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两类边值问题正解的存在性
Existence of Positive Solutions for Boundary Value Problems of Two Types
【作者】 王波;
【导师】 张国伟;
【作者基本信息】 东北大学 , 基础数学, 2015, 硕士
【摘要】 近年来,非线性常微分方程边值问题正解的存在性受到广泛的关注.本文第一章主要概述一些文章对非线性常微分方程边值问题正解存在性问题的研究.第二章讨论非线性二阶两点边值问题正解的存在性.通过构造锥P,定义全连续算子T:P→P,利用推广的Leggett-Williams不动点定理,分别在拉伸条件和压缩条件下证明上述边值问题至少存在一个正解.第三章讨论(k,n-k)多点边值问题正解的存在性,其中h:(0,1)→[0,+∞)连续,h(x)不恒为0,允许h(x)在x = 0和x = 1处奇异.通过定义全连续算子A:P→P,利用不动点指数理论来证明该边值问题至少存在一个正解.
【Abstract】 In recent years,there has been considerable interest in the existence of positive solutions to boundary value problems of nonlinear ordinary differential equations.This thesis has mainly summarized some articles that studied the existence of positive solutions to boundary value problems of nonlinear ordinary differential equations in Chapter One.In the second chapter,we study the existence of positive solution to the nonlinear second-order two-point boundary value problem By constructing a cone P,define completely continuous operator T:P→P.According to the extended theorem of Leggett-Williams fixed point theorem,which respectively under expansion and compression conditions,we prove the existence of at least one positive solution to the above boundary value problem.In the third chapter,we study the existence of positive solution to the multi-point boundary problem where h:(0,1)→[0,+∞)is continuous,h(x)is not identically equal to 0 and allows it singular at x = 0 and x = 1.By defining completely continuous operator A:P→P,according to the fixed point index we prove that the boundary value problem possesses at least one positive solution.
【Key words】 fixed point theorem; the fixed point index; boundary value problem; positive solution;
- 【网络出版投稿人】 东北大学 【网络出版年期】2019年 01期
- 【分类号】O175.8
- 【下载频次】37