Oscillation theorems for second-order nonlinear neutral delay dynamic equations on time scales

Abstract

By employing the generalized Riccati transformation technique, we will establish some new oscillation criteria and study the asymptotic behavior of the nonoscillatory solutions of the second-order nonlinear neutral delay dynamic equation $$ [r(t)[y(t) + p(t)y(\tau (t))]^\Delta ]^\Delta + q(t)f(y(\delta (t))) = 0 $$ , on a time scale \( \mathbb{T} \). The results improve some oscillation results for neutral delay dynamic equations and in the special case when \( \mathbb{T} \) = ℝ our results cover and improve the oscillation results for second-order neutral delay differential equations established by Li and Liu [Canad. J. Math., 48 (1996), 871–886]. When \( \mathbb{T} \) = ℕ, our results cover and improve the oscillation results for second order neutral delay difference equations established by Li and Yeh [Comp. Math. Appl., 36 (1998), 123–132]. When \( \mathbb{T} \) =hℕ, \( \mathbb{T} \) = {t: t = qk, k ∈ ℕ, q > 1}, \( \mathbb{T} \) = ℕ2 = {t2: t ∈ ℕ}, \( \mathbb{T} \) = \( \mathbb{T}_n \) = {tn = Σk=1n\( \tfrac{1} {k} \), n ∈ ℕ0}, \( \mathbb{T} \) ={t2: t ∈ ℕ}, \( \mathbb{T} \) = {√n: n ∈ ℕ0} and \( \mathbb{T} \) ={\( \sqrt[3]{n} \): n ∈ ℕ0} our results are essentially new. Some examples illustrating our main results are given.