【作者】 王峰；

【导师】 张国伟；

【作者基本信息】 东北大学 ， 基础数学， 2008， 硕士

【摘要】 微分方程边值问题来源于应用数学和应用物理的多个分支,这类课题引起了广大学者的关注,本文第1章对这类问题的现状进行了简要的概述.第2章研究了高阶(k,n-k)多点边值问题分别在边值条件下正解的存在性,其中0<ξ1<ξ2<…<ξm-2<1,ai∈[0,+∞),且允许h(x)在x=0和x=1奇异.利用推广的锥拉伸与锥压缩不动点定理得出了多点边值问题正解的存在性结论,推广了张国伟、孙经先有关文献的相应结果.第3章研究了二阶超线性奇异半正方程组在边值条件下正解的存在性,其中f,g：(0,1)×[0,+∞)×[0,+∞)→[0,+∞)是连续的,并可能在t=0,1处奇异.p,q：(0,1)→(-∞,+∞)是L-可积的,0<ξ1<ξ2<…<ξm-1<1,ai∈[0,+∞).利用的方法是锥拉伸与锥压缩不动点定理,该工作推广了刘立山等的结果.更多还原

【Abstract】 Boundary value problems of differential equations arise in a variety of applied mathematics and physics problems. The problems have been attached much attention by many scholars. In chapter 1 of this thesis, the author explains the present condition of this question.In chapter 2 the author considers the existence of positive solution in the paper for the multi-point boundary value problem of the higher order (k,n-k) differential equation subject to the boundary value conditions respectively, where0<ξ1<ξ2<…<ξm-2< 1, ai∈[0,+∞), and h(x) is allowed to be singular at x= 0 and x= 1.The existence of positive solution is obtained in the paper for the multi-point boundary value problem by means of the extended fixed point theorem concerning cone compression and expansion. The conclusions extend and improve the main results of Zhang Guowei and Sun Jingxian.In chapter 3 the author is concerned with the existence of positive solution for the following second order superlinear singular semipositone differential system subject to the boundary value conditions where f and g:(0, 1)×[0,+∞) x [0,+∞)→[0,+∞) are continuous and allowed to be singular at t=0,1. p and q:(0,1)→(-∞,+∞) are Lebesgue integrable,0<ξ1<ξ2< …<ξm-1 < 1,ai∈(0,+∞).The method is the fixed point theorem concerning cone compression and expansion. The conclusions extend and improve the main results of Liu Lishan.更多还原