International Journal of Differential Equations

A Stability Result for the Solutions of a Certain System of Fourth-Order Delay Differential Equation

A. M. A. Abou-El-Ela, A. I. Sadek, A. M. Mahmoud, and R. O. A. Taie

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Abstract

The main purpose of this work is to give sufficient conditions for the uniform stability of the zero solution of a certain fourth-order vector delay differential equation of the following form: X(4)+F(X˙,X¨)X+Φ(X¨)+G(X˙(t-r))+H(X(t-r))=0. By constructing a Lyapunov functional, we obtained the result of stability.

Article information

Source
Int. J. Differ. Equ., Volume 2015 (2015), Article ID 618359, 11 pages.

Dates
Received: 25 September 2014
Revised: 11 February 2015
Accepted: 13 February 2015
First available in Project Euclid: 20 January 2017

Permanent link to this document
https://projecteuclid.org/euclid.ijde/1484881430

Digital Object Identifier
doi:10.1155/2015/618359

Mathematical Reviews number (MathSciNet)
MR3335444

Zentralblatt MATH identifier
1326.34113

Citation

Abou-El-Ela, A. M. A.; Sadek, A. I.; Mahmoud, A. M.; Taie, R. O. A. A Stability Result for the Solutions of a Certain System of Fourth-Order Delay Differential Equation. Int. J. Differ. Equ. 2015 (2015), Article ID 618359, 11 pages. doi:10.1155/2015/618359. https://projecteuclid.org/euclid.ijde/1484881430


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References

  • A. M. Lyapunov, Stability of Motion, Academic Press, New York, NY, USA, 1966.MR0208093
  • N. N. Krasovskii, “On conditions of inversion of A. M. Lyapunov's theorems on instability for stationary systems of differential equations,” Doklady Akademii Nauk SSSR (N.S.), vol. 101, pp. 17–20, 1955 (Russian).MR0068064
  • T. Yoshizawa, Stability Theory by Lyapunov's Second Method, The Mathematical Society of Japan, 1966.MR0208086
  • R. Reissig, G. Sansone, and R. Conti, Nonlinear Differential Equations of Higher-Order, Noordhoff International Publishing, Leyden, The Netherlands, 1974, translated from the German.
  • A. M. Abou-El-Ela and A. I. Sadek, “A stability result for the solutions of a certain system of fourth-order differential equations,” Annals of Differential Equations, vol. 6, no. 2, pp. 131–141, 1990.MR1070404
  • A. M. A. Abou-El-Ela and A. I. Sadek, “A stability theorem for a certain fourth-order vector differential equation,” Annals of Differential Equations, vol. 10, no. 2, pp. 125–134, 1994.MR1296540
  • A. M. A. Abou-El-Ela and A. I. Sadek, “On the asymptotic behaviour of solutions of certain non-autonomous differential equations,” Journal of Mathematical Analysis and Applications, vol. 237, no. 1, pp. 360–375, 1999.MR1708179
  • H. Bereketoglu and C. Kart, “Some results for a certain third-order nonlinear ordinary differential equation,” Bulletin Mathématique de la Société des Sciences Mathématiques de Roumanie, vol. 39, no. 1–4, pp. 77–83, 1996.
  • A. I. Sadek, “On the stability of solutions of certain fourth-order delay differential equations,” Applied Mathematics and Computation, vol. 148, no. 2, pp. 587–597, 2004.MR2015393
  • C. Tunç, “Some stability results for the solutions of certain fourthorder delay differential equations,” Journal of Difference Equations and Applications, vol. 4, pp. 165–174, 2005.
  • C. Tunç, “On stability of solutions of certain fourth-order delay differential equations,” Applied Mathematics and Mechanics. English Edition, vol. 27, no. 8, pp. 1141–1148, 2006.MR2286115
  • C. Tunç, “On asymptotic stability of solutions to third-order nonlinear differential equations with retarded argument,” Communications in Applied Analysis, vol. 11, no. 3-4, pp. 515–527, 2007.MR2368199
  • C. Tunç, “On the stability of solutions to a certain fourth-order delay differential equation,” Nonlinear Dynamics, vol. 51, no. 1-2, pp. 71–81, 2008.MR2362370
  • A. M. A. Abou-El-Ela, A. I. Sadek, and A. M. Mahmoud, “On the stability of solutions to a certain fourth-order vector delay differential equation,” Annals of Differential Equations, vol. 28, no. 1, pp. 1–10, 2012.MR2962498
  • T. A. Burton, Stability and Periodic Solutions of Ordinary and Functional Differential Equations, Academic Press, New York, NY, USA, 1985.MR837654
  • R. Bellman, Introduction to Matrix Analysis, Classics in Applied Mathematics 19, reprint of the second edition (1970), with a foreword by Gene Golub, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, Pa, USA, 1997. \endinput