Positive Solutions For Nonlinear Third-Order Three-Point Boundary Value Problems

Recently, third-order boundary value problems for ordinary differential equations have received much attention. The used tools are the fixed point theorem of cone expansion and compression (or called Guo-Krasnoselskii fixed point theorem), Leggett-Williams fixed point theorem, the five functional fixed point theorem and so on.This dissertation mainly discusses the existence and multiplicity of positive solutions for a class of nonlinear third-order three-point boundary value problem. In this dissertation, firstly, the Green's function for the associated linear boundary value problem is constructed, and then, some useful properties of the Green's function are obtained by a new method. Finally, existence and multiplicity results of positive solutions for the boundary value problem are established by using the different fixed point theorems.The paper is divided into four chapters: in chapter one we introduce a survey to the development of boundary value problems, the scientific background, theoretical and practical significance of this research topic and the source and main contents of this paper.In chapter two, we obtain the existence results of at least one positive solution and at least two positive solutions for the boundary value problem using the Guo-Krasnoselskii fixed point theorem and the fixed index theorem, respectively.In chapter three, we establish the existence result of at least three positive solutions by using the well-known Leggett-Williams fixed point theorem. Furthermore, for arbitrary positive integer m, we prove the existence of at least 2m-1 positive solutions.In chapter four, we study the existence result of the positive solution for the boundary value problem when the nonlinear term satisfies the singularity assumption.This dissertation is different from the others at the following aspects: no one involved the boundary value problem studied in this dissertation; The method to study the properties of the Green's function is new. All of these will give an illumination to people who are interested in the boundary value problem. In fact, the results of this dissertation have been expanded and cited.