Consider the delay difference equation \((1)\quad y_{n+1}-y_ n+p_ ny_{n-k}=0,\) \(n=0,1,2,...\), where \(\{p_ n\}\) is a sequence of nonnegative real numbers and k is a positive integer. The aim in this paper is to obtain respectively sufficient conditions for all solutions of (1) to oscillate and the existence of a nonoscillatory solution of (1). The oscillation of a discrete delay logistic equation is also studied.