Existence And Uniqueness Of Solutions For Impulsive Boundary Value Problems Of Several Fractional Differential Equations

In recent years,in many scientific fields,the boundary value theory of fractional differential equations plays an extremely important role.It lays the foundation for the development of chaos and turbulence,chemical physics,speech signals,cybernetics,porous media and other theories.The practical experience of human beings has further realized its application value.Therefore,the boundary value problem of fractional differential equations has attracted the attention of many domestic and foreign scholars,and has carried out profound research on it.At the same time,many research results have emerged.In the boundary value problem of the differential equations,the impulsive system is an important application of such problems,which makes many mathematicians have a strong interest in exploring the existence and uniqueness of their solutions and have obtained many excellent results.But for the singular fractional differential equations impulsive boundary value problems,singular semi-positive fractional differential equations with impulsive boundary value problems and differential inclusion of impulsive boundary value problems are rarely studied.This thesis will discuss the impulsive boundary value problems for singular fractional differential equations,singular semi-positive fractional differential equations with impulsive boundary value problems and fractional-order impulsive differential inclusion with four-point boundary value problem,explore the existence and uniqueness of its solution under impulsive conditions in accordance with their corresponding Green's function,to find their special nature,and select the appropriate fixed point theorem,positive solutions to its expansion in detail.The content of the thesis is arranged as follows:The first chapter mainly introduces the related knowledge of the fractional integral and fractional differentials and its research background and research status.The following part introduces some basic knowledge and fixed point theorems needed in this thesis.In the second chapter,based on the research of the impulsive boundary value problem of fractional differential equations,a class of singular fractional differential equations with impulsive boundary value problems is studied.Firstly,the expression of the solution is calculated,and then the corresponding Green is explored.Function(in-depth discussion of the nature of the Green function becomes a key step to prove the existence of the solution to the boundary value problem),then transforms the differential equation into an integral equation,and uses the Arzela-Ascoli theorem to prove that the operator has a fixed point,thatis,the problem is obtained.Finally,an example is given to verify the main conclusion.In the third chapter,based on the previous chapter,the existence of solutions for a class of fractional-order semi-positive differential equations with impulsive boundary value problems is discussed.In the proof process,the upper and lower solutions are determined by the upper and lower solutions.And combined with the Arzela-Ascoli theorem,the sufficient conditions for the positive solution of the boundary value problem are obtained.In the fourth chapter,using Bohenblust-Karlin fixed point theorem and upper and lower solution method,the existence of solutions for a class of fractional differentials containing four-point boundary value problems is investigated.Finally,a sufficient condition for the existence of at least one solution for the boundary value problem is obtained.