J. Indones. Math. Soc. Vol. 17, No. 2 (2011), pp. 123–128.
INSTABILITY FOR A CERTAIN FUNCTIONAL DIFFERENTIAL EQUATION OF SIXTH ORDER Cemil Tunc ¸ Department of Mathematics, Faculty of Sciences, Y¨ uz¨ unc¨ u Yıl University, 65080, Van -Turkey,
[email protected]
Abstract. Sufficient conditions are obtained for the instability of the zero solution of a certain sixth order nonlinear functional differential equation by the LyapunovKrasovskii functional approach. Key words: Instability, Lyapunov-Krasovskii functional, delay differential equation, sixth order.
Abstrak. Dalam paper ini diperoleh syarat cukup untuk ketakstabilan solusi nol dari persamaan diferensial fungsional nonlinear orde keenam tertentu dengan pendekatan fungsional Lyapunov-Krasovskii. Kata kunci: Ketakstabilan, fungsional Lyapunov-Krasovskii, persamaan diferensial tunda, orde keenam.
1. Introduction First, in 1982, Ezeilo [2] discussed instability of the zero solution of the sixth order nonlinear differential equation without delay, x(6) (t) + a1 x(5) (t) + a2 x(4) (t) + e(x(t), x0 (t), x00 (t), x000 (t), x(4) (t), x(5) (t))x000 (t) +f (x0 (t))x00 (t) + g(x(t), x0 (t), x00 (t), x000 (t), x(4) (t), x(5) (t))x0 (t) + h(x(t)) = 0. Later, in 1990, Tiryaki [4] proved an instability theorem for the sixth order nonlinear differential equation without delay, x(6) (t) + a1 x(5) (t) + f1 (x(t), x0 (t), x00 (t), x000 (t), x(4) (t), x(5) (t))x(4) (t) + f2 (x00 (t))x000 (t) +f3 (x(t), x0 (t), x00 (t), x000 (t), x(4) (t), x(5) (t))x00 (t) + f4 (x0 (t)) + f5 (x(t)) = 0. (1) 2000 Mathematics Subject Classification: 34K20. Received: 11-07-2011, revised: 18-11-2011, accepted: 18-11-2011. 123
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Cemil Tunc ¸
On the other hand, recently, Tun¸c [10] established some sufficient conditions to ensure the instability of the zero solution of the sixth order nonlinear delay differential equation, x(6) (t) + a1 x(5) (t) + a2 x(4) (t) +e(x(t − r), x0 (t − r), x00 (t − r), x000 (t − r), x(4) (t − r), x(5) (t − r))x000 (t) + f (x0 (t))x00 (t) +g(x(t − r), x0 (t − r), x00 (t − r), x000 (t − r), x(4) (t − r), x(5) (t − r))x0 (t) + h(x(t − r)) = 0. In this paper, instead of Eq. (1), we consider the sixth order nonlinear delay differential equation x(6) (t) + a1 x(5) (t) + f2 (x(t − r), x0 (t − r), x00 (t − r), x000 (t − r), x(4) (t − r), x(5) (t − r))x(4) (t) +f3 (x00 (t))x000 (t) + f4 (x(t − r), x0 (t − r), x00 (t − r), x000 (t − r), x(4) (t − r), x(5) (t − r))x00 (t) +f5 (x0 (t − r)) + f6 (x(t − r)) = 0. (2) We write Eq. (2) in system form as follows x01 = x2 , x02 = x3 , x03 = x4 , x04 = x5 , x05 = x6 , x06 = −a1 x6 − f2 (x1 (t − r), x2 (t − r), x3 (t − r), x4 (t − r), x5 (t − r), x6 (t − r))x5 −f3 (x3 )x4 − f4 (x1 (t − r), x2 (t − r), x3 (t − r), x4 (t − r), x5 (t − r), x6 (t − r))x3 −f5 (x2 ) +
Rt t−r
f50 (x2 (s))x3 (s)ds − f6 (x1 ) +
Rt
f60 (x1 (s))x2 (s)ds,
t−r
(3) which was obtained as usual by setting x = x1 , x0 = x2 , x00 = x3 , x000 = x4 , x(4) = x5 and x(5) = x6 from (2), where r is a positive constant, a1 is a constant, the primes in Eq. (2) denote differentiation with respect to t, t ∈ 0. Let (x1 , x2 , x3 , x4 , x5 , x6 ) = (x1 (t), x2 (t), x3 (t), x4 (t), x5 (t), x6 (t)) be an arbitrary solution of (3). By an elementary differentiation, time derivative of the Lyapunov functional V in (4) along the solutions of (3)yields
Instability for a Certain Functional Dif. Eq.
