Multiple Positive Solutions For A Nonlinear2n-th Order M-point Boundary Value Problems

The boundary value problems for ordinary differential equations often encounter in our real life, it is applied in mathematics, physics, chemistry and many other scientific fields. It has been widely studied in recent years. The research methods include:monotone iterative technique, upper and lower solution methods, degree theory, such as Guo-Krasnoselskii, Leggett-Williams fixed-point theorems, and so on.In this paper, we consider the existence of multiple positive solutions for the2n-th order m-point boundary value problems: where By using Leggett-Williams fixed point theorem, we provide sufficient conditions for the existence of at least three positive solutions to the above boundary value problem.In chapter one, we describe the contents of the boundary value problem, the background, and the main problems which studied in this article.In chapter two, we are focused on the relevant preliminaries, the theory of cones in Banach spaces, Leggett-Williams Fixed Point Theorem and properties of Green’s function.In chapter three, we give the proof of the existence of multiple positive solutions for problem (1), it is the main contents of this paper which is divided into two parts..And in chapter four, we present an example to demonstrate the application of related theorem (theorem3.2.1). In summary part, we show the overall framework and possible extension of this thesis.