stability in m -dimensional linear delay difference system

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Stability in m-dimensional linear delay differencesystemX. H. Tang a & Zhiyuan Jiang aa School of Mathematical Science and Computing Technology, Central South University,Changsha, 410083, People's Republic of ChinaPublished online: 19 Sep 2007.

To cite this article: X. H. Tang & Zhiyuan Jiang (2007): Stability in m-dimensional linear delay difference system, Journal ofDifference Equations and Applications, 13:10, 927-944

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Stability in m-dimensional linear delaydifference system†

X. H. TANG* and ZHIYUAN JIANG

School of Mathematical Science and Computing Technology, Central South University,Changsha 410083, People’s Republic of China

(Received 30 October 2006; revised 12 February 2007; in final form 9 April 2007)

In this paper, we give some sufficient conditions for the zero solution of an m-dimensional delaydifference equation of the form

xðnþ 1Þ2 xðnÞ ¼ AðnÞxðn2 kÞ; n ¼ 0; 1; 2; . . .

to be uniformly stable and asymptotically stable.

Keywords: Stability; m-Dimensional; Delay difference equation; Uniformly stable

2000 Mathematics Subject Classification: 39 A

1. Introduction

Consider the m-dimensional linear delay difference system

xðnþ 1Þ2 xðnÞ ¼ AðnÞxðn2 kÞ; n ¼ 0; 1; 2; . . . ; ð1Þ

where {x(n)} is an m-dimensional vector sequence, {A(n)} is an m £ m matrix sequence, k is

a nonnegative integer.

In the one-dimensional case of (1), the stability of the zero solution of equation (1) has

been much studied [1,2,4–6,8,10]. In the multi-dimensional case, the results dealt with the

stability of the zero solution of equation (1) are relatively scarce. We only find a paper [7]

which gives a necessary and sufficient condition for the asymptotic stability of equation (1)

in terms of the eigenvalues of the coefficient matrix A when AðnÞ ; A is a constant matrix.

However, when A(n) is not a constant matrix, the theory of characteristic equation fails to

apply equation (1). In this case, it seems a common method to construct a working

“Liapunov” function. But this method is also difficult in general. Motivated by the work on

stability in higher dimensional delay differential equations in [3] and [9], we will give

sufficient conditions for the uniform stability and asymptotic stability of the zero solution of

equation (1) by a “geometric method” instead of “Liapunov method”.

For x; y [ Rm, jxj is the Euclidean norm, and x·y is the inner product of x and y. For

A [ Rm£m, jAj denotes the matrix norm induced by the above vector norm. Let f :

{ 2 k;2k þ 1; . . . ; 0} ! Rm and Ekm be the set of m-dimensional vector system

Journal of Difference Equations and Applications

ISSN 1023-6198 print/ISSN 1563-5120 online q 2007 Taylor & Francis

http://www.tandf.co.uk/journals

DOI: 10.1080/10236190701388419

†This work was supported by NNSF (No: 10471153) of China.*Corresponding author. Email: [email protected]

Journal of Difference Equations and Applications,

Vol. 13, No. 10, October 2007, 927–944

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ðfð2kÞ;fð2k þ 1Þ; . . . ;fð0ÞÞ, and let

kfk ¼ max{jfðsÞj : s ¼ 2k;2k þ 1; . . . ; 0}:

For any m-dimensional vector sequence {y(n)} and any integer n . 0, we define ynðsÞ ¼

yðnþ sÞ for s [ { 2 k;2k þ 1; . . . ; 0}. For any initial value f [ Ekm and integer n0 $ 0,

we denote by xðn; n0;fÞ the solution of equation (1) with initial condition xn0¼ f.

Definition 1. We say the zero solution of equation (1) is uniformly stable, if for any 1 . 0

there exists d(1) . 0 such that if n0 $ 0 and f [ Ekm, then kfk , dð1Þ implies

jxðn; n0;fÞj , 1 for all n $ n0 2 k.

2. Main results

First, we state a lemma which is taken from [9].

Lemma 1[9]. Let a, b and c be points in Rm with jaj ¼ jbj ¼ jcj ¼ 1. Then

2ða·bÞða·cÞðb·cÞ $ ða·bÞ2 þ ða·cÞ2 2 1:

Lemma 2. Let K [ (0,1] and

qKðxÞ ¼ð1 2 xÞ2

2þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 þ x2 2 2x

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 2 K 2

pq

:

Then

(i) there exists a unique a [ ð0; 1Þ such that

1 2 a ¼a2


p

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 þ a2 2 2a


pp ; ð2Þ

(ii) min0#x#1qKðxÞ ¼ qKðaÞ.

