Applied Mathematics and Computation 217 (2011) 6663–6670

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Applied Mathematics and Computation

journal homepage: www.elsevier .com/ locate/amc

Sensitivity of Schur stability of monodromy matrix

Ahmet Duman a, Kemal Aydın b,⇑a Kahramanmaras� Sütçü _Imam University, Faculty of Education, Department of Mathematics, Kahramanmaras�, Turkeyb Selçuk University, Faculty of Science, Department of Mathematics, Konya, Turkey

a r t i c l e i n f o

Keywords:Difference equation systemsMonodromy matrixPeriodic coefficientsSchur stability parametersSensitivity

0096-3003/$ - see front matter Crown Copyright �doi:10.1016/j.amc.2011.01.052

⇑ Corresponding author.E-mail addresses: aduman@ksu.edu.tr (A. Duman

a b s t r a c t

For Schur stable linear difference equation system with periodic coefficients, we prove con-tinuity theorems on monodromy matrix which show how much change is permissiblewithout disturbing the Schur stability, and some examples illustrating the efficiency ofthe theorems are given.

Crown Copyright � 2011 Published by Elsevier Inc. All rights reserved.

1. Introduction

In this study, we shall be focused on the sensitivity of Schur stability of the linear difference equation system with peri-odic coefficients

xðnþ 1Þ ¼ AðnÞxðnÞ; n 2 Z: ð1:1Þ

Here, A(n) is an N � N dimensional matrix with a period T, i.e., A(n + T) = A(n). The Schur stability of the monodromy matrixX(T) = A(T � 1)A(T � 2) . . .A(1)A(0) of the system (1.1) is equivalent to the Schur stability of the system (1.1) Schur stability isknown as asymptotic stability of the difference equations (see, for example, [1,2]). According to the spectral criterion, if alleigenvalues of the monodromy matrix X(T) belong to the unit disc, i.e. jki(X(T))j < 1 (i = 1,2, . . . ,N), then the monodromy ma-trix X(T) is called to be a Schur stable matrix (see, for example, [3,4]). In [5], they are defined the parameters x1 and x2 whichshow the quality of the Schur stability of a linear difference equation system with periodic coefficients.

Predicting the behaviour of solutions of a problem and knowing under which conditions similar properties are protectedunder perturbations is important for the problem not to cause any chaos. The question ‘‘how much perturbation is ignorablefor preserving the characteristic properties?’’ is known as the sensitivity problem. In recent years, there are few papersstudying the sensitivity of the linear difference equations (see, for example, [6–9]).

In the literature, some results are given under which the perturbated system

yðnþ 1Þ ¼ ðAðnÞ þ BðnÞÞyðnÞ; Bðnþ TÞ ¼ BðnÞ; n 2 Z; ð1:2Þ

preserves the Schur stability when the system (1.1) is Schur stable (see, for example, [7,8]). Some of these results give explicitconditions for bounds of the Schur stability parameters of the system (1.1) and its perturbated system (1.2), while some ofthe results provide bounds for the difference between the monodromy matrices X(T) and Y(T) of the systems (1.1) and (1.2),respectively. Such theorems in general known as the continuity theorems. Recently there has been some interest also onnonlinear difference equations with periodic coefficients. In such studies, the mentioned equations have been convertedto linear equations by the change of convenient variable or linearized. For the some results see, for example, [10,11] andreferences therein.

2011 Published by Elsevier Inc. All rights reserved.

), kaydin@selcuk.edu.tr, kaydin29@gmail.com (K. Aydın).

6664 A. Duman, K. Aydın / Applied Mathematics and Computation 217 (2011) 6663–6670

In this study, we shall provide new results which show the durability of the Schur stability of difference equation systemsby means of the monodromy matrices. The consistency of the new results with the existing ones in the literature will becompared. Moreover, the results will be discussed on numerical examples.

2. Preliminaries

This section is dedicated to the preliminary results for periodic difference equation systems. Since for the case T = 1, sys-tems with periodic coefficients reduce to autonomous difference equation systems, we shall also recall some results onautonomous systems to show the compatibility of the results.

