Oscillation theorems for second-order nonlinear neutral delay dynamic equations on time scales. (English) Zbl 1153.34040

Saker, Samir H.; O’Regan, Donal; Agarwal, Ravi P.

Acta Math. Sin., Engl. Ser. 24, No. 9, 1409-1432 (2008).

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 \(\mathbb{T}\). The results improve some oscillation results for neutral delay dynamic equations and in the special case when \(\mathbb{T}= \mathbb R\) 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 \(\mathbb{T} = \mathbb N\), 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 \(\mathbb{T} =h\mathbb N, \mathbb{T} = \{t: t = q k , k \in \mathbb N, q > 1\}\), \(\mathbb{T} = \mathbb N^{2} = \{t ^{2}: t \in \mathbb N\}\), \(\mathbb{T} = \mathbb{T}_n = \{t_n = \Sigma _{k=1}^n \tfrac{1}{k}, n \in \mathbb N_{0}\}\), \(\mathbb{T} =\{t ^{2}: t \in \mathbb N\}\), \(\mathbb{T} = \{\surd n: n \in \mathbb N_{0}\}\) and \(\mathbb{T} =\{\root 3\of {n}: n \in \mathbb N_{0}\}\) our results are essentially new. Some examples illustrating our main results are given.

MSC:

34K11	Oscillation theory of functional-differential equations
34K40	Neutral functional-differential equations
39A10	Additive difference equations

Keywords:

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

Citations:

Zbl 0859.34055; Zbl 0933.39027

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