Proof. Set a ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 2 K 2

p. Then qKðxÞ ¼ ð1=2Þð1 2 xÞ2 þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 þ x2 2 2ax

p,

q 0KðxÞ ¼ 21 þ xþ

x2 affiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 þ x2 2 2ax

p ; q 0Kð0Þ ¼ 2a , 0; q 0

Kð1Þ ¼

ffiffiffiffiffiffiffiffiffiffiffi1 2 a

2

r. 0;

and

q 00KðxÞ ¼ 1 þ

1 2 a2

ð1 þ x 2 2 2axÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 þ x2 2 2ax

p . 0; for x [ ½0; 1�:

Therefore, the equation q 0KðxÞ ¼ 0 has a unique foot a [ (0,1), i.e. there exists a unique

a [ (0,1) such that (2) holds. It follows that (ii) holds also. The proof is complete. A

Theorem 1. Assume that there exists a K [ ð0; 1� such that

2AðnÞx

jAðnÞxj·x

jxj$ K ð3Þ

X. H. Tang et al.928

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for all ðn; xÞ [ {0; 1; 2; . . . ; } £ Rm with AðnÞx – 0, and that

Xni¼n2k

jAðiÞj #ð1 2 aÞ2

2þ



pq

; n $ 0; ð4Þ

where a [ (0,1) is defined by (2). Then the zero solution of equation (1) is uniformly stable.

Proof. By (2), we have

M U qKðaÞ ¼ð1 2 aÞ2

2þ

a2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 2 K 2

p

1 2 a¼

ð1 2 aÞ2

2þ



pq

:

Then from (1), we have

jxðnÞj # jxðn0Þj þXn21

i¼n0

jAðiÞj jxði2 kÞj # jxðn0Þj þXn21

i¼n0

jAðiÞj kxik:

It follows that

kxnk # kxn0k þ

Xn21

i¼n0

jAðiÞj kxik; n $ n0:

By Discrete Grownwall’s inequality, we have

kxnk # kxn0kYn21

i¼n0

ð1 þ jAðiÞjÞ # kfkexpXn21

i¼n0

jAðiÞj

!; n $ n0;

which implies that

jxðnÞj # kfke2M ; n0 # n # n0 þ 2k: ð5Þ

Let

r ¼ max{jxðiÞj : n0 2 k # i # n0 þ 2k}:

Assume that r . 0. In what follows, we will show that

jxðnÞj # r for n $ n0 þ 2k: ð6Þ

Suppose (6) is false. Let N þ 1 ¼ min{n : jxðnÞj . r}. Then jxðN þ 1Þj . r;

N þ 1 . n0 þ 2k, and jxðnÞj # r for n0 # n # N. Hence, from (1), we have

jxðnþ 1Þ2 xðnÞj # jAðnÞj jxðn2 kÞj # jAðnÞj jxðN þ 1Þj; n0 # n # N þ 1: ð7Þ

For N 2 k # n # N, from (7), we have

jxðn2 kÞj # jxðn2 kÞ2 xðN 2 kÞj þ jxðN 2 kÞj

¼XN2k21

i¼n2k

½xðiþ 1Þ2 xðiÞ�

��þ jxðN 2 kÞj

# jxðN þ 1ÞjXN2k21

i¼n2k

jAðiÞj þ jxðN 2 kÞj:

Linear delay difference system 929

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Substituting this into the first inequality in (7), then we have

jxðnþ 1Þ2 xðnÞj # jAðnÞj jxðN þ 1ÞjXN2k21

i¼n2k

jAðiÞj þ jxðN 2 kÞj

" #; N 2 k # n # N:

On combining this and (7), we obtain

jxðnþ 1Þ2 xðnÞj # jxðN þ 1Þj jAðnÞjmin 1;XN2k21

i¼n2k

jAðiÞj þjxðN 2 kÞj

jxðN þ 1Þj

( );

N 2 k # n # N:

ð8Þ

Note that

jxðN þ 1Þj2 jxðN 2 kÞj # jxðN þ 1Þ2 xðN 2 kÞj # jxðN þ 1ÞjXN

i¼N2k

jAðiÞj # MjxðN þ 1Þj:

It follows that 1 2 jxðN 2 kÞj=jxðN þ 1Þj # M. Then there are two possible cases to

consider:

Case 1. M $PN

i¼N2kjAðiÞj . 1 2 jxðN 2 kÞj=jxðN þ 1Þj. In this case, there exist an integer

l with 0 # l # k2 1 and j [ ½0; 1Þ such that

XNi¼N2l

jAðiÞj þ jjAðN 2 l2 1Þj ¼ 1 2jxðN 2 kÞj

jxðN þ 1Þj:

Hence, from (4) and (8), we have

jxðN þ 1Þ2 xðN 2 kÞj

#XN2l22

n¼N2k

jxðnþ 1Þ2 xðnÞj þ ð1 2 jÞjxðN 2 lÞ2 xðN 2 l2 1Þj

þ jjxðN 2 lÞ2 xðN 2 l2 1Þj þXN

n¼N2l

jxðnþ 1Þ2 xðnÞj

# jxðN þ 1ÞjXN2l22

i¼N2k

jAðiÞj þ ð1 2 jÞjAðN 2 l2 1Þj

"

þjjAðN 2 l2 1ÞjXN2k21

i¼N2l2k21


jxðN þ 1Þj

!