2.1. System of the linear difference equations with constant coefficients

Consider the Cauchy problem

xðnþ 1Þ ¼ AxðnÞ; n 2 Z; ð2:1Þ

where A is a matrix with N � N dimensions and x(n) is an N dimensional vector. The Schur stability of the system (2.1) iswell-known in the literature (see, for example, [3,4,12]). With respect to the criterion known as spectral criterion, if eigen-values of the matrix A lie in the unit disc, in other words if jki(A)j < 1 (i = 1,2, . . . ,N), then the matrix A is called to be the Schurstable matrix. Therefore, the system (2.1) is also called as Schur stable system.

Schur stability parameter x(A) for systems with constant coefficients is defined as

xðAÞ ¼ kHk; H ¼X1k¼0

ðA�ÞkAk; H ¼ H� > 0; A�HA� H þ I ¼ 0:

The system (2.1) is Schur stable if and only if x (A) <1 holds (see, for example, [3,6,12,13]). Here, A⁄ is adjoint of the matrixA, H is positive definite matrix the spectral norm of A is defined as

kAk ¼maxkxk¼1

kAxk; kxk ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiXN

i¼1

x2i

vuut ; x ¼ ðx1; x2; . . . ; xNÞT :

Now let us give the following result which shows how much perturbation is permissible for the autonomous differenceequation system

yðnþ 1Þ ¼ ðAþ BÞyðnÞ; n 2 Z; ð2:2Þ

where B is a constant matrix with N � N dimensional, which is the perturbated system of (2.1).

Theorem 1. Let (2.1) be a Schur stable system, i.e., (x(A) <1). If B satisfies the inequality

kBk <ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffikAk2 þ 1

xðAÞ

s� kAk

then the system (2.2) is Schur stable [8,9].

2.2. System of the linear difference equations with periodic coefficients

Consider the system (1.1)

xðnþ 1Þ ¼ AðnÞxðnÞ; n 2 Z

where A(n) is a T-periodic matrix with N � N dimensions and x(n) is a N dimensional vector. The solution X(n) of the Cauchyproblem

Xðnþ 1Þ ¼ AðnÞXðnÞ; Xð0Þ ¼ I

is called as fundamental matrix of (1.1) and the X(T) matrix defined as

XðTÞ ¼YT�1

j¼0

AðjÞ ¼ AðT � 1ÞAðT � 2Þ . . . Að1ÞAð0Þ

is called to be the monodromy matrix. Similar to determinating the Schur stability of the coefficient matrix A of the systemswith constant coefficients given above, according to spectral criterion; if jki(X(T))j < 1 (i = 1,2, . . . ,N), then the matrix X(T) isSchur stable. Schur stability of the monodromy matrix X(T) implies that the linear difference Cauchy problem with periodiccoefficients (1.1) is Schur stable (see, for example, [3,4]).

A. Duman, K. Aydın / Applied Mathematics and Computation 217 (2011) 6663–6670 6665

Schur stability parameters for the systems with periodic coefficients consist of two different parameters [5];First one of the parameters is x1(A,T) which is given by

x1ðA; TÞ ¼ kFk; F ¼X1k¼0

ðX�ðTÞÞkðXðTÞÞk; F ¼ F� > 0; X�ðTÞFXðTÞ � F þ I ¼ 0:

Second of the parameters is x2(A,T) which is given by

x2ðA; TÞ ¼ kUk; U ¼X1k¼0

X�ðkÞXðkÞ; U ¼ U� > 0

where

U ¼X1k¼0

ðX�ðTÞÞkCðXðTÞÞk; C ¼XT�1

i¼0

X�ðiÞXðiÞ; X�ðTÞUXðTÞ �Uþ C ¼ 0:

The system (1.1) is Schur stable if and only if x1(A,T) <1 (x2(A,T) <1) holds [5]. For the case T = 1, x1(A,T) = x2(A,T) = x(A)holds [5].

The solution Y(n) of the Cauchy problem

Yðnþ 1Þ ¼ ðAðnÞ þ BðnÞÞYðnÞ; Yð0Þ ¼ I; n 2 Z

is the fundamental matrix of (1.2), and the monodromy matrix for the perturbated system (1.2) is therefore Y(T).Below, we quote a continuity theorem for the sensitivity of difference equation systems with periodic coefficients by

means of the monodromy matrix.

Theorem 2. Let X(T) and Y(T) be the monodromy matrices of (1.1) and (1.2), respectively, then the system (1.2) is Schur stable forthe matrix B(n) satisfying the inequality

kYðTÞ � XðTÞk <ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffikXðTÞk2 þ 1

x1ðA; TÞ

s� kXðTÞk

[7].

