Advertisement

Mediterranean Journal of Mathematics

, Volume 11, Issue 4, pp 1115–1127 | Cite as

Comparison Theorems for Oscillation of Second-Order Neutral Dynamic Equations

  • Ravi P. Agarwal
  • Martin Bohner
  • Tongxing Li
  • Chenghui Zhang
Article

Abstract

We study oscillation of certain second-order neutral dynamic equations under the assumptions that allow applications to dynamic equations with both delayed and advanced arguments. Some new comparison criteria are presented that can be used in cases where known theorems fail to apply.

Mathematics Subject Classification (2010)

34K11 34N05 39A10 39A21 

Keywords

Oscillation comparison theorem neutral dynamic equation second-order equation time scale 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Agarwal R.P., Bohner M., O’Regan D., Peterson A.: Dynamic equations on time scales: a survey. J. Comput. Appl. Math. 141, 1–26 (2002)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Agarwal, R.P., Grace, S.R., O’Regan, D.: Oscillation Theory for Second Order Dynamic Equations. Series in Mathematical Analysis and Applications, vol. 5. Taylor and Francis Ltd., London (2003)Google Scholar
  3. 3.
    Agarwal R.P., Grace S.R., O’Regan D.: The oscillation of certain higher-order functional differential equations. Math. Comput. Modell. 37, 705–728 (2003)MathSciNetCrossRefMATHGoogle 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. 300, 203–217 (2004)MathSciNetCrossRefMATHGoogle 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. 27, 1–12 (2007)MathSciNetMATHGoogle Scholar
  6. 6.
    Bohner, M., Peterson, A.: Dynamic Equations on Time Scales: An Introduction with Applications. Birkhäuser, Boston (2001)Google Scholar
  7. 7.
    Bohner, M., Peterson, A.: Advances in Dynamic Equations on Time Scales. Birkhäuser, Boston (2003)Google Scholar
  8. 8.
    Chen D.: Oscillation of second-order Emden–Fowler neutral delay dynamic equations on time scales. Math. Comput. Modell. 51, 1221–1229 (2010)CrossRefMATHGoogle Scholar
  9. 9.
    Erbe L.: Oscillation criteria for second order linear equations on a time scale. Can. Appl. Math. Q. 9, 1–31 (2001)MathSciNetGoogle Scholar
  10. 10.
    Erbe L., Hassan T.S., Peterson A.: Oscillation criteria for nonlinear functional neutral dynamic equations on time scales. J. Differ. Equ. Appl. 15, 1097–1116 (2009)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Erbe L., Peterson A.: Some oscillation results for second order linear delay dynamic equations. Adv. Stud. Pure Math. 53, 237–245 (2009)MathSciNetGoogle Scholar
  12. 12.
    Grace S.R.: On the oscillation of nth order dynamic equations on time-scales. Mediterr. J. Math. 10, 147–156 (2013)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Grace S.R., Agarwal R.P., Bohner M., O’Regan D.: Oscillation of second-order strongly superlinear and strongly sublinear dynamic equations. Commun. Nonlinear Sci. Numer. Simul. 14, 3463–3471 (2009)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Grace S.R., Agarwal R.P., Kaymakçalan B., Sae-jie W.: On the oscillation of certain second order nonlinear dynamic equations. Math. Comput. Modell. 50, 273–286 (2009)CrossRefMATHGoogle Scholar
  15. 15.
    Han Z., Li T., Sun S., Zhang C.: On the oscillation of second-order neutral delay dynamic equations on time scales. Afri. Dias. J. Math. 9, 76–86 (2010)MathSciNetMATHGoogle Scholar
  16. 16.
    Hassan T.S.: Kamenev-type oscillation criteria for second order nonlinear dynamic equations on time scales. Appl. Math. Comput. 217, 5285–5297 (2011)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Hilger S.: Analysis on measure chains—a unified approach to continuous and discrete calculus. Results Math. 18, 18–56 (1990)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Karpuz B.: Asymptotic behavior of bounded solutions of a class of higher-order neutral dynamic equations. Appl. Math. Comput. 215, 2174–2183 (2009)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Li T., Agarwal R.P., Bohner M.: Some oscillation results for second-order neutral dynamic equations. Hacet. J. Math. Stat. 41, 715–721 (2012)MathSciNetMATHGoogle Scholar
