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Science China Mathematics

, Volume 58, Issue 7, pp 1445–1452 | Cite as

Kamenev-type criteria for nonlinear damped dynamic equations

  • Martin Bohner
  • TongXing LiEmail author
Articles

Abstract

We establish a new Kamenev-type theorem for a class of second-order nonlinear damped delay dynamic equations on a time scale by using the generalized Riccati transformation technique. The criterion obtained improves related contributions to the subject. An example is provided to illustrate assumptions in our theorem are less restrictive.

Keywords

oscillation second-order dynamic equation nonlinear damped delay equation time scale 

MSC(2010)

34K11 34N05 39A10 

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References

  1. 1.
    Agarwal R P, Bohner M, Li T, et al. Hille and Nehari type criteria for third-order delay dynamic equations. J Difference Equ Appl, 2013, 19: 1563–1579zbMATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    Agarwal R P, Bohner M, Saker S H. Oscillation of second order delay dynamic equations. Can Appl Math Q, 2005, 13: 1–17zbMATHMathSciNetGoogle Scholar
  3. 3.
    Agarwal R P, Bohner M, Tang S, et al. Oscillation and asymptotic behavior of third-order nonlinear retarded dynamic equations. Appl Math Comput, 2012, 219: 3600–3609MathSciNetCrossRefGoogle Scholar
  4. 4.
    Agarwal R P, O’Regan D, Saker S H. Oscillation criteria for second-order nonlinear neutral delay dynamic equations. J Math Anal Appl, 2004, 300: 203–217zbMATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    Akın-Bohner E, Bohner M, Saker S H. Oscillation criteria for a certain class of second order Emden-Fowler dynamic equations. Electron Trans Numer Anal, 2007, 27: 1–12zbMATHMathSciNetGoogle Scholar
  6. 6.
    Bohner M, Erbe L, Peterson A. Oscillation for nonlinear second order dynamic equations on a time scale. J Math Anal Appl, 2005, 301: 491–507zbMATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    Bohner M, Peterson A. Dynamic Equations on Time Scales: An Introduction with Applications. Boston: Birkhäuser, 2001CrossRefGoogle Scholar
  8. 8.
    Erbe L, Hassan T S, Peterson A. Oscillation criteria for nonlinear damped dynamic equations on time scales. Appl Math Comput, 2008, 203: 343–357zbMATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    Erbe L, Peterson A, Saker S H. Oscillation criteria for second-order nonlinear delay dynamic equations. J Math Anal Appl, 2007, 333: 505–522zbMATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    Grace S R, Bohner M, Agarwal R P. On the oscillation of second-order half-linear dynamic equations. J Difference Equ Appl, 2009, 15: 451–460zbMATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    Hassan T S. Kamenev-type oscillation criteria for second order nonlinear dynamic equations on time scales. Appl Math Comput, 2011, 217: 5285–5297zbMATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    Hassan T S, Erbe L, Peterson A. Oscillation of second order superlinear dynamic equations with damping on time scales. Comput Math Appl, 2010, 59: 550–558zbMATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    Hassan T S, Erbe L, Peterson A. Oscillation criteria for second order sublinear dynamic equations with damping term. J Difference Equ Appl, 2011, 17: 505–523zbMATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    Karpuz B. Li type oscillation theorem for delay dynamic equations. Math Methods Appl Sci, 2013, 36: 993–1002zbMATHMathSciNetCrossRefGoogle Scholar
  15. 15.
    Karpuz B, Öcalan Ö. Necessary and sufficient conditions on asymptotic behaviour of solutions of forced neutral delay dynamic equations. Nonlinear Anal TMA, 2009, 71: 3063–3071zbMATHCrossRefGoogle Scholar
  16. 16.
    Li T, Saker S H. A note on oscillation criteria for second-order neutral dynamic equations on isolated time scales. Commun Nonlinear Sci Numer Simul, 2014, 19: 4185–4188MathSciNetCrossRefGoogle Scholar
  17. 17.
    Saker S H. Oscillation Theory of Dynamic Equations on Time Scales, Second and Third Orders. Berlin: Lambert Academic Publishing, 2010Google Scholar
  18. 18.
    Saker S H, Agarwal R P, O’Regan D. Oscillation of second-order damped dynamic equations on time scales. J Math Anal Appl, 2007, 330: 1317–1337zbMATHMathSciNetCrossRefGoogle Scholar
  19. 19.
    Şahiner Y. Oscillation of second-order delay differential equations on time scales. Nonlinear Anal TMA, 2005, 63: 1073–1080CrossRefGoogle Scholar
  20. 20.
    Şenel M T. Kamenev-type oscillation criteria for the second-order nonlinear dynamic equations with damping on time scales. Abstr Appl Anal, 2012, 2012: 1–18Google Scholar
  21. 21.
    Zhang C, Li T, Agarwal R P, et al. Oscillation results for fourth-order nonlinear dynamic equations. Appl Math Lett, 2012, 25: 2058–2065zbMATHMathSciNetCrossRefGoogle Scholar
  22. 22.
    Zhang Q. Oscillation of second-order half-linear delay dynamic equations with damping on time scales. J Comput Appl Math, 2011, 235: 1180–1188zbMATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Science China Press and Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsMissouri University of Science and TechnologyRollaUSA
  2. 2.School of Mathematical SciencesUniversity of JinanJinanChina
  3. 3.Department of MathematicsLinyi UniversityLinyiChina

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