Instability of a Nonlinear Differential Equation of Fifth Order with Variable Delay

Authors: Cemil Tunc

Abstract:

In this paper, we study the instability of the zero solution to a nonlinear differential equation with variable delay. By using the Lyapunov functional approach, some sufficient conditions for instability of the zero solution are obtained.

Keywords: Instability, Lyapunov-Krasovskii functional, delay differential equation, fifth order.

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1333969

References:

[1] El-sgol-ts, L. E., 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.
[2] Ezeilo, J. O. C., Instability theorems for certain fifth-order differential equations. Math. Proc. Cambridge Philos. Soc., no. 2, 84 (1978), 343-350.
[3] Ezeilo, J. O. C., A further instability theorem for a certain fifth-order differential equation. Math. Proc. Cambridge Philos. Soc., no. 3, 86 (1979), 491-49
[4] Ezeilo, J. O. C., Extension of certain instability theorems for some fourth and fifth order differential equations. Atti Accad. Naz. Lincei Rend.Cl. Sci. Fis. Mat. Natur. (8) 66 (1979), no. 4, 239-242.
[5] LaSalle, J.; Lefschetz, S., Stability by Liapunov-s direct method, with applications. Mathematics in Science and Engineering, Vol. 4 Academic Press, New York-London 1961.
[6] Krasovskii, N. N., Stability of motion. Applications of Lyapunov-s second method to differential systems and equations with delay. Translated by J. L. Brenner Stanford University Press, Stanford, Calif. 1963.
[7] Li, Wen-jian; Duan, Kui-chen, Instability theorems for some nonlinear differential systems of fifth order. J. Xinjiang Univ. Natur. Sci. 17 (2000), no. 3, 1-5.
[8] Li, W. J.; Yu, Y. H., Instability theorems for some fourth-order and fifthorder differential equations, (Chinese) J. Xinjiang Univ. Natur. Sci., no. 2, 7(1990), 7-10.
[9] Sadek, A. I., Instability results for certain systems of fourth and fifth order differential equations. Appl. Math. Comput. 145 (2003), no. 2-3, 541-549.
[10] Sun, W. J.; Hou, X., New results about instability of some fourth and fifth order nonlinear systems. (Chinese) J. Xinjiang Univ. Natur. Sci. 16 (1999), no. 4, 14-17.
[11] Tiryaki, A., Extension of an instability theorem for a certain fifth order differential equation. National Mathematics Symposium (Trabzon, 1987). J. Karadeniz Tech. Univ. Fac. Arts Sci. Ser. Math.-Phys. 11 (1988), 225- 227 (1989).
[12] Tunc¸, C., On the instability of solutions of certain nonlinear vector differential equations of fifth order. Panamer. Math. J. 14 (2004), no. 4, 25-30.
[13] Tunc¸, C., An instability result for a certain non-autonomous vector differential equation of fifth order. Panamer. Math. J. 15 (2005), no. 3, 51-58.
[14] Tunc¸, C., Further results on the instability of solutions of certain nonlinear vector differential equations of fifth order. Appl. Math. Inf. Sci. 2 (2008), no. 1, 51-60.
[15] Tunc¸, C., Recent advances on instability of solutions of fourth and fifth order delay differential equations with some open problems. World Scientific Review, Vol. 9, World Scientific Series on Nonlinear Science Series B (Book Series), (2011), 105-116.
[16] Tunc¸, C.,On the Instability of Solutions of Nonlinear Delay Differential Equations of Fourth and Fifth Order. Sains Malaysiana 40(12), (2011), 1455-1459.
[17] Tunc¸, C.,An instability theorem for a certain fifth-order delay differential equation, Filomat, 25 (2011), no. 3, 145-151.
[18] Tunc¸, C., Instability result of a fifth order non-linear delay system. Optoelectronics and Advanced Materials-Rapid Communications. 5 ( 2011), no. 9, 891-894.
[19] Tunc¸, C.; Erdogan, F., On the instability of solutions of certain nonautonomous vector differential equations of fifth order. SUT J. Math. 43 (2007), no. 1, 35-48.
[20] Tunc¸, C.; Karta, M., A new instability result to nonlinear vector differential equations of fifth order. Discrete Dyn. Nat. Soc. 2008, Art. ID 971534, 6 pp.
[21] Tunc¸, C.; S¸ evli, H., On the instability of solutions of certain fifth order nonlinear differential equations. Mem. Differential Equations Math. Phys. 35 (2005), 147-156. Cemil