Research Article | Open Access

Yang-Cong Qiu, Qi-Ru Wang, "New Oscillation Results of Second-Order Damped Dynamic Equations with -Laplacian on Time Scales", *Discrete Dynamics in Nature and Society*, vol. 2015, Article ID 709242, 9 pages, 2015. https://doi.org/10.1155/2015/709242

# New Oscillation Results of Second-Order Damped Dynamic Equations with -Laplacian on Time Scales

**Academic Editor:**Wei Nian Li

#### Abstract

By employing a generalized Riccati technique and functions in some function classes for integral averaging, we derive new oscillation criteria of second-order damped dynamic equation with -Laplacian on time scales of the form , where the coefficient function may change sign. Two examples are given to demonstrate the obtained results.

#### 1. Introduction

In 1988, the theory of time scales was introduced by Hilger in his Ph.D. thesis [1] in order to unify continuous and discrete analysis; see also [2]. In recent years, there has been much research activity concerning the oscillation of solutions of dynamic equations on time scales; for example, see [3–19] and the references therein.

Došlý and Hilger [10] considered the second-order dynamic equation and they gave some necessary and sufficient conditions for the oscillation of all solutions as (1) on unbounded time scales. Erbe et al. [11] studied the second-order nonlinear dynamic equation By means of generalized exponential functions and generalized Riccati transformation techniques, some oscillation criteria were given for (2). Del Medico and Kong [8, 9] employed the Riccati transformation and gave sufficient conditions for Kamenev-type oscillation criteria of (1) on a measure chain. Wang [18] considered the second-order nonlinear damped differential equation used the generalized Riccati transformations, and established new oscillation criteria of (3). Saker et al. [16] studied the second-order damped dynamic equation with damping and they gave some oscillation criteria. Agarwal et al. [5] studied the second-order half-linear dynamic equation and established Philos-type oscillation criteria. Huang and Wang [13] considered the second-order nonlinear dynamic equation The authors extended the results in [8, 9], and established some new Kamenev-type oscillation criteria. Qiu and Wang [14] studied the dynamic equations of a more general form and established some similar oscillation criteria.

Şenel [17] discussed the oscillation of second-order damped dynamic equations of the following form: on a time scale satisfying and , where . However, it seemed that several mistakes had been made and the obtained theorems and corollaries were incorrect. Qiu and Wang [15] corrected some mistakes in [17] and established some Kamenev-type oscillation criteria of (8) by employing functions in some function classes and a similar generalized Riccati transformation as used in [13, 18, 19].

Hassan and Kong [12] considered the second-order dynamic equation with -Laplacian and damping where , . The authors used the notations where , by which (9) can be written into the form and established some oscillation criteria of (9).

In this paper, we will consider the second-order damped dynamic equation with -Laplacian on a time scale satisfying and , where the coefficient function may change sign. Throughout this paper, we will assume that(C1), , ;(C2), is positively regressive, which means , and (C3), and for any , (C4), and there exists a function such that .

The rest of this paper is organized as follows. In Section 2, we give some basic lemmas. In Section 3, we derive new oscillation criteria for (12). In Section 4, two examples are included to show the significance of the results.

#### 2. Basic Lemmas

Preliminaries about time scale calculus can be found in [3, 4, 6, 7] and hence we omit them here. Note that we have the following properties for some typical time scales, respectively:(1), (2), (3),

*Definition 1. *A solution of (12) is said to have a generalized zero at if , and it is said to be nonoscillatory on if there exists such that for all . Otherwise, it is oscillatory. Equation (12) is said to be oscillatory if all solutions of (12) are oscillatory.

For simplicity, throughout this paper, we denote , where , and are denoted similarly.

To establish new oscillation criteria of (12), we give three lemmas in this section.

Lemma 2. *Assume that (C1)–(C4) hold, and there exists a sufficiently large such that is a solution of (12) satisfying for . Then for , we have
*

*Proof. *Let such that is a solution of (12) satisfying for ; then we also have . It is obvious that (12) is equivalent to
which is simplified as
By (12) and (C4), we have

We claim that for . Assume the contrary; that is, there exists such that . Also, for we have
Integrating (22) from to , we obtain
That is,
which implies that
Integrating (25) from to and letting , by (C2) we obtain
which contradicts . Hence for , and . By (21), we obtain

Lemma 2 is proved.

