Discrete Dynamics in Nature and Society

Volume 2010, Article ID 907130, 15 pages

http://dx.doi.org/10.1155/2010/907130

## Oscillation Behavior of a Class of Second-Order Dynamic Equations with Damping on Time Scales

^{1}School of Science, University of Jinan, Jinan, Shandong 250022, China^{2}School of Control Science and Engineering, Shandong University, Jinan, Shandong 250061, China^{3}Department of Mathematics and Statistics, Missouri University of Science and Technology, Rolla, MO 65409-0020, USA

Received 21 May 2010; Accepted 16 September 2010

Academic Editor: Guang Zhang

Copyright © 2010 Weisong Chen et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

By using a Riccati transformation and inequality, we present some new oscillation theorems for the second-order nonlinear dynamic equation with damping on time scales. An example illustrating the importance of our results is also included.

#### 1. Introduction

The theory of time scales, which has recently received a lot of attraction, was introduced by Hilger in his Ph.D. Thesis in 1990 [1] in order to unify continuous and discrete analysis. The books on the subjects of time scale, that is, measure chain, by Bohner and Peterson [2, 3] summarize and organize much of time scale calculus.

We are concerned with second-order nonlinear dynamic equations with damping on a time scale here and are real-valued positive rd-continuous positive functions defined on , and is a quotient of odd positive integers. We assume that , , and define .

In recent years, there has been much research activity concerning the oscillation and nonoscillation of solutions of various dynamic equations [4–13]. However, there are few papers dealing with the oscillation of dynamic equations with damping term [14–17].

Saker [18] presented several oscillation criteria for the nonlinear second-order dynamic equation where and .

Hassan [19] studied the oscillation behavior of the second-order half-linear dynamic equation and obtained several new results.

Bohner et al. [20] established some oscillation criteria for the second-order nonlinear dynamic equation

Erbe et al. [16] considered the second-order nonlinear dynamic equations with damping and established some sufficient conditions for oscillation of (1.5).

Saker et al. [17] investigated the oscillation of second-order dynamic equations with damping term of the form and obtained some new oscillation criteria for (1.6).

Zafer [21] studied the second-order nonlinear dynamic equations on time scales and presented some oscillation and nonoscillation criteria. Obviously, (1.7) is the special situation of (1.1).

Note that in the special case when (1.1) becomes the second-order nonlinear damped differential equation and when , (1.1) becomes the second-order nonlinear damped difference equation where .

This paper is organized as follows: in Section 2, we give some preliminaries and lemmas. In Section 3, we will establish some oscillation criteria for (1.1). In Section 4, we give an example to illustrate the main results.

#### 2. Preliminaries

It will be convenient to make the following notations:

Lemma 2.1. *Assume that is -differentiable. Then from Keller's chain rule [2, Theorem ],
*

Lemma 2.2 (see [22]). *If , , then attains its maximum value at , and .*

Lemma 2.3. *Suppose that is an eventually positive solution of equation (1.1), and
**
Then there exists a , such that for ,
*

*Proof. *Pick such that on . From (1.1), we have
So, we get
Therefore,
We claim that . If not, there exist and a constant such that
hence
Integrating the above inequality from to , we obtain
which is a contradiction. Hence,
Obviously, by (2.7) and (2.11), we can see that
From (2.11) and (2.12), we have
It follows from (2.13) that
In view of (2.14) and , it is easy to get that

#### 3. Main Results

In this section, we will give some new oscillation criteria for (1.1).

Theorem 3.1. *Assume that (2.3) holds. Further, suppose that , and there exists a positive -differentiable function , such that for all sufficiently large ,
**
where , . Then every solution of (1.1) oscillates on .*

*Proof. *Let be a nonoscillatory solution of (1.1) on . Without loss of generality, we assume , for . Consider the generalized Riccati substitution
then , and by the product rule and then the quotient
Using (1.1) and (3.2), we find
If , from Lemma 2.1, we get
hence
In view of Lemma 2.3 and (3.2), we obtain
If , from Lemma 2.1, we get
hence
In view of Lemma 2.3, we have
Therefore,
From Lemma 2.3, we get
Integrating the above inequality from to , we have
which leads to a contradiction to (3.1). This completes the proof.

