The Existence Of Solutions For Some Higher-Order Nonlinear Boundary Value Problems

The boundary value problem of nonlinear differential equation that arises from applied mathematics, physics, cybernetics and other applied sciences is an important branch of modern analysis mathematics. A growing number of mathematicians are concerned about it because well explain a variety of natural phenomena. Among them, higher order nonlinear differential equation boundary value problem in physics has a very rich field and a wide range of applications. Therefore, the study of the existence of solutions both in theory and practice has very important significance. This paper studies a class of 2m-order nonlinear differential equation boundary value problem that is studied for non-singular and singular points of two cases, and a class of third-order boundary value problems, and has proved that the existence of positive solutions by using main application cone theory and the fixed-point theorem.The main purpose of this paper is to use non-linear functional analysis of the cone theory, fixed point theorem and so on, on some higher-order non-linear differential equations existence and uniqueness. According to the contents, this paper is divided into the following two chapters.The first chapter is divided into four sections. Section I Introduction, describes a number of related issues facing the status quo and the question we are studying. Section II gives some basic definitions, theorems, and gives the following higher-order nonlinear differential equation boundary value problem: Green function expression, and the upper and lower bound estimation; Section III, using the cone tension and compression fixed point theorem to prove existence of solutions of these problems and uniqueness;Section IV,using Schauder fixed point theorem and perturbation techniques, the comparison principle, and establishing the existence of solutions of the above-mentioned boundary value problem and uniqueness of the results when x = 0, x=1orThe second chapter is divided into three sections. It is a major study of the conditions of over-determined third-order boundary value problem: g is Odd function,when x∈(0 ,1), g ( x)≥0. This chapter primarily through the dual extension of the above-mentioned third-order boundary value problems for transformation, using Schauder fixed point theorem to discuss the issue transforms the existence of positive solutions, which proves the existence of positive solutions of the original problem.Finally, all research content is summarized.