Lemma 3. *Assume that (C1)–(C4) hold and is a solution of (12) satisfying for with . Then for , if , we have
**
and if , we have
**
where
*

*Proof. *Since is a solution of (12) satisfying for with , by Lemma 2 we have
From (21) it follows that
Integrating (32) from to with , we obtain
Hence when , we have

Since , , for , we get
and we obtain

Lemma 3 is proved.

*Remark 4. *In Lemma 3, when and for any , the integral in must be convergent, which means

Lemma 5. *Assume that (C1)–(C4) hold and is a solution of (12) satisfying for with . For , define
**
where , . Then u(t) satisfies
**
where
*

*Proof. *Without loss of generality we may assume there exists such that for ; then Lemmas 2 and 3 hold. Let be defined by (38). Then, differentiating (38) and using (12), it follows that
Using the fact that , we obtain

When , using the Pötzsche chain rule, we have
and it follows that
By Lemmas 2 and 3, for , we obtain
So (42) becomes

When , we have
and it follows that
By Lemmas 2 and 3, for , we obtain
So (42) becomes

By (46) and (50), (39) holds. Lemma 5 is proved.

#### 3. Main Results

In this section, we will derive new oscillation criteria of (12). Our approach to oscillation results of (12) is based on the application of the generalized Riccati transformation. Firstly, we give some definitions.

Let and . For any function , denote by the partial derivative of with respect to . Define These function classes will be used throughout this paper. Now, we are in a position to give our first theorem.

Theorem 6. *Assume that (C1)–(C4) hold and that there exist and such that, for any ,
**
where and are defined as before. Then, (12) is oscillatory.*

*Proof. *Assume that (12) is not oscillatory. Without loss of generality we may assume there exists such that for . Let be defined by (38). Then by Lemma 5, (39) holds.

For simplicity in the following, we let , , , and omit the arguments in the integrals.

Multiplying (39), where is replaced by , by , and integrating it with respect to from to , we obtain
Noting that , by the integration by parts formula we have

Since
by (54) we have
Hence
which contradicts (52) and completes the proof.

When , (38) is simplified as Now we have the following theorem.

Theorem 7. *Assume that (C1)–(C4) hold and that there exists such that, for any ,
**
where is defined as before. Then, (12) is oscillatory.*

*Proof. *Assume that (12) is not oscillatory. Without loss of generality we may assume there exists such that for . Let be defined by (58). Then by Lemma 5, we have
which implies that
So we obtain
Letting be replaced by , and integrating (62) with respect to from to , we obtain
which contradicts (59) and completes the proof.

When , it is easy to obtain , so we have Then, Theorems 6 and 7 can be simplified as the following corollaries, respectively.

Corollary 8. *Assume that (C1)–(C4) hold and that there exist and such that, for any ,
**
Then, (12) is oscillatory.*

Corollary 9. *Assume that (C1)–(C4) hold and that there exists such that, for any ,
**
Then, (12) is oscillatory.*

*Remark 10. *Compared to the theorems and corollaries in [15], the conclusions of Theorems 6 and 7 and Corollaries 8 and 9 are much simpler, and the proofs of Theorems 6 and 7 are more convenient. Furthermore, may change sign in Theorems 6 and 7, for the employment of the function .

*Remark 11. *Consider the equations as the following form:
It is obvious that (67) is equivalent to
which is simplified as
We can get the similar oscillation results as before, where
are replaced by
respectively, in Lemmas 2, 3, and 5, Theorems 6 and 7, and Corollaries 8 and 9.

#### 4. Examples

In this section, we will show the application of our oscillation results in two examples. Firstly, we give an example to demonstrate Theorem 6.

*Example 12. *Consider the equation
where , , , , so we have . Letting , we have the following results.

(i) . In this case, there exists such that Hence That is, (52) holds. By Theorem 6 we see that (72) is oscillatory.

(ii) . In this case, there exists such that Hence That is, (52) holds. By Theorem 6 we see that (72) is oscillatory.

The second example illustrates Theorem 7 (or Corollary 9).

*Example 13. *Consider the equation
where , , , , where is a fixed constant and satisfies , so we have . Let , , when ; then there exists such that
So we obtain

(i) , . Since it follows that

(ii) , . Since it follows that

That is, (66) holds. By Corollary 9 we see that (77) is oscillatory.

Similarly, we can get the same conclusion when .

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgment

This project was supported by the NNSF of China (no. 11271379).

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