Oscillation Criteria for Second-Order Delay, Difference, and Functional Equations

L K Kikina,I P Stavroulakis

doi:10.1155/2010/598068

Abstract

Consider the second-order linear delay differential equationx′′(t)+p(t)x(τ(t))=0,t≥t0, wherep∈C([t0,∞),ℝ+),τ∈C([t0,∞),ℝ),τ(t)is nondecreasing,τ(t)≤tfort≥t0andlimt→∞τ(t)=∞, the (discrete analogue) second-order difference equationΔ2x(n)+p(n)x(τ(n))=0, whereΔx(n)=x(n+1)−x(n),Δ2=Δ∘Δ,p:ℕ→ℝ+,τ:ℕ→ℕ,τ(n)≤n−1, andlimn→∞τ(n)=+∞, and the second-order functional equationx(g(t))=P(t)x(t)+Q(t)x(g2(t)),t≥t0, where the functionsP,Q∈C([t0,∞),ℝ+),g∈C([t0,∞),ℝ),g(t)≢tfort≥t0,limt→∞g(t)=∞, andg2denotes the 2th iterate of the functiong, 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 whereliminft→∞∫τ(t)tτ(s)p(s)ds≤1/eandlimsupt→∞∫τ(t)tτ(s)p(s)ds<1for the second-order linear delay differential equation, and0<liminft→∞{Q(t)P(g(t))}≤1/4andlimsupt→∞{Q(t)P(g(t))}<1, for the second-order functional equation, are presented.

Highlights

The problem of establishing sufficient conditions for the oscillation of all solutions to the second-order delay differential equation xtptxτt 0, t ≥ t0, where p ∈ C t0,∞, R here R 0, ∞, τ ∈ C t0, ∞, R, τ t is nondecreasing, τ t ≤ t for t ≥ t0, and limt → ∞τ t ∞, has been the subject of many investigations; see, for example, 1–21 and the references cited therein.International Journal of Differential EquationsBy a solution of 1 we understand a continuously differentiable function defined on τ T0, ∞ for some T0 ≥ t0 and such that 1 is satisfied for t ≥ T0
Oscillation results are obtained for 1 by reducing it to a first-order equation. Since for the latter the oscillation is due solely to the delay, the criteria hold for delay equations only and do not work in the ordinary case
Theorem 2.5 reduces the question of oscillation of 1 to that of the absence of eventually positive solutions of the differential inequality τt x t τt ξτ ξ p ξ dξ ptxτt ≤ 0

Summary

Introduction

The problem of establishing sufficient conditions for the oscillation of all solutions to the second-order delay differential equation xtptxτt 0, t ≥ t0, where p ∈ C t0,∞ , R here R 0, ∞ , τ ∈ C t0, ∞ , R , τ t is nondecreasing, τ t ≤ t for t ≥ t0, and limt → ∞τ t ∞, has been the subject of many investigations; see, for example, 1–21 and the references cited therein. By a solution of 1 we understand a continuously differentiable function defined on τ T0 , ∞ for some T0 ≥ t0 and such that 1 is satisfied for t ≥ T0. Such a solution is called oscillatory if it has arbitrarily large zeros, and otherwise it is called nonoscillatory.

Oscillation Criteria for 1