Acta Mathematica Hungarica

, Volume 100, Issue 1–2, pp 37–62

Oscillation of second order neutral delay differential equations of Emden-Fowler type

  • S. H. Saker
Article

Abstract

We present new oscillation criteria for the second order nonlinear neutral delay differential equation [y(t)-py(t-τ)]''+ q(t)yλ(g(t)) sgn y(g(t)) = 0, tt0. Our results solve an open problem posed by James S.W . Wong [24]. The relevance of our results becomes clear due to a carefully selected example.

oscillation second order nonlinear neutral delay differential equations 

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References

  1. [1]
    R. P. Agarwal, S. R. Grace and D. O'Regan, Oscillation Theory for Difference and functional Differential Equations, Kluwer Academic Publishers (Dordrecht, 2000).Google Scholar
  2. [2]
    R. P. Agarwal, S. R. Grace and D. O'Regan, Oscillation Theory for Second order Dynamic Equations, to appear.Google Scholar
  3. [3]
    F. V. Aitkinson, On second order nonlinear oscillation, Pacific J. Math., 5 (1955), 643–647.MathSciNetGoogle Scholar
  4. [4]
    S. Belohorec, Oscillatory solutions of certain nonlinear differential equations of the second order, Math.-Fyz. Casopis Sloven. Acad. Vied, 11 (1961), 250–255.MATHGoogle Scholar
  5. [5]
    J. Dzurina and J. Ohriska, Asymptotic and oscillatory properties of differential equations with deviating arguments, Hiroshima Math. J., 22 (1992), 561–571.MATHMathSciNetGoogle Scholar
  6. [6]
    J. Dzurina and B. Mihalikova, Oscillation criteria for second order neutral differential equations, Math. Boh., 125 (2000), 145–153.MATHMathSciNetGoogle Scholar
  7. [7]
    L. H. Erbe, Q. King and B. Z. Zhang, Oscillation Theory for Functional Differential Equations, Marcel Dekker (New York, 1995).Google Scholar
  8. [8]
    J. R. Graef, M. K. Grammatikopoulos and P. W. Spikes, Asymptotic properties of solutions of nonlinear neutral delay differential equations of the second order, Radovi Mat., 4 (1988), 133–149.MATHMathSciNetGoogle Scholar
  9. [9]
    J. R. Graef, M. K. Grammatikopoulos and P. W. Spikes, On the asymptotic behavior of solutions of the second order nonlinear neutral delay differential equations, J. Math. Anal. Appl., 156 (1991), 23–39.MATHMathSciNetCrossRefGoogle Scholar
  10. [10]
    J. R. Graef, M. K. Grammatikopoulos and P. W. Spikes, Some results on the on the asymptotic behavior of the solutions of a second order nonlinear neutral delay differential equations, Contemporary Mathematics, 129 (1992), 105–114.MATHMathSciNetGoogle Scholar
  11. [11]
    S. R. Grace and B. S. Lalli, Oscillation criteria for forced neutral differential equations, Czech. Math. J., 44 (1994), 713–724.MATHMathSciNetGoogle Scholar
  12. [12]
    S. R. Grace, Oscillations criteria of comparison type for nonlinear functional differential equations, Math. Nachr., 173 (1995), 177–192.MATHMathSciNetGoogle Scholar
  13. [13]
    S. R. Grace and B. S. Lalli, Comparison and oscillations theorems for functional differential equations with deviating arguments, Math. Nachr., 144 (1989), 65–79.MATHMathSciNetGoogle Scholar
  14. [14]
    M. K. Grammatikopoulos, G. Ladas and A. Meimaridou, Oscillation of second order neutral delay differential equations, Radovi Mat., 1 (1985), 267–274.MATHMathSciNetGoogle Scholar
  15. [15]
    M. K. Grammatikopoulos, G. Ladas and A. Meimaridou, Oscillation and asymptotic behavior of second order neutral differential equations, Annali di Matematica Pura ed Applicata, CXL (1987), 20–40.Google Scholar
  16. [16]
    M. K. Grammatikopoulos and P. Marusiak, Oscillatory properties of second order nonlinear neutral differential inequalities with oscillating coefficients, Arch. Math., 31 (1995), 29–36.MATHMathSciNetGoogle Scholar
  17. [17]
    I. Győri and G. Ladas, Oscillation Theory of Delay Differential Equations With Applications, Oxford Univ. Press (London-New York, 1991).Google Scholar
  18. [18]
    J. K. Hale, Theory of Functional Differential Equations, Springer-Verlag (New York, 1977).Google Scholar
  19. [19]
    I. V. Kamenev, Integral criterion for oscillation of linear differential equations of second order, Math. Zemetki (1978), 249–251 (in Russian).Google Scholar
  20. [20]
    G. S. Ladde, V. Lakshmikantham and B. Z. Zhang, Oscillation Theory of Differential Equations with Deviating Arguments, Marcel Dekker (New York, 1987).Google Scholar
  21. [21]
    H. J. Li, and W. L. Liu, Oscillation criteria for second order neutral differential equations, Can. J. Math., 48 (1996), 871–886.MATHGoogle Scholar
  22. [22]
    W. T. Li, Classification and existence of nonoscillatory solutions of second order nonlinear neutral differential equations, Ann. Polon Math., LXV (1997), 283–302.Google Scholar
  23. [23]
    C. G. Philos, Oscillation theorems for linear differential equation of second order, Arch. Math., 53 (1989), 483–492.MathSciNetCrossRefGoogle Scholar
  24. [24]
    J. S. W. Wong, Necessary and sufficient conditions for oscillation of second order neutral differential equations, J. Math. Anal. Appl., 252 (2000), 342–352.MATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Kluwer Academic Publishers/Akadémiai Kiadó 2003

Authors and Affiliations

  • S. H. Saker
    • 1
  1. 1.Mathematics Department, Faculty of ScienceMansoura UniversityMansouraEgypt

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