Opuscula Math. 37, no. 6 (2017), 839-852
http://dx.doi.org/10.7494/OpMath.2017.37.6.839

 
Opuscula Mathematica

Oscillatory and asymptotic behavior of a third-order nonlinear neutral differential equation

John R. Graef
Ercan Tunҫ
Said R. Grace

Abstract. This paper discusses oscillatory and asymptotic properties of solutions of a class of third-order nonlinear neutral differential equations. Some new sufficient conditions for a solution of the equation to be either oscillatory or to converges to zero are presented. The results obtained can easily be extended to more general neutral differential equations as well as to neutral dynamic equations on time scales. Two examples are provided to illustrate the results.

Keywords: third order, neutral differential equations, asymptotic behavior, nonoscillatory, oscillatory solution.

Mathematics Subject Classification: 34K10, 34K11, 34K15, 34C10.

Full text (pdf)

  1. R.P. Agarwal, S.R. Grace, D. O'Regan, The oscillation of certain higher-order functional differential equations, Math. Comput. Modelling 37 (2003), 705-728.
  2. B. Baculíková, J. Dzurina, Oscillation of third-order neutral differential equations, Math. Comput. Modelling 52 (2010), 215-226.
  3. A. Domoshnitskii, Extension of Sturm's theorem to equations with time-lag, Differ. Uravn. 19 (1983), 1475-1482.
  4. J.R. Graef, S.H. Saker, Oscillation theory of third-order nonlinear functional differential equations, Hiroshima Math. J. 43 (2013), 49-72.
  5. J.R. Graef, M.K. Grammatikopoulos, P.W. Spikes, Asymptotic behavior of nonoscillatory solutions of neutral delay differential equations of arbitrary order, Nonlinear Anal. 21 (1993), 23-42.
  6. J.R. Graef, R. Savithri, E. Thandapani, Oscillatory properties of third order neutral delay differential equations, Proceedings of the Fourth International Conference on Dynamical Systems and Differential Equations, May 24-27, 2002, Wilmington, NC, USA, pp. 342-350.
  7. J.K. Hale, Theory of Functional Differential Equations, Springer, New York, 1977.
  8. G.H. Hardy, J.E. Littlewood, G. Polya, Inequalities, Reprint of the 1952 edition, Cambridge University Press, Cambridge, 1988.
  9. I.T. Kiguradze, T.A. Chanturia, Asymptotic Properties of Solutions of Nonautonomous Ordinary Differential Equations, Kluwer, Dordrecht 1993.
  10. R. Koplatadze, On oscillatory properties of solutions of functional differential equations, Publishing House, Tbilisi, 1995.
  11. T. Li, E. Thandapani, Oscillation of solutions to odd-order nonlinear neutral functional differential equations, Electron. J. Differential Equations 23 (2011), 1-12.
  12. T. Li, E. Thandapani, Oscillation theorems for odd-order neutral differential equations, Funct. Differ. Equ. 19 (2012), 147-155.
  13. T. Li, C. Zhang, G. Xing, Oscillation of third-order neutral delay differential equations, Abstr. Appl. Anal. 2012 (2012), Article ID 569201.
  14. B. Mihaliková, E. Kostiková, Boundedness and oscillation of third order neutral differential equations, Tatra Mt. Math. Publ. 43 (2009), 137-144.
  15. H.A. Mohamad, Oscillation of linear neutral differential equation of third order, Iraqi J. Sci. 50 (2009) 4, 543-547.
  16. A.A. Soliman, R.A. Sallam, A. Elbitar, A.M. Hassan, Oscillation criteria of third order nonlinear neutral differential equations, Int. J. Appl. Math. Res. 1 (2012), 268-281.
  17. E. Thandapani, T. Li, On the oscillation of third-order quasi-linear neutral functional differential equations, Arch. Math. (Brno) 47 (2011), 181-199.
  • John R. Graef
  • University of Tennessee at Chattanooga, Department of Mathematics, Chattanooga, TN 37403, USA
  • Ercan Tunҫ
  • Gaziosmanpasa University, Department of Mathematics, Faculty of Arts and Sciences, 60240, Tokat, Turkey
  • Said R. Grace
  • Department of Engineering Mathematics, Faculty of Engineering, Cairo University Orman, Giza 12221, Egypt
  • Communicated by Alexander Domoshnitsky.
  • Received: 2016-09-01.
  • Revised: 2017-01-01.
  • Accepted: 2017-02-07.
  • Published online: 2017-09-28.
Opuscula Mathematica - cover

Cite this article as:
John R. Graef, Ercan Tunҫ, Said R. Grace, Oscillatory and asymptotic behavior of a third-order nonlinear neutral differential equation, Opuscula Math. 37, no. 6 (2017), 839-852, http://dx.doi.org/10.7494/OpMath.2017.37.6.839

Download this article's citation as:
a .bib file (BibTeX),
a .ris file (RefMan),
a .enw file (EndNote)
or export to RefWorks.

In accordance with EU legislation we advise you this website uses cookies to allow us to see how the site is used. All data is anonymized.
All recent versions of popular browsers give users a level of control over cookies. Users can set their browsers to accept or reject all, or certain, cookies.