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Discrete Dynamics in Nature and Society
Volume 2010, Article ID 907130, 15 pages
http://dx.doi.org/10.1155/2010/907130
Research Article

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

1School of Science, University of Jinan, Jinan, Shandong 250022, China
2School of Control Science and Engineering, Shandong University, Jinan, Shandong 250061, China
3Department 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??????(??)???????+??(??)??(??)??+??(??)??(????(??))=0(1.1) 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 ??(??)/????=??>0, ???0, sup??=8, and define [??0,8)??:=[??0,8)n??.

In recent years, there has been much research activity concerning the oscillation and nonoscillation of solutions of various dynamic equations [413]. However, there are few papers dealing with the oscillation of dynamic equations with damping term [1417].

Saker [18] presented several oscillation criteria for the nonlinear second-order dynamic equation???(??)????(??)?[],+??(??)??(??(??(??)))=0,?????,??(1.2) where ??,????? and ??<??.

Hassan [19] studied the oscillation behavior of the second-order half-linear dynamic equation??????(??)??(??)????+??(??)????(??)=0,(1.3) and obtained several new results.

Bohner et al. [20] established some oscillation criteria for the second-order nonlinear dynamic equation????(??)+??(??)?????(??)+??(??)(??°????)=0.(1.4)

Erbe et al. [16] considered the second-order nonlinear dynamic equations with damping??????(??)??(??)???????+??(??)????(??)??+??(??)??(??(??(??)))=0,?????,(1.5) 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???(??)????(??)?+??(??)?????(??)+??(??)??(??(??(??)))=0,?????,(1.6) and obtained some new oscillation criteria for (1.6).

Zafer [21] studied the second-order nonlinear dynamic equations on time scales????+??(??)???+??(??)????=0,?????,(1.7) and presented some oscillation and nonoscillation criteria. Obviously, (1.7) is the special situation of (1.1).

Note that in the special case when ??=R, (1.1) becomes the second-order nonlinear damped differential equation ??????(??)???????+??(??)??(??)??+??(??)??(????(??))=0,???R,(1.8) and when ??=Z, (1.1) becomes the second-order nonlinear damped difference equation ?((???(??))??)+??(??)(???(??))??+??(??)??(????(??))=0,???Z,(1.9) where ???(??)=??(??+1)-??(??).

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:??+(??):=max{0,??(??)},??-???(??):=max{0,-??(??)},??(??):=??(??),0<??=1,??(??),??>1,??(??):=??-??*??-??*+??(??),??(??):=????/(1-????)???,??*?.(2.1)

Lemma 2.1. Assume that ?? is ?-differentiable. Then from Keller's chain rule [2, Theorem 1.90], ((??(??))??)??=??10[h????](??)+(1-h)??(??)??-1???(??)dh.(2.2)

Lemma 2.2 (see [22]). If ??(??)=-????(??+1)/??+????, ??>0, then ??(??) attains its maximum value at ??0=(????/(??+1)??)??, and ??(??0)=(????/(??+1)??+1)(????+1/????).

Lemma 2.3. Suppose that ?? is an eventually positive solution of equation (1.1), 1-??(??)??(??)>0, and ?8??0?????1/??(??)=8.(2.3) Then there exists a ??*>??0, such that for ??>??*, ??????(??)????<0,???(??)>0,?????(??)<0,??(??)>??-??*????(??),??(??)????(??)>??(??).(2.4)

Proof. Pick ??1?[??0,8)?? such that ????(??)>0 on [??1,8)??. From (1.1), we have ??????(??)???????+??(??)??(??)?????<0,???1?,8??.(2.5) So, we get 1??1-??(??)??(??)???(???)????+??(??)???1-??(??)??(??)?(???)?????<0,???1?,8??.(2.6) Therefore, ??????(??)??(??)???????<0,???1?,8??.(2.7) We claim that ???(??)>0. If not, there exist ??1=??0 and a constant ??<0 such that ?????(??)??(??)??=??<0,(2.8) hence ??????(??)=???(??)1/??.(2.9) Integrating the above inequality from ??1 to ??, we obtain ?????(??)=??1?+??1/???????11??1/??(??)????-8,???8,(2.10) which is a contradiction. Hence, ???(??)>0.(2.11) Obviously, by (2.7) and (2.11), we can see that ??????(??)????<0.(2.12) From (2.11) and (2.12), we have ????(??)<0.(2.13) It follows from (2.13) that ?????(??)>??(??)-??*?=?????*???(??)???=???(???)??-??*?.(2.14) In view of (2.14) and ????(??)=??(??)+??(??)???(??), it is easy to get that ??(??)????(??)>??(??).(2.15)

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 1-??(??)??(??)>0, and there exists a positive ?-differentiable function ??, such that for all sufficiently large ??*, limsup???8?????*?????(??)??????(??)-??(??+1)??+1????+1(??)????(???)???=8,(3.1) where ??(??)=??????(??)??(??)/(??(??))(??+1)/??, ??(??)=(???(??)-??(??)????(??)????(??))/??(??). Then every solution ?? of (1.1) oscillates on [??0,8)??.

