Oscillation For Neutral Differential And Difference Equations

Differential equations and difference equations are powerful tools that describe the variation of natural phenomenon, we usually regard the corresponding difference equation as the discrete form of a differential equation, but it also has its own particularity. In recent decades, in the many field of natural science including physics, biology, population dynamics, medicine, economies, etc and edging field, lots of mathematical models which are described by differential equations and difference equations were proposed. So the researches of differential equations and difference equations are of great importance. The theory of oscillation is an important branch of qualitative theory of differential equations, which stemmed from the second order linear ordinary differential equation x"(t)+a(t)x(t)=0by Sturm in1836. Nowadays Oscillatory behaviors are the basic problems of differential equations and difference equations.This dissertation focuses on two sides:one is the oscillation of differential equations, the other is the oscillation of difference equations. This paper is made up of three chapters. Main content are as follows:In chapter one, we introduces briefly the research background of differential equations and difference equations.In chapter two, oscillation of the following first order neutral delay differential equation is studied. By constructing several suitable functions, the necessary and sufficient conditions, which guar-antee the oscillation of this equation, are obtained systematically for0<p<1and p>1, respectively. And on the basis of our results, several easily verifiable sufficient conditions are established, thereby improve and extend some existing results in the literatures.In chapter three, oscillation of the following first order neutral delay difference equation is studied. By constructing several suitable functions, the necessary and sufficient conditions, which guar-antee the oscillation of this equation, are obtained systematically for0<p<1and p>1, respectively. And then several easily verifiable sufficient conditions are given.