*Remark 3.2. *From Theorem 3.1, we can obtain different conditions for oscillation of all solutions of (1.1) with different choice of .

Theorem 3.3. *Assume that (2.3) holds. Further, suppose that , and there exist positive -differentiable functions and , such that for all sufficiently large ,
**
where , . Then every solution of (1.1) oscillates on .*

*Proof. *Let be a nonoscillatory solution of (1.1) on . Without loss of generality, we assume , for . Consider the generalized Riccati substitution as in (3.2). Then , and by the product rule and then the quotient
it follows from (1.1) and (3.2) that
If , from Lemma 2.1, we get
hence
In view of Lemma 2.3, we see that
If , from Lemma 2.1, we get
So,
In view of Lemma 2.3, we find
Therefore,
From Lemma 2.2, we obtain
Integrating the above inequality from to , we get
which leads to a contradiction to (3.14). This completes the proof.

*Remark 3.4. *From Theorem 3.3, we can obtain different conditions for oscillation of all solutions of (1.1) with different choice of and .

In the following, we will establish Kamenev-type oscillation criteria for (1.1).

Theorem 3.5. *Assume that (2.3) holds. Further, suppose that , and there exists a positive -differentiable function , such that for and all sufficiently large ,
**
where , . Then every solution of (1.1) is oscillatory on .*

*Proof. *We may assume that (1.1) has a nonoscillatory solution such that . Define by (3.2) as before, then we get (3.24). From (3.24), we have
Thus
Upon integration, we arrive at
Note that (see Saker [11]); then using (3.28), we have
Therefore,
Hence,
which contradicts (3.26). This completes the proof.

Theorem 3.6. *Assume that (2.3) holds. Further, suppose that , and there exists a positive -differentiable function , such that for and all sufficiently large ,
**
where , . Then every solution of (1.1) oscillates on .*

*Proof. *In view of Theorem 3.3, the proof is similar to that of [18, Theorem ].

In the following, we will establish the Philos-type oscillation criteria for (1.1).

Theorem 3.7. *Assume that (2.3) holds. Further, suppose that , there exists a positive -differentiable function and , where such that
** has a continuous and nonpositive -partial derivative with respect to the second variable and satisfies
**
and for sufficiently large ,
**
where
**
Then every solution of (1.1) oscillates on .*

*Proof. *Let be a nonoscillatory solution of (1.1) on . Without loss of generality, we assume , for . Define by (3.2) as before, then we have (3.11). From (3.11), we have
Thus,
Integrating the right side by parts, we have
and then by using (3.34) and (3.35), we arrive at

Define
By employing the inequality
we obtain
Therefore,
which contradicts (3.36). The proof is complete.

Theorem 3.8. *Assume that (2.3) holds. Further, suppose that , there exists a positive -differentiable function and , where such that (3.28) holds, and has a continuous and nonpositive -partial derivative with respect to the second variable and satisfies
**
If for sufficiently large **
where
**
then every solution of (1.1) oscillates on . *

*Proof. *In view of Theorem 3.3, the proof is similar to [16, Theorem ].

Theorem 3.9. *Assume that (2.3) holds. Further, suppose that , and for all sufficiently large ,
**
Then every solution of (1.1) oscillates on .*

*Proof. *Let be a nonoscillatory solution of (1.1) on . Without loss of generality, we assume , for . From (1.1) and Lemma 2.3, we get for ,
Letting , we obtain
In view of Lemma 2.3, we obtain
Thus
which is a contradiction. This completes the proof.

#### 4. Example

In this section, we will give an example to illustrate our results.

*Example 4.1. *Consider the second-order damped dynamic equation on time scales
where
Obviously, .

It is easy to see that (2.3) holds. For , one has and for , Hence, by Theorem 3.1, every solution of (4.1) is oscillatory.

*Remark 4.2. *It is easy to see that the results in [16–21] cannot be applied in (4.1), and to the best of our knowledge nothing is known regarding the oscillatory behavior of (1.1), so our results are new.

#### Acknowledgments

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), also supported by University of Jinan Research Funds for Doctors (XBS0843).

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