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Applied Mathematics,2012, 3, 997-1000doi:10.4236/am.2012.39147 Published Online September 2012 (http://www.SciRP.org/journal/am)

An Instability Result to a Certain Vector DifferentialEquation of the Sixth Order

Cemil TunDepartment of Mathematics, Faculty of Sciences, Yznc Y l University, Van, Turkey

Email: cemtunc@yahoo.com

Received August 12, 2012; revised September 12, 2012; accepted September 17, 2012

ABSTRACT

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

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

1. Introduction

In 2008,E. Tun and C. Tun [1] proved a theorem onthe instability of the zero solution of the sixth ordernonlinear vector differential equation

(6) (5) (4)

(5)

(5)

, , ,

, , , 0

X t AX t BX t

E X t X t X t X t F X t X t

G X t X t X t X t H X t

.

(1)

The objective of this article is to investigate the insta-bility of the zero solution of the sixth order nonlinearvector differential equation with constant delay, 0,

(6) (5) (4)

(5)

(5)

, , , ,

, , , ,

0,

X t AX t BX t

E X t X t X t X t X t

F X t X t

G X t X t X t X t X t

H X t

(2)

by the Lyapunov-Krasovskii functional approach underassumptions A and B are constant;nX n n -symmetric matrices; E, F and G are continuous n n -symmetric matrix functions depending, in each case, onthe arguments shown; and His continuous. Let

: ,n nH

0H 0H X

,denote the Jacobian matrix

corresponding to H X that is,

,iHj

hJ X

x

, 1,2, ,i j n ,

where 1 2, , , nx x x and are the com-ponents of X and H, respectively. We also assume that

the Jacobian matrix

1 2, , , nh h h

HJ X

(4)

(5)1

, , ,

, , ,

ik k

k

b x

x t

x

x

t

, ,

, ,

t

H

t Y

t Y

J X

exists and is continuous.It should be noted that Equation (2) is the vector ver-

sion for systems of real nonlinear differential equationsof the sixth order

(6) (5)

1 1

(5) (5)1 1 1

1

11

(5)1 1

1

1

, , , ,

, ,

, ,

, , 0, 1,2, , .

n n

i ik kk k

n

ik k kk

n

ik k kk

n

ik k kk

i n

x a x

e x x t x x t x

f x x

g x x t x t x

h x t x i n

We can write Equation (2) in the system form

, ,

, , ,

d ,t

S AS BU

E X X S t W F Y Z

G X X U S t Y

H X s Y s s

(3)

which is obtained from (2) by setting ,X Y ,X Z ,X W (4)X U and .(5)X S Throughout what fol-

lows ,, ,X t Y t S t are abbreviated as , , , ,X Y S respectively.

Consider, in the case the linear differentialequation of the sixth order:

1,n

(4)2x a

(6) (5)1 3 4 5 6 0,x a x a x a x a x a x (4)

where are real constants.1 2 6, , ,a a a

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It is known from the qualitative properties of solutionsof Equation (4) that the zero solution of this equation isunstable if and only if the associated auxiliary equation

6 5 4 3 21 2 3 4 5 6 0a a a a a a (5)

has at the least one root with a positive real part. Theexistence of such a root depends on (though not alwaysall of) the coefficients 1 2 6 in Equation (5).Basing on the relations between the roots and the coeffi-cients of Equation (5) it can be said that if

, , ,a a a

2 11 5 3

10,

4a a a a 1

or

24 2 6

1,

2a a a 0, (6)

then at the least one root of Equation (5) has a positivereal part for arbitrary values of and or

and respectively.

1,a 3a 5a 2,a

4a 6It should be noted that Equation (2) is an n-dimen-sional generalization of Equation (4), and when we es-tablish our assumptions, we will take into considerationthe estimates in (6). The symbol

,a

,X Y correspondingto any pair X,Y in stands for the usual scalar prod-n

uct1

,n

i ii

xy

and ,i A are the ei- 1, 2, , ,i n

genvalues of the -matrixn n .A It is worth mentioning that using theLyapunov func-

tions or Lyapunov-Krasovskii functionals and based onthe Krasovskii properties [2], the instability of the solu-

tions of the sixth order nonlinear scalar differential equa-tions and the sixth order vector differential equationswithout delay were discussed by Ezeilo [3], Tejumola [4],Tiryaki [5] and Tun [6-13]. The aim of this paper is toimprove the results of ([1,3]) form the scalar and vectordifferential equations without delay to the sixth ordernonlinear vector differential equation with delay, Equa-tion (2).

