# Oscillation Behavior of Third-Order Neutral Emden-Fowler Delay Dynamic Equations on Time Scales

- Zhenlai Han
^{1, 2}, - Tongxing Li
^{1}, - Shurong Sun
^{1, 3}Email author and - Chenghui Zhang
^{2}

**2010**:586312

**DOI: **10.1155/2010/586312

© Zhenlai Han et al. 2010

**Received: **14 September 2009

**Accepted: **10 December 2009

**Published: **2 February 2010

## Abstract

We will establish some oscillation criteria for the third-order Emden-Fowler neutral delay dynamic equations on a time scale , where is a quotient of odd positive integers with , and real-valued positive rd-continuous functions defined on . To the best of our knowledge nothing is known regarding the qualitative behavior of these equations on time scales, so this paper initiates the study. Some examples are considered to illustrate the main results.

## 1. Introduction

The study of dynamic equations on time-scales, which goes back to its founder Hilger [1], is an area of mathematics that has recently received a lot of attention. It has been created in order to unify the study of differential and difference equations. Many results concerning differential equations carry over quite easily to corresponding results for difference equations, while other results seem to be completely different from their continuous counterparts. The study of dynamic equations on time-scales reveals such discrepancies, and helps avoid proving results twice—once for differential equations and once again for difference equations.

Several authors have expounded on various aspects of this new theory; see the survey paper by Agarwal et al. [2], Bohner and Guseinov [3], and references cited therein. A book on the subject of time-scales, by Bohner and Peterson [4], summarizes and organizes much of the time-scale calculus; see also the book by Bohner and Peterson [5] for advances in dynamic equations on time-scales.

In the recent years, there has been increasing interest in obtaining sufficient conditions for the oscillation and nonoscillation of solutions of various equations on time-scales; we refer the reader to the papers [6–38]. To the best of our knowledge, it seems to have few oscillation results for the oscillation of third-order dynamic equations; see, for example, [14–16, 21, 35]. However, the paper which deals with the third-order delay dynamic equation is due to Hassan [21].

Hassan [21] considered the third-order nonlinear delay dynamic equations

where is required, and the author established some oscillation criteria for (1.1) which extended the results given in [16].

To the best of our knowledge, there are no results regarding the oscillation of the solutions of the following third-order nonlinear neutral delay dynamic equations on time-scales up to now:

We assume that is a quotient of odd positive integers, and are positive real-valued rd-continuous functions defined on such that the delay functions are rd-continuous functions such that and

As we are interested in oscillatory behavior, we assume throughout this paper that the given time-scale is unbounded above. We assume and it is convenient to assume We define the time-scale interval of the form by .

For the oscillation of neutral delay dynamic equations on time-scales, Mathsen et al. [26] considered the first-order neutral delay dynamic equations on time-scales

and established some new oscillation criteria of (1.3) which as a special case involve some well-known oscillation results for first-order neutral delay differential equations.

Agarwal et al. [7], Şahíner [28], Saker [31], Saker et al. [33], Wu et al. [34] studied the second-order nonlinear neutral delay dynamic equations on time-scales

by means of Riccati transformation technique, the authors established some oscillation criteria of (1.4).

Saker [32] investigated the second-order neutral Emden-Fowler delay dynamic equations on time-scales

and established some new oscillation for (1.5).

Our purpose in this paper is motivated by the question posed in [26]: What can be said about higher-order neutral dynamic equations on time-scales and the various generalizations? We refer the reader to the articles [23, 24] and we will consider the particular case when the order is 3, that is, (1.2). Set By a solution of (1.2), we mean a nontrivial real-valued function satisfying and and satisfying (1.2) for all

The paper is organized as follows. In Section 2, we apply a simple consequence of Keller's chain rule, devoted to the proof of the sufficient conditions which guarantee that every solution of (1.2) oscillates or converges to zero. In Section 3, some examples are considered to illustrate the main results.

## 2. Main Results

In this section we give some new oscillation criteria for (1.2). In order to prove our main results, we will use the formula

where is delta differentiable and eventually positive or eventually negative, which is a simple consequence of Keller's chain rule (see Bohner and Peterson [4, Theorem ]).

Before stating our main results, we begin with the following lemmas which are crucial in the proofs of the main results.

For the sake of convenience, we denote: for Also, we assume that

there exists such that and

Lemma 2.1.

then there are only the following three cases for sufficiently large:

, , ,

or

,

Proof.

If then there are two possible cases:

(1) eventually; or

(2) eventually.

If there exists a such that case (2) holds, then exists, and We claim that Otherwise, We can choose some positive integer such that for Thus, we obtain

which implies that so hence,

Assume that We claim that eventually. Otherwise, we have or By there exists we can choose some positive integer such that for and we obtain

which implies that so hence, This completes the proof.

In [4, Section ] the Taylor monomials are defined recursively by

It follows from [4, Section ] that for any time-scale, but simple formulas in general do not hold for

Lemma 2.2 (see [15, Lemma ]).

