The Existence And Uniqueness Of Solution For Boundary Value Problem Of Semi-linear Impulsive Fractional Differential Equations

With the development of science and technology,the mathematical model of differential equations has been widely applied,and the research of differential equation is becoming more and more.As an important branch of differential equations,the impulsive differential equation has been paid more and more attention by researchers.The boundary value problems of integer order impulsive differential equations have obtained fruitful results.In recent years,fractional differential equations have played an important role in many disciplines.The fractional impulsive differential equations have great advantages in mathematical modeling and can reflect the law of change deeply and accurately.Therefore,the various problems of fractional order impulsive differential equations have attracted more and more attention from scholars and gradually become a hot issue.However,in comparison,there are few studies on the boundary value problems of semi-linear fractional differential equations.In this thesis,the existence and uniqueness of the solution of the boundary value problem of the semilinear impulsive fractional differential equations are mainly studied.The results of the existing literature are improved and generalized,and the validity of the conclusions is illustrated by specific examples.The thesis consists of five chapters,the main contents are as follows:In chapter 1,the research background and present situation of fractional differential equations at home and abroad are introduced,and some basic definitions,theorems and necessary properties of fractional calculus to obtain the main conclusions are also introduced.In chapter 2,based on the principle of Banach contractive mapping,the following boundary value problems of semi-linear fractional differential equations with impulses is deduced,the existence of the solution is given,and two examples are given.In chapter 3,by using the Guo-Krasnosellskii's cone expansion and compression fixed point theorem,the following boundary value problem of semi-linear fractional differential equations with impulses is deduced,the existence of positive solutions is given,and two concrete examples are given.In chapter 4,by means of Arzela-Ascoli theorem?Banach contractive mapping principle and Krasnosellskii fixed point theorem,the following boundary value problem of semi-linear fractional differential equations with non-instantaneous impulses is discussed the existence and uniqueness of the solution are given,and two concrete examples are given to illustrate the validity of the obtained conclusions.In chapter 5,we made a brief summary and prospect for the study in the thesis.