Next, by the means of Floquet theory, we obtain some stability results.. In this study, we first give a theorem on the reducibility of 1.1 into the form of 1.5 and then obtain asymptotic
Trang 1Volume 2008, Article ID 867635, 6 pages
doi:10.1155/2008/867635
Research Article
Reducibility and Stability Results for
Linear System of Difference Equations
Aydin Tiryaki 1 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,adilm@gazi.edu.tr
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
xn 1 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
Copyrightq 2008 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
xn 1 Anxn, n ∈ N {0, 1, 2, }, 1.1
where An a ij n is a k × k nonsingular matrix with real entries and xn x1n,
x2n, , x k n T ∈ Rk
If for somen0≥ 0,
is specified, then1.1 is called an initial value problem IVP The solution of this IVP is given by
xn, n0, x0
n−1
in
Ai
Trang 2
whereΦn is the fundamental matrix defined by
n−1
in0
Ai
⎧
⎨
⎩
An − 1An − 2 · · · An0
, if n > n0,
However,1.1 is called reducible to equation
yn 1 Bnyn, 1.5
if there is a nonsingular matrixHn with real entries such that
LetSn be a k × k matrix function whose entries are real-valued functions defined for
n ≥ n0 Consider the system
zn 1 Snzn, n ≥ n0. 1.7 LetHn be a fundamental matrix of 1.7 satisfying Hn0 I This Hn can be used
to transform1.1 into 1.5
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
1.1 into the form of 1.5 and then obtain asymptotic stability of the zero solution of 1.1
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
transformation1.6 if and only if there exists a k × k regular real matrix Sn such that
An 1Sn Sn 1An Sn 1SnHnΔBnH−1n,
An0
Sn0
Bn0
hold.
Proof Let Sn and Hn be defined as above Under the transformation 1.6, 1.1 becomes
Hn 1yn 1 AnHnyn, 2.2 and after reorganizing, we get
yn 1 H−1nS−1nAnHnyn. 2.3 Thus,1.1 is reducible to 1.5 with
Bn H−1nS−1nAnHn. 2.4 Clearly,Bn is the unique solution of the IVP:
ΔBn Fn,
Bn0
S−1
n0
An0
whereFn : ΔH−1nS−1nAnHn.
This problem is equivalent to solving2.1
Trang 3Corollary 2.2 The homogeneous linear system of difference equation 1.1 is reducible to
with a constant matrix B under transformation 1.6 if and only if there exists a k × k regular real
matrix Sn defined for n ≥ n0, such that
An 1Sn Sn 1An, 2.7
An0
Sn0
hold.
Below, we give an example forCorollary 2.2 in the special casek 2 To obtain the
matrixHn, we choose a suitable form of the matrix Sn.
Example 2.3 Consider the system
xn 1 a11n a12n
a21n a22n
where
i a ij n are real-valued functions defined for n ≥ n0such thata ij n / 0 for all i, j
1, 2,
ii det A / 0 for all n ≥ n0,
iii Θn : a12n 1a22n a12na11n 1 / 0.
We also assume that for alln ≥ n0,
Θn − 1
a
21n 1
a12n − 1−
a22n 1a11n 1
a12n 1a12n − 1 Θn
a
11na22n 1
a12na12n 1
a11na11n 2
a12na12n 2
−Θn 1
a
11na11n 1
a12n 1a12n 2
a21n
a12n 2 0.
2.10
It is easy to verify that if we take
Sn s11n 0
s21n s22n
where
s11n a Θn − 1
s22n a Θn
s21n a11nΘn
a12na12n 1−
a11n 1Θn − 1
a12n − 1a12n 1 , 2.14
Trang 4then2.7 holds Moreover, from 2.8 we have
B S−1n0An0. 2.15
In cases21n 0 for every n ≥ n0, that is,
a11nΘn
a12na12n 1−
a11n 1Θn − 1
a12n − 1a12n 1 0, n ≥ n0, 2.16 the relations2.10, 2.12, and 2.13 take the form
a12n 1a22n
a11n 1a12n
a22n 1a21n
a11na21n 1 α,
s11n α1a11n,
s22n α2a22n,
2.17
whereα / 0 is a real constant and α1,α2are arbitrary real constants such thatα1/α2 α.
Corollary 2.4 If there exists a k × k regular constant matrix S such that
An 1S SAn, 2.18
then1.1 reduces to 2.6 with B S−1An0.
It should be noted that in case the constant matricesS and B commute, that is, SB BS,
thenAn must 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 matrix 2 Indeed, this allows using the Floquet theory for linear periodic system1.7
We need the following three well-known theorems3 5
Theorem 3.1 Let Φn be the fundamental matrix of 1.1 with Φn0 I.
The zero solution of 1.1 is
i stable if and only if there exists a positive constant M such that
ii asymptotically stable if and only if
lim
where · is a norm inR k×k
Trang 5Theorem 3.2 Consider system 1.1 with An A, a constant regular matrix Then its zero
solution is
i stable if and only if ρA ≤ 1 and the eigenvalues of unit modulus are semisimple;
ii asymptotically stable if and only if ρA < 1, where ρA max{|λ| : λ is an eigenvalue
of A} is the spectral radius of A.
Consider the linear periodic system
zn 1 Snzn, 3.