2. Main Result

First, we give an algebraic result.Lemma. Let D be a real symmetric -matrix.

Then for any

n nn

X 2 2, ,d dX DX X X

where d and d are the least and greatest eigenval-ues of respectively (Bellman14).

,D

Let be given, and let0r ,0 , nC C r with

0maxr s

, .s C

For define by0H HC C

: .HC C H,

If is continuous, then,for eacht in

: , nx r A 0, ,

0 A A tx inC is defined by

, 0,tx s x t s r s t 0.

Let G be an open subset ofC and consider the generalautonomous delay differential system with finite delay

, 0 0,

0, 0,

t tx F x F x x t

r t

,

n

where is continuous and maps closed andbounded sets into bounded sets. It follows from theseconditions onF that each initial value problem

:F G

,tx F x 0x G

has a unique solution defined on some interval 0, ,A 0 A . This solution will be denoted by .x so that .0x

Definition. The zero solution, of0,x tx F x is stable if for each 0 there exists 0 such that implies that x t for all

The zero solution is said to be unstable if it is notstable.

0.t

The result to be proved is the following theorem.Theorem. In addition to the basic assumptions im-

posed onA, B, E, F, G andH that appear in Equation (2),we suppose that there are constants 1 and,a 2,a 6a such that the following conditions hold:The matricesA, B, E, F, G and H X aresymmetric

and 1 1 1sgn sgn ,i A a a a 2,B ai 0 0,H 0,H X when 0,X and i Ha J . 06

2

1

1

. sgn ( . 0,4

i iG a Ea

1

1,2, , .i n If

6

,na

then the zero solution of Equation (2) is unstable.Remark. It is worth mentioning that there is no sign

restriction on eigenvalues ofF, and it is obvious that forthe delay case our assumptions also have a very simpleform and their applicability can be easily verified.

Proof.Define a Lyapunov-Krasovskii functional

1 1. , , , , ,t t t t t tV V X Y Z W U S :

0 2

1 0 1sgn d d ,t

t s

V V a Y s

where

1 1

0

0 0

02

, d , d

, , ,

1, , ,

2

1, d d ,

2

t

t s

V F Y Y Y H X X

S Y AU Y BW Y

BZ Z Z U AW Z

W W Y s

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where is a certain positive constant and will be de-termined later in the proof.

It follows that

1 0,0,0,0,0,0 0V

and

1 1 2

1 1 1 2

1 2

2 2 2

2 2 2

0,0, sgn ,0, 1 ,0

1sgn , sgn sgn , 1 sgn

2

1, sgn , 1

2

1 11 1 0

2 2

V a a

B a a a a

B a a

a a a

1a

for all arbitrary 0, so that the propertyof Krasovskii [2] holds.

n 1P

Using a basic calculation, the time derivative ofalong solutions of (3)results in

1V

1

1

0

1

1

0

02

d. , ,

d

, . , .

d( ) , d

d

d, sgn

d

d ,

dd d .

d

t

H

t

t

t s

V AW W F Y Z Yt

,H X Y E W Y G Y Y

F Y Y Yt

H X X d at

J X s Y s s Y

Y st

The following estimates can be easily calculated:

1

0

d, d ,

d,H X X H X Y

t

1

0

d, d ,

d,F Y Y Y F Y Z Y

t

6 6

22

6 6

d ,

d

d d

1 1d

2 2

t

H

t

t

H

t

t t

t t

t

t

J X s Y s s Y

Y J X s Y s s

na Y Y s s na Y Y s s

na Y na Y s s

and

0

2 22dd d d

d

t t

t s t

Y s Y Yt

so that

1 1

22

6 6

d

. , . , . , sgd

1 1d .

2 2

t

t

V AW W E W Y G Y Yt

na Y na Y

na

Using the assumptions of the theorem, we get

1 1 1

22

6 6

2

1 1

1 1

2

1 6

2

6

1

1

2

6

d. , (.) , . , sgn

d

1 1d

2 2

1 1. sgn . , .

2 4

1. , sgn

2

1d

2

1. , . . , sgn

4

1

2

t

t

t

t

V aW W E W Y G Y Y at

na Y na Y

a W E Y a E Y E Ya a

G Y Y a na Y

na Y

E Y E Y G Y Y aa

na Y

2

6

22

6 6

1d

2

1 1d .

