Research Article | Open Access

Yang-Cong Qiu, "Oscillation Criteria of Third-Order Nonlinear Damped Dynamic Equations on Time Scales", *Discrete Dynamics in Nature and Society*, vol. 2015, Article ID 231261, 8 pages, 2015. https://doi.org/10.1155/2015/231261

# Oscillation Criteria of Third-Order Nonlinear Damped Dynamic Equations on Time Scales

**Academic Editor:**Allan C. Peterson

#### Abstract

We establish oscillation criteria of third-order nonlinear damped dynamic equations on time scales of the form by employing functions in some function classes and the generalized Riccati transformation. Two examples are given to show the significance of the conclusions.

#### 1. Introduction

In this paper, we study third-order nonlinear damped dynamic equationon a time scale satisfying and .

Throughout this paper we shall assume that(C1) such that(C2) is a quotient of odd positive integers;(C3), and for any ,(C4) and there exists a function such that for with a same sign,(C5)when , it always satisfies

In 1988, Hilger introduced the theory of time scales in his Ph.D. thesis [1] in order to unify continuous and discrete analysis; see also [2]. Preliminaries about time scale calculus can be found in [3–6] and omitted here.

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

There has been much research achievement about the oscillation of dynamic equations on time scales in the last few years; see the papers [1–14] and the references therein.

Wang [14] discussed the even order nonlinear damped differential equationand obtained some oscillation criteria for (6). Hassan [8] gave some oscillation criteria for the third order nonlinear delay dynamic equationAfterwards, Erbe et al. [7] established some new oscillation criteria for (7). Saker et al. [11] studied the second-order damped dynamic equationand they gave some oscillation criteria.

Qiu and Wang [9] considered the second-order nonlinear dynamic equationBy using a generalized Riccati transformationthe authors established some Kamenev-type oscillation criteria. Şenel [12] had tried to establish Kamenev-type oscillation criteria for the second-order nonlinear dynamic equation of the formHowever, it seemed that the obtained theorems and corollaries are incorrect. Qiu and Wang [10] corrected some mistakes in [12] and established correct oscillation criteria for (11) by employing functions in some function classes and the generalized Riccati transformation. Şenel [13] considered the third-order nonlinear dynamic equationand established some sufficient conditions which guarantee that every solution of (12) oscillates or converges to zero on an arbitrary time scale . In this paper, we shall establish new oscillation criteria of (1), which is more general than (12), and give two examples to show the significance of the results.

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

#### 2. Preliminary Results

To establish oscillation criteria of (1), we give four lemmas in this section.

Lemma 2. *Assume that (C1)–(C5) hold and there exists a sufficiently large such that is a solution of (1) satisfying for . Then there exists such that for , we have*

*Proof. *Let such that is a solution of (1) satisfying for ; then we also have . By (1) and (C4), it follows that, for ,Hence, is strictly decreasing on . We claim thatAssume not, then there exists such that for . So there exists a constant and such that for , which means thatIntegrating (16) from to , we obtainLetting , by (C1) we have . Then there exists such that for , which implies thatIntegrating (18) from to , we obtainLetting , by (C1) we have , which contradicts . So (15) holds, which implies thatTherefore, is strictly increasing on . It follows that is either eventually positive or eventually negative. Then, there exists such that, for , we have (13) holding. Lemma 2 is proved.

Lemma 3. *For , assume that (C1)–(C5) hold and is a solution of (1) satisfying , for with . Then, we have**where*

*Proof. *Since is a solution of (1) satisfying for with , by Lemma 2 we haveBy , it follows that, for ,Integrating (24) from to , by (23) we obtainAs is strictly decreasing on , for we haveHence when , we havewhich implies thatLemma 3 is proved.

Lemma 4. *For , assume that (C1)–(C4) hold and is a solution of (1) satisfying , for with , and**Then .*

*Proof. *Since is a solution of (1) satisfying for with , by Lemma 2 we haveBy , there exists such that . Assume , by (24) and , we obtainLetting , , we have , andIntegrating (32) from to , we obtainBy (29), there exists a sufficiently large such that , , which contradicts . So . Lemma 4 is proved.

