An Instability Result to a Certain Vector Differential Equation of the Sixth Order

Cemil Tunç

Department of Mathematics, Faculty of Sciences, Yüzüncü Y?l University, Van, Turkey.

DOI: 10.4236/am.2012.39147                

Abstract

The nonlinear vector differential equation of the sixth order with constant delay is considered in this article. New criteria for instability of the zero solution are established using the Lyapunov-Krasovskii functional approach and the differential inequality techniques. The result of this article improves previously known results.

Keywords

Vector; Nonlinear Differential Equation; Sixth Order; Lyapunov-Krasovskii Functional; Instability; Delay

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	E. Tun? and C. Tun?, “On the Instability of Solutions of Certain Sixth-Order Nonlinear Differential Equations,” Nonlinear Stud, Vol. 15, No. 3, 2008, pp. 207-213.
[2]	N. N. Krasovskii, “Stability of Motion. Applications of Lyapunov’s Second Method to Differential Systems and Equations with Delay,” Stanford University Press, Stanford, 1963.
[3]	J. O. C. Ezeilo, “An Instability Theorem for a Certain Sixth Order Differential Equation,” Journal of the Australian Mathematical Society, Vol. 32, No. 1, 1982, pp. 129-133. doi:10.1017/S1446788700024460
[4]	H. O. Tejumola, “Instability and Periodic Solutions of Certain Nonlinear Differential Equations of Orders Six and Seven,” Proceedings of the National Mathematical Centre, National Mathematical Center, Abuja, 2000.
[5]	A. Tiryaki, “An Instability Theorem for a Certain Sixth Order Differential Equation,” Indian Journal of Pure and Applied Mathematics, Vol. 21, No. 4, 1990, pp. 330-333.
[6]	C. Tun?, “An Instability Result for Certain System of Sixth Order Differential Equations,” Applied Mathematics and Computation, Vol. 157, No. 2, 2004, pp. 477-481. doi:10.1016/j.amc.2003.08.046
[7]	C. Tun?, “On the Instability of Certain Sixth-Order Nonlinear Differential Equations,” Electronic Journal of Differential Equations, No. 117, 2004, p. 6.
[8]	C. Tun?, “On the Instability of Solutions to a Certain Class of Non-Autonomous and Non-Linear Ordinary Vector Differential Equations of Sixth Order,” Albanian Journal of Mathematics, Vol. 2, No. 1, 2008, pp. 7-13.
[9]	C. Tun?, “A Further Result on the Instability of Solutions to a Class of Non-Autonomous Ordinary Differential Equations of Sixth Order,” Applications & Applied Mathematics, Vol. 3, No. 1, 2008, pp. 69-76.
[10]	C. Tun?, “New Results about Instability of Nonlinear Ordinary Vector Differential Equations of Sixth and Seventh Orders,” Dynamics of Continuous, Discrete and Impulsive Systems, Series A: Mathematical Analysis, Vol. 14, No. 1, 2007, pp. 123-136.
[11]	C. Tun?, “Instability for a Certain Functional Differential Equation of Sixth Order,” Journal of the Indonesian Mathematical Society, Vol. 17, No. 2, 2011, pp. 123-128.
[12]	C. Tun?, “Instability Criteria for Solutions of a Delay Differential Equation of Sixth Order,” Journal of Advanced Research in Applied Mathematics, Vol. 4, No. 2, 2012, pp. 1-7.
[13]	C. Tun?, “An Instability Theorem for a Certain Sixth Order Nonlinear Delay Differential Equation,” Journal of the Egyptian Mathematical Society, 2012, in press.
[14]	R. Bellman, “Introduction to Matrix Analysis,” 2nd Edition, In: G. Golub, Eds., Classics in Applied Mathematics, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, 1997.