Research On The Existence Of Solutions For Several Classes Of Fractional Order Differential Equation Boundary Value Problems

Fractional order differential equation plays an important role in describing natural sci-ence,and has become an important research field,and has been favored by many experts and scholars.In this paper,we study three kinds of fractional differential equations boundary value problems by using the cone theory in Banach space and some fixed point theorems,we obtain the existence of positive solutions.Some of them are the uniqueness of posi-tive solutions and the corresponding iterative sequences are given for converging to unique solutions.This thesis has four chapters:In Chapter 1,we give the introduction,which briefly introduces the background and significance of the research,and also briefly introduces the research results of three kinds of fractional differential boundary value problems.In Chapter 2,we discuss the following Caputo fractional boundary value problem(?) where CDa? is Caputo fractional derivatives of order ? with 1<??2,?>0,?<?<?,and f,g:[a,b]×[0,?)?R are continuous functions.In this chapter,the existence of positive solutions for the above problem is obtained by a for the sum operator fixed point theorem.In Chapter 3,we investigate the the following system of Hadamard fractional boundary value problem subject to integral boundary conditions(?) where ?,??(n-1,n]are real numbers with n?3,i=0,1,2,…,n-2,and HD?,HD?are the Hadamard fractional derivatives.The nonlinearities f,g ?C([1,e]×R+×R+,R+).By using a fixed point theorem for concave operators in ordered Banach space,the local existence and uniqueness of positive solutions for the system is obtained,and an iterative sequence is constructed to approximate the unique positive solution.In Chapter 4,we consider the following p-Laplacian fractional differential equation involving Riemann-Stieltjes integral boundary condition(?) where Dt?,Dt?,Dt? are the Riemann-Liouville fractional derivatives of orders ?,?,? with 0<??1<??2<?<3,?-?>1,(?)denotes a Riemann-Stieltjes integral,and A is a function of bounded variation.The p-Laplacian operator is defined as?p(s)=|s|p-2s,p>2,?p(s)is invertible and its inverse operator is ?q(s),where q=p/p-1 is the conjugate index of p.In this chapter,two fixed point theorems are used to obtain the uniqueness of the solution under different conditions,and an iterative sequence is constructed to approximate the unique positive solution.