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.

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