Remark 1. It should be noticed that the upper bound on the right-hand side of the inequality in Theorem 2 does dependupon the perturbation. In the case T = 1, the constant upper bound reduces to

kBk <

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffikAk2 þ 1

xðAÞ

s� kAk

because of A(n) = A, B(n) = B, X(T)jT=1 = X(1) = A, Y(T)jT=1 = Y(1) = A + B.

3. Main results

In this section, we present new continuity theorems for the durability of the Schur stability by means of the monodromymatrices.

3.1. Symbols

Before introducing our continuity theorems, we need to give the following definitions.

a ¼XT�1

i¼0

kXðiÞk2; Qðn; sÞ ¼

Yn�1

j¼s

AðjÞ; Wðn; sÞ ¼Yn�1

j¼s

BðjÞ;

c ¼ ðT � 1Þmax16k6T

kQðT; kÞk; b ¼ max16k6T

kQðT; kÞk 1þ ðT � 1Þ max16k6T�1

kQðk;0Þk� �

;

l ¼ max16k6T

kQðT; kÞk �

max06k6T�2

kAðjÞk; max06k6T�2

kAðjÞk 6 1;

max06k6T�2

kAðjÞk� �T�2

; max06k6T�2

kAðjÞk > 1;

8>><>>:

6666 A. Duman, K. Aydın / Applied Mathematics and Computation 217 (2011) 6663–6670

D ¼ kYðTÞ � XðTÞk ¼XT�1

k¼0

YT�1

j¼kþ1

AðjÞ !

BðkÞYðkÞ�����

�����;

D1 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffikXðTÞk2 þ 1

x1ðA; TÞ

s� kXðTÞk; D2 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffikXðTÞk2 þ a

x2ðA; TÞ

r� kXðTÞk;

D3 ¼ max06k6T�1

kBðkÞk bþ c max16k6T�1

kWðT; kÞk þ l max16k6T

kQðT; kÞkXT�1

k¼2

Xk�1

l¼1

k!

l!ðk� lÞ! max06j6k�1

kBðjÞk� �l

!" #;

D4 ¼max16j;k6TkQðj; kÞk

�1þ ðT � 1Þmax16k6T�1kXðkÞk

�1� ðT � 1Þmax16j;k6TkQðj; kÞkmax06k6T�1kBðkÞk

max06k6T�1kBðkÞk:

3.2. Continuity Theorems for Sensitivity of Monodromy Matrix

We give the following lemma which is essential for the proof of our continuity theorem.

Lemma 1. Let Y(n) be the fundamental solution of (1.2), then

XT�1

k¼0

kYðkÞk 6 1þ ðT � 1Þ max16k6T�1

kQðk;0Þk þ ðT � 1Þ max16k6T�1

kWðk;0Þk

þXT�1

k¼2

Xk�1

l¼1

k!

l!ðk� lÞ! max06j6k�1

kAðjÞk� �k�l

max06j6k�1

kBðjÞk� �l

" #:

Proof. It is clear that

XT�1

k¼0

kYðkÞk ¼ 1þ kYð1Þk þ � � � þ kYðT � 1Þk

¼ 1þ kAð0Þ þ Bð0Þk þ � � � þ k½AðT � 2Þ þ BðT � 2Þ� � � � ½Að1Þ þ Bð1Þ�½Að0Þ þ Bð0Þ�k;

which yields by rearranging that

XT�1

k¼0

kYðkÞk 6 1þXT�1

k¼1

kQðk;0Þk þXT�1

k¼1

kWðk; 0Þk þXT�1

k¼2

Xk�1

l¼1

k!

l!ðk� lÞ! max06j6k�1

kAðjÞk� �k�l

max06j6k�1

kBðjÞk� �l

" #:

Therefore, we get the inequality

XT�1

k¼0

kYðkÞk 6 1þ ðT � 1Þ max16k6T�1

kQðk; 0Þk þ max16k6T�1

kWðk;0Þk� �

þXT�1

k¼2

Xk�1

l¼1

k!

l!ðk� lÞ! max06j6k�1

kAðjÞk� �k�l

max06j6k�1

kBðjÞk� �l

" #: �

Theorem 3. Let X(T) and Y(T) be the monodromy matrices of the systems (1.1) and (1.2), respectively, then

kYðTÞ � XðTÞk 6 D3:

Moreover, if (1.1) is Schur stable, then for the perturbation matrix B(n) satisfying D3 < D1, the system (1.2) is Schur stable too.

