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

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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 = q k, k ∈ ℕ, q > 1}, \( \mathbb{T} \) = ℕ2 = {t 2: t ∈ ℕ}, \( \mathbb{T} \) = \( \mathbb{T}_n \) = {t n = Σ n k=1 \( \tfrac{1} {k} \), n ∈ ℕ0}, \( \mathbb{T} \) ={t 2: 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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Authors and Affiliations

  1. Department of Mathematics, College of Science, King Saud University, Riyadh, 11451, Saudi Arabia

    Samir H. Saker

  2. Department of Mathematics, National University of Ireland, Galway, Ireland

    Donal O’regan

  3. Department of Mathematical Sciences, Florida Institute of Technology, Melbourne, Florida, 32901, USA

    Ravi P. Agarwal

Authors
  1. Samir H. Saker
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  2. Donal O’regan
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  3. Ravi P. Agarwal
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Correspondence to Samir H. Saker.

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Saker, S.H., O’regan, D. & Agarwal, R.P. Oscillation theorems for second-order nonlinear neutral delay dynamic equations on time scales. Acta. Math. Sin.-English Ser. 24, 1409–1432 (2008). https://doi.org/10.1007/s10114-008-7090-7

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  • DOI: https://doi.org/10.1007/s10114-008-7090-7

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