þXN

n¼N2l

jAðnÞjXN2k21

i¼n2k


jxðN þ 1Þj

!#

¼ jxðN þ 1ÞjXN2l21

i¼N2k

jAðiÞj2 jjAðN 2 l2 1Þj

!"


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£XN

n¼N2l

jAðnÞj þ jjAðN 2 l2 1Þj þjxðN 2 kÞj

jxðN þ 1Þj

!

þjjAðN 2 l2 1ÞjXN2k21

i¼N2l2k21


jxðN þ 1Þj

!

þXN

n¼N2l

jAðnÞjXN2k21

i¼n2k


jxðN þ 1Þj

!#

¼ jxðN þ 1ÞjXN

n¼N2l

jAðnÞjXN2l21

i¼n2k


!" #

þjjAðN 2 l2 1ÞjXN2l21

i¼N2l2k21


!

þjxðN 2 kÞj

jxðN þ 1Þj

XNi¼N2k

jAðiÞj

#

# jxðN þ 1Þj MXNi¼N2l

jAðiÞj2XN

n¼N2l

jAðnÞjXni¼N2l

jAðiÞj

"

þjjAðN 2 l2 1Þj M 2XNi¼N2l


!þ

jxðN 2 kÞj

jxðN þ 1ÞjM

#

¼ jxðN þ 1Þj MXNi¼N2l

jAðiÞj21

2

XNi¼N2l

jAðiÞj

!2

21

2

XNi¼N2l

jAðiÞj2

24

þjjAðN 2 l2 1Þj M 2XNi¼N2l


!þ

jxðN 2 kÞj

jxðN þ 1ÞjM

#

¼ jxðN þ 1Þj MXNi¼N2l

jAðiÞj þ jjAðN 2 l2 1Þj

!þ

jxðN 2 kÞj

jxðN þ 1ÞjM

"

21

2

XNi¼N2l


!2

21

2

XNi¼N2l

jAðiÞj2þ j2jAðN 2 l2 1Þj

2

!#


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# jxðN þ 1Þj MXNi¼N2l


!þ

jxðN 2 kÞj

jxðN þ 1ÞjM

"

21

2

XNi¼N2l


!235

¼ jxðN þ 1Þj M 21

21 2

jxðN 2 kÞj

jxðN þ 1Þj

� �2" #

:

Case 2.PN

i¼N2kjAðiÞj # 1 2 jxðN 2 kÞj=jxðN þ 1Þj # M. Then from (4) and (8), we have

jxðN þ 1Þ2 xðN 2 kÞj

#XN

n¼N2k


# jxðN 2 kÞjXN

n¼N2k

jAðnÞj þ jxðN þ 1ÞjXN

n¼N2k

jAðnÞjXN2k21

i¼n2k

jAðiÞj

¼ jxðN 2 kÞjXN

n¼N2k

jAðnÞj þ jxðN þ 1ÞjXN

n¼N2k

jAðnÞjXni¼n2k

jAðiÞj2Xn

i¼N2k

jAðiÞj

!

# jxðN 2 kÞjM þ jxðN þ 1Þj MXN

n¼N2k

jAðnÞj21

2

XNi¼N2k

jAðiÞj

!2

21

2

XNi¼N2k

jAðiÞj2

24

35

# jxðN 2 kÞjM þ jxðN þ 1Þj MXN

i¼N2k

jAðiÞj21

2

XNi¼N2k

jAðiÞj

!224

35

# jxðN þ 1Þj MjxðN 2 kÞj

jxðN þ 1ÞjþM 1 2

jxðN 2 kÞj

jxðN þ 1Þj

� �2

1

21 2

jxðN 2 kÞj

jxðN þ 1Þj

� �2" #

¼ jxðN þ 1Þj M 21

21 2

jxðN 2 kÞj

jxðN þ 1Þj

� �2" #

:

Combining Case 1 and Case 2, we have

jxðN þ 1Þ2 xðN 2 kÞj # jxðN þ 1Þj M 21

21 2

jxðN 2 kÞj

jxðN þ 1Þj

� �2" #

: ð9Þ

From (1), we have

jxðN þ 1Þj2 jxðNÞj ¼2xðN þ 1Þ · ½xðN þ 1Þ2 xðNÞ�2 jxðN þ 1Þ2 xðNÞj

2

jxðN þ 1Þj þ jxðNÞj

¼2xðN þ 1Þ ·AðNÞxðN 2 kÞ2 jAðNÞxðN 2 kÞj

2

jxðN þ 1Þj þ jxðNÞj

#2xðN þ 1Þ ·AðNÞxðN 2 kÞ

jxðN þ 1Þj þ jxðNÞj:


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It follows from the fact that jxðN þ 1Þj2 jxðNÞj . 0 that

AðNÞxðN 2 kÞ

jAðNÞxðN 2 kÞj

xðN þ 1Þ

jxðN þ 1Þj. 0: ð10Þ

From (3), we have

2AðNÞxðN 2 kÞ

jAðNÞxðN 2 kÞj

xðN 2 kÞ

jxðN 2 kÞj$ K: ð11Þ

Hence, by (10), (11) and Lemma 1, we have

xðN þ 1Þ

jxðN þ 1Þj

xðN 2 kÞ

jxðN 2 kÞj,


p: ð12Þ

Set u ¼ jxðN 2 kÞj=jxðN þ 1Þj and use the fact that

2xðN þ 1ÞxðN 2 kÞ ¼ jxðN þ 1Þj2þ jxðN 2 kÞj

22 jxðN þ 1Þ2 xðN 2 kÞj

2:

It follows from (9) that

xðN þ 1Þ

jxðN þ 1Þj

xðN 2 kÞ

jxðN 2 kÞj$

1

2u1 þ u2 2 M 2

ð1 2 uÞ2

2

� �2( )

: ð13Þ

Combining (12) and (13), we have

2uffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 2 K 2

p. 1 þ u2 2 M 2

ð1 2 uÞ2

2

� �2

:

It follows Lemma 2 (ii) that

M .ð1 2 uÞ2

2þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 þ u2 2 2u


pq

$ min0#u#1

ð1 2 uÞ2

2þ



pq� �

¼ð1 2 aÞ2

2þ



pq

¼ M:

This is a contradiction, hence, (6) holds. Combining (5) and (6), we have

jxðnÞj # kfke2M for n $ n0 2 k:

Let 1 . 0 be any given and choose d ¼ ð1=2Þ1e22M . Then f [ Ekm and kfk , d implies

jxðnÞj ¼ jxðn; n0;fÞj , 1. The proof is complete. A

Theorem 2. Assume that there exist a K [ ð0; 1�, N0 . 0 and a nonnegative sequence

{mðnÞ} with S1mðnÞ ¼ 1 such that (4) holds and

jAðnÞxj $ mðnÞjxj ð14Þ

for all ðn; xÞ [ {N0;N0 þ 1; . . . } £ Rm with AðnÞx – 0, and that

lim supn!1

Xni¼n2k

jAðiÞj ,ð1 2 aÞ2

2þ



pq

; ð15Þ

where a [ ½0; 1Þ is defined by (2). Then every solution of equation (1) tends zero as n!1.


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Proof. Set M ¼ ð1 2 aÞ2=2 þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 þ a2 2 2a


pp. By (15), there exist M1 , M and

N1 . N0 such that Xni¼n2k

jAðiÞj # M1; n $ N1: ð16Þ

From (1), we have

jxðnþ 1Þj2 jxðnÞj ¼2xðnþ 1ÞAðnÞxðn2 kÞ2 jAðnÞxðn2 kÞj

2

jxðnþ 1Þj þ jxðnÞj: ð17Þ

In view of Theorem 1, jxðnÞj is bounded. In what follows, we will show

limn!1

jxðnÞj ¼ 0 ð18Þ

in two possible cases.

Case 1. {jxðnþ 1Þj2 jxðnÞj} is eventually non-negative or eventually non-positive. In this

case, {jxðnÞj} is eventually monotone and the limit limn!1jxðnÞj ¼ c exists. If c . 0, then

limn!1

jxðn2 kÞj

jxðnÞj¼ lim

n!1

jxðn2 2kÞj

jxðnÞj¼ 1: ð19Þ

From (1) and (16), we have

jxðnþ1Þ2 xðn2 kÞj#Xni¼n2k

jAðiÞj jxði2 kÞj#M1max{jxðnþ1Þj; jxðn22kÞj}; n$N1;

which, together with (19), implies that there exists N2 .N1 such that

xðnþ1Þ

jxðnþ1Þj

xðn2 kÞ

jxðn2 kÞj

¼jxðnþ1Þj

2þjxðn2 kÞj

22 jxðnþ1Þ2 xðn2 kÞj

2

2jxðnþ1Þj jxðn2 kÞj

$jxðnþ1Þj

2þjxðn2 kÞj

22M2

1max{jxðnþ1Þj2; jxðn22kÞj

2}

2jxðnþ1Þj jxðn2 kÞj

$ 121

2M 2; n$N2:

On the other hand, by (2), we have

2AðnÞxðn2 kÞ

jAðnÞxðn2 kÞj

xðn2 kÞ

jxðn2 kÞj$K; jAðnÞxðn2 kÞj– 0; n$N0:

