Existence Of Positive Solutions For Boundary Value Problems Of Integer And Fractional Nonlinear Differential Equations

In this thesis, we establish the existence of triple positive solutions for a integer order m-point boundary value problem with a delayed argument, the existence and uniqueness of positive solutions for Sturm-Liouville-type boundary value problem of fractional order impulsive differential equations, and existence and multiplicity of positive solutions for higher-order fractional differential equation eigenvalue problem,which is based on the theory of nonlinear functional analysis and boundary value problems of nonlinear differential equations and applying fixed point theorem of operator. In this paper, the main conclusions and proof of the existence to boundary value problems are presented in detail. Some relevant examples are included to illustrate our main results. And the method and results of some existing literature are improved and generalized.This paper is divided into six chapters and the organization is as follows.In Chapter 1, we mainly give the background and development of the boundary value problems for nonlinear differential equations, as well as the main research contents.In Chapter 2, we describe the definition and theorem which will be used in the following chapters.Chapter 3 establishes the new expression and properties of Green’s function for an integer order m-point boundary value problem with a delayed argument.Furthermore, using Holder??’s inequality and a fixed point theorem due to Leggett and Williams, the existence of at least three positive solutions is given.In Chapter 4, the expression and properties of Green’s function for boundary value problems of nonlinear Sturm-Liouville fractional order impulsive differential equations are considered. At the same time, we obtain the sufficient conditions of the existence and the uniqueness of solution by using Schauder fixed point theorem and Lerray-Schauder fixed point theorem.In Chapter 5, we study the existence of solutions for higher-order fractional differential equation eigenvalue problem. Using Guo-Krasnoselski fixed point theorem, the existence and multiplicity results for positive solutions are derived in terms of different values of λ.In Chapter 6, we give the main conclusions and prospects.