Skip to main content
Log in

Oscillation of third order nonlinear functional dynamic equations on time scales

  • Original Research
  • Published:
Differential Equations and Dynamical Systems Aims and scope Submit manuscript

Abstract

It is the purpose of this paper to give oscillation criteria for the third order nonlinear functional dynamic equation

$$ \left( {a\left( t \right)\left[ {\left( {r\left( t \right)x^\Delta \left( t \right)} \right)^\Delta } \right]^\gamma } \right)^\Delta + f\left( {t,x\left( {g\left( t \right)} \right)} \right) = 0 $$

on a time scale \( \mathbb{T} \), where γ is the quotient of odd positive integers, a and r are positive rd-continuous functions on \( \mathbb{T} \), and the function g: \( \mathbb{T} \to \mathbb{T} \) satisfies limt→∞ g(t) = ∞ and fC \( \left( {\mathbb{T} \times \mathbb{R}, \mathbb{R}} \right) \). Our results are new for third order delay dynamic equations and extend many known results for oscillation of third order dynamic equations. Some examples are given to illustrate the main results.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Agarwal R., Bohner M. and Saker S. H., Oscillation of second order delay dynamic equations, Canad. Appl. Math. Quart., 13, 1–17, (2005)

    MATH  MathSciNet  Google Scholar 

  2. Bohner M. and Peterson A., Dynamic Equations on Time Scales: An Introduction with Applications, Birkhäuser, Boston, (2001)

    MATH  Google Scholar 

  3. Bohner M. and Peterson A., Advances in Dynamic Equations on Time Scales, Birkhäuser, Boston, (2003)

    MATH  Google Scholar 

  4. Bohner M. and Saker S. H., Oscillation of second order half-linear dynamic equations on discrete time scales, Internat. J. Difference Equs., 1, 208–218, (2006)

    MathSciNet  Google Scholar 

  5. Gera M., Graef J. R. and Gregus M., On oscillatory and asymptotic properties of solutions of certain nonlinear third order differential equations, Nonlinear Anal., 32, 417–425, (1998)

    Article  MATH  MathSciNet  Google Scholar 

  6. Došlý O. and Hilger E., A necessary and sufficient condition for oscillation of the Sturm-Liouville dynamic equation on time scales, Special Issue on Dynamic Equations on Time Scales (Agarwal P. P., Bohner M. and O’Regan D., eds.), J. Comp. Appl. Math., 141(1–2), 571–585, (2002)

    Google Scholar 

  7. Elabbasy E. M. and Hassan T. S., Oscillation of third order nonlinear functional differential equations, Diff. Eq. Appl., submitted

  8. Erbe L., Hassan T. S. and Peterson A., Oscillation criteria for nonlinear damped dynamic equations on time scales, Appl. Math. Comp., 203, 343–357, (2008)

    Article  MATH  MathSciNet  Google Scholar 

  9. Erbe L., Hassan T. S. and Peterson A., Oscillation criteria for nonlinear functional neutral dynamic equations on time scales, J. Diff. Eq. Appl., 15, 1097–1115, (2009)

    Article  MATH  MathSciNet  Google Scholar 

  10. Erbe L., Hassan T. S., Peterson A. and Saker S. H., Oscillation criteria for half-linear delay dynamic equations on time scales, Nonlinear Dynam. Sys. Th., 9, 51–68, (2009)

    MATH  MathSciNet  Google Scholar 

  11. Erbe L., Hassan T. S., Peterson A. and Saker S. H., Oscillation criteria for sublinear half-linear delay dynamic equations on time scales, Int. J. Diff. Equ., 3, 227–245, (2008)

    MathSciNet  Google Scholar 

  12. Erbe L., Peterson A. and Saker S. H., Asymptotic behavior of solutions of a third-order nonlinear dynamic equation on time scales, J. Comp. Appl. Math., 181, 92–102, (2005)

    Article  MATH  MathSciNet  Google Scholar 

  13. Erbe L., Peterson A. and Saker S. H., Hille and Nehari type criteria for third order dynamic equations, J. Math. Anal. Appl., 329, 112–131, (2007)

    Article  MATH  MathSciNet  Google Scholar 

  14. Erbe L., Peterson A. and Saker S. H., Oscillation and asymptotic behavior a third-order nonlinear dynamic equation, Canad. Quart. Appl. Math., 14, 2, (2006)

    MathSciNet  Google Scholar 

  15. Hardy G. H., Littlewood J. E. and Polya G., Inequalities, second ed., Cambridge University Press, Cambridge, (1988)

    MATH  Google Scholar 

  16. Hassan T. S., Oscillation criteria for half-linear dynamic equations on time scales, J. Math. Anal. Appl., 345, 176–185, (2008)

    Article  MATH  MathSciNet  Google Scholar 

  17. Hilger S., Analysis on measure chains — a unified approach to continuous and discrete calculus, Results Math., 18, 18–56, (1990)

    MATH  MathSciNet  Google Scholar 

  18. Kac V. and Cheung P., Quantum Calculus, Universitext, (2002)

  19. Zhang B. G. and Deng X., Oscillation of delay differential equations on time scales, Math. Comp. Mod., 36, 1307–1318, (2002)

    Article  MATH  MathSciNet  Google Scholar 

  20. Şahiner Y. and Stavroulakis I. S., Oscillation of first order delay dynamic equations, Dynam. Systems Appl., 15, 645–655, (2006)

    MathSciNet  Google Scholar 

  21. Wu H., Zhuang R. and Mathsen R. M., Oscillation criteria of second-order nonlinear neutral variable delay dynamic equations, Appl. Math. Comp., 178, 321–331, (2006)

    MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Lynn Erbe.

Additional information

This paper is dedicated to Bernd Aulbach.

Supported by the Egyptian Government while visiting the University of Nebraska-Lincoln.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Erbe, L., Hassan, T.S. & Peterson, A. Oscillation of third order nonlinear functional dynamic equations on time scales. Differ Equ Dyn Syst 18, 199–227 (2010). https://doi.org/10.1007/s12591-010-0005-y

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12591-010-0005-y

Keywords

Mathematics Subject Classification (2000)

Navigation