Proof. The difference of the fundamental matrices of (1.1) and (1.2) is

YðnÞ � XðnÞ ¼Xn�1

k¼0

Yn�1

j¼kþ1

AðjÞ !

BðkÞYðkÞ: ð3:1Þ

In (3.1), letting n = T, we get

YðTÞ � XðTÞ ¼XT�1

k¼0

YT�1

j¼kþ1

AðjÞ !

BðkÞYðkÞ:

A. Duman, K. Aydın / Applied Mathematics and Computation 217 (2011) 6663–6670 6667

From this equality, we get

kYðTÞ � XðTÞk 6 max16k6T

kQðT; kÞk max06k6T�1

kBðkÞkXT�1

k¼0

kYðkÞk: ð3:2Þ

Substituting the inequality in Lemma 1 to (3.2), we have

kYðTÞ�XðTÞk6 max06k6T�1

kBðkÞk bþc max16k6T�1

kWðk;0Þkþmax06k6T

kQðT;kÞkXT�1

k¼2

Xk�1

l¼1

k!

l!ðk�lÞ! max06j6k�1

kAðjÞk� �k�l

max06j6k�1

kBðjÞk� �l

" # !

For the latter inequality above, we have the following cases.If max06k6T�2kA(j)k 6 1;

kYðTÞ �XðTÞk6 max06k6T�1

kBðkÞk bþ c max16k6T�1

kWðk;0Þkþmax06k6T

kQðT;kÞk max06j6T�2

kAðjÞkXT�1

k¼2

Xk�1

l¼1

k!

l!ðk� lÞ! max06j6k�1

kBðjÞk� �l

!;

If max06k6T�2kA(j)k > 1;

kYðTÞ�XðTÞk6 max06k6T�1

kBðkÞk bþc max16k6T�1

kWðk;0Þkþmax06k6T

kQðT;kÞk max06j6T�2

kAðjÞk� �T�2XT�1

k¼2

Xk�1

l¼1

k!

l!ðk�lÞ! max06j6k�1

kBðjÞk� �l

!:

It is clear that Theorem 2 implies the system (1.2) remain Schur stable for the perturbation matrix B(n) satisfying theinequality D3 < D1 provided the system (1.1) is Schur stable system. h

Remark 2. Taking T = 1, one can see that Theorem 3 which is a continuity theorem for the periodic system, reduces to The-orem 1, which is given for the autonomous system (2.1) This shows that theorem given above is compatible with the existingones in the literature.

Indeed, in the case T = 1, we have A(n) = A, B(n) = B, X(T)jT=1 = A, Y(T)jT=1 = A + B, x1(A,T)jT=1 = x(A), bjT¼1 ¼ 1; cjT¼1 ¼ 1;

ljT¼1 ¼ 0; max06k60kBðkÞk ¼ kBk; D3jT¼1 ¼ kBk; D1jT¼1 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffikAk2 þ 1

xðAÞ

q� kAk. Here, for convenience, we assume that

maxi6k6j,j<i{.} = 0 andPj

k¼i;j<ið:Þ ¼ 0.

Theorem 4. Let X(n) and Y(n) be the fundamental matrices of the systems (1.1) and (1.2), respectively, and (n � 1)max16j,k6nkQ(j,k)kmax06k6n�1kB(k)k < 1, then we have

kYðnÞ � XðnÞk 6max16j;k6nkQðj; kÞk

�1þ ðn� 1Þmax16k6n�1kXðkÞk

�1� ðn� 1Þmax16j;k6nkQðj; kÞkmax06k6n�1kBðkÞk

max06k6n�1

kBðkÞk:

In this inequality, letting n = T, we get

kYðTÞ � XðTÞk 6max06j;k6TkQðj; kÞk

�1þ ðT � 1Þmax16k6T�1kXðkÞk

�1� ðT � 1Þmax16j;k6TkQðj; kÞkmax06k6T�1kBðkÞk

max06k6T�1

kBðkÞk:

Moreover, if (1.1) is Schur stable system,then (1.2) is Schur stable too provided that B(n) satisfies

kBðnÞk < D1

max16j;k6TkQðj; kÞk 1þ ðT � 1Þ max16k6T�1kXðkÞk þ D1ð Þ½ � :

