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We establish some new oscillation criteria for the second-order neutral delay dynamic equations of Emden-Fowler type,

The study of dynamic equations on time scales, which goes back to its founder Hilger [

The three most popular examples of calculus on time scales are differential calculus, difference calculus, and quantum calculus (see Kac and Cheung [

In recent years, there has been much research activity concerning the oscillation and nonoscillation of solutions of various dynamic equations on time scales unbounded above and neutral differential equations; we refer the reader to the papers [

Agarwal et al. [

By employing different Riccati transformation technique, the authors established some oscillation criteria for all solutions of (

Recently, some authors have been interested in obtaining sufficient conditions for the oscillation and nonoscillation of solutions of Emden-Fowler type dynamic equations on time scales, differential equations, and difference equations; see, for example, [

Han et al. [

Saker [

The author assumes that

the delay functions

The main result for the oscillation of (

Assume that

We note that in [

In [

The author assumes that

The main result for the oscillation of (

Assume that (

We find that the conclusion of this theorem is wrong. The following is a counter example of this theorem.

Consider the second-order differential equation

Let

If

Abdalla [

Lin [

Wong [

The main results for the oscillation of (

Suppose that

Suppose that

Li and Saker [

The main result for the oscillation of (

Assume that there exists a positive sequence

Yildiz and Öcalan [

The main results for the oscillation of (

Assume that

Assume that

Cheng [

Following this trend, in this paper, we are concerned with oscillation of the second-order neutral delay dynamic equations of Emden-Fowler type

As we are interested in oscillatory behavior, we assume throughout this paper that the given time scales

We assume that

We note that if

If

In the case of

In this section, we give some new oscillation criteria of (

For the sake of convenience, we assume that

Assume that (

From (

Assume that

Let

Assume that

Let

Assume that

The proof is similar to that of the proof Lemmas

Assume that (

Suppose to the contrary that (

By [

If

Assume that (

Suppose to the contrary that (

From Lemma

If (ii) holds, by Lemma

By [

Assume that

Theorems

From Theorem

Assume that (

We assume that (

Assume that (

By Lemma

Assume that

By Lemma

Assume that

By using Lemma

Assume that (

Suppose to the contrary that (

Assume that (

By using Lemma

Assume that

Suppose to the contrary that (

Assume that (

By using Lemma

Assume that (

We assume that (

Assume that (

By using Lemma

In the following, we use a Riccati transformation technique to establish new oscillation criteria for (

Assume that

We assume that (

Assume that

By Lemma

Assume that (

We assume that (

Assume that (

By Lemma

Assume that

We assume that (

Assume that

By Lemma

In this paper, we consider the oscillation of second-order Emden-Fowler neutral delay dynamic equations (

The main results in this paper require that

The authors sincerely thank the reviewers for their valuable suggestions and useful comments that have lead to the present improved version of the original manuscript. This research is supported by the Natural Science Foundation of China (11071143, 60904024), China Postdoctoral Science Foundation funded project (20080441126, 200902564), Shandong Postdoctoral funded project (200802018) and supported by the Natural Science Foundation of Shandong (Y2008A28, ZR2009AL003), and by University of Jinan Research Funds for Doctors (XBS0843).