2 2

t

t

t

t

na Y

na Y na Y

Let

6

1

2na

so that

21 6d

. .d

V nat

Y

If6

,na

then, for a positive constant we

have

1,k

2

1 1

d. 0

dV k Y

t

so that the property 2P of Krasovskii [2] holds.It is seen that

1d

. 0 , 0, 0d

0, 0for all 0

V Y X Z Y W Zt

U W S U t

,

so that

, 0X Y Z W U S .

Using these estimates in (3) and the assumptions of thetheorem, we get 0H 0. Thus, we have

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1000

0X Y Z W U T

for all So that theproperty

0.t 3P ofKrasovskii [2] holds.

The proof of the theorem is complete.Example. For the particular case in Equation

(2), we have2n

10 1,

1 10A

1 9,A 2 11,A

1 1sgn 9 sgn ,i 1A a a a

2 21 1

2 22 2

42

1.

0

40 2

1

x sE ,

x s

1 2 21 1

4. 21

E ,x s

2 2 22 2

4. 2

1E ,

x s

2

. 3i E 6,

,

2 21 1

2 22 2

6 0.

0 6

x sG

x s

1 ,s

2,s

2 21 1. 6G x

2 22 2. 6G x

2

1

1

1. sgn . 5 0,

4i iG a E

a

1

2

9,

9

x tH X t

x t

9 0

,0 9

HJ X

. 9i HJ ,

6 9 .i Ha J 0, 1,2 .i

If

5,

9 2

then all the assumptions of the theorem hold.

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 ofLyapunovs Second Method to Differential Systems andEquations with Delay, Stanford University Press, Stan-

ford, 1963.[3] J. O. C. Ezeilo, An Instability Theorem for a Certain

Sixth Order Differential Equation,J ournal of the Aus-tralian Mathematical Society, Vol. 32, No. 1, 1982, pp.129-133. doi:10.1017/S1446788700024460

[4] H. O. Tejumola, Instability and Periodic Solutions ofCertain Nonlinear Differential Equations of Orders Sixand Seven, Proceedings of the National MathematicalCentre, National Mathematical Center, Abuja, 2000.

[5] A. Tiryaki, An Instability Theorem for a Certain SixthOrder Differential Equation, Indian J ournal of Pure andApplied Mathematics, Vol. 21, No. 4, 1990, pp. 330-333.

[6] C. Tun, An Instability Result for Certain System ofSixth Order Differential Equations, Applied Mathematicsand 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-OrderNonlinear Differential Equations, Electronic Journal ofDifferential Equations, No. 117, 2004, p. 6.

[8] C. Tun, On the Instability of Solutions to a CertainClass of Non-Autonomous and Non-Linear OrdinaryVector 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 Solutionsto a Class of Non-Autonomous Ordinary DifferentialEquations of Sixth Order, Applications & AppliedMathematics, Vol. 3, No. 1, 2008, pp. 69-76.

[10] C. Tun, New Results about Instability of NonlinearOrdinary Vector Differential Equations of Sixth andSeventh Orders, Dynamics of Continuous, Discrete andImpulsive Systems, Series A: Mathematical Analysis, Vol.14, No. 1, 2007, pp. 123-136.

[11] C. Tun, Instability for a Certain Functional DifferentialEquation of Sixth Order, J ournal of the IndonesianMathematical Society, Vol. 17, No. 2, 2011, pp. 123-128.

[12] C. Tun, Instability Criteria for Solutions of a DelayDifferential Equation of Sixth Order, J ournal of Ad-vanced Research in Applied Mathematics, Vol. 4, No. 2,2012, pp. 1-7.

[13] C. Tun, An Instability Theorem for a Certain SixthOrder Nonlinear Delay Differential Equation,J ournal ofthe Egyptian Mathematical Society, 2012, in Press.

[14] R. Bellman, Introduction to Matrix Analysis, 2ndEdi-tion, In: G. Golub, Ed., Classics in Applied Mathematics,Society for Industrial and Applied Mathematics (SIAM),Philadelphia, 1997.

http://dx.doi.org/10.1017/S1446788700024460http://dx.doi.org/10.1017/S1446788700024460http://dx.doi.org/10.1016/j.amc.2003.08.046http://dx.doi.org/10.1016/j.amc.2003.08.046http://dx.doi.org/10.1016/j.amc.2003.08.046http://dx.doi.org/10.1017/S1446788700024460