Lemma 5. *Assume that (C1)–(C5) hold and is a solution of (1) satisfying , for with . For , define**where and . Then, 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 (34). Then, differentiating (34) and using (1), it follows thatUsing the fact thatwe obtainWhen , using the Pötzsche chain rule, we haveand it follows thatBy Lemmas 2 and 3, for , we obtainSo (39) becomesWhen , we haveand it follows thatBy Lemmas 2 and 3, for , we obtainwhich implies thatSo (39) becomesBy (43) and (48), (35) holds. Lemma 5 is proved.

#### 3. Main Results

In this section, we establish oscillation criteria of (1) by 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 give our first theorem.

Theorem 6. *Assume that (C1)–(C5) hold and that there exist and such that, for any , **where**Then, (1) is oscillatory or exists.*

*Proof. *Assume that (1) is not oscillatory. Without loss of generality we may assume there exists such that for . By Lemma 2, for , either or holds. Assume , . Let be defined by (34). Then by Lemma 5, (35) holds.

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

Multiplying (35), where is replaced by , by and integrating it with respect to from to with , we obtainNoting that , by the integration by parts formula we haveWhen , we haveWhen , we have Using the inequalitylet , andthen we haveTherefore, for all , by (53) we havewhich implies thatHence,which contradicts (50). So , , and it is clearly that exists. The proof is completed.

When , if (29) holds, we have the following corollary on the basis of Lemma 4 and Theorem 6.

Corollary 7. *When , assume that (C1)–(C4) and (29) hold. If there exist and such that, for any , **where**Then, (1) is oscillatory or .*

*Remark 8. *In Corollary 7, letting , we can simplify (62) asWhen , (34) is simplified asNow we have the following theorem.

Theorem 9. *Assume that (C1)–(C5) hold and that there exists such that, for any ,**where**Then, (1) is oscillatory or exists.*

*Proof. *Assume that (1) is not oscillatory. Without loss of generality we may assume that there exists such that for . By Lemma 2, for , either or holds. Assume , . Let be defined by (65). Then by Lemma 5, we havewhere is simplified asWhen , we haveWhen , we haveUsing the inequalitylet , andthen we haveTherefore, for all , we always havewhich implies thatLetting be replaced by , and integrating (76) with respect to from to , we obtainwhich is a contradiction of (66). So , , and, as before, exists. The proof is completed.

When , if (29) holds, from Lemma 4 and Theorem 9, we have the following result.

Corollary 10. *When , assume that (C1)–(C4) and (29) hold. If there exists such that, for any ,**then (1) is oscillatory or .*

*Remark 11. *Actually, letting , by (29) we have (78). Hence, it is easy to satisfy the conditions in Corollary 10.

*Remark 12. *When , for with a same sign, if we assume thatorinstead of (4) in (C4), by (31) in Lemma 4, the conclusions above are also applicable. Furthermore, if , , andby (79) it is clear that the conclusions above include the results in Şenel [13].

*Remark 13. *When , if , the conclusions above are not applicable. Similarly, the assumption that in Şenel [12] and Qiu and Wang [10] should be changed to (C3) in this paper.

#### 4. Examples

In this section, the application of our oscillation criteria will be shown in two examples. The first example is given to demonstrate Theorem 6.

*Example 1. *Consider the equationwhere , is a constant and , , , , , and , by (C4) we get . Letting , , we haveSince , when ,so we havewhere . Hence,When ,where . Hence,That is, (50) holds. By Theorem 6 we see that (82) is oscillatory or exists.

The second example illustrates Corollary 10.

*Example 2. *Consider the equation (see Şenel [13, Example 3.4])where , , , and . Letting , by (79) we have . Then, letting , we obtainTherefore,

That is, (78) holds. By Corollary 10 we see that (89) is oscillatory or . This result is consistent with the conclusion in Şenel [13, Example 3.4].

#### Conflict of Interests

The author declares 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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#### Copyright

Copyright © 2015 Yang-Cong Qiu. 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.