  20. 20.
    Li T., Han Z., Sun S., Yang D.: Existence of nonoscillatory solutions to second-order neutral delay dynamic equations on time scales. Adv. Differ. Equ. 2009, 1–10 (2009)MathSciNetGoogle Scholar
  21. 21.
    Li T., Rogovchenko Y.V., Zhang C.: Oscillation of second-order neutral differential equations. Funkc. Ekvacioj 56, 111–120 (2013)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Řehák P.: How the constants in Hille–Nehari theorems depend on time scales. Adv. Differ. Equ. 2006, 1–15 (2006)Google Scholar
  23. 23.
    Şahiner Y.: Oscillation of second order neutral delay and mixed type dynamic equations on time scales. Adv. Differ. Equ. 2006, 1–9 (2006)Google Scholar
  24. 24.
    Saker S.H.: Oscillation of nonlinear dynamic equations on time scales. Appl. Math. Comput. 148, 81–91 (2004)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Saker S.H.: Oscillation of second-order nonlinear neutral delay dynamic equations on time scales. J. Comput. Appl. Math. 187, 123–141 (2006)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Saker S.H.: Oscillation criteria for a second-order quasilinear neutral functional dynamic equation on time scales. Nonlinear Oscil. 13, 407–428 (2011)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Saker S.H., Agarwal R.P., O’Regan D.: Oscillation theorems for second-order nonlinear neutral delay dynamic equations on time scales. Acta Math. Sin. 24, 1409–1432 (2008)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Saker S.H., Agarwal R.P., O’Regan D.: Oscillation results for second-order nonlinear neutral delay dynamic equations on time scales. Appl. Anal. 86, 1–17 (2007)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Saker S.H., O’Regan D.: New oscillation criteria for second-order neutral functional dynamic equations via the generalized Riccati substitution. Commun. Nonlinear Sci. Numer. Simul. 16, 423–434 (2011)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Tang S., Li T., Thandapani E.: Oscillation theorems for second-order half-linear advanced dynamic equations on time scales. Int. J. Differ. Equ. 2011, 1–16 (2011)MathSciNetMATHGoogle Scholar
  31. 31.
    Thandapani E., Veeraraghavan P., Sandra P.: Oscillation criteria for second order neutral delay dynamic equations with mixed nonlinearities. Adv. Differ. Equ. 2011, 1–14 (2011)CrossRefGoogle Scholar
  32. 32.
    Tripathy A.K.: Some oscillation results for second order nonlinear dynamic equations of neutral type. Nonlinear Anal. TMA 71, 1727–1735 (2009)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Wu H., Zhuang R., Mathsen R.M.: Oscillation criteria for second-order nonlinear neutral variable delay dynamic equations. Appl. Math. Comput. 178, 321–331 (2006)MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    Zhang C., Agarwal R.P., Bohner M., Li T.: New oscillation results for second-order neutral delay dynamic equations. Adv. Differ. Equ. 2012, 1–14 (2012)MathSciNetCrossRefMATHGoogle Scholar
  35. 35.
    Zhang, C., Agarwal, R.P., Bohner, M., Li, T.: Oscillation of second-order nonlinear neutral dynamic equations with noncanonical operators. Bull. Malays. Math. Sci. Soc. (2013, in press)Google Scholar
  36. 36.
    Zhang S., Wang Q.: Oscillation of second-order nonlinear neutral dynamic equations on time scales. Appl. Math. Comput. 216, 2837–2848 (2010)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Basel 2013

Authors and Affiliations

  • Ravi P. Agarwal
    • 1
  • Martin Bohner
    • 2
  • Tongxing Li
    • 3
  • Chenghui Zhang
    • 3
  1. 1.Department of MathematicsTexas A&M University-KingsvilleKingsvilleUSA
  2. 2.Department of Mathematics and StatisticsMissouri S&TRollaUSA
  3. 3.School of Control Science and EngineeringShandong UniversityJinanPeople’s Republic of China

Personalised recommendations