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

*Advances in Difference Equations*
**volume 2010**, Article number: 586312 (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.

Assume that holds. Further, assume that is an eventually positive solution of (1.2). If

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

, , ,

or

,

Proof.

Let be an eventually positive solution of (1.2). Then there exists such that and for all From (1.2) we have

Hence is strictly decreasing on We claim that eventually. Assume not, then there exists such that

Then we can choose a negative and such that

Dividing by and integrating from to we have

Letting then by (2.2). Thus, there is a such that for

Integrating the previous inequality from to we obtain

Therefore, there exist and such that

We can choose some positive integer such that for Thus, we obtain

The above inequality implies that for sufficiently large which contradicts the fact that eventually. Hence we get

It follows from this that either or Since

which yields

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 and from the definition of we have which contradicts Now, we assert that is bounded. If it is not true, there exists with as such that

From

which implies that it contradicts that Therefore, we can assume that

By we get

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 and from the definition of we have which contradicts or Now, we have that here is finite. We assert that is bounded. If it is not true, there exists with as such that

From

which implies that it contradicts that Therefore, we can assume that

By we get

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 ]).

Assume that satisfies case (i) of Lemma 2.1. Then

Lemma 2.3.

Assume that is a solution of (1.2) satisfying case (i) of Lemma 2.1. If

then satisfies eventually

Proof.

Let be a solution of (1.2) such that case of Lemma 2.1 holds for Define

Thus

We claim that eventually. Otherwise, there exists such that for Therefore,

which implies that is strictly increasing on Pick such that for Then we have

then for By Lemma 2.2, for any there exists such that

Hence there exists so that

By the definition of we have that

From (1.2), we obtain

Integrating both sides of (2.35) from to we get

which yields that

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

Lemma 2.4.

Assume that holds and is a solution of (1.2) which satisfies case (iii) of Lemma 2.1. If

where for then

Proof.

Let be a solution of (1.2) such that case of Lemma 2.1 holds for Then , Next we claim that Otherwise, there exists such that for all By the definition of we have that (2.35) holds. Integrating both sides of (2.35) from to we get

Integrating again from to we have

Integrating again from to we obtain

which contradicts (2.38), since by [23, Lemma ] and [3, Remark ], we get

Hence and completes the proof.

Theorem 2.5.

Assume that (2.2), (2.26), and (2.38) hold, Furthermore, assume that there exists a positive function such that for some and for all constants

where Then every solution of (1.2) oscillates or

Proof.

Suppose that (1.2) has a nonoscillatory solution We may assume without loss of generality that and for all Then by Lemma 2.1, satisfies three cases. Assume that satisfies case Define the function by

Then Using the product rule, we have

By the quotient rule, we get

By the definition of and (1.2), we obtain (2.35). From (2.35) and (2.44), we have

from (2.25) and (2.27), for any we obtain

hence by (2.48), we have

In view of from (2.1) and of Lemma 2.1, we have

where By (2.49), we have

from we have by we have

so we get

by (2.44), we have

Therefore, we obtain

Integrating inequality (2.55) from to , we obtain

which yields

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.

Assume that (2.2), (2.26), and (2.38) hold, If

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.

Assume that (2.2), (2.26), and (2.38) hold, and If

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.

Assume that (2.2), (2.26), and (2.38) hold, Let and be as defined in Theorem 2.5. If for some and for all constants

where and

then every solution of (1.2) oscillates or

Proof.

Suppose that (1.2) has a nonoscillatory solution We may assume without loss of generality that and for all Then by Lemma 2.1, satisfies three cases. Assume that satisfies case We proceed as in the proof of Theorem 2.5 to get (2.54) for all sufficiently large. Multiplying (2.54) by and integrating from to we have

Integration by parts, we obtain

Next, we show that if and then

If it is easy to see that (2.64) is an equality. If then we get

Using the inequality

we obtain for

and from this we see that (2.64) holds. From (2.62)–(2.64), we get

Thus

which implies that

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.

Assume that (2.2), (2.26), and (2.38) hold, Let and be as defined in Theorem 2.5. Furthermore, assume that there exist functions , , where such that

and has a nonpositive continuous -partial derivation with respect to the second variable and satisfies

If for some and for all constants

where

where then every solution of (1.2) oscillates or

Proof.

Suppose that (1.2) has a nonoscillatory solution We may assume without loss of generality that and for all Then by Lemma 2.1, satisfies three cases. Assume that satisfies case We proceed as in the proof of Theorem 2.5 to get (2.54) for all sufficiently large. Multiplying both sides of (2.54), with replaced by by integrating with respect to from to we have

Integrating by parts and using (2.71) and (2.72), we obtain

Therefore, we get

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.

Assume that (2.2), (2.26), and (2.38) hold, If

then every solution of (1.2) is either oscillatory or

Proof.

Suppose that (1.2) has a nonoscillatory solution We may assume without loss of generality that and for all Then by Lemma 2.1, satisfies three cases. Assume that satisfies case Define the function

Using the product rule, (2.25) and (2.27), for any we have that

By (1.2), we have that (2.35) holds, then from (2.80), we calculate

where the last inequality is true because due to (2.1) and because

Upon integration we arrive at

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.

Assume that (2.2), (2.26), and (2.38) hold, Furthermore, assume that there exists a positive function such that for some and for all constants

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

Proof.

Suppose that (1.2) has a nonoscillatory solution We may assume without loss of generality that and for all Then by Lemma 2.1, satisfies three cases. Assume satisfies case Define the function as (2.44). We proceed as in the proof of Theorem 2.5 and we get (2.49). In view of from (2.1) and of Lemma 2.1, we have

from (2.27), there exists a constant such that so

By (2.49), we have

Therefore, we obtain

Integrating inequality (2.88) from to , we obtain

which yields

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.

Assume that (2.2), (2.26), and (2.38) hold, If

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.

Assume that (2.2), (2.26), and (2.38) hold, Let and be as defined in Theorem 2.12. If for some and for all constants

where and

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.

Assume that (2.2), (2.26), and (2.38) hold, Let and be as defined in Theorem 2.12. Furthermore, assume that there exist functions , , where such that (2.71) holds, and has a nonpositive continuous -partial derivation with respect to the second variable and satisfies (2.72). If

holds for some and for all constants where

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.

Assume that (2.2), (2.26), and (2.38) hold If

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

Proof.

Suppose that (1.2) has a nonoscillatory solution We may assume without loss of generality that and for all Then by Lemma 2.1, satisfies three cases. Assume that satisfies case From and (2.1) we have

so

By (1.2), we have that (2.35) holds. Using (2.25) and (2.27), for any we obtain after dividing (2.35) by for all large

So,

Upon integration we arrive at

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.

Consider the third-order neutral delay dynamic equations on time-scales

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.

Consider the third-order neutral delay differential equation

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.

Consider the third-order delay dynamic equation

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

Hence (2.26) holds. Also

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

Example 3.4.

Consider the third-order delay dynamic equation

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

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## 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).

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Han, Z., Li, T., Sun, S. *et al.* Oscillation Behavior of Third-Order Neutral Emden-Fowler Delay Dynamic Equations on Time Scales.
*Adv Differ Equ* **2010, **586312 (2010). https://doi.org/10.1155/2010/586312

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DOI: https://doi.org/10.1155/2010/586312

### Keywords

- Nonoscillatory Solution
- Oscillation Criterion
- Partial Derivation
- Neutral Delay
- Oscillation Result