Hence, in view of Lemma 1, we have

xðnþ1Þ

jxðnþ1Þj

AðnÞxðn2 kÞ

jAðnÞxðn2 kÞj#

xðnþ1Þjxðnþ1Þj

xðn2kÞjxðn2kÞj

2

þ AðnÞxðn2kÞjAðnÞxðn2kÞj

xðn2kÞjxðn2kÞj

2

21

2 xðnþ1Þjxðnþ1Þj

xðn2kÞjxðn2kÞj

AðnÞxðn2kÞjAðnÞxðn2kÞj

xðn2kÞjxðn2kÞj

#12 12M 2

2

2

2K 2


xðn2kÞjxðn2kÞj


xðn2kÞjxðn2kÞj

;jAðnÞxðn2 kÞj– 0; n$N2:


Dow

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Let

b¼ 12M 2

2

� �2

þK 2 21:

Note that

M¼ð12aÞ2

2þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þa2 22a

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi12K 2

pq

¼ min0#u#1

ð12uÞ2

2þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þu2 22u


pq� �

,

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi222


pq

:

It follows that

1

4M 4 2M 2 þK 2 . 0;

and so b. 0. Hence, for n$N2 and jAðnÞxðn2 kÞj– 0,

xðnþ1Þ

jxðnþ1Þj

AðnÞxðn2 kÞ

jAðnÞxðn2 kÞj#2

b


xðn2kÞjxðn2kÞj


xðn2kÞjxðn2kÞj

#2b

2; n$N2:

ð20Þ

Combining (14), (17) and (20), we obtain

jxðnþ1Þj2 jxðnÞj ¼2xðnþ1ÞAðnÞxðn2 kÞ2 jAðnÞxðn2 kÞj

2

jxðnþ1Þjþ jxðnÞj

#2xðnþ1ÞAðnÞxðn2 kÞ

jxðnþ1Þj

#2bjAðnÞxðn2 kÞj

#2bmðnÞjxðn2 kÞj; n$N2:

The above shows jxðnÞj nonincreasing when n$N2. Summing the above from N2 to 1,

we get

c2 jxðN2Þj#2cbX1n¼N2

mðnÞ ¼21:

This is a contradiction. Hence, c ¼ 0, and so (18) holds.

Case 2. {jxðnþ 1Þj2 jxðnÞj} is oscillatory and jxðnþ 1Þj2 jxðnÞj � 0 eventually.

Let v ¼ lim supn!1jxðnÞj. It suffices to show v ¼ 0. Note that M1 , M. If v . 0, then we

can choose 1 . 0 is sufficient small so that

M1 þ21

vþ 1¼ M2 , M 2

21

vþ 1: ð21Þ

Choose a sequence {nj} of integers with n1 . N0 þ 2k such that

jxðnj þ 1Þj2 jxðnjÞj . 0; jxðnj þ 1Þj . v2 1;

limj!1

jxðnj þ 1Þj ¼ v; and jxðnÞj , vþ 1; n $ n1 þ k:

8<: ð22Þ


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Then from (1) and (22), we have

jxðnþ 1Þ2 xðnÞj # jAðnÞj jxðn2 kÞj # jAðnÞjðvþ 1Þ; n $ n1: ð23Þ

Hence, for large j

jxðn2 kÞj # jxðn2 kÞ2 xðnj 2 kÞj þ jxðnj 2 kÞj

¼Xnj2k21

i¼n2k

½xðiþ 1Þ2 xðiÞ�

��þ jxðnj 2 kÞj

# ðvþ 1ÞXnj2k21

i¼n2k

jAðiÞj þ jxðnj 2 kÞj; nj 2 k # n # nj:

Substituting this into the first inequality in (23), then for nj 2 k # n # nj we have

jxðnþ 1Þ2 xðnÞj # jAðnÞj ðvþ 1ÞXnj2k21

i¼n2k

jAðiÞj þ jxðnj 2 kÞj

" #:

On combining this and (23), we obtain

jxðnþ 1Þ2 xðnÞj # ðvþ 1ÞjAðnÞjmin 1;Xnj2k21

i¼n2k

jAðiÞj þjxðnj 2 kÞj

vþ 1

( );

nj 2 k # n # nj:

ð24Þ

Note that

jxðnj þ 1Þj2 jxðnj 2 kÞj # jxðnj þ 1Þ2 xðnj 2 kÞj # ðvþ 1ÞXnj

i¼nj2k

jAðiÞj # M1ðvþ 1Þ:

It follows that 1 2 jxðnj 2 kÞj=ðvþ 1Þ # M2. There are two possible cases to consider:

Case 2.1. M2 $Pnj

i¼nj2kjAðiÞj . 1 2 jxðnj 2 kÞj=ðvþ 1Þ. Then there exist an integer l with

0 # l # k2 1 and a j [ ½0; 1Þ such that

Xnji¼nj2l

jAðiÞj þ jjAðnj 2 l2 1Þj ¼ 1 2jxðnj 2 kÞj

vþ 1:

