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

In recent years, with the development of science and technology, people have put forward a great many new neutral difference equations in natural science and social science as biology, economics, demology, physics and control theory ,etc.., it describes movement process of dynamic system well and truly. Because of applied universality and a mass of mathematics problems relate to itself, which starve for us to solve them with related mathematical theories. Moreover, oscillation and nonoscillation theory being the key content of qualitative study of the neutral difference equations, it gets prevalent attention of people and it 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 equations, the second order nonlinear neutral delay difference equations and the neutral difference equations with positive and negative coefficients. Some results which expand and improved the related conclusions in previous literatures are obtained.Firstly, this thesis studies the oscillation for two kinds of first order neutral delay difference equations. The oscillation of first order neutral delay difference equations with constant coefficients is studied by using characteristic equation method and a oscillatory criteria for the equation is established and at the same time, we expands this conclusion to the cases with several delays. Later, the linearized oscillation for first order nonlinear neutral delay equations with variable coefficients is discussed, a simple oscillatory criteria of this equation is obtained successfully by using the fixed point theorem.Secondly, this thesis studies the oscillation for a class of second order nonlinear neutral delay difference equations, the equation which is discussed by us is more complicated than the ones in the previous literatures, two sufficient conditions of this equation to be oscillatory are obtained finally.Finally this thesis researches the oscillation and nonoscillation for neutral difference equations with positive and negative coefficients. Under the hypothetic conditions different from that in the existed literatures, the oscillation and nonoscillation for a class of neutral difference equations with positive and negative coefficients are studied, two sufficient conditions of this equation to be oscillatory are obtained, a sufficient condition for the existence of positive solution of this equation and a sufficient and necessary condition for the existence of bounded positive solution of this equation are obtained; The oscillation of a class of forced neutral difference equations with positive and negative coefficients is discussed and two sufficient conditions of this equation to be oscillatory are obtained.