Proof. Let ??(??) be a nonoscillatory solution of (1.1) on [??0,8)??. Without loss of generality, we assume ??(??)>0, for ??=??*=??0. Consider the generalized Riccati substitution ?????(??)=??(??)??(??)??????(??),??=??*=??0,(3.2) then ??(??)>0, and by the product rule and then the quotient???(??)=??????(??)??(??)??????(??)+????????(??)??(??)???????(??)?=??????(??)??(??)??????(??)+??????(??)????(??)????(????(??))??-???????(??)??(??)??((??(??))??)?(??(??))??(????(??))??.(3.3) Using (1.1) and (3.2), we find ???(????)=??(??)?(??)??(??)-??(??)????(?????)?(??)???(??)?????(??)?????(??)??-????(??)????(??)-????(?????)??(??)??((??(??))??)?(??(??))??(????(??))??.(3.4) If 0<??=1, from Lemma 2.1, we get ((??(??))??)?=??(????(??))??-1???(??),(3.5) hence ?????(??)=??(??)?(??)??(??)-??(??)???????(??)?(??)???(??)?????(??)?????(??)??-????(??)????(??)-?????????(??)?(??)???(??)??+1??(??)????.(??)(3.6) In view of Lemma 2.3 and (3.2), we obtain ???(??)=-????(??)???????(??)+?(??)????(??)-??(??)??(??)????(??)???(??)??(??)-??????(??)??(??)(??(??))(??+1)/??(??(??))(??+1)/??.(3.7) If ??>1, from Lemma 2.1, we get ((??(??))??)?=??(??(??))??-1???(??),(3.8) hence ?????(??)=??(??)?(??)??(??)-??(??)???????(??)?(??)???(??)?????(??)?????(??)??-????(??)????(??)-?????????(??)?(??)???(??)??+1???(??)?????(??)??.(3.9) In view of Lemma 2.3, we have ???(??)=-????(??)???????(??)+?(??)????(??)-??(??)??(??)????(??)???(??)??(??)-??????(??)????(??)(??(??))(??+1)/??(??(??))(??+1)/??.(3.10) Therefore, ???(??)=-????(??)???????(??)+?(??)????(??)-??(??)??(??)????(??)???(??)??(??)-??????(??)??(??)(??(??))(??+1)/??(??(??))(??+1)/??.(3.11) From Lemma 2.3, we get ???(??)=-????(??)??????(??)+??(??+1)??+1????+1(??)????(.??)(3.12) Integrating the above inequality from ??* to ??, we have ?????*?????(??)??????(??)-??(??+1)??+1????+1(??)????(??????)???=??*?(3.13) 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 1-??(??)??(??)>0, and there exist positive ?-differentiable functions ?? and ??, such that for all sufficiently large ??*, limsup???8?????*???????(??)??(??)??(??)-??(??+1)??+1????+1(??)????(???)???=8,(3.14) where ??(??)=????(??)??(??)/(????(??))(??+1)/??, ??(??)=??(??)(???(??)-??(??)??(??))/????(??)+???(??). Then every solution ?? of (1.1) oscillates on [??0,8)??.