Proof. Using (3.1), we have

kYðTÞ � XðTÞk ¼Xn�1

k¼0

Qðn; kþ 1ÞBðkÞYðkÞ�����

����� 6 max16k6n

kQðn; kÞk max06k6n�1

kBðkÞkXn�1

k¼0

kYðkÞk: ð3:3Þ

We have

Xn�1

k¼0

kYðkÞk¼Xn�1

k¼0

kYðkÞ�XðkÞþXðkÞk6Xn�1

k¼0

kYðkÞ�XðkÞkþXn�1

k¼0

kXðkÞk61þðn�1Þ max16k6n�1

kYðkÞ�XðkÞkþ max16k6n�1

kXðkÞk� �

:

Substituting the latter inequality into (3.3), we are led to

6668 A. Duman, K. Aydın / Applied Mathematics and Computation 217 (2011) 6663–6670

kYðnÞ � XðnÞk 6 max16k6n

kQðn; kÞk max06k6n�1

kBðkÞk 1þ ðn� 1Þ max16k6n�1

kYðkÞ � XðkÞk þ max16k6n�1

kXðkÞk� ��

Since the inequality holds for all n, and the terms on the right-hand side are maximum, we may replace kY(n) � X(n)k bymax16k6nkY(k) � X(k)k. Thus, letting

g ¼ max16j;k6n

kQðj; kÞk max06k6n�1

kBðkÞk;

we have

max16k6n

kYðkÞ � XðkÞk 6 g 1þ ðn� 1Þ max16k6n�1

kYðkÞ � XðkÞk þ max16k6n�1

kXðkÞk� ��

:

Making arrangements after noting here that

max16k6n�1

kYðkÞ � XðkÞk 6 max16k6n

kYðkÞ � XðkÞk

holds, we get the inequality

max16k6n

kYðkÞ � XðkÞk 6 g1� ðn� 1Þg 1þ ðn� 1Þ max

16k6n�1kXðkÞk

� �:

Thus, we obtain

kYðnÞ � XðnÞk 6 g1� ðn� 1Þg 1þ ðn� 1Þ max

16k6n�1kXðkÞk

� �:

Letting n = T in the last inequality, we have kY(T) � X(T)k 6 D4.From Theorem 2, we get that for the perturbation matrix B(n) satisfying D4 < D1, the system (1.2) is Schur stable. And

making some arrangements, we get

kBðnÞk < D1

max16j;k6T

kQðj; kÞk 1þ ðT � 1Þ max16k6T�1

kXðkÞk þ D1

� �� :

The proof is therefore completed. h

Remark 3. Taking T = 1, one can see that Theorem 4 reduces to Theorem 1. This shows the compatibility of Theorem 4.Indeed, when T = 1, we have A(n) = A, B(n) = B, X(T)jT=1 = A, Y(T)jT=1 = A + B, x1(A,T)jT=1 = x(A) and

D4jT¼1 ¼ kBk;D1

max16j;k6T

kQðj; kÞk 1þ ðT � 1Þ max16k6T�1

kXðkÞk þ D1

� ��

T¼1

¼ D1jT¼1:

The parameter D1 in Theorems 3 and 4 depend upon the parameter x1(A,T). The parameter D2 is obtained by the inequal-ity 1

x1ðA;TÞ6

ax2ðA;TÞ

between the parameters x1(A,T) and x2(A,T). Therefore, we can state Theorems 3 and 4 for the parameterD2 as follows.

Theorem 5. Let the system (1.1) be Schur stable, and B(n) be a perturbation matrix satisfying each of the following conditions:

(i) D3 < D2

(ii) kBðnÞk < D2

max16j;k6TkQðj;kÞk 1þðT�1Þ max16k6T�1kXðkÞkþD2ð Þ½ �

then the perturbated system (1.2) is Schur stable too.

4. Numerical examples

In this section, we give some numerical examples showing the efficiency of the results in Section 3.