Hence, from (24), we have

jxðnjþ1Þ2xðnj2kÞj

#Xnj2l22

n¼nj2k

jxðnþ1Þ2xðnÞjþð12jÞjxðnj2 lÞ2xðnj2 l21Þj

þjjxðnj2 lÞ2xðnj2 l21ÞjþXnj

n¼nj2l

jxðnþ1Þ2xðnÞj


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# ðvþ1ÞXnj2l22

i¼nj2k

jAðiÞjþð12jÞjAðnj2 l21Þj

24

þjjAðnj2 l21ÞjXnj2k21

i¼nj2l2k21

jAðiÞjþjxðnj2kÞj

vþ1

0@

1A

þXnj

n¼nj2l

jAðnÞjXnj2k21

i¼n2k


vþ1

!35

¼ðvþ1ÞXnj2l21

i¼nj2k

jAðiÞj2jjAðnj2 l21Þj

0@

1A Xnj

n¼nj2l

jAðnÞjþjjAðnj2 l21Þjþjxðnj2kÞj

vþ1

0@

1A

24

þjjAðnj2 l21ÞjXnj2k21

i¼nj2l2k21


vþ1

0@

1A

þXnj

n¼nj2l

jAðnÞjXnj2k21

i¼n2k


vþ1

!35

¼ðvþ1ÞXnj

n¼nj2l

jAðnÞjXnj2l21

i¼n2k


!24

þjjAðnj2 l21ÞjXnj2l21

i¼nj2l2k21


0@

1Aþ

jxðnj2kÞj

vþ1

Xnji¼nj2k

jAðiÞj

35

# ðvþ1Þ M2

Xnji¼nj2l

jAðiÞj2Xnj

n¼nj2l

jAðnÞjXni¼nj2l

jAðiÞjþjjAðnj2 l21Þj

24

M22Xnji¼nj2l


0@

1Aþ

jxðnj2kÞj

vþ1M2

35

¼ðvþ1Þ M2

Xnji¼nj2l

jAðiÞj21

2

Xnji¼nj2l

jAðiÞj

0@

1A

2

21

2

Xnji¼nj2l

jAðiÞj2þjjAðnj2 l21Þj

24

M22Xnji¼nj2l


0@

1Aþ

jxðnj2kÞj

vþ1M2

35


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# ðvþ1Þ M2

Xnji¼nj2l


0@

1Aþ

jxðnj2kÞj

vþ1M2

24

21

2

Xnji¼nj2l


0@

1A

235

¼ðvþ1Þ M221

212

jxðnj2kÞj

vþ1

� �2" #

:

Case 2.2.Pnj

i¼nj2kjAðiÞj # 1 2 jxðnj 2 kÞj=ðvþ 1Þ # M2. Then from (24), we have

jxðnj þ 1Þ2 xðnj 2 kÞj

#Xnj

n¼nj2k


# jxðnj 2 kÞjXnj

n¼nj2k

jAðnÞj þ ðvþ 1ÞXnj

n¼nj2k

jAðnÞjXnj2k21

i¼n2k

jAðiÞj

¼ jxðnj 2 kÞjXnj

n¼nj2k

jAðnÞj þ ðvþ 1ÞXnj

n¼nj2k

jAðnÞjXni¼n2k

jAðiÞj2Xn

i¼nj2k

jAðiÞj

0@

1A

# jxðnj 2 kÞjM2 þ ðvþ 1Þ M2

Xnjn¼nj2k

jAðnÞj21

2

Xnji¼nj2k

jAðiÞj

0@

1A

2

21

2

Xnji¼nj2k

jAðiÞj2

24

35

# jxðnj 2 kÞjM2 þ ðvþ 1Þ M2

Xnji¼nj2k

jAðiÞj21

2

Xnji¼nj2k

jAðiÞj

0@

1A

224

35

# ðvþ 1Þ M2

jxðnj 2 kÞj

vþ 1þM2 1 2

jxðnj 2 kÞj

vþ 1

� �2

1

21 2

jxðnj 2 kÞj

vþ 1

� �2" #

¼ ðvþ 1Þ M2 21

21 2

jxðnj 2 kÞj

vþ 1

� �2" #

:

Combining Case 2.1 and Case 2.2, we have

jxðnj þ 1Þ2 xðnj 2 kÞj # ðvþ 1Þ M2 21

21 2

jxðnj 2 kÞj

vþ 1

� �2" #

: ð25Þ


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Similar the proof of Theorem 1, we have

xðnj þ 1Þ

jxðnj þ 1Þj·xðnj 2 kÞ

jxðnj 2 kÞj,


p: ð26Þ

Set u ¼ jxðnj 2 kÞj=ðvþ 1Þ. Then from (25),

xðnj þ 1Þ

jxðnj þ 1Þj

xðnj 2 kÞ

jxðnj 2 kÞj

¼jxðnj þ 1Þj

2þ jxðnj 2 kÞj

22 jxðnj þ 1Þ2 xðnj 2 kÞj

2

2jxðnj þ 1Þj jxðnj 2 kÞj

$jxðnj þ 1Þj

2þ jxðnj 2 kÞj

22 ðvþ 1Þ2 M2 2

12ð1 2 uÞ2

� �2jxðnj þ 1Þj jxðnj 2 kÞj

a

¼

jxðnjþ1Þj

vþ1

2

þu2 2 M2 212ð1 2 uÞ2

� �22u

jxðnjþ1Þj

vþ1

$vþ 1

2uðv2 1Þ

v2 1

vþ 1

� �2

þu2 2 M2 2ð1 2 uÞ2

2

� �2( )

