The Behavior Of Solutions Of Some Differential Equations And Difference Equations

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.