Example 1. Consider the periodic linear difference equation system

xðnþ 1Þ ¼0:99 0

0 ð�1Þn2

!xðnÞ:

A. Duman, K. Aydın / Applied Mathematics and Computation 217 (2011) 6663–6670 6669

The condition numbers for the system are x1(A,2) = 25.3781 and x2(A,2) = 50.2513. The system persists on its Schur stabil-ity if perturbated with

– kB(n)k < 0.00995026 (Theorem 3)– kB(n)k < 0.00990101 (Theorem 4)– kB(n)k < 0.00995024 (Theorem 5 (i))– kB(n)k < 0.00990098 (Theorem 5 (ii))

Perturbate the system with the following matrices:

B1ðnÞ ¼0:0099 0

0 ð�1Þn0:0099

� �; max

06k61kB1ðkÞk ¼ 0:0099;

B2ðnÞ ¼0:01 0

0 ð�1Þn0:01

� �; max

06k61kB2ðkÞk ¼ 0:01;

B3ðnÞ ¼0:011 0

0 ð�1Þn0:011

� �; max

06k61kB3ðkÞk ¼ 0:011:

While the system is perturbated by B1 we have x1 (A + B1,2) = 2500.38 hence it is Schur stable, the Schur stability of the sys-tem perturbated by the matrices B2 and B3 can not be guarantied by the bounds of Theorems 3–5. Also for these systems, theparameters turn out to be x1(A + B2,2) =1 and x1(A + B3,2) =1, which implies that the system can not be Schur stable.

Remark 4. From the example above, when we check the values of the perturbation matrices B1(n) and B2(n), we see clearlythat the theorems give sharp bounds. Indeed, due to the spectral criteria, we have k1(X(T)) = �0.25 and k2(X(T)) = 0.9801,k1(Y1(T)) = �0.259998 and k2(Y1(T)) = 0.9998, k1(Y2(T)) = �0.2601 and k2(Y2(T)) = 1, k1(Y3(T)) = �0.261121 andk2(Y3(T)) = 1.002 which shows that the bounds are sharp. This situation can be seen from Fig. 1, which considers the spectralcriterion.

Remark 5. Related to the given system the perturbation bounds allowed by Theorems 3–5 can be sometimes less or greaterthan the others. As is seen from Example 1, the bound provided by Theorem 3 is the greatest, however, for the systemxðnþ 1Þ ¼ sin p

2 n� �

14

14 sin p

2 n� �� �

xðnÞ with T = 4 the bound allowed by Theorem 3 can not be easily computable and that

kBðnÞk < 0:158162 ðTheorem4Þ; kBðnÞk < 0:158167 ðTheorem5ðiiÞÞ:

Example 2. Consider the periodic linear difference equation system

xðnþ 1Þ ¼ð�1Þn

2 00 1

2

!xðnÞ:

Let the perturbation matrices Bi(n)(i = 1,2,3,4) be given as

ð�1Þn4 00 1

4

!;

ð�1Þn6 00 1

6

!;

ð�1Þn16 00 1

16

!;

ð�1Þn24 00 1

24

!;

Fig. 1. Criteria results for Example 1.

Table 1This table illustrates the admisible perturbation matrices (Bi, i = 1,2,3,4), which do not change the Schur stability of the system, the norm difference of themonodromy matrices of the systems (D) and the upper bounds of the difference (Di, i = 1,2,3,4).

B(n) D D1 D2 D 3 D4

B1 0.3125 0.749999 0.75 0.4375 0.5B2 0.194445 0.749999 0.75 0.277778 0.3B3 0.0664062 0.749999 0.75 0.0976562 0.1B4 0.0434028 0.749999 0.75 0.0642361 0.0652174

6670 A. Duman, K. Aydın / Applied Mathematics and Computation 217 (2011) 6663–6670

respectively. Perturbating the system with these matrices, we obtain the following table which allows us to compare theupper bounds of kY(T) � X(T)k.

As is seen from Table 1, the upper bounds D1 and D2 are constant, in this work, the upper bounds D3 and D4 depend onthe perturbation. Therefore, the upper bounds D3 and D4 give closer values to the accured value D. For the perturbationmatrix B3 of the system, we get D = 0.0664062, and the constant upper bounds are as D1 = 0.749999 and D2 = 0.75, butthe upper bounds of this work are found as D3 = 0.0976562 and D4 = 0.1.

The numerical examples have been computed by using matrix vector calculator MVC [14].

5. Conclusions

In this work, we presented theorems, which replace the constant upper bound provided by the well-known result The-orem 2 for kY(T) � X(T)k with varying and sharper ones. Since the upper bounds given in this paper are closer to the accuredvalue, we are able to get better information about the behaviour of the term kY(T) � X(T)k. Moreover, examples showing theefficiency of the upper bounds are given too. As is obvious from the examples, the upper bounds can be computed easer thanthe others for the tolerable perturbation in Theorems 4 and 5 (ii) for those systems with T P 3.

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