Discrete Dynamics in Nature and Society

Discrete Dynamics in Nature and Society / 2015 / Article

Research Article | Open Access

Volume 2015 |Article ID 231261 | https://doi.org/10.1155/2015/231261

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
Received04 Feb 2015
Accepted31 Mar 2015
Published14 Apr 2015

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 [36] 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 [114] 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 havewhere

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 , andThen .

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 , definewhere and . Then, satisfieswhere

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 , whereThen, (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 , whereThen, (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 ,whereThen, (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).

References

  1. S. Hilger, Ein Maßkettenkalkäul mit Anwendung auf Zentrumsmannigfaltigkeiten [Ph.D. thesis], Universität Wäurzburg, 1988.
  2. S. Hilger, “Analysis on measure chains—a unified approach to continuous and discrete calculus,” Results in Mathematics, vol. 18, no. 1-2, pp. 18–56, 1990. View at: Publisher Site | Google Scholar | MathSciNet
  3. R. P. Agarwal and M. Bohner, “Basic calculus on time scales and some of its applications,” Results in Mathematics. Resultate der Mathematik, vol. 35, no. 1-2, pp. 3–22, 1999. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  4. R. Agarwal, M. Bohner, D. O'Regan, and A. Peterson, “Dynamic equations on time scales: a survey,” Journal of Computational and Applied Mathematics, vol. 141, no. 1-2, pp. 1–26, 2002. View at: Publisher Site | Google Scholar | MathSciNet
  5. M. Bohner and A. Peterson, Dynamic Equations on Time Scales: An Introduction with Applications, Birkhäuser, Boston, Mass, USA, 2001. View at: Publisher Site | MathSciNet
  6. M. Bohner and A. Peterson, Eds., Advances in Dynamic Equations on Time Scales, BirkhÄauser, Boston, Mass, USA, 2003. View at: Publisher Site | MathSciNet
  7. L. Erbe, T. S. Hassan, and A. Peterson, “Oscillation of third order nonlinear functional dynamic equations on time scales,” Differential Equations and Dynamical Systems, vol. 18, no. 1-2, pp. 199–227, 2010. View at: Publisher Site | Google Scholar | MathSciNet
  8. T. S. Hassan, “Oscillation of third order nonlinear delay dynamic equations on time scales,” Mathematical and Computer Modelling, vol. 49, no. 7-8, pp. 1573–1586, 2009. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  9. Y.-C. Qiu and Q.-R. Wang, “Kamenev-type oscillation criteria of second-order nonlinear dynamic equations on time scales,” Discrete Dynamics in Nature and Society, vol. 2013, Article ID 315158, 12 pages, 2013. View at: Publisher Site | Google Scholar | MathSciNet
  10. Y.-C. Qiu and Q.-R. Wang, “Oscillation criteria of second-order dynamic equations with damping on time scales,” Abstract and Applied Analysis, vol. 2014, Article ID 964239, 11 pages, 2014. View at: Publisher Site | Google Scholar | MathSciNet
  11. S. H. Saker, R. P. Agarwal, and D. O'Regan, “Oscillation of second-order damped dynamic equations on time scales,” Journal of Mathematical Analysis and Applications, vol. 330, no. 2, pp. 1317–1337, 2007. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  12. M. T. Şenel, “Kamenev-type oscillation criteria for the second-order nonlinear dynamic equations with damping on time scales,” Abstract and Applied Analysis, vol. 2012, Article ID 253107, 18 pages, 2012. View at: Publisher Site | Google Scholar | MathSciNet
  13. M. T. Şenel, “Behavior of solutions of a third-order dynamic equation on time scales,” Journal of Inequalities and Applications, vol. 2013, article 47, 2013. View at: Publisher Site | Google Scholar
  14. Q. R. Wang, “Oscillation criteria for even order nonlinear damped differential equations,” Acta Mathematica Hungarica, vol. 95, no. 3, pp. 169–178, 2002. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet

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.


More related articles

551 Views | 357 Downloads | 0 Citations
 PDF Download Citation Citation
 Download other formatsMore
 Order printed copiesOrder