Acta Mathematica Sinica, English Series

, Volume 24, Issue 9, pp 1409–1432

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

• Samir H. Saker
• 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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© 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