Abstract and Applied Analysis

Oscillation Criteria of Third-Order Nonlinear Impulsive Differential Equations with Delay

Xiuxiang Liu

Full-text: Open access

Abstract

This paper deals with the oscillation of third-order nonlinear impulsive equations with delay. The results in this paper improve and extend some results for the equations without impulses. Some examples are given to illustrate the main results.

Article information

Source
Abstr. Appl. Anal., Volume 2013 (2013), Article ID 405397, 8 pages.

Dates
First available in Project Euclid: 18 April 2013

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1366306752

Digital Object Identifier
doi:10.1155/2013/405397

Mathematical Reviews number (MathSciNet)
MR3034957

Zentralblatt MATH identifier
1267.34064

Citation

Liu, Xiuxiang. Oscillation Criteria of Third-Order Nonlinear Impulsive Differential Equations with Delay. Abstr. Appl. Anal. 2013 (2013), Article ID 405397, 8 pages. doi:10.1155/2013/405397. https://projecteuclid.org/euclid.aaa/1366306752


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References

  • D. D. Bainov and P. S. Simeonov, Impulsive Differential Equations: Periodic Solutions and Applications, Longman Scientific and Technical, 1993.
  • G. Ballinger and X. Liu, “Existence, uniqueness and boundedness results for impulsive delay differential equations,” Applicable Analysis, vol. 74, no. 1-2, pp. 71–93, 2000.
  • V. Lakshmikantham, D. D. Baĭnov, and P. S. Simeonov, Theory of Impulsive Differential Equations, vol. 6, World Scientific Publishing, New Jersey, NJ, USA, 1989.
  • A. M. Samoĭlenko and N. A. Perestyuk, Impulsive Differential Equations, vol. 14, World Scientific Publishing, New Jersey, NJ, USA, 1995.
  • Y. S. Chen and W. Z. Feng, “Oscillations of second order nonlinear ODE with impulses,” Journal of Mathematical Analysis and Applications, vol. 210, no. 1, pp. 150–169, 1997.
  • R. P. Agarwal, F. Karakoc, and A. Zafer, “A survey on oscillation of impulsive ordinary differential equations,” Advances in Differential Equations, vol. 2010, pp. 1–52, 2010.
  • W.-H. Mao and A.-H. Wan, “Oscillatory and asymptotic behavior of solutions for nonlinear impulsive delay differential equations,” Acta Mathematicae Applicatae Sinica, vol. 22, English Series, no. 3, pp. 387–396, 2006.
  • L. Erbe, A. Peterson, and S. H. Saker, “Asymptotic behavior of solutions of a third-order nonlinear dynamic equation on time scales,” Journal of Computational and Applied Mathematics, vol. 181, no. 1, pp. 92–102, 2005.
  • L. Erbe, A. Peterson, and S. H. Saker, “Oscillation and asymptotic behavior of a third-order nonlinear dynamic equation,” The Canadian Applied Mathematics Quarterly, vol. 14, no. 2, pp. 124–147, 2006.
  • L. Erbe, A. Peterson, and S. H. Saker, “Hille and Nehari type criteria for third-order dynamic equations,” Journal of Mathematical Analysis and Applications, vol. 329, no. 1, pp. 112–131, 2007.
  • S. H. Saker, “Oscillation criteria of third-order nonlinear delay differential equations,” Mathematica Slovaca, vol. 56, no. 4, pp. 433–450, 2006.
  • S. H. Saker and J. Džurina, “On the oscillation of certain class of third-order nonlinear delay differential equations,” Mathematica Bohemica, vol. 135, no. 3, pp. 225–237, 2010.
  • J. Yu and J. Yan, “Positive solutions and asymptotic behavior of delay differential equations with nonlinear impulses,” Journal of Mathematical Analysis and Applications, vol. 207, no. 2, pp. 388–396, 1997.
  • T. Candan and R. S. Dahiya, “Oscillation of third order functional differential equations with delay,” in Proceedings of the 5th Mississippi State Conference on Differential Equations and Computational Simulations, vol. 10, p. 7988, 2003.
  • I. V. Kamenev, “An integral criterion for oscillation of linear differential equations of second order,” Matematicheskie Zametki, vol. 23, pp. 136–138, 1978.
  • G. Ladas, Y. G. Sficas, and I. P. Stavroulakis, “Necessary and sufficient conditions for oscillations of higher order delay differential equations,” Transactions of the American Mathematical Society, vol. 285, no. 1, pp. 81–90, 1984.
  • B. Baculíková, E. M. Elabbasy, S. H. Saker, and J. Džurina, “Oscillation criteria for third-order nonlinear differential equations,” Mathematica Slovaca, vol. 58, no. 2, pp. 201–220, 2008.