【作者】 梁海燕；

【导师】 张振国；

【作者基本信息】 河北师范大学 ， 基础数学， 2008， 博士

【摘要】 随着科学技术的进步与发展,微分方程和差分方程出现在许多重要的应用领域,包括物理学、种群动力学、自动控制、生物学、医学和经济学等.微分方程及差分方程是用来描述自然现象变化规律的一种有力工具,由于寻求其通解十分困难,有时甚至是不可能的,故从理论上探讨解的性态一直是近年来研究的热点问题.我们的工作主要集中在三个方面:一是微分方程的振动性及非振动解的存在性;一是时标上微分动力系统的振动性;一是差分方程解的周期性及渐近性.本文由五章组成,主要内容如下:第一章概述了微分方程及差分方程的应用背景和国内、外研究状况,这一章也包括一些预备知识,如有关微分方程、时标及差分方程的基本概念和重要的不动点定理.第二章讨论了高阶非线性中立型微分方程,通过建立一些重要的微分不等式,得出了方程振动的充分条件.第三章通过构造映射及运用不动点定理,给出了高阶中立型微分方程有界非振动解存在的充分条件.第四章研究了时标上二阶方程的振动性,给出了方程有界解振动的充分必要条件,我们的结果改进和推广了文献中的一些结果.第五章考虑一类有理型差分方程,通过构造一个函数,使其满足一定的条件,得到了方程解的周期性及渐近性的结果.更多还原

【Abstract】 With the increasing development of natural science, the differential and difference equations arise in numbers of important applications, including physics, population dynamics, theory of controls, biologies, medicines and economies. The differential and difference equations are a kind of powerful means to investigate the rule of natural phenomena. Since it is difficult to find their general solutions, therefore, there has been an increasing interest in the study of the nature of solutions of differential equations and difference equations in theory.This dissertation focuses on three sides: The first one is the oscillation and existence of nonoscillatory solutions of differential equations; The second one is the oscillatory theory of differential dynamic systems on time scales; The last one is the periodicity and asymptotic behavior of the solutions of the difference equation. The paper is made up of five chapters. Main contents are as follows:In chapter one, we give a survey to the development and current state of the oscillation of differential equations and difference equations. We also introduce some preliminary material, including some basic concepts of the oscillation theory and some important fixed point theorems.In chapter two, we consider the higher order nonlinear neutral differential equation, by establishing some important differential inequalities, we obtain sufficient conditions of oscillation of the solutions of the equation.In chapter three, through establishing mappings and using fixed point theorems, we get the sufficient conditions for the existence of the bounded nonoscillatory solutions of the higher order neutral differential equations.In chapter four, we investigate the oscillation of the two order equations on time scales. We obtain the sufficient and necessary conditions of oscillation of the bounded solution, our results improve and extend a number of results in the literature.In chapter five, we consider a kind of rational difference equation, by establishing a function which satisfies some conditions, we obtain the results for periodicity and asymptotic behavior of the solutions.更多还原