Proof. Let ??(??) be a nonoscillatory solution of (1.1) on [??0,8)??. Without loss of generality, we assume ??(??)>0, for ??=??*=??0. Consider the generalized Riccati substitution as in (3.2). Then ??(??)>0, and by the product rule and then the quotient ???(??)=???????(??)??(??)???????(??)??????+??(??)??(??)???????(??)?=???????(??)??(??)???????(??)????+??(??)????(??)????(????(??))?????-??(??)??(??)??((??(??))??)?(??(??))??(????(??))??,(3.15)it follows from (1.1) and (3.2) that ???(??)=????(????)?(??)???????(??)-??(??)??(??)???(??)?????(??)??????(??)??????(??)?????-????(??)??(??)-??(??)??(??)??((??(??))??)?(??(??))??(????(??))??.(3.16) If 0<??=1, from Lemma 2.1, we get ((??(??))??)?=??(????(??))??-1???(??),(3.17) hence ???(??)=??????(??)?(??)????(?????)-??(??)??(??)???(??)????(???)??????(??)??????(??)?????-????(??)??(??)-????(??)???(??)?????(??)??+1?????(??)???(??)??????(??)??????(??)??+1.(3.18) In view of Lemma 2.3, we see that ??????(??)=-????(??)??(??)+?(??)????(??)-??(??)??(??)???????(??)??(??)-????(??)(????(??))(??+1)/??(????(??))(??+1)/??.(3.19) If ??>1, from Lemma 2.1, we get ((??(??))??)?=??(??(??))??-1???(??).(3.20) So, ???(??)=??????(??)?(??)????(?????)-??(??)??(??)???(??)????(???)??????(??)??????(??)?????-????(??)??(??)-????(??)???(??)?????(??)??+1????(??)?????(??)?(??)??????(??)??+1.(3.21) In view of Lemma 2.3, we find ??????(??)=-????(??)??(??)+?(??)????(??)-??(??)??(??)???????(??)??(??)-????(??)(????(??))(??+1)/??(????(??))(??+1)/??.(3.22) Therefore, ??????(??)=-????(??)??(??)+?(??)????(??)-??(??)??(??)???????(??)??(??)-????(??)(????(??))(??+1)/??(????(??))(??+1)/??.(3.23) From Lemma 2.2, we obtain ???(??)=-????(??)??????(??)+??(??+1)??+1????+1(??)????(.??)(3.24) Integrating the above inequality from ??* to ??, we get ?????*???????(??)??(??)??(??)-??(??+1)??+1????+1(??)????(??????)???=??*??????*?(3.25) 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 1-??(??)??(??)>0, and there exists a positive ?-differentiable function ??, such that for ??>1 and all sufficiently large ??*, limsup???81?????????*(??-??)???????(??)??????(??)-??(??+1)??+1????+1(??)????(???)???=8,(3.26) where ??(??)=??????(??)??(??)/(??(??))(??+1)/??, ??(??)=(???(??)-??(??)????(??)????(??))/??(??). Then every solution ?? of (1.1) is oscillatory on [??0,8)T.

Proof. We may assume that (1.1) has a nonoscillatory solution ??(??) such that ??(??)>0. Define ?? by (3.2) as before, then we get (3.24). From (3.24), we have ????(??)??????(??)-??(??+1)??+1????+1(??)????(??)=-???(??).(3.27) Thus ?????*(??-??)???????(??)??????(??)-??(??+1)??+1????+1(??)????(????)???=-????*(??-??)?????(??)???.(3.28) Upon integration, we arrive at -?????*(??-??)?????(??)???=(??-??)????(??)|????*-?????*((??-??)??)?????(??(??))???.(3.29) Note that ((??-??)??)???=-??(??-??(??))??-1=0,??=??(??),??=1 (see Saker [11]); then using (3.28), we have ?????*(??-??)???????(??)??????(??)-??(??+1)??+1????+1(??)????(????)???=??-??*????????*?.(3.30) Therefore, 1?????????*(??-??)???????(??)????(????)-??(??+1)??+1????+1(??)??????(??)???=??-??*????????????*?.(3.31) Hence, limsup???81?????????*(??-??)???????(??)??????(??)-??(??+1)??+1????+1(??)????(???)???=8,(3.32) which contradicts (3.26). This completes the proof.

Theorem 3.6. Assume that (2.3) holds. Further, suppose that 1-??(??)??(??)>0, and there exists a positive ?-differentiable function ??, such that for ??>1 and all sufficiently large ??*, limsup???81?????????*(??-??)?????????(??)??(??)-??(??+1)??+1????+1(??)????(???)???=8,(3.33) where ??(??)=????(??)/(????(??))(??+1)/??, ??(??)=(???(??)-??(??)??(??))/????(??). Then every solution ?? of (1.1) oscillates on [??0,8)??.

Proof. In view of Theorem 3.3, the proof is similar to that of [18, Theorem 3.2].

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 1-??(??)??(??)>0, there exists a positive ?-differentiable function ??, and ??,h???rd(??,R), where ??={(??,??):??=??=??0} such that ??(??,??)=0,??=??0,??(??,??)>0,??>??=??0.(3.34)?? has a continuous and nonpositive ?-partial derivative ?????(??,??) with respect to the second variable and satisfies ?????(?????(??),??)+??(??(??),??(??))?(??)-??(??)????(??)????(??)???(??)=-h(??,??)(??(??)??(??(??),??(??)))??/(??+1),(3.35) and for sufficiently large ??*, limsup???81?????(??),??*??????(??)*??(??,??)???=8,(3.36) where ??(??,??)=????(??(??),??(??))?????h(??)??(??)--?(??,??)??+1(??+1)??+1(????(??)??(??))??.(3.37) Then every solution ?? of (1.1) oscillates on [??0,8)??.

