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Acta Mathematica Sinica, English Series

, Volume 24, Issue 9, pp 1409–1432 | Cite as

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

  • Samir H. SakerEmail author
  • Donal O’regan
  • Ravi P. Agarwal
Article

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 k=1 n \( \tfrac{1} {k} \), 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.

Keywords

oscillation neutral delay dynamic equation generalized Riccati technique time scales 

MR(2000) Subject Classification

34B10 39A10 34K11 34C10 

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Copyright information

© Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Chinese Mathematical Society and Springer-Verlag GmbH 2008

Authors and Affiliations

  • Samir H. Saker
    • 1
    Email author
  • Donal O’regan
    • 2
  • Ravi P. Agarwal
    • 3
  1. 1.Department of Mathematics, College of ScienceKing Saud UniversityRiyadhSaudi Arabia
  2. 2.Department of MathematicsNational University of IrelandGalwayIreland
  3. 3.Department of Mathematical SciencesFlorida Institute of TechnologyMelbourneUSA

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