Reducibility and Stability Results for Linear System of Difference Equations

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doi:10.1155/2008/867635

Research Article

Reducibility and Stability Results for

Linear System of Difference Equations

Aydin Tiryaki1_{and Adil Misir}2

1_{Department of Mathematics and Computer Sciences, Faculty of Arts and Science, Izmir University,}

35340 Izmir, Turkey

2_{Department of Mathematics, Faculty of Arts and Science, Gazi University, Teknikokullar,}

06500 Ankara, Turkey

Correspondence should be addressed to Adil Misir,[email protected]

Received 8 August 2008; Revised 22 October 2008; Accepted 29 October 2008

Recommended by Martin J. Bohner

We first give a theorem on the reducibility of linear system of diﬀerence equations of the form

xn1 Anxn. Next, by the means of Floquet theory, we obtain some stability results. Moreover, some examples are given to illustrate the importance of the results.

Copyrightq2008 A. Tiryaki and A. Misir. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

Consider the homogeneous linear system of diﬀerence equations

xn1 Anxn, n∈N{0,1,2, . . .}, 1.1

where An a_ijn is ak×k nonsingular matrix with real entries and xn x1n,

x2n, . . . , xknT ∈Rk.

If for somen0≥0,

xn0 x0 1.2

is specified, then1.1is called an initial value problemIVP. The solution of this IVP is given by

xn, n0, x0

_n₋₁

in0

Ai

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whereΦnis the fundamental matrix defined by

However,1.1is called reducible to equation

yn1 Bnyn, 1.5 if there is a nonsingular matrixHnwith real entries such that

xn Hnyn. 1.6 LetSnbe ak×kmatrix function whose entries are real-valued functions defined for

n≥n0. Consider the system

zn1 Snzn, n≥n0. 1.7

LetHnbe a fundamental matrix of1.7satisfyingHn0 I. ThisHncan be used

to transform1.1into1.5.

Stability properties of 1.1 can be deduced by considering the reduced form 1.5 under some additional conditions. In this study, we first give a theorem on the reducibility of

1.1into the form of1.5and then obtain asymptotic stability of the zero solution of1.1.

2. Reducible systems

In this section, we give a theorem on the structure of the matrixSn, and provide an example for illustration. The results in this section are discrete analogues of the ones given in1 .

Theorem 2.1. The homogeneous linear diﬀerence system 1.1 is reducible to 1.5 under the transformation1.6if and only if there exists ak×kregular real matrixSnsuch that

An1Sn Sn1An Sn1SnHnΔBnH−1n, Thus,1.1is reducible to1.5with

Bn H−1_n_S−1_n_A_n_H_n_. _2.4

Clearly,Bnis the unique solution of the IVP:

ΔBn Fn,

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Corollary 2.2. The homogeneous linear system of diﬀerence equation1.1is reducible to

yn1 Byn, 2.6

with a constant matrixBunder transformation1.6if and only if there exists ak×kregular real matrixSndefined forn≥n0,such that

Below, we give an example forCorollary 2.2 in the special casek 2. To obtain the matrixHn, we choose a suitable form of the matrixSn.

It is easy to verify that if we take

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then2.7holds. Moreover, from2.8we have

BS−1_n

0An0. 2.15

In cases21n 0 for everyn≥n0, that is,

a11nΘn

a12na12n1−

a11n1Θn−1

a12n−1a12n1

0, n≥n0, 2.16

the relations2.10,2.12, and2.13take the form

a12n1a22n

a11n1a12n

a22n1a21n

a11na21n1 α,

s11n α1a11n,

s22n α2a22n,

2.17

whereα /0 is a real constant andα1,α2are arbitrary real constants such thatα1/α2α.

Corollary 2.4. If there exists ak×kregular constant matrixSsuch that

An1SSAn, 2.18

then1.1reduces to2.6withBS−1_A_n 0.

It should be noted that in case the constant matricesSandBcommute, that is,SBBS, thenAnmust be a constant matrix as well.

3. Stability of linear systems

It turns out that to obtain a stability result, one needs takeSn, a periodic matrix2 . Indeed, this allows using the Floquet theory for linear periodic system1.7.

We need the following three well-known theorems3–5 .

Theorem 3.1. LetΦnbe the fundamental matrix of 1.1withΦn0 I.

The zero solution of 1.1is

istable if and only if there exists a positive constant M such that

Φn ≤M forn≥n0≥0; 3.1

iiasymptotically stable if and only if

lim

n→ ∞Φn0, 3.2

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Theorem 3.2. Consider system 1.1 with An A, a constant regular matrix. Then its zero solution is

istable if and only ifρA≤1and the eigenvalues of unit modulus are semisimple;

iiasymptotically stable if and only ifρA<1, whereρA max{|λ|:λis an eigenvalue ofA}is the spectral radius ofA.

Consider the linear periodic system

zn1 Snzn, 3.3

wheren∈Z,SnN Sn,for some positive integerN.

From the literature, we know that ifΨn,with Ψn0 I is a fundamental matrix

of 3.3, then there exists a constant C matrix, whose eigenvalues are called the Floquet exponents, and periodic matrixPnwith periodNsuch thatΨn PnCn−n0.

Theorem 3.3. The zero solution of 3.3is

istable if and only if the Floquet exponents have modulus less than or equal to one; those with modulus of one are semisimple;

iiasymptotically stable if and only if all the Floquet exponents lie inside the unit disk.

In view of Theorems3.1,3.2, and3.3, we obtain fromCorollary 2.2the following new stability criteria for1.1.

Theorem 3.4. The zero solution of 1.1is stable if and only if there exists ak×kregular periodic matrixSnsatisfying2.8such that

ithe Floquet exponents ofSnhave modulus less than or equal to one; those with modulus of one are semisimple;

iiρS−1_n

0An0≤1; those eigenvalues ofS−1n0An0of unit modulus are semisimple. Theorem 3.5. The zero solution of 1.1is asymptotically stable if and only if there exists ak×k regular periodic matrixSnsatisfying2.8such that either

iall the Floquet exponents ofSnlie inside the unit disk andρS−1_n

0An0≤1; those

eigenvalues ofS−1_n

0An0of unit modulus are semisimple; or

iithe Floquet exponents ofSnhave modulus less than or equal to one; those with modulus of one are semisimple; andρS−1_n

0An0<1.

Remark 3.6. Let Sn be periodic with period N. The Floquet exponents mentioned in

Theorem 3.3are the eigenvalues ofC,whereCNSN−1SN−2· · ·S0.

Example 3.7. Consider the system

xn1

−1n βn1

β−n ₋₁n

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Note that the conditions ofExample 2.3are all satisfied. It follows that

Applying Theorems3.4and3.5, we see that the zero solution of3.4is asymptotically stable if 2/3< β <1,and is stable ifβ2/3.

In fact, the unique solution of3.4satisfyingx0 x0is

xn HnBn_x

The authors would like to thank to Professor A ˘gacık Zafer for his valuable contributions to

Section 3.

References

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