A further result on the oscillation of delay difference equations

In this paper, we are concerned with the delay difference equations of the form yn+1 − yn + pnyn−k = 0, N = 0, 1, 2, …, (*) where pn ≥ 0 and k is a positive integer. We prove by using a new technique that guarantees that all solutions of equation (*) oscillate, which improves many previous well-known results. In particular, our theorems also fit the case where Σn−1i=n−kpi ≤ kk+1/(k + 1)k+1. In addition, we present a nonoscillation sufficient condition for equation (*).