关于非线性两点边值问题解的存在性与唯一性

On the Existence and Uniqueness of Solutions of Two-Point Boundary Value Problems for Nonlinear Differential Equations

非线性常微分方程边值问题,是常微分方程研究领域中一个很为实际,其发展也很为活跃的一个分支。从Picard经典工作开始,特别是从本世纪六十年代以来,它一直吸引着众多的数学工作者的注意,因而,它得到了较为迅速的发展。另一方面,现代数学也为深入研究非线性边值问题提供了可能,例如,拓扑学理论,非线性泛函分析等。 一般说来,非线性边值问题,它包括有限区间上的边值问题与无限区间上的边值问题两大类。方程类型比较繁多。对于有限区间上边值问题的研究,主要有解的存在与唯一性。

Since the early part of this century, two point boundary value problems for nonlinear second order equations have been a main subject of research in the area of the boundary value problems for nonlinear ordinary differential equations.Due to the special difficulty, even impossibility of solving the boundary value problems, the study of the existence and uniqueness of the solutions has been extremely important not only in solving numerical solutions but in applied fields, and consequently an important subject of research of nonlinear boundary value problems for nonlinear second order ordinary differential equations.This paper is generally concerned with important results and the methods which have been developed by mathematicians in the world since Picard discussed the existence and uniqueness of the solutions ofy'+f(t,y,y')=0,y(a)=A,y(b)=Bin 1893.It is hoped that our discussion will bring readers the interest in the field.