Positive Solutions For Some Kinds Of Boundary Value Problems Of Nonlinear Differential Equations

Based upon the fixed point theorem on Banach spaces such as the fixed point theorems of cone expansion and cone compression, Leray-Schauder fixed point theorem and two new fixed point theorems on cone, this dissertation gives a study to the existence and multiplicity of positive solutions for two-point and multi- point boundary value problems of nonlinear differential equations. In light of the content and the methods used, our dissertation is divided into three chapters.In Chapter 1, we introduce a survey to the development of topological methods and their applications in boundary value problems of ordinary differential equation. We also summarize main results of the dissertation.In Chapter 2, we study the existence of positive solutions for two-point and three-point boundary value problems of nonlinear differential equations. Firstly in§2.1, by applying the fixed point theorem the existence of positive solutions for second-order three-point boundary value problem is discussed. Secondly in§2.2 ,some fourth-order two-point boundary value problem is studied by the Leray-Schauder fixed point theorem on Banach space. And the results generalize and improve the corresponding work in the past. Finaly§2.3 ,we consider the existence of positive solutions for the singular fourth order p-Laplacian equation. By using the upper and lower solution method and fixed point theorems, the existence of positive solutions to the above the boundary value problem is obtained.In Chapter 3, the existence of multiple positive solutions of two-point boundary value problem are researched. Firstly in§3.1, after giving corresponding opeartor and some important properities, we obtain the existence of multiple positive solutions by using the fixed point theorem. Secondly in§3.2 ,firstly we prove a new fixed point theorem on Banach space,and study the Sturm-Liourille boundary value problem with the theorem and some properities of concave function, then the existence of three positive solutions is gotten. Finaly in§3.3, applying another new fixed point theory to three-order p-Laplacian boundary value problem depending on the first-order derivative the existence of three positive solutions is established. The key point here is that the nonlinear term f depended on the first order derivative. Compared with the past work, the study is new.