【作者】 刘斌；

【导师】 庚建设；

【作者基本信息】 湖南大学 ， 应用数学， 2001， 博士

【摘要】 本篇博士论文由五章组成，主要讨论了具时滞微分方程周期正解的存在性，具偏差变元微分方程边值问题解的存在性，具共振条件下二阶非线性微分方程多点边值问题解的存在性和具高维二阶脉冲微分方程多点边值问题解的存在性。 第一章简述了问题产生的历史背景和本文的主要工作。 第二章讨论了一类具时滞微分方程周期正解的存在性，通过利用Krasnoselskii不动点定理，建立了存在一个或多个周期正解的若干充分条件，同时也给出了两个著名生态数学模型存在周期正解的条件。 第三章考虑了一类具偏差变元微分方程边值问题，利用Leray-Schauder度理论，在对阻尼力没有任何限制的前提下，获得了解存在的充分条件，这些结果大大改进和推广了一些已知的结果。 第四章研究了具共振条件下二阶非线性微分方程多点边值问题解的存在性，通过使用重合度理论和一些分析技巧，建立了一系列解存在的充分条件，其中在第一节考虑了同号情形，在第二节考虑了异号情形。 第五章考虑了具高维二阶脉冲微分方程多点边值问题，经过使用重合度理论和自治曲率界集概念，首先建立了一般性存在定理，然后给出了解存在的一些简明条件，这些结果即使没有脉冲点也是新的。更多还原

【Abstract】 This Ph.D.Thesis is composed of five chapters, which mainly investigated the posi- tive periodic solutions for a nonautonomous delay differential equations, the existence of solutions for boundary value problems for differential equations with deviating ar- guments and p-Laplacian, solvability of multi-point boundary value problems at reso- nance, and the existence of solutions for rn-point boundary value problems of second order differential systems with impulses. In the first chapter, we introduce the historical background of problems which will be investigated and the main results of this paper. In Chapter 2, by using the Krasnoselskii axed point theorem, we study the existence of positive periodic solutions of a nonautonomous delay differential equation, and some sufficient conditions will be given. Two known dynemic mathematical models will be studied too. Chapter 3 mainly considers the boundary value problems for differential equations with deviating arguments. An existence result, obtained by the help of Leray-Schauder degree theory, have no restriction on the damping forces. Our result improves and generalizes the some known results. The purpose of Chapter 4 is to study the solvability of multi-point boundary value problems at resonance, and establishs some existence theorems under nonlinear growth restriction. Our method is based upon the coincidence degree theory. In Section 1, we consider the same sign case, and in Section 2, discuss the non-same sign case. In the last chapter, we consider the existence of solutions for rn-point boundary value problems of second order differential systems with impulses. We first obtain a general existence result, and give some sufficient conditions. The key tool in our approach is based on the coincidence degree theory and the concept of autonomouse curvature bound set. Our result is new even for the case without impulsive point.更多还原