Advertisement

Journal of Mathematical Sciences

, Volume 201, Issue 3, pp 296–309 | Cite as

Oscillation of Fourth-Order Delay Differential Equations

  • C. ZhangEmail author
  • T. Li
  • S. H. Saker
Article

This article deals with the oscillation of a certain class of fourth-order delay differential equations. Some new oscillation criteria (including Hille- and Nehari-type criteria) are presented. The results obtained in the paper improve some results from [C. Zhang, T. Li, B. Sun, and E. Thandapani, Appl. Math. Lett., 24, 1618 (2011)]. Two examples are presented to illustrate our main results.

Keywords

Functional Differential Equation Delay Differential Equation Oscillation Theory Nonoscillatory Solution Oscillation Criterion 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    R. P. Agarwal, M. Bohner, and W.-T. Li, Nonoscillation and oscillation: theory for functional differential equations, Monogr. Textbooks Pure Appl. Math., 267, Marcel Dekker Inc., New York (2004).Google Scholar
  2. 2.
    R. P. Agarwal, S. R. Grace, and J.V. Manojlovic, “Oscillation criteria for certain fourth order nonlinear functional differential equations,” Math. Comput. Model., 44, 163–187 (2006).CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    R. P. Agarwal, S. R. Grace, and D. O’Regan, Oscillation theory for difference and functional differential equations, Kluwer Acad. Publ., Dordrecht (2000).CrossRefzbMATHGoogle Scholar
  4. 4.
    R. P. Agarwal, S. R. Grace, and D. O’Regan, Oscillation theory for second order linear, half-linear, superlinear and sublinear dynamic equations, Kluwer Acad. Publ., Dordrecht (2002).CrossRefzbMATHGoogle Scholar
  5. 5.
    R. P. Agarwal, S. R. Grace, and D. O’Regan, Oscillation theory for second order dynamic equations, Ser. Math. Analysis Appl., 5, Taylor & Francis, London (2003).CrossRefzbMATHGoogle Scholar
  6. 6.
    R. P. Agarwal, S. R. Grace, and D. O’Regan, “Oscillation criteria for certain nth order differential equations with deviating arguments,” J. Math. Anal. Appl., 262, 601–622 (2001).CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    R. P. Agarwal, S. R. Grace, and D. O’Regan, “The oscillation of certain higher-order functional differential equations,” Math. Comput. Model., 37, 705–728 (2003).CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    R. P. Agarwal, S. L. Shieh, and C. C. Yeh, “Oscillation criteria for second order retarded differential equations,” Math. Comput. Model., 26, 1–11 (1997).CrossRefMathSciNetGoogle Scholar
  9. 9.
    B. Baculíková and J. Džurina, “Oscillation of third-order nonlinear differential equations,” Appl. Math. Lett., 24, 466–470 (2011).CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    S. R. Grace, R. P. Agarwal, and J. R. Graef, “Oscillation theorems for fourth order functional differential equations,” J. Appl. Math. Comput., 30, 75–88 (2009).CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    K. I. Kamo and H. Usami, “Oscillation theorems for fourth order quasilinear ordinary differential equations,” Stud. Sci. Math. Hung., 39, 385–406 (2002).zbMATHMathSciNetGoogle Scholar
  12. 12.
    K. I. Kamo and H. Usami, “Nonlinear oscillations of fourth order quasilinear ordinary differential equations,” Acta Math. Hung., 132, 207–222 (2011).CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    I. T. Kiguradze and T. A. Chanturiya, Asymptotic properties of solutions of nonautonomous ordinary differential equations, Kluwer Acad. Publ., Dordrecht (1993).CrossRefzbMATHGoogle Scholar
  14. 14.
    T. Kusano, J. Manojlović, and T. Tanigawa, “Sharp oscillation criteria for a class of fourth order nonlinear differential equations ,” Rocky Mountain J. Math., 41, 249–274 (2011).CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Ladde G. S., Lakshmikantham V., and Zhang B. G., Oscillation theory of differential equations with deviating arguments, Marcel Dekker, New York (1987).Google Scholar
  16. 16.
    T. Li, C. Zhang, B. Baculíková, and J. Džurina, “On the oscillation of third-order quasilinear delay differential equations,” Tatra Mt. Math. Publ., 48, 117–123 (2011).zbMATHMathSciNetGoogle Scholar
  17. 17.
    J. Manojlović, “Oscillation criteria for second-order half-linear differential equations,” Math. Comput. Model., 30, 109–119 (1999).CrossRefzbMATHGoogle Scholar
  18. 18.
    Z. Nehari, “Oscillation criteria for second-order linear differential equations,” Trans. Amer. Math. Soc., 85, 428–445 (1957).CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    Ch. G. Philos, “A new criterion for the oscillatory and asymptotic behavior of delay differential equations,” Bull. Acad. Pol. Sci., Sér. Sci. Math., 39, 61–64 (1981).Google Scholar
  20. 20.
    P. Řehák, “How the constants in Hille–Nehari theorems depend on time scales,” Adv. Difference Equat., 2006, 1–15 (2006).Google Scholar
  21. 21.
    S. H. Saker, Oscillation theory of delay differential and difference equations. — Second and third orders, Verlag Dr Müller, Germany (2010).Google Scholar
  22. 22.
    C. Zhang, T. Li, B. Sun, and E. Thandapani, “On the oscillation of higher-order half-linear delay differential equations,” Appl. Math. Lett., 24, 1618–1621 (2011).CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.School Control Sci. Eng.Shandong Univ.JinanP. R. China
  2. 2.Mansoura Univ.MansouraEgypt

Personalised recommendations