¼1

2aua2 þ u2 2 M2 2

ð1 2 uÞ2

2

� �2( )

;

where a ¼ ðv2 1Þ=ðvþ 1Þ. Combining this and (26), we have

2auffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 2 K 2

p. a2 þ u2 2 M2 2

ð1 2 uÞ2

2

� �2

:

It follows that

M2 .ð1 2 uÞ2

2þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia2 þ u2 2 2au


pq

$ min0#u#1

ð1 2 uÞ2

2þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia2 þ u2 2 2au


pq� �

¼ð1 2 aðaÞÞ2

2þ

aðaÞ2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 2 K 2

p

1 2 aðaÞ;

where a(a) [ (0,1) satisfy that

1 2 aðaÞ ¼aðaÞ2


p

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia2 þ ½aðaÞ�2 2 2aaðaÞ


pq : ð27Þ

Let 1! 0þ in (27), then lim1!0þaðaÞ ¼ lima!1aðaÞ ¼ a and so

M1 $ð1 2 aÞ2

2þ

a2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 2 K 2

p

1 2 a¼ M:

This is a contradiction, hence, v ¼ 0 and so (18) holds. The proof is complete. A


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3. Remarks and examples

In the last section, we give some remarks and examples to illustrate our results. Note that

qKðaÞ ¼ð1 2 aÞ2

2þ



pq

¼ min0#u#1

ð1 2 uÞ2

2þ



pq� �

. min0#u#1



pq

¼ K;

and

qKðaÞ ¼ð1 2 aÞ2

2þ



pq

¼ min0#u#1

ð1 2 uÞ2

2þ



pq� �

,

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 2 2


pq

:

Hence,

K , qKðaÞ ,

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 2 2


pq

; 0 , K # 1: ð28Þ

For comparing K with qKðaÞ, we give some values of K and qKðaÞ as follows:

Example 1. Consider the 2-dimensional delay difference equation

xðnþ 1Þ2 xðnÞ ¼2aðnÞ bðnÞ

2bðnÞ 2aðnÞ

!xðn2 kÞ; ð29Þ

where {aðnÞ}; {bðnÞ} are two real sequences and aðnÞ . 0. Let

AðnÞ ¼2aðnÞ bðnÞ

2bðnÞ 2aðnÞ

!:

K qK ðaÞ

0.1 0.1000114110.2 0.2001700850.3 0.3008159410.4 0.4024885550.5 0.5059790810.6 0.6124816530.7 0.7239634970.8 0.8442805660.9 0.9828663891 1.242217666


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Then for any x [ R2,

jAðnÞxj ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia2ðnÞ þ b2ðnÞ

pjxj;

and for x – 0,

2AðnÞx

jAðnÞxj

x

jxj¼

aðnÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia2ðnÞ þ b2ðnÞ

p :

Let

N ¼ supn$0

Xns¼n2k

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia2ðsÞ þ b2ðsÞ

p; and K ¼ inf

n$0

aðnÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia2ðnÞ þ b2ðnÞ

p :

By Theorem 1, if N # qKðaÞ, then the zero solution of (29) is uniformly stable.

Furthermore, if

lim supn!1

Xns¼n2k


p, qKðaÞ and

X1s¼0


p¼ 1:

Then by Theorem 2, every solution of (29) tends zero as n!1.

In particular, let aðnÞ ¼ rðnÞ cos u and bðnÞ ¼ rðnÞ sin u, where juj , ðp=2Þ and

rðnÞ $ 0; n ¼ 0; 1; 2; . . . . Then if

lim supn!1

Xns¼n2k

rðsÞ , f ðuÞ andX1s¼0

rðsÞ ¼ 1; ð30Þ

then every solution of the equation

xðnþ 1Þ2 xðnÞ ¼ rðnÞ2cos u sin u

2sin u 2cos u

!xðn2 kÞ; ð31Þ

tends to zero as n!1, where

f ðuÞ ¼ð1 2 aÞ2

2þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 þ a2 2 2ajsin uj

p;

and a [ ½0; 1Þ satisfies the equation

1 2 a ¼a2 jsin ujffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1 þ a2 2 2ajsin ujp : ð32Þ

