Abstract and Applied Analysis

Asymptotic Behavior of Higher-Order Quasilinear Neutral Differential Equations

Tongxing Li and Yuriy V. Rogovchenko

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Abstract

We study asymptotic behavior of solutions to a class of higher-order quasilinear neutral differential equations under the assumptions that allow applications to even- and odd-order differential equations with delayed and advanced arguments, as well as to functional differential equations with more complex arguments that may, for instance, alternate indefinitely between delayed and advanced types. New theorems extend a number of results reported in the literature. Illustrative examples are presented.

Article information

Source
Abstr. Appl. Anal., Volume 2014, Special Issue (2013), Article ID 395368, 11 pages.

Dates
First available in Project Euclid: 26 March 2014

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1395858041

Digital Object Identifier
doi:10.1155/2014/395368

Mathematical Reviews number (MathSciNet)
MR3166608

Zentralblatt MATH identifier
07022303

Citation

Li, Tongxing; Rogovchenko, Yuriy V. Asymptotic Behavior of Higher-Order Quasilinear Neutral Differential Equations. Abstr. Appl. Anal. 2014, Special Issue (2013), Article ID 395368, 11 pages. doi:10.1155/2014/395368. https://projecteuclid.org/euclid.aaa/1395858041


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References

  • C. H. Ou and J. S. W. Wong, “Oscillation and non-oscillation theorems for superlinear Emden-Fowler equations of the fourth order,” Annali di Matematica Pura ed Applicata. Series IV, vol. 183, no. 1, pp. 25–43, 2004.
  • J. S. W. Wong, “On the generalized Emden-Fowler equation,” SIAM Review, vol. 17, pp. 339–360, 1975.
  • R. P. Agarwal, L. Berezansky, E. Braverman, and A. Domoshnitsky, Nonoscillation Theory of Functional Differential Equations with Applications, Springer, New York, NY, USA, 2012.
  • R. P. Agarwal, M. Bohner, and W.-T. Li, Nonoscillation and Oscillation: Theory for Functional Differential Equations, vol. 267, Marcel Dekker, New York, NY, USA, 2004.
  • R. P. Agarwal, S. R. Grace, and D. O'Regan, Oscillation Theory for Difference and Functional Differential Equations, Kluwer Academic, Dordrecht, The Netherlands, 2000.
  • G. S. Ladde, V. Lakshmikantham, and B. G. Zhang, Oscillation Theory of Differential Equations with Deviating Arguments, vol. 110, Marcel Dekker, New York, NY, USA, 1987.
  • B. Baculíková, “Properties of third-order nonlinear functional differential equations with mixed arguments,” Abstract and Applied Analysis, vol. 2011, Article ID 857860, 15 pages, 2011.
  • B. Baculíková and J. Džurina, “Oscillation theorems for second order neutral differential equations,” Computers & Mathematics with Applications, vol. 61, no. 1, pp. 94–99, 2011.
  • B. Baculíková and J. Džurina, “Oscillation theorems for second-order nonlinear neutral differential equations,” Computers & Mathematics with Applications, vol. 62, no. 12, pp. 4472–4478, 2011.
  • B. Baculíková and J. Džurina, “Oscillation theorems for higher order neutral differential equations,” Applied Mathematics and Computation, vol. 219, no. 8, pp. 3769–3778, 2012.
  • B. Baculíková, J. Džurina, and T. Li, “Oscillation results for even-order quasilinear neutral functional differential equations,” Electronic Journal of Differential Equations, no. 143, pp. 1–9, 2011.
  • J. Džurina and B. Baculíková, “Oscillation and asymptotic behavior of higher-order nonlinear differential equations,” International Journal of Mathematics and Mathematical Sciences, vol. 2012, Article ID 951898, 9 pages, 2012.
  • M. Hasanbulli and Yu. V. Rogovchenko, “Asymptotic behavior of nonoscillatory solutions to $n$-th order nonlinear neutral differential equations,” Nonlinear Analysis: Theory, Methods & Applications, vol. 69, no. 4, pp. 1208–1218, 2008.
  • M. Hasanbulli and Yu. V. Rogovchenko, “Oscillation criteria for second order nonlinear neutral differential equations,” Applied Mathematics and Computation, vol. 215, no. 12, pp. 4392–4399, 2010.
  • Y. Kitamura and T. Kusano, “Oscillation of first-order nonlinear differential equations with deviating arguments,” Proceedings of the American Mathematical Society, vol. 78, no. 1, pp. 64–68, 1980.
  • T. Li, R. P. Agarwal, and M. Bohner, “Some oscillation results for second-order neutral differential equations,” The Journal of the Indian Mathematical Society, vol. 79, no. 1-4, pp. 97–106, 2012.
  • T. Li, Z. Han, P. Zhao, and S. Sun, “Oscillation of even-order neutral delay differential equations,” Advances in Difference Equations, vol. 2010, Article ID 184180, 2010.
  • T. Li, Yu. V. Rogovchenko, and C. Zhang, “Oscillation of second-order neutral differential equations,” Funkcialaj Ekvacioj, vol. 56, no. 1, pp. 111–120, 2013.
  • T. Li and E. Thandapani, “Oscillation of solutions to odd-order nonlinear neutral functional differential equations,” Electronic Journal of Differential Equations, no. 23, pp. 1–12, 2011.
  • Ch. G. Philos, “A new criterion for the oscillatory and asymptotic behavior of delay differential equations,” Bulletin de l'Académie Polonaise des Sciences, vol. 39, pp. 61–64, 1981.
  • Ch. G. Philos, “On the existence of nonoscillatory solutions tending to zero at $\infty $ for differential equations with positive delays,” Archiv der Mathematik, vol. 36, no. 2, pp. 168–178, 1981.
  • G. Xing, T. Li, and C. Zhang, “Oscillation of higher-order quasi-linear neutral differential equations,” Advances in Difference Equations, vol. 2011, article 45, p. 10, 2011.
  • C. Zhang, R. P. Agarwal, M. Bohner, and T. Li, “New results for oscillatory behavior of even-order half-linear delay differential equations,” Applied Mathematics Letters, vol. 26, no. 2, pp. 179–183, 2013.
  • C. Zhang, T. Li, R. P. Agarwal, and M. Bohner, “Oscillation results for fourth-order nonlinear dynamic equations,” Applied Mathematics Letters, vol. 25, no. 12, pp. 2058–2065, 2012.
  • C. Zhang, T. Li, B. Sun, and E. Thandapani, “On the oscillation of higher-order half-linear delay differential equations,” Applied Mathematics Letters, vol. 24, no. 9, pp. 1618–1621, 2011. \endinput