Proof. Let ??(??) be a nonoscillatory solution of (1.1) on [??0,8)??. Without loss of generality, we assume ??(??)>0, for ??=??*=??0. Define ?? by (3.2) as before, then we have (3.11). From (3.11), we have ????(??)????(??)=-??????(??)+?(??)????(??)-??(??)??(??)????(??)???(??)??(??)-??????(??)??(??)(??(??))(??+1)/??(??(??))(??+1)/??.(3.38) Thus, ???????(??)*??(??(??),??(??))??(??)?????(??)???=-????(??)*??(??(??),??(??))???+?(??)???????(??)*?????(??(??),??(??))?(??)????(??)-??(??)??(??)????(??)??(?-???)??(??)???????(??)*??(??(??),??(??))??????(??)??(??)(??(??))(??+1)/??(??(??))(??+1)/?????.(3.39) Integrating the right side by parts, we have -?????(??)*??(??(??),??(??))????(??)???=????(??),??*??????*?+?????(??)*?????(??(??),??)??(??)???,(3.40) and then by using (3.34) and (3.35), we arrive at ???????(??)*??(??(??),??(??))??(??)????=?(??)???????(??)*?h-(??,??)??(??)(??(??(??),??(??)))??/(??+1)??(??)-??(??(??),??(??))??????(??)??(??)(??(??))(??+1)/??(??(??))(??+1)/???????+????(??),??*??????*?.(3.41)
Define ??=??+1??,????=??(??(??),??(??))??????(??)??(??)????(??),????-1=h-(??,??)??(??????(??)??(??))1/??.(3.42) By employing the inequality ????????-1-????=(??-1)????,??=1,(3.43) we obtain h-(??,??)??(??)(??(??(??),??(??)))??/(??+1)??(??)-??(??(??),??(??))??????(??)??(??)(??(??))(??+1)/??(??(??))(??+1)/??=?h-?(??,??)??+1(??+1)??+1(????(??)??(??))??.(3.44) Therefore, 1??(??(??),??*)?????(??)*?????(??,??)???=??*?,(3.45) which contradicts (3.36). The proof is complete.

Theorem 3.8. Assume that (2.3) holds. Further, suppose that 1-??(??)??(??)>0, there exists a positive ?-differentiable function ??, and ??,h???rd(??,R), where ??={(??,??):??=??=??0} such that (3.28) holds, and ?? has a continuous and nonpositive ?-partial derivative ?????(??,??) with respect to the second variable and satisfies ????????(??,??)+??(??,??)?(??)????(??)-??(??)??(??)???(??)=-h(??,??)????(??)(??(??,??))??/(??+1).(3.46) If for sufficiently large ??*limsup???81?????,??*??????*??(??,??)???=8,(3.47) where ?h??(??,??)=????(??,??)??(??)??(??)--?(??,??)??+1(??+1)??+1(????(??))??,(3.48) then every solution ?? of (1.1) oscillates on [??0,8)??.

Proof. In view of Theorem 3.3, the proof is similar to [16, Theorem 2.2].

Theorem 3.9. Assume that (2.3) holds. Further, suppose that 1-??(??)??(??)>0, and for all sufficiently large ??*, limsup???8???-??*????8??1??(??)???>??.(3.49) Then every solution ?? of (1.1) oscillates on [??0,8)??.

Proof. Let ??(??) be a nonoscillatory solution of (1.1) on [??0,8)??. Without loss of generality, we assume ??(??)>0, for ??=??*=??0. From (1.1) and Lemma 2.3, we get for ??=??=??*, ?????????(??)?????(??)???<????????(??)?????????(??)???<??(??)??-?????(??)??<?????(??)??.(3.50) Letting ???8, we obtain ???8????(??)???????(??)???<??(??)??.(3.51) In view of Lemma 2.3, we obtain ???8???????(??)???<?(??)???(??)??<?1??-??*???.(3.52) Thus limsup???8???-??*????8??1??(??)???=??,(3.53) 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 ??????(??)????+1???????(??)??+??(????(??))??=0,(4.1) where 1??(??)<??,??(??)=??,??(??)=??,??(??)=1,??=1,??(??)=????.(4.2) Obviously, ??(??)/????=1=??=1.

It is easy to see that (2.3) holds. For 0<??=1, one has limsup???8?????*?????-??(??+1)??+1((1/??)????(??))??+1(??????(??))??????=limsup???8?????*?1??-(??+1)??+11????+1????2?(??)???=8,(4.3) and for ??>1, limsup???8?????*?????-??(??+1)??+1((1/??)????(??))??+1(????(??))??????=limsup???8?????*?1??-(??+1)??+11????+1?????(??)???=8.(4.4) Hence, by Theorem 3.1, every solution ?? of (4.1) is oscillatory.

Remark 4.2. It is easy to see that the results in [1621] 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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