The Oscillation Behavior For Neutral Delay Difference Equations

In recent years, with the development of science and technology, the neutral difference equations with universality of application in natural science and social science as biology, economics, demology, physics and control theory, etc., get people's prevalent attention. Moreover, being the key content of qualitative study of the neutral difference equations, oscillation theory attracts scholars'attention. It can describe the motion process of various dynamic systems objectively, so it has great theoretical significance and practical value to research the oscillation theory of the neutral difference equations.This paper studies the oscillation for four kinds of neutral delay difference equations and gives several sufficient conditions of oscillation of solutions to the equations which expand and improve the related conclusions in previous literatures.Firstly, this paper studies the oscillation for a class of second-order neutral multi-delay difference equation. Using the Riccati transformation techniques, we obtain some sufficient conditions of oscillation of solutions to this equation and give examples of actual applications. The results obtained expand and improve the conclusions in previous literatures.Secondly, we discuss the oscillation for a class of second-order neutral multi-delay difference equation with continuous arguments and provide some sufficient conditions of oscillation of solutions to this equation by means of reduction to absurdity and mathematics induction.Thirdly, this paper researches the oscillation for a class of higher-order neutral delay difference equation. Some sufficient conditions of oscillation of solutions to this equation are obtained by inequalities and characteristic equations which extend the known results from first-order to higher-order.Finally, we study the oscillation for a class of higher-order nonlinear neutral multi-delay difference equation with an oscillating coefficient and obtain several sufficient conditions of oscillation of this equation.