A SHARP OSCILLATION CRITERION FOR A DIFFERENCE EQUATION WITH CONSTANT DELAY

Abstract

It is known that all solutions of the difference equation

Δx(n)+p(n)x(n−k)=0,n≥0, where {p(n)}∞n=0 is a nonnegative sequence of reals and k is a natural number, oscillate if lim infn→∞∑n−1i=n−kp(i)>(kk+1)k+1. In the case that ∑n−1i=n−kp(i) is slowly varying at infinity, it is proved that the above result can be essentially improved by replacing the above condition with lim supn→∞∑n−1i=n−kp(i)>(kk+1)k+1. An example illustrating the applicability and importance of the result is presented.