Three Types Of Fractional Differential Equations With Measure Integral Boundary Conditions

The boundary value problem of fractional differential equations is a major branch of the qualitative theory of differential equations.The related research theories are constantly improved and mature,and gradually formed a complete system theory with strong application background.In the study of its related boundary value problems,the integer order differential equation has certain limitations,while the fractional order differential equation is more practical in practical problems,especially the study of the fractional order boundary value problem has attracted much attention,and many novel and accurate theoretical results about the boundary value problem have been obtained.Based on the previous theoretical research,this paper further deepens the common boundary conditions into measure integral boundaries.The existence,positive and uniqueness of solutions of Riemann-Liouville,Hadamard and Caputo fractional differential equations are studied by using single-point iteration,non-negative matrix and fixed point theorem.The theoretical results further broaden the research content of fractional differential equations.The paper consists of five parts:Chapter 1,introduce the background,current situation,concepts and lemmas of the research content.Chapter 2,the existence of extremum solutions for Remann-Liouville fractional differential equations with measure integral boundary conditions is studied.In this part,the fixed point theorem and appropriate non-negative matrices are used to describe the nonlinear coupling behavior,and the positive solutions of coupled nonlinear Hadamard fractional differential equations with measure integral boundary conditions are obtained.Compared with the general theoretical model of the first part,this part extends to the coupling theoretical model,which not only widens the complexity of the research problem,but also complements the depth and breadth of the research of the coupling model.Chapter 3,the positive solutions of coupled nonlinear Hadamard fractional differential equations with measure integral boundary conditions are studied.In this part,the fixed point theorem and appropriate non-negative matrices are used to describe the nonlinear coupling behavior,and the positive solutions of coupled nonlinear Hadamard fractional differential equations with measure integral boundary conditions are obtained.Compared with the general theoretical model of the first part,this part extends to the coupling theoretical model,which not only widens the complexity of the research problem,but also complements the depth and breadth of the research of the coupling model.Chapter 4,the existence and uniqueness of solutions of Caputo fractional differential equations combining measure impulse integrals with hybrid systems are studied,and the solution of a new system combining measure impulse integrals with hybrid systems is obtained.The existence and uniqueness of solutions are proved by using the fixed point theorem and Leray-Schauder choice theorem.This part of the research content is based on the previous in depth research,it is found that a complex system and process can not be described by a single differential equation,so the coupled hybrid system of fractional differential equation is studied.Compared with the previous two parts,this part has broader content and stronger practical significance.At present,there are relatively few researches on this kind of problems,and the theories obtained are relatively novel,which further enrich the research results of boundary value problems.Chapter 5,the main research contents of three types of fractional differential equations with measure integral boundary are summarizedand the prospect of future work.