Lemma 2.3.

Proof.

which contradicts (2.26). Hence and is nonincreasing. The proof is complete.

Lemma 2.4.

where for then

Proof.

Hence and completes the proof.

Theorem 2.5.

where Then every solution of (1.2) oscillates or

Proof.

for all large which contradicts (2.43). If holds, from Lemma 2.1, then If case holds, by Lemma 2.4, then The proof is complete.

Remark 2.6.

From Theorem 2.5, we can obtain different conditions for oscillation of all solutions of (1.2) with different choices of .

For example, let Now Theorem 2.5 yields the following result.

Corollary 2.7.

holds for some and for all constants then every solution of (1.2) is either oscillatory or

For example, let From Theorem 2.5, we have the following result which can be considered as the extension of the Leighton-Wintner Theorem.

Corollary 2.8.

then every solution of (1.2) is either oscillatory or

In the following theorem, we present a new Kamenev-type oscillation criteria for (1.2).

Theorem 2.9.

then every solution of (1.2) oscillates or

Proof.

This easily leads to a contradiction of (2.60). If holds, from Lemma 2.1, then If holds, by Lemma 2.4, then The proof is complete.

In the following theorem, we present a new Philos-type oscillation criteria for (1.2).

Theorem 2.10.

where then every solution of (1.2) oscillates or

Proof.

This easily leads to a contradiction of (2.73). If case holds, from Lemma 2.1, then If case holds, by Lemma 2.4, then The proof is complete.

The following result can be considered as the extension of the Atkinson's theorem [39].

Theorem 2.11.

then every solution of (1.2) is either oscillatory or

Proof.

from This contradicts (2.78). If case holds, from Lemma 2.1, then If case holds, by Lemma 2.4, then The proof is complete.

Theorem 2.12.

where is as defined as in Theorem 2.5. Then every solution of (1.2) is either oscillatory or

Proof.

for all large which contradicts (2.84). If case holds, from Lemma 2.1, then If case holds, by Lemma 2.4, then The proof is complete.

Remark 2.13.

From Theorem 2.12, we can obtain different conditions for oscillation of all solutions of (1.2) with different choices of .

For example, let Now Theorem 2.12 yields the following results.

Corollary 2.14.

holds for some and for all constants then every solution of (1.2) is either oscillatory or

For example, let From Theorem 2.12, we have the following result which can be considered as the extension of the Leighton-Wintner theorem.

Corollary 2.15.

Assume that (2.2), (2.26), and (2.38) hold, If (2.59) holds, then every solution of (1.2) is either oscillatory or

In the following theorem, we present a new Kamenev-type oscillation criteria for (1.2).

Theorem 2.16.

then every solution of (1.2) oscillates or

The proof is similar to that of Theorem 2.9 using inequality (2.88), so we omit the details.

In the following theorem, we present a new Philos-type oscillation criteria for (1.2).

Theorem 2.17.

where Then every solution of (1.2) oscillates or

The proof is similar to that of the proof of Theorem 2.10 using inequality (2.88), so we omit the details.

The following result can be considered as the extension of the Belohorec's theorem [40].

Theorem 2.18.

then every solution of (1.2) is either oscillatory or satisfies

Proof.

This contradicts (2.96). If case holds, from Lemma 2.1, then If case holds, by Lemma 2.4, then The proof is complete.

Remark 2.19.

One can easily see that the results obtained in [14–16, 21, 23, 24, 35] cannot be applied in (1.2), so our results are new.

## 3. Examples

In this section we give the following examples to illustrate our main results.

Example 3.1.

where is a quotient of odd positive integers,

Let . It is easy to see that (2.2), (2.26), and (2.38) hold. Also

Hence by Corollary 2.8, every solution of (3.1) is either oscillatory or

Example 3.2.

Let . It is easy to see that all the conditions of Corollary 2.8 hold. Then by Corollary 2.8, every solution of (3.3) is either oscillatory or satisfies In fact, is a solution of (3.3).

Example 3.3.

where is a quotient of odd positive integers.

For , we have . Let . It is easy to see that (2.2) and (2.38) hold, and

so (2.78) holds. By Theorem 2.11, every solution of (3.4) is either oscillatory or satisfies .

Example 3.4.

where is a quotient of odd positive integers.

Let It is easy to see that (2.2), (2.26), and (2.38) hold. Also we have

Hence (2.96) holds. By Theorem 2.18, every solution of (3.7) is either oscillatory or satisfies

## Declarations

### Acknowledgments

The authors sincerely thank the reviewers for their valuable suggestions and useful comments that have to the present improved version of the original manuscript. This research is supported by the Natural Science Foundation of China (60774004, 60904024), China Postdoctoral Science Foundation Funded Project (20080441126, 200902564), Shandong Postdoctoral Funded Project (200802018), Shandong Research Funds (Y2008A28), and also supported by University of Jinan Research Funds for Doctors (B0621, XBS0843).

## Authors’ Affiliations

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