Journal of Applied Mathematics and Computing

, Volume 20, Issue 1–2, pp 133–147

Oscillation criteria for nonlinear perturbed dynamic equations of second-order on time scales

Article

Abstract

In this paper, by using the Riccati transformation technique we establish some new oscillation criteria for second-order nonlinear perturbed dynamic equation on time scales. An example illustrating our main results is also given.

AMS Mathematics Subject Classification

34K11 39A10 39A99 (34A99 34C10 39A11) 

Key words and phrases

Oscillation second-order nonlinear dynamic equation time scale Riccati transformation technique 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    R. P. Agarwal, M. Bohner, D. O’Regan and A. Peterson,Dynamic equations on time scales: A survey, Special Issue on “Dynamic Equations on Time Scales”, edited by R. P. Agarwal, M. Bohner, and D. O’Regan, (Preprint in Ulmer Seminare 5) J. Comp. Appl. Math.141 (2002), 1–26.MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    R. P. Agarwal, M. Bohner and S. H. Saker,Oscillation of second order delay dynamic equations, Canadian Appl. Math. Quart. (accepted).Google Scholar
  3. 3.
    R. P. Agarwal, D. O’Regan and S. H. Saker,Oscillation criteria for second-order nonlinear neutral delay dynamic equations, J. Math. Anal. Appl.300 (2004), 203–217.MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    R. P. Agarwal, D. O’Regan and S. H. Saker,Philos-type oscillation criteria of second-order half-linear dynamic equations on time scales, submitted.Google Scholar
  5. 5.
    R. P. Agarwal, D. O’Regan and S. H. Saker,Properties of bounded solutions of nonlinear dynamic equations on time scales, submitted.Google Scholar
  6. 6.
    E. A. Bohner and J. Hoffacker,Oscillation properties of an Emden-Fowler type Equations on Discrete time scales, J. Diff. Eqns. Appl.9 (2003), 603–612.MATHCrossRefGoogle Scholar
  7. 7.
    M. Bohner and A. Peterson,Dynamic Equations on Time Scales: An Introduction with Applications, Birkhäuser, Boston, 2001.MATHGoogle Scholar
  8. 8.
    M. Bohner, A. Peterson,Advances in Dynamic Equations on Time Scales, Birkhäuser, Boston, 2003.MATHGoogle Scholar
  9. 9.
    M. Bohner and S. H. Saker,Oscillation of second order nonlinear dynamic equations on time scales, Rocky Mountain J. Math.34 (2004), 1239–1254.MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    M. Bohner and S. H. Saker,Oscillation criteria for perturbed nonlinear dynamic equations, Math. Comp. Modelling40 (2004), 249–260.MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    E. A. Bohner, M. Bohner and S. H. Saker,Oscillation criteria for a certain class of second order Emden-Fowler dynamic Equations, Elect. Transc. Numerical. Anal. (accepted).Google Scholar
  12. 12.
    O. Doslý and S. Hilger,A necessary and sufficient condition for oscillation of the Sturm-Liouville dynamic equation on time scales, Special Issue on “Dynamic Equations on Time Scales”, edited by R. P. Agarwal, M. Bohner, and D. O’Regan, J. Comp. Appl. Math.141 (2002) 147–158.MATHCrossRefGoogle Scholar
  13. 13.
    L. Erbe,Oscillation criteria for second order linear equations on a time scale, Canadian Appl. Math. Quart.9 (2001), 1–31.MathSciNetGoogle Scholar
  14. 14.
    L. Erbe and A. Peterson,Positive solutions for a nonlinear differential equation on a measure chain, Mathl. Comp. Modelling, Boundary Value Problems and Related Topics32(5-6), (2000), 571–585.MATHMathSciNetGoogle Scholar
  15. 15.
    L. Erbe and A. Peterson,Riccati equations on a measure chain, In G. S. Ladde, N. G. Medhin, and M. Sambandham, editors, Proceedings of Dynamic Systems and Applications 3, 193–199, Atlanta, 2001, Dynamic Publishers.Google Scholar
  16. 16.
    L. Erbe and A. Peterson,Oscillation criteria for second order matrix dynamic equations on a time scale, Special Issue on “Dynamic Equations on Time Scales”, edited by R. P. Agarwal, M. Bohner, and D. O’Regan, J. Comput. Appl. Math.141 (2002), 169–185.MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    L. Erbe and A. Peterson,Boundedness and oscillation for nonlinear dynamic equations on a time scale, Proc. Amer. Math. Soc.132 (2004), 735–744.MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    L. Erbe, A. Peterson and S. H. Saker,Oscillation criteria for second-order nonlinear dynamic equations on time scales, J. London Math. Soc.67 (2003), 701–714.MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    G. H. Hardy, J. E. Littlewood and G. Polya,Inequalities, 2nd Ed. Cambridge Univ. Press 1952.Google Scholar
  20. 20.
    S. Hilger,Analysis on measure chains —a unified approach to continuous and discrete calculus, Results Math.18 (1990), 18–56.MathSciNetMATHGoogle Scholar
  21. 21.
    S. H. Saker,Oscillation of nonlinear dynamic equations on time scales, Appl. Math. Comp.148 (2004), 81–91.MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    S. H. Saker,Oscillation criteria of second-order half-linear dynamic equations on time scales, J. Comp. Appl. Math.177 (2005), 375–387.MATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    S. H. Saker,Boundedness of solutions of second-order forced nonlinear dynamic equations, Rocky Mount. J. Math (accepted).Google Scholar

Copyright information

© Korean Society for Computational & Applied Mathematics and Korean SIGCAM 2006

Authors and Affiliations

  • Ravi P. Agarwal
    • 1
  • Donal O’regan
    • 2
  • S. H. Saker
    • 3
  1. 1.Department of Mathematical SciencesFlorida Institute of TechnologyMelbourneUSA
  2. 2.Department of MathematicsNational University of IrelandGalwayIreland
  3. 3.Department of Mathematics, Faculty of ScienceMansoura UniversityMansouraEgypt

Personalised recommendations