Copyright © 2010 L. K. Kikina and I. P. Stavroulakis. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Consider the second-order linear delay differential equation x′′(t)+p(t)x(τ(t))=0, t≥t0, where p∈C([t0,∞),ℝ+), τ∈C([t0,∞),ℝ), τ(t) is nondecreasing, τ(t)≤t for t≥t0 and limt→∞τ(t)=∞, the (discrete analogue) second-order difference equation Δ2x(n)+p(n)x(τ(n))=0, where Δx(n)=x(n+1)−x(n), Δ2=Δ∘Δ, p:ℕ→ℝ+, τ:ℕ→ℕ, τ(n)≤n−1, and limn→∞τ(n)=+∞, and the second-order functional equation x(g(t))=P(t)x(t)+Q(t)x(g2(t)), t≥t0, where the functions P, Q∈C([t0,∞),ℝ+), g∈C([t0,∞),ℝ), g(t)≢t for t≥t0, limt→∞g(t)=∞, and g2 denotes the 2th iterate of the function g, that is, g0(t)=t, g2(t)=g(g(t)), t≥t0. The most interesting oscillation criteria for the second-order linear delay differential equation, the second-order difference equation and the second-order functional equation, especially in the case where liminft→∞∫τ(t)tτ(s)p(s)ds≤1/e and limsupt→∞∫τ(t)tτ(s)p(s)ds<1 for the second-order linear delay differential equation, and 0<liminft→∞{Q(t)P(g(t))}≤1/4 and limsupt→∞{Q(t)P(g(t))}<1, for the second-order functional equation, are presented.