Abstract and Applied Analysis

Volume 2014 (2014), Article ID 351256, 7 pages

http://dx.doi.org/10.1155/2014/351256

Research Article

Oscillation Results for Second-Order Nonlinear Damped Dynamic Equations on Time Scales

Yang-Cong Qiu¹ and Qi-Ru Wang²

¹School of Humanities and Social Science, Shunde Polytechnic, Foshan, Guangdong 528333, China

²School of Mathematics and Computational Science, Sun Yat-Sen University, Guangzhou, Guangdong 510275, China

Received 13 January 2014; Accepted 16 February 2014; Published 15 April 2014

Academic Editor: Tongxing Li

Copyright © 2014 Yang-Cong Qiu and Qi-Ru Wang. 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.

This paper is concerned with second-order nonlinear damped dynamic equations on time scales of the following more general form . New oscillation results are given to handle some cases not covered by known criteria. An illustrative example is also presented.

1. Introduction

Let denote the set of real numbers and a time scale, that is, a nonempty closed subset of with the topology and ordering inherited from . The theory of time scales was introduced by Hilger in his Ph.D. thesis [1] in 1988, and for a comprehensive treatment of the subject, see [2]. Much recent attention has been concerned with the oscillation of dynamic equations on time scales; see, for example, [1–15]. In [9], Došlý and Hilger studied the second-order dynamic equation

The authors gave a necessary and sufficient condition for the oscillation of all solutions of (1) on time scales. In [7, 8], Del Medico and Kong used the Riccati transformation as

and obtained some sufficient conditions for oscillation of (1). In [14], Wang considered the nonlinear second-order damped differential equation

and established new oscillation criteria. In [13], Tiryaki and Zafer considered the second-order nonlinear differential equation with nonlinear damping

and gave interval oscillation criteria of (4). In [10], Huang and Wang considered the second-order nonlinear dynamic equation

The authors gave some new oscillation criteria of (5) and extended the results in [7, 8]. In [11], Qiu and Wang studied the second-order nonlinear dynamic equation

By employing the Riccati transformation as

where , , the authors established interval oscillation criteria for (6). And in [12], Qiu and Wang obtained some new Kamenev-type oscillation criteria for dynamic equations of the following more general form:

by using the transformation

In this paper, we consider second-order nonlinear damped dynamic equations of the form

on a time scale . We will employ functions of the form and a generalized Riccati transformation as (7) and (9) which was used in [14, 15] and derive oscillation criteria for (10) in Section 2. An example is presented to demonstrate the obtained results in the final section.

Definition 1. A solution of (10) 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 (10) is said to be oscillatory if all solutions of (10) are oscillatory.

2. Main Results

In this section, we establish some oscillation criteria for (10). Our work is based on the application of the Riccati transformation. Throughout this paper we will assume that and

 (C1);(C2);(C3), and there exist and such that and for all ;(C4)for , , , above, we always have ;(C5).

Preliminaries about time scale calculus can be found in [3–6] and are omitted here. For simplicity, we denote throughout this paper, where and , , are denoted similarly.

Now, we give the first theorem.

Theorem 2. Assume that (C1)–(C5) hold and that there exists a function such that . Also, suppose that is a solution of (10) satisfying for with . For , define

where , , and for . Then, satisfies

where

Proof. By (C3) we see that and are both positive or both negative or both zero. When , which implies that , it follows that

When , which implies that , it follows that

When , which implies that and , it follows that

Hence, we always have

so (12) holds. Then differentiating (11) and using (10), it follows that

so (13) holds. Theorem 2 is proved.

Remark 3. In Theorem 2, the condition ensures that the coefficient of in (13) is always negative. The condition is obvious and easy to be fulfilled. For example, when for all , we have ; by (C4) we see that

Let and . For any function : , denote by the partial derivatives of with respect to . For , denote by the space of functions which are integrable on any compact subset of . Define

These function classes will be used throughout this paper. Now, we are in a position to give the second theorem.

Theorem 4. Assume that (C1)–(C5) hold and that there exists a function such that . Also, suppose that there exist and such that and for any ,

where is defined as before, and

Then, (10) is oscillatory.

Proof. Assume that (10) is not oscillatory. Without loss of generality we may assume that there exists such that for . Let be defined by (11). Then by Theorem 2, (12) and (13) hold.

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

Multiplying (13), where is replaced by , by and integrating it with respect to from to with and , we obtain

where , , are defined as before.

Noting that , by the integration by parts formula we have

Since on , from (12) we see that, for ,

Since on , we see that . For , , from and (C4), we have

For , , and , from (27) we have

For , , and , from (27) we have

Therefore, for all , , we have

Then, from (25), (26), and (30) we obtain that, for and ,

Hence,

which contradicts (22) and completes the proof.

Remark 5. If we change the condition in the definition of to a stronger one , (27) in the proof of Theorem 4 will be changed to

Then the definition of can be simplified as

where

When , Theorem 4 can be simplified as Corollary 6.

Corollary 6. Assume that (C1)–(C5) hold and that there exists a function such that . Also, suppose that there exists such that, for any ,

Then, (10) is oscillatory.

When , (10) will be simplified as

Then Theorem 4 can be simplified as Corollary 7.

Corollary 7. Assume that (C1)–(C5) hold and that there exists a function such that . Also, suppose that there exist and such that, for any ,

where

Then, (37) is oscillatory.

Remark 8. When , , and (C3) is replaced by

 (C6), where is a fixed positive constant;(C7), and there exists such that for all .

Theorem 4 is reduced to [12, Theorem 4].

3. Example

In this section, we will give an example to demonstrate Corollary 7.

Example 1. Consider the equations

where , , , in (40), and in (41), so we have both , . Letting , we have

 (1), ,

That is, (38) holds. By Corollary 7 we see that (40) and (41) are oscillatory. Consider

 (2), ,

That is, (38) holds. By Corollary 7 we see that (40) and (41) are oscillatory.

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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