Study On The Oscilation And Nonoscillation For Some Kinds Of Neutral Difference Equations

In recent years, as the development of science and technology, people have put forward a great many new neutral difference equations in natural science and social science such as biology, economics, demology, physics and control theory, etc.., and they can describe the movement process of dynamic system well and truly. Because of applied universality and relating to a mass of mathematics problems, we need to solve them with related mathematical theories. Moreover, being the key content of qualitative study of the neutral difference equations, oscillation and nonoscillation theory gets prevalent attention of people and is a new investigative realm with blooming life-force. Because of the progress in modern science and technology, the studies about this new academic branch are not only a demand of the development of mathematics theories themselves, but also a demand of actual applications.This paper is devoted to the problems of qualitative behavior of the first order neutral delay difference equation, the first order neutral several delays difference equation, the first order delay difference equation with maxima and the third order delay difference equation with maxima. Some results which expand and improve the related conclusions in previous literatures are obtained.At first the paper studies the oscillation and nonoscillatoion for two kinds of first order neutral delay difference equations. The oscillation and nonoscillation of first order neutral delay difference equations with constant coefficients are studied by using characteristic equation method and some oscillatory and nonoscillatory criteria for the equation are established. At the same time, we expands these conclusions to the cases of several delays and some oscillatory and nonoscillatory criteria for the equation are established too.And then the paper studies the oscillation for the delay difference equations with maxima. At first the condition of the difference equation in previous literatures is changed, and two new oscillatory criteria are established about the first order delay difference equation with maxima. And then the ways about studying the second order delay difference equation with maxima are used, and a new oscillatory criteria is established about the third order delay difference equation with maxima.