Existence Of Maximal And Minimal Solutions And Uniqueness Of Solutions For Four-points Boundary Value Problems Of Two Order Ordinary Differential Equation

The multiple points boundary value problem of ordinary differential equa-tion stems from fields of application mathematics and physics research, pro-viding important theoretical values and significance. It is always applied to many models in physics, cybernetics, and biologic domains.The study on boundary value problem of ordinary differential equation be-gan in the early years when differential and integral calculus was established. The two-points BVPs of second order ordinary differential equation was inves-tigated firstly. Many deep and complete results were obtained under the efforts devoted by lots of scholars across a long period. However, study the two-points BVPs is not enough according to the human's further theoretical and applied needs. Therefore, people start to study the multiple points boundary value problems, which have a more wide theoretical and applied background. For example, the investigation of multiple points BVPs embrace many classical results of two-points BVPs, and the models of porous medium flow, or the density of insect pest and so on.Recently, there are mainly two methods of studying the multiple points boundary value problems, by Upper-Lower solutions method, or by applying some fixed point theorems. We can obtain the existence of solutions of some problems by the corresponding fixed point theorems, but it is hard to get the uniqueness results with this method. However, the Upper-Lower solutions method can solve this problem effectively.The purpose of this work is to consider the following four-points BVPs: andThis work consists four chapters and the main contents are as follows:In chapter 1, we introduce the theoretical values and significance of bound-ary value problems, especially multiple points boundary value problems of ordi-nary differential equation. We also recall some existing criteria of the existence and uniqueness of solution for multiple points BVPs of ordinary differential equation.In chapter 2, we give some preliminaries for the following arguments in this paper, including the Upper-Lower solutions method and corresponding concepts involved in this method, and some basic lemmas that will be used in chapter 3. In the second section of this chapter, we will calculate the Green functions for BVP (1.1) and BVP (1.2) using the approaches provided by Ge Weigao [12].In chapter 3, applying Upper-Lower solutions method, firstly we give an iterative scheme for approximating solutions, and consequently obtain the ex-istence of the maximal and minimal solutions of the problems by studying the monotonicity and sequentially compactness of the corresponding iterative op-erator. Then, combining the properties of Green functions of the two BVPs under the non-resonant situations that acquired in chapter 2, we can also show the results of the uniqueness of the solutions for the above BVPs.In chapter 4, we provide an example. We can find its upper and lower solutions, and verify the problem satisfy conditions of the theorems obtained in chapter 3. So we can obtain the uniqueness of solution for the given problem.