A sharp oscillation criterion for a difference equation with constant delay

Abstract

The paper establishes a refined oscillation criterion for the first-order linear difference equation Δx(n) + p(n) x(n−k) = 0, where p(n) ≥ 0 and k is a positive integer. It shows that if the partial sums ∑_{i=n−k}^{n−1} p(i) are slowly varying, then oscillation of all solutions occurs under the sharp condition lim sup_{n→∞} ∑_{i=n−k}^{n−1} p(i) > (k/(k+1))^(k+1), improving the standard lim inf-based result. An illustrative example demonstrates applicability of the improved criterion.