摘要
随着科学技术的进步与发展,在物理学、种群动力学、自动控制、生物学、医学和经济学等许多自然科学和边缘学科的领域中提出了大量由微分方程和差分方程描述的具体数学模型。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,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两个方面:一方面是微分方程的振动性理论;另一方面是差分方程的振动性理论。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、差分方程理论的基本概念和重要的不动点定理。
第二章讨论了二阶自共轭脉冲微分方程,通过建立与之振动性等价的微分方程,或考虑脉冲点与时滞间的关系,我们得出方程振动的一些充分条件。
第三章研究了带有阻尼项的二阶微分方程的振动性,利用Riccati变换,给出了方程有界解振动准则,且对一些特殊情形得到非振动解的一些性质。
第四章给出了具有连续变量的一阶中立型、二阶中立型及高阶非线性差分方程的振动准则。我们的结果改进和推广了文献中的一些结果。
第五章考虑高阶中立型非线性差分方程,通过构造函数,得到方程解振动的充分条件,同时还研究了方程存在不以零为界的正解的条件。
With the increasing development of natural science, in the field of natural science including physics, population dynamics, theory of control, biology, medicine, economies and edging field, many mathematical models which are described by differential equations and difference equations are proposed. Differential equations and difference equations are powerful tools that describe the law of nature, but 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 equation and difference equation in theory.This dissertation focuses on two sides: one is the oscillation of differential equations, the other is the oscillation of difference equations. 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 investigate the oscillation of second order self-conjugate differential equation with impulses. Through creating a differential equation whose oscillation is equivalent to the impulsive equation, or considering the relation between impulsive points and delay, we get some sufficient conditions for the oscillation of the equation.
In chapter three, we discuss the oscillation of second order nonlinear differential equation with forced terms. Using Riccati transformation, some criteria are given for the oscillation of bounded solution of the equation. We also obtain some results about the nonoscillatory solution of it in particular case.In chapter four, we establish oscillatory criteria for one order, second order and higher order difference equations with continuous variables . Our results improve and extend a number of results in the literature.In chapter five, we concern oscillation and nonoscillation of higher order neutral nonlinear difference equation. By introducing a function, we obtain sufficient con-ditons for the oscillation of the solutions of the equation. Some conditions for the existence of bounded nonoscillatory solution which is bounded away from zero are obtained.
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