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## Proyecciones (Antofagasta)

##
*Print version* ISSN 0716-0917

### Proyecciones (Antofagasta) vol.32 no.3 Antofagasta Sept. 2013

#### http://dx.doi.org/10.4067/S0716-09172013000300004

Proyecciones Journal of Mathematics

Vol. 32, N^{o} 3, pp. 245-258, September 2013.

Universidad Católica del Norte Antofagasta - Chile

**On the instability of a kind of vector functional differential equations of the eighth order with multiple deviating arguments**

**Cemil Tunç**

*Yüzüncü Yil University, Turkey *

**ABSTRACT**

*In this paper, we investigate the instability of solutions to a certain class of nonlinear vector functional differential equations of the eighth order with n-deviating arguments. We employ the Lyapunov-Krasovskii functional approach and base on the Krasovskii criteria to prove two new theorems on the topic. Our results improve certain results in the literature from scalar functional differential equations to their vectorial forms.*

**Subjclass[2010] :** *34K20.*

**Keywords :** *Instability, Lyapunov functional, vector functional differential equation, eighth order, multiple deviating arguments.*

**REFERENCES**

[1] R. Bellman, Introduction to matrix analysis. Reprint of the second (1970) edition. With a foreword by Gene Golub. Classics in Applied Mathematics, 19. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, (1997).

[2] H. Bereketoglu, On the instability of trivial solutions of a class of eighth-order differential equations, Indian J. Pure Appl. Math., 22, no. 3, pp. 199—202, (1991).

[3] L. E. El'sgol'ts, Introduction to the theory of differential equations with deviating arguments. Translated from the Russian by Robert J. McLaughlin Holden-Day Inc., San Francisco, Calif.-London-Amsterdam, (1966).

[4] S. A. Iyase, Periodic solutions of certain eighth order differential equations, J. Nigerian Math. Soc., 14/15 (1995/96), 31—39.

[5] N. N. Krasovskii, On conditions of inversion of A. M. Lyapunov's theorems on instability for stationary systems of differential equations, (Russian) Dokl. Akad. Nauk SSSR (N.S.) 101, pp. 17—20, (1955).

[6] C. Tunc, Instability of solutions of a certain non-autonomous vector differential equation of eighth-order, Ann. Differential Equations 22, no. 1, pp. 7—12, (2006).

[7] C. Tunc, Nonexistence of nontrivial periodic solutions to a class of nonlinear differential equations of eighth order, Bull. Malays. Math.Sci. Soc., (2) 32, no. 3, pp. 307—311, (2009).

[8] C. Tunc, On the instability of solutions of an eighth order nonlinear differential equation of retarded type, Proyecciones 30, no. 1, pp. 43—50, (2011).

[9] C. Tunc, An instability theorem for solutions of a kind of eighth order nonlinear delay differential equations, World Applied Sciences Journal 12 (5), pp. 619—623, (2011).

[10] C. Tunc, Instability of solutions for nonlinear differential equations of eighth order with multiple deviating arguments, J. Appl. Math.Inform., 31, no. 4-5, (2012).

[11] C. Tunç and E. Tunc, An instability theorem for a class of eighth-order differential equations, (Russian) Differ. Uravn., 42, no. 1, 135-138, 143; translation in Differ. Equ. 42, no. 1, pp. 150—154; (2006).

**Cemil Tunç**

Department of Mathematics,

Faculty of Sciences, Yüzmicü YilUniversity,

65080, van-Turkey

e-mail : cemtunc@yahoo.com

*Received : August 2012. Accepted : May 2013*