【作者】 何希萍；

【导师】 孙红蕊；

【作者基本信息】 兰州大学 ， 应用数学， 2008， 硕士

【摘要】 近年来,三阶常微分方程边值问题受到了人们的广泛关注,许多作者已作过一系列的研究,主要的工具是不动点理论和上下解方法等。这篇硕士论文主要讨论非线性微分方程组多点边值问题正解的存在性,采用的工具主要是Leray-Schauder非线性抉择、Guo-Krasnoselskii不动点定理和不动点指数理论、Leggett-Williams不动点定理。全文由四部分组成。第一章简述了问题产生的历史背景和本文的主要工作,并给出了本文用到的一些预备知识。第二章主要讨论了下列非线性三阶微分方程组三点边值问题正解的存在性通过利用Leray-Schauder非线性抉择、Guo-Krasnoselskii不动点定理和Leggett-Williams不动点定理,建立了单个及多个正解存在的若干充分条件,所得结果是已有相关文献的推广和改进,且所用方法对非线性微分方程组边值问题正解的存在性的讨论具有一定的普遍意义。第三章主要借助于Krasnoselskii不动点定理,讨论了下列三阶三点的微分方程组边值问题一个、两个正解的存在性,并举例说明。第四章通过运用锥上不动点指数理论研究了下列含有参数的非线性微分方程组三点边值问题正解的存在性、多重性以及不存在性。这些结果改进和推广了一些已有的结果.更多还原

【Abstract】 Recently, third-order boundary value problems for ordinary differential equations have received much attention, Many authors have made a series of studies, the main tools are the fixed point theorems and the upper and lower solution method. This thesis consists of four chapters,which mainly investigated the existence of positive solutions of three-pointboundary value problems for nonlinear differential equations.In chapter 1 we introduce the historical background of problems which will be investigatedstate the main results of this thesis. In addition, we list some preliminary knowledge which is needed later.In chapter 2 we study the existence of positive solutions of the following three-order three-point boundary value problems for nonlinear differential equationsand establish some suficient conditions for the existence of single, twin and three positivesolutions by using nonlinear alternative of Leray-Schauder type, Guo-Krasnoselskii’s fixed point theorem and Leggett-Williams fixed point theorem.Chapter 3 is concerned with the following three-order three-point boundary value problemsthe existence of one and two Positive Solutions.In chapter 4 we deal with the following nonlinear boundary value problem with parameter. The nonexistence, existence and multiplicity of positive solutions are obtained by means of fixed-point index theory on the cone. In particular, our results extend and improve some relate conclusions in recent literatures.更多还原