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

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

${\left[r\left(t\right){\left[y\left(t\right)+p\left(t\right)y\left(\tau \left(t\right)\right)\right]}^{{\Delta }}\right]}^{{\Delta }}+q\left(t\right)f\left(y\left(\delta \left(t\right)\right)\right)=0,$

on a time scale $𝕋$. The results improve some oscillation results for neutral delay dynamic equations and in the special case when $𝕋=ℝ$ our results cover and improve the oscillation results for second-order neutral delay differential equations established by Li and Liu [Canad. J. Math. 48, No. 4, 871–886 (1996; Zbl 0859.34055)]. When $𝕋=ℕ$, our results cover and improve the oscillation results for second order neutral delay difference equations established by Li and Yeh [Comp. Math. Appl., 36, No. 10–12, 123–132 (1998; Zbl 0933.39027)]. When $𝕋=hℕ,𝕋=\left\{t:t=qk,k\in ℕ,q>1\right\}$, $𝕋={ℕ}^{2}=\left\{{t}^{2}:t\in ℕ\right\}$, $𝕋={𝕋}_{n}=\left\{{t}_{n}={{\Sigma }}_{k=1}^{n}\frac{1}{k},n\in {ℕ}_{0}\right\}$, $𝕋=\left\{{t}^{2}:t\in ℕ\right\}$, $𝕋=\left\{\surd n:n\in {ℕ}_{0}\right\}$ and $𝕋=\left\{\sqrt[3]{n}:n\in {ℕ}_{0}\right\}$ 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
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