The Oscillation Asymptotic Behavior For Neutral Delay Difference Equations

The qualitative theory of difference equation,including oscillation, existence of positive solution, asymptotic behavior,etc..,is an important part of the theory of difference equation. In recent years, with the development of science and technology, the neutral difference equation with universality of application,gets people's prevalent attention and it is a new investigative realm with vigorous vitality in many subjects of natural science and social science. It can describe the motion process of various dynamic systems objectively and accurately in ecology, biology, economics, demology and control theory, etc…Therefore, it has great theoretical significance and practical value to research the qualitative theory of neutral difference equation.This paper studies the oscillation for two kinds of constant coefficient neutral delay difference equations, the oscillation and the existence of positive solutions for the second order neutral delay difference equations, the oscillation and the asymptotic behavior for high-order neutral delay difference equations. The relative topic extends the problems studied in the document. For these equations mentioned above, it gives separately some sufficient conditions for oscillation of solutions, asymptotic behavior of solutions and existence of positive solutions.Firstly, this paper studies the oscillation for first order constant coefficient neutral delay difference equations, and obtains four sufficient conditions of oscillatory of solutions to this equation. On the basis of this,we study the oscillation for second order constant coefficient neutral delay difference equations further, and give several sufficient conditions of oscillatory and nonoscillatory of solutions,and give examples of actual applications.Then, we discuss the oscillation for second order neutral multi-delay difference equations,provide three sufficient conditions of oscillatory of solutions to this equation, and study the existence of ultimate positive solution using fixed point theorem, and obtain a sufficient condition of the existence of final positive solution.Finally, we study the oscillation and the asymptotic behavior for high-order neutral variable delay difference equations using reduction to absurdity and mathematics induction,and obtain three sufficient conditions of the asymptotic behavior and five sufficient conditions of the oscillation to this equation.