Oscillation And Nonoscillation For Some Kinds Of Neutral Difference Equations

In this paper ,firstly,we discuss the neutral difference equation with variable coefficientsBy establishing some new lemmas, we obtain several new sufficient conditions for the oscillation of all solutions of equation (I).Secondly,by building some Riccati type difference inequalities,we investigate the oscillation of second-order neutral difference equationFinally, we discuss the odd-order neutral difference equationand obtain some new sufficient conditions for the oscillation of all solutions of equation (III).Our results improve and extend the all known results in related literature. Moreover the theorems for the existence of a positive solution of equtions(II)and (III) are firstly obtained. Some examples given in the paper illustrate the effect of our results.