【作者】 王海华；

【导师】 陈海波；

【作者基本信息】 中南大学 ， 应用数学， 2009， 博士

【摘要】 本篇博士学位论文研究几类微分方程边值问题解的存在性.论文借助不动点定理(包括锥上泛函不动点定理)研究几类非线性微分方程(系统)边值问题正解的存在性;应用单调迭代技巧、上下解方法和拟线性方法分别研究两类非线性微分方程边值问题解的存在性;利用度理论研究半直线上一类边值共振问题解的存在性.全文由如下六部分组成.第一章是绪论,简述问题的产生和研究的意义.边值问题普遍存在于自然科学的各个研究领域.边值问题解的存在性一直是广大学者和专家关注的问题.我们对与本文相关的非线性微分方程边值问题解的存在性研究现状进行回顾,同时对本文工作的背景知识做了简要的介绍.第二章借助于Krasnosel’skii不动点定理、锥上不动点指数理论和一些分析技巧,我们讨论四类二阶非线性微分方程多点边值问题正解的存在性,获得了一系列新的结果,相应地推广和改进了已知文献的结论.第三章首先应用拟线性方法研究一类含非线性边界条件的微分方程边值问题解的存在性.其次,使用单调迭代技巧和广义上下解方法讨论一类二阶非线性脉冲泛函微分方程边值问题解的存在性.所得结果包含和推广了已有文献的结果.第四章利用泛函不动点定理研究时间测度上一类具p-Laplacian算子非线性脉冲动力方程边值问题多个正解的存在性和一类具p-Laplacian算子非线性脉冲泛函动力方程边值问题多个正解的存在性.我们得到了一些新的结果.第五章研究非线性微分系统边值问题正解的存在性.利用Krasnosel’skii不动点定理研究一类含变参数二阶非线微分系统多点边值问题多个正解的存在性.利用Leggett-Williams泛函不动点定理研究一类二阶微分系统Sturm-Liouville非局部边值问题多个正解的存在性.我们得到了一些新的结果.第六章利用葛渭高教授对Mawhin延拓定理的推广定理,研究半直线上一类具p-Laplacian算子多点边值问题在共振情形下解的存在性,得到了新的结果.更多还原

【Abstract】 This thesis investigates the existence of solutions for some class of boundary value problems of differential equations. The existence of positive solutions for several class of nonlinear differential equations (systems) boundary value problems are obtained by using the fixed point theorems (including functional fixed point theorems on cone); We investigate the existence of solutions for two kinds of boundary value problems of nonlinear differential equations by applying the monotone iterative technique、the method of upper and lower solutions and the quasilinearization method, respectively; We study the existence of solution for a class of boundary value problem at resonance on the half-line by the degree theory. The thesis consists of six chapters.Chapter 1 is preface, the historical background of the problems and the significance of this thesis are introduced. Boundary value problems occur in various fields of natural science. The existence of solutions for boundary value problems attracts much attention of many famous mathematicians all around the world. The recent developments on the existence of solutions for boundary value problems of nonlinear differential equations in this thesis are given. And some knowledge needed in this thesis are also summarized.In Chapter 2, using the Krasnosel’skii fixed point theorem、the fixed-point index theory on cone and some techniques of analysis, we discuss the existence of positive solutions for four kinds of multi-point boundary value problems of second-order nonlinear differential equations, and obtain a set of new results which generalize and improve some known results in literatures.In Chapter 3, firstly, applying the quasilinearization method, we study the existence of solutions to a class of boundary value problem of differential equation with nonlinear boundary conditions. Secondly, we discuss the existence of solutions to a class of boundary value problem of second-order nonlinear impulsive functional differential equation by using the monotone iterative technique and the method of general upper and lower solutions. We obtain some results which extend and improve the related known works in literatures.Chapter 4 deals with boundary value problems on time scales. By using functional fixed point theorems, we study the existence of multiple positive solutions for a class of boundary value problem of nonlinear impulsive dynamic equation with p -Laplacian operator on time scales and the existence of multiple positive solutions for a type of boundary value problem of nonlinear impulsive functional dynamic equation with p -Laplacian operator on time scales. We obtain some news results.In Chapter 5, we study the existence of positive solutions for boundary value problems of nonlinear differential system. Using the Krasnosel’skii fixed point theorem, we investigate the existence of multiple positive solutions for a nonlinear differential system of second-order multi-point boundary value problem with variable parameters. By the Leggett-Williams functional fixed point theorem, we study the existence of multiple positive solutions for a class of second-order differential system of Sturm-Liouville nonlocal boundary value problem. We obtain some news results.In Chapter 6, we investigate the existence of solution to a kind of multi-point boundary value problem with p -Laplacian operator at resonance on the half line. The main tool is the Ge theorem, which is more general than Mawhin continuous theorem. New results are obtained.更多还原