Remark 1. When rðnÞ ; r . 0, it is shown in [7] that every solution of the equation of

equation (31) tends to zero as n!1 if and only if

r , gðuÞ ¼ 2 coskpþ juj

2k þ 1: ð33Þ

In this case, condition (30) reduces to

r ,f ðuÞ

k þ 1¼

1

k þ 1

ð1 2 aÞ2

2þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 þ a2 2 2ajsin uj

p� �: ð34Þ


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To compare our result (34) with (33) in obtained in [7], we let k ¼ 2, and give some values of

f ðuÞ=3 and gðuÞ as follows:

Remark 2. When rðnÞ � constant, for example

rðnÞ ¼½1 þ ð21Þn� f ðuÞ

k þ 3þ

1

nþ 1; n ¼ 0; 1; 2; . . . :

It is easy to see that rðnÞ satisfies condition (30), therefore, every solution of the equation of

equation (31) tends to zero as n!1. However, the results in [7] can not be used for the

above rðnÞ.

Example 2. Consider more general 2-dimensional delay difference equation

xðnþ 1Þ2 xðnÞ ¼2aðnÞ 2bðnÞ

2cðnÞ 2dðnÞ

!xðn2 kÞ; ð35Þ

where {aðnÞ}; {bðnÞ}; {cðnÞ} and {dðnÞ} are four sequences of real numbers and

aðnÞ; dðnÞ . 0. Let

AðnÞ ¼2aðnÞ 2bðnÞ

2cðnÞ 2dðnÞ

!:

Then

jAðnÞj ¼

ffiffiffi2

p

2

�a2ðnÞ þ b2ðnÞ þ c2ðnÞ þ d 2ðnÞ

þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi½a2ðnÞ þ b2ðnÞ þ c 2ðnÞ þ d 2ðnÞ�2 2 4½aðnÞdðnÞ2 bðnÞcðnÞ�2

q �1=2

;

and for any x [ R2,

2AðnÞx · x ¼ aðnÞx21 þ dðnÞx2

2 þ ½bðnÞ þ cðnÞ�x1x2

$1

2aðnÞ þ dðnÞ2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi½aðnÞ2 dðnÞ�2 þ ½bðnÞ þ cðnÞ�2

q� �jxj

2;

u f(u)/3 g(u)0 0.4141 0.6174p6

0.3108 0.4157p3

0.1687 0.2086p2

0 0


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and

jAðnÞxj2¼ ½a2ðnÞ þ c2ðnÞ�x2

1 þ ½b2ðnÞ þ d 2ðnÞ�x22 þ 2½aðnÞbðnÞ þ cðnÞdðnÞ�x1x2

$1

2a2ðnÞ þ b2ðnÞ þ c2ðnÞ þ d 2ðnÞ

�

2


q �jxj

2

; m2ðnÞjxj2:

It follows that for x – 0,

2AðnÞx

jAðnÞxj·x

jxj$

2AðnÞxx

jAðnÞj jxj2

$1

2jAðnÞjaðnÞ þ dðnÞ2


q� �:

Set

K ¼ infn$0

1

2jAðnÞjaðnÞ þ dðnÞ2


q� �� :

Then by Theorem 2, if

2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiaðnÞdðnÞ

p$ bðnÞ þ cðnÞ;

X1s¼0

mðsÞ ¼ 1;

and

lim supn!1

Xns¼n2k

jAðsÞj , qKðaÞ;

then every solution of (35) tends to zero as n!1, where a [ ð0; 1Þ is defined by (2) and

mðnÞ ¼

ffiffiffi2

p

2a2ðnÞ þ b2ðnÞ þ c2ðnÞ þ d 2ðnÞ

�

2


q �1=2

:

Acknowledgements

The authors are grateful to the referees for their valuable comments and suggestions.

References

[1] Elaydi, S.N., 1996, An Introduction to Difference Equations (New York: Springer-Verlag).[2] Erbe, L.H., Xia, H. and Yu, J.S., 1995, Global stability of a linear nonautonomous delay difference equation.

Journal of Difference Equations and Applications, 1, 151–161.


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[6] Levin, S.A. and May, R.M., 1976, A note on difference-delay equations. Theoretical Population Biology,9, 178–187.

[7] Matsunaga, H. and Hara, T., 1999, The asymptotic stability of a two-dimesional linear delay differenceequation. Dynamics of Continuous, Discrete and Impulsive Systems, 6, 465–473.

[8] Papanicolaou, V.G., 1996, On the asymptotic stability of a class of linear difference equations. MathematicsMagazine, 69, 34–43.

[9] Tang, X.H., 2004, Stability in n-dimensional delay differential equations. Journal of Mathematical Analysisand Applications, 295, 485–501.

[10] Yu, J.S., 1998, Asymptotic stability for a linear difference equation with variable delay. Computers andMathematics with Applications, 36, 203–210.


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