Higher Order Neutral Differential Difference Equations And Impulsive Delay Differential Equations Nature

This paper is divided into two parts. In the first part, by using the integral averaging technique, oscillation criteria for certain even order neutral differential equation with deviating arguments are established. These results extend some known oscillation criteria due to [19]. In the second part, higher-order neutral delay difference equation is considered. the existence of a nonoscillatory solution is obtained by Kra-noselskii's fixed point theorem . The results obtained here are discrete analogue of what in [34].We consider the oscillation and asymptotic behaviors of a certain delay difference equation with impulsive, and conditions are obtained for the oscillation for all nontrivial solutions and asymptotic behaviors of all nonoscillatory solutions of the equation. The results obtained here are development and generalization of the oscillation of delay differential equation(see [26]).