1<α≤n), are considered.This article is as follows.In Chapter Ⅰ, we will introduce the backgrounds of the study, as well as the pre-liminary definitions and lemmas which will be used in the following chapters.In Chapter Ⅱ, we will study the existence of positive solutions for fractional im- pulsive integro-differential equations of Caputo type with order0<α≤1: where G Dα is Caputo fractional differential,J=[0,1],O=t0<t1<t2<…<tm<tm+1=1,J0=[0,t1],Jk=(tk,tk+1],k=1,2,3…,m,R+:={x∈R|x≥0),f∈C(J×R+×R+×R+,R+),(Tu)(t)=∫0tK(t,s)u(s)ds,(su)(t)=∫01H(t,s)u(s)ds K∈C(D,R+),D={(t,s)∈J×J|t≥s},H∈C(J×J,R+),△u|t=tk=u(tk+)-u(tk-),Ik∈C(R+,R+),k=1,2,…,m,β>1. We obtain the suffcient conditions for the existence of positive solutions by uing fixed point theorem of cone expansion and compression with norm type.In Chapter Ⅲ,we will study the fractional impulsive diffetential equations of Caputo type with order n-1<α≤n,where n∈Z+: where G Dα is Caputo fractional differential,J=[0,1],0=t0<t1<t2<…<tm<tm+1=1,J0=[0,t1],Jp=(tp,tp+1],p=1,2,3….,m,△u(q)|t=tp=u(q)(tp+)-u(q)(tp-),f∈C(J×R,R),Iqp∈C(R,R),βq≠1,q=0,1,2,…,n-1,p=1,2,3,…,m.By studying and calculalting,we get the equivalent integral equation of the prob-lem.Moreover,we obtain a special function T,by which the integral equation can be represented explicitly. Then, we get several conditions for the existence of solutions as well as unique solution by using fixed point theorems such as contraction mapping principle, Leray-Schauder theorem, the corollary of Altman theorem. This can be recognized as a expansion of the conclusions in [22] and some other previous works.