Suppose that {pn} is a nonnegative sequence of real numbers
and let k be a positive integer. We prove that
limn→∞inf [1k∑i=n−kn−1pi]>kk(k+1)k+1
is a sufficient condition for the oscillation of all solutions of the
delay difference equation
An+1−An+pnAn−k=0, n=0,1,2,….
This result is sharp in that the lower bound kk/(k+1)k+1 in
the condition cannot be improved. Some results on difference
inequalities and the existence of positive solutions are also presented.