V˙
≡
d dt V
127
(x1t , x2t , x3t , x4t , x5t , x6t )
= x25 + f2 (x1 (t − r), ..., x6 (t − r))x3 x5 + f4 (x1 (t − r), ..., x6 (t − r))x23 − f60 (x1 )x22 −x3
Rt
f50 (x2 (s))x3 (s)ds − x3
t−r
Rt
f60 (x1 (s))x2 (s)ds
t−r
−λ1 rx22 + λ1
Rt
x22 (s)ds − λ2 rx23 + λ2
t−r
Rt
x23 (s)ds
t−r
= x5 + 21 f2 (x1 (t − r), ..., x6 (t − r))x3
2
+{f4 (x1 (t − r), ..., x6 (t − r)) − 41 f22 (x1 (t − r), ..., x6 (t − r))}x23 Rt
−f60 (x1 )x22 − x3
f50 (x2 (s))x3 (s)ds − x3
t−r
−λ1 rx22 + λ1
Rt
Rt
f60 (x1 (s))x2 (s)ds
t−r
x22 (s)ds − λ2 rx23 + λ2
t−r
Rt
x23 (s)ds.
t−r
Making use of the assumptions −a5 ≤ f50 (x2 ) ≤ −¯ a5 < 0, −a6 ≤ f60 (x1 ) ≤ −¯ a6 < 0 2 2 and the estimate 2 |mn| ≤ m + n , we get −x3
Rt
f60 (x1 (s))x2 (s)ds ≥ |x3 |
t−r
Rt
f60 (x1 (s)) |x2 (s)| ds
t−r
≥ − 21 a6 rx23 − 21 a6
Rt
x22 (s)ds
t−r
and −x3
Rt
f50 (x2 (s))x3 (s)ds ≥ |x3 |
t−r
Rt
f50 (x2 (s)) |x3 (s)| ds
t−r
≥
− 12 a5 rx23
− 12 a5
Rt
x23 (s)ds.
t−r
Hence V˙
≥ (¯ a6 − λ1 r)x22 + {δ − (λ2 + 21 a5 + 21 a6 )r}x23 Rt 2 Rt 2 + λ1 − 21 a6 x2 (s)ds + λ2 − 21 a5 x3 (s)ds. t−r
Let λ1 =
1 2 a6 ,
λ2 =
1 2 a5
and r
0 2 2
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Cemil Tunc ¸
On the other hand, it follows that x2 = 0, which implies that
d dt V
(x1t , x2t , x3t , x4t , x5t , x6t ) = 0 if and only if
x1 = ξ1 (constant), x2 = x3 = x4 = x5 = x6 = 0.
(5)
The substitution of the estimate (5) in system (3) leads f6 (ξ1 ) = 0. In view of the assumption of the theorem, f6 (0) = 0, f6 (x1 ) 6= 0, (x1 6= 0), it is seen that f6 (ξ1 ) = 0 if and only if ξ1 = 0. Hence, we can easily conclude that the only solution (x1 , ..., x6 ) = (x1 (t), ..., x6 (t)) of system (3) which satisfies d (t ≥ 0), is the trivial solution (0, 0, 0, 0, 0, 0). Thus, the dt V (x1t , x2t , ..., x6t ) = 0, d estimate dt V (x1t , x2t , x3t , x4t , x5t , x6t ) = 0 implies x1 = x2 = x3 = x4 = x5 = x6 = 0. Hence, we see that the functional V satisfies the property (P3 ). In view of the whole discussion made above, it follows that the functional V has all Krasovskii properties, (P1 ), (P2 ) and (P3 ). Thus, one can conclude that the zero solution of Eq. (2) is unstable. The proof of the theorem is completed.
References ` ` Introduction to the theory of differential equations with deviating arguments, [1] El’sgol’ts, L. E., Translated from the Russian by Robert J. McLaughlin Holden-Day, Inc., San Francisco, Calif.-London-Amsterdam, 1966. [2] Ezeilo, J. O. C., ”An instability theorem for a certain sixth order differential equation”, J. Austral. Math. Soc. Ser. A 32 (1982), no. 1, 129-133. [3] 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. [4] Tiryaki, A., ”An instability theorem for a certain sixth order differential equation”, Indian J. Pure Appl.Math. 21 (1990), no. 4, 330-333. [5] Tun¸c, C., ”An instability result for certain system of sixth order differential equations”, Appl. Math. Comput. 157 (2004), no. 2, 477-481. [6] Tun¸c, C., ”On the instability of certain sixth-order nonlinear differential equations”, Electron. J. Differential Equations (2004), no. 117, 6 pp. [7] Tun¸c, C., ”On the instability of solutions to a certain class of non-autonomous and non-linear ordinary vector differential equations of sixth order”, Albanian J. Math. 2 (2008), no. 1, 7-13. [8] Tun¸c, C., ”A further result on the instability of solutions to a class of non-autonomous ordinary differential equations of sixth order”, Appl.Appl. Math. 3 (2008), no. 1, 69-76. [9] Tun¸c, C., ”New results about instability of nonlinear ordinary vector differential equations of sixth and seventh orders”, Dyn. Contin. Discrete Impuls.Syst. Ser. A Math. Anal. 14 (2007), no. 1, 123-136. [10] Tun¸c, C., ”An instability theorem for a certain sixth order nonlinear delay differential equation”, J. Egyptian Math. Soc., (2011), (in press). [11] Tun¸c, E. and Tun¸c, C., ”On the instability of solutions of certain sixth-order nonlinear differential equations”, Nonlinear Stud. 15 (2008), no. 3, 207-213.
