Opuscula Mathematica
Opuscula Math. 39, no. 1 (2019), 39-47

Oscillatory behavior of even-order nonlinear differential equations with a sublinear neutral term

John R. Graef
Said R. Grace
Ercan Tunç
Abstract. The authors present a new technique for the linearization of even-order nonlinear differential equations with a sublinear neutral term. They establish some new oscillation criteria via comparison with higher-order linear delay differential inequalities as well as with first-order linear delay differential equations whose oscillatory characters are known. Examples are provided to illustrate the theorems.
Keywords: oscillatory behavior, neutral differential equation, even-order.
Mathematics Subject Classification: 34C10, 34K11.
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References ▼
  1. [1] R.P. Agarwal, S.R. Grace, The oscillation of higher-order differential equations with deviating arguments, Comput. Math. Appl. 38 (1999), 185-199.
  2. [2] R.P. Agarwal, S.R. Grace, Oscillation theorems for certain functional differential equations of higher order, Math. Comput. Model. 39 (2004), 1185-1194.
  3. [3] R.P. Agarwal, M. Bohner, T. Li, C. Zhang, Oscillation of second-order differential equations with a sublinear neutral term, Carpathian J. Math. 30 (2014), 1-6.
  4. [4] .P. Agarwal, S.R. Grace, I.T. Kiguradze, D. O'Regan, Oscillation of functional differential equations, Math. Comput. Model. 41 (2005), 417-461.
  5. [5] R.P. Agarwal, S.R. Grace, D. O'Regan, Oscillation criteria for certain \(n\)th order differential equations with deviating arguments, J. Math. Anal. Appl. 262 (2001), 601-622.
  6. [6] R.P. Agarwal, S.R. Grace, D. O'Regan, Oscillation of certain fourth-order functional differential equations, Ukrainian Math. J. 59 (2007), 315-342.
  7. [7] R.P. Agarwal, S.R. Grace, D. O'Regan, Oscillation Theory for Difference and Functional Differential Equations, Kluwer Academic Publishers, Dordrecht, 2000.
  8. [8] R.P. Agarwal, S.R. Grace, P.J.Y. Wong, On the bounded oscillation of certain fourth order functional differential equations, Nonlinear Dyn. Syst. Theory 5 (2005), 215-227.
  9. [9] B. Baculíková, J. Džurina, Oscillation theorems for second-order nonlinear neutral differential equations, Comput. Math. Appl. 62 (2011), 4472-4478.
  10. [10] S.R. Grace, Oscillation theorems for nth-order differential equations with deviating arguments, J. Math. Anal. Appl. 101 (1984), 268-296.
  11. [11] S.R. Grace, B.S. Lalli, On oscillation and nonoscillation of general functional differential equations, J. Math. Anal. Appl. 109 (1985), 522-533.
  12. [12] S.R. Grace, B.S. Lalli, A comparison theorem for general nonlinear ordinary differential equations, J. Math. Anal. Appl. 120 (1986), 39-43.
  13. [13] S.R. Grace, R.P. Agarwal, M. Bohner, D. O'Regan, Oscillation of second-order strongly superlinear and strongly sublinear dynamic equations, Commun. Nonlinear Sci. Numer. Simul. 14 (2009), 3463-3471.
  14. [14] I. Györi, G. Ladas, Oscillation Theory of Delay Differential Equations with Applications, Clarendon Press, Oxford,1991.
  15. [15] G.H. Hardy, I.E. Littlewood, G. Polya, Inequalities, Reprint of the 1952 edition, Cambridge University Press, Cambridge, 1988.
  16. [16] I.T. Kiguradze, On the oscillatory character of solutions of the equation \(d^{m}u/dt^{m} +a(t)|u|^{n}sgn u = 0\), Mat. Sb. (N.S.) 65 (1964), 172-187.
  17. [17] T. Li, Y.V. Rogovchenko, Oscillation criteria for even-order neutral differential equations, Appl. Math. Lett. 61 (2016), 35-41.
  18. [18] Ch.G. Philos, On the existence of nonoscillatory solutions tending to zero at \(\infty\) for differential equations with positive delays, Arch. Math. (Basel) 36 (1981), 168-178.
  19. [19] Ch.G. Philos, A new criterion for the oscillatory and asymptotic behavior of delay differential equations, Bull. Acad. Pol. Sci. Ser. Sci. Mat. 30 (1981), 61-64.
  20. [20] Q. Zhang, J. Yan, L. Gao, Oscillation behavior of even-order nonlinear neutral differential equations with variable coefficients, Comput. Math. Appl. 59 (2010), 426-430.
About the authors ▼
John R. Graef
https://orcid.org/0000-0002-8149-4633
University of Tennessee at Chattanooga, Department of Mathematics, Chattanooga, TN 37403, USA
Said R. Grace
https://orcid.org/0000-0001-8783-5227
Cairo University, Department of Engineering Mathematics, Faculty of Engineering, Orman, Giza 12221, Egypt
Ercan Tunç
https://orcid.org/0000-0001-8860-608X
Gaziosmanpasa University, Department of Mathematics, Faculty of Arts and Sciences, 60240, Tokat, Turkey
About the article ▼
Communicated by Josef Diblík.
Received: 2018-02-08.
Accepted: 2018-04-01.
Published online: 2018-08-07.
Cite this article as ▼
John R. Graef, Said R. Grace, Ercan Tunç, Oscillatory behavior of even-order nonlinear differential equations with a sublinear neutral term, Opuscula Math. 39, no. 1 (2019), 39-47, https://doi.org/10.7494/OpMath.2019.39.1.39.
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