非线性微分方程的解及其应用

【作者】 孙国玲；

【导师】 赵增勤；

【作者基本信息】 曲阜师范大学 ， 应用数学， 2007， 硕士

【摘要】 非线性泛函分析是现代分析数学的一个重要分支，它能够清楚的解释自然界中很多自然现象，因而受到了越来越多的数学家与数学工作者的关注。其中，非线性初值与边值问题来源于应用数学和物理的多个分支，是目前分析数学中研究最为活跃的领域之一。本文利用锥理论，不动点定理，不动点指数方法，上下解方法等研究了几类积微分方程奇异边值问题解的情况，获得了一些新的结果。根据内容本文分为以下四章：第一章第一章是本文的绪论部分。主要介绍了本文的研究课题。第二章本文允许h(t)在t=0和t=1奇异，利用Krasnoselskii不动点定理和Leggett-Williams不动点定理，得到了二阶三点边值问题多个正解的存在性结果。第三章本文允许h(t)在t=0和t=1奇异，在与相应的线性算子第一特征值有关的条件下，利用不动点指数方法，获得了次线性脉冲Sturm-Liouville边值问题正解的存在性结果。第四章本文在不要求f连续的前提下，利用较为宽松的条件研究了Banach空间中非线性混合型微分-积分方程初值问题解的存在唯一性以及解的迭代逼近，改进和推广了最近的一些结果，并有新的应用。更多还原

【Abstract】 Nonlinear functional analysis is an important branch of mordern analysis mathematics. It can explain a lot of natural phenomena clearly,so more and more mathematicans and mathematical researchers are devoting their time to it. Among them, the nonlinear initial value problem and boundary value problem comes from a lot of branches of applied mathematics and physics, it is at present one of the most active fields that is studied in analyse mathematics. The present paper employs the cone theory, fixed point theory, fixed point index methord and upper and lower methord and so on, to investigate the existence of solutions to initial value problem and boundary value problem of several kinds of nonlinear systems of differential equations. The obtained results are either new or intrinsically generalize and improve the previous relevant ones under weaker conditions.The thesis is divided into four sections according to contents.In chapter 1, it is the introduction of this paper, which introduces the main contents of this paper.In chapter 2, Allowing h(t) is singular at t=0 and t=1,the existence of multiple positive solutions of second-order three-point boundary value problems is attained.The existence results are given by using Krasnoselskii fixed point theorem and Leggett-Williams fixed point theorem.In Chapter 3, Allowing h(t) is singular at t=0 and t=1,the existence of positive solutions of sublinear impulsive Sturm-Liouville boundary value problems is attained.The existence results are given by means of the fixed point index under some conditions concerning the first eigenvalues corresponding to the relevant linear operator .In Chapter 4, In this paper,the initial value problem of the nonlinear integro-differential equations in Banach spaces is investigated under the condi- tion weaker than the continuity of f.The existence and uniqueness and iterative approximation of solution for the initial value problem is abtained.The results presented here not only improve and extend many results,but also apply to other aspects.更多还原