【作者】 吕海燕；

【导师】 刘衍胜；

【作者基本信息】 山东师范大学 ， 基础数学， 2005， 硕士

【摘要】 本文第一章考虑Banach空间中半直线上一阶积微分方程初值问题解的存在性。该问题源于[1]中第三章的第三节,这里去掉了原文要求的Lipschits条件,代之以紧性条件,然后利用M(?)nch不动点定理得到了解的存在性。 第二章考虑Banach空间中一类带脉冲的奇异积分微分方程边值问题正解的存在性。利用锥上的不动点定理得到了一个正解和两个正解的存在性。 第三章研究了由二阶和四阶常微分方程奇异边值问题耦合的系统。该系统来源于Lazer和Mckenna提出的吊桥非线性振动模型的静态问题。使用方法不同于[2]中的Schauder不动点定理和[3]中的临界点理论,利用锥拉伸与锥压缩不动点定理得到了两个正解的存在性。 最后一章考虑非线性四阶奇异边值问题的正解。在非线性项拟齐次的条件下,得到了C~2正解和C~3正解存在的充要条件。更多还原

【Abstract】 In the first chapter, the existence of solutions of initial value problems for first order integro-differential equations on half-line in Banach spaces is considered.This consideration is resulted from the third section of the third chapter in [1]. This dissertation refers to the measure of noncompactness instead of the Lipschits condition, then obtains the existence of solutions by using Monch fixed point theorem.In the second chapter, we consider the existence of positive solutions of a class singular boundary value problems with impulse in Banach spaces. Using cone fixed point theorem we get the existence of one and two positive solutions.In the third chapter, we investigate singular boundary value problems of coupled system of second and fourth order ordinary differential equations. This system can be seen as the steady state from the nonlinear perturbation model of suspension bridge equations which are presented by Lazer and Mckenna. Differing from both Schauder fixed point theorem in [2] and critical point theory in [3]. this paper gets the existence of two positive solutions by using fixed point theorem of cone expansion and compression.In the last chapter, nonlinear fourth order singular boundary value problems are dealt with. Under the hypothesis of quasi-homogeneous, we obtain the necessary and sufficient conditions for the existence of C~2 and C~3 positive solution respectively.更多还原