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Zbl 1153.34040
Saker, Samir H.; O'Regan, Donal; Agarwal, Ravi P.
Oscillation theorems for second-order nonlinear neutral delay dynamic equations on time scales. (English)
[J] Acta Math. Sin., Engl. Ser. 24, No. 9, 1409-1432 (2008). ISSN 1439-8516; ISSN 1439-7617/e

Summary: 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 $\Bbb{T}$. The results improve some oscillation results for neutral delay dynamic equations and in the special case when $\Bbb{T}= \Bbb R$ our results cover and improve the oscillation results for second-order neutral delay differential equations established by Li and Liu [Canad. J. Math. 48, No.~4, 871--886 (1996; Zbl 0859.34055)]. When $\Bbb{T} = \Bbb N$, our results cover and improve the oscillation results for second order neutral delay difference equations established by Li and Yeh [Comp. Math. Appl., 36, No.~10--12, 123--132 (1998; Zbl 0933.39027)]. When $\Bbb{T} =h\Bbb N, \Bbb{T} = \{t: t = q k , k \in \Bbb N, q > 1\}$, $\Bbb{T} = \Bbb N^{2} = \{t ^{2}: t \in \Bbb N\}$, $\Bbb{T} = \Bbb{T}_n = \{t_n = \Sigma _{k=1}^n \tfrac{1}{k}, n \in \Bbb N_{0}\}$, $\Bbb{T} =\{t ^{2}: t \in \Bbb N\}$, $\Bbb{T} = \{\surd n: n \in \Bbb N_{0}\}$ and $\Bbb{T} =\{\root 3\of {n}: n \in \Bbb N_{0}\}$ our results are essentially new. Some examples illustrating our main results are given.

MSC 2000:: *34K11 Oscillation theory of functional-differential equations
34K40 Neutral equations
39A10 Difference equations

Keywords: oscillation; neutral delay dynamic equation; generalized Riccati technique; time scales

Citations: Zbl 0859.34055; Zbl 0933.39027

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