The Oscillation Of Neutral Difference Equations

Nowadays, the development of science and technology is very fast, people putforward the large number of neutral difference equations from natural science and socialscience such as biology, economics, cybernetics, physics and digital signal, etc.. Asneutral difference equation and high order difference equations are used in real lifeapplication more and more extensively. Neutral difference equation and high orderdifference equations are needed of being analyzed and studied by mathematics theoryurgently. Oscillation theory is the key content of qualitative study of the neutraldifference equations, so it attracts scholars’ prevalent attention. Neutral differenceequations with damping and maxima can describe the motion process of various dynamicsystems objectively, so they has great theoretical significance and practical value toresearch the oscillation theory of the neutral difference equations.This paper studies the oscillation for three kinds of neutral difference equations withdamping and one kind of neutral difference with maxima, and then some sufficientconditions for oscillation and bounded oscillation of the solutions of the equation areobtained.Firstly, In this paper, we consider a kind of damped neutral difference equations.Some sufficient conditions for oscillation and bounded oscillation of the solutions of theequation are obtained, by using the suitable inequality technique, reduction to absurdityand generalized Riccati transformation techniques.Secondly, we consider two kinds of second order damped neutral differenceequations with continuous arguments. Some sufficient conditions for oscillation of thesolutions and bounded oscillation of the solutions of the equation are obtained, by usingthe Reductio ad absurdum, mathematical induction, integral transform and generalizedRiccati transformation techniques.Finally, we study the oscillation for a kind of higher-order neutral differenceequations with maxima. Some sufficient conditions for oscillation of the solutions of thesuitable equation are obtained, by using the suitable inequality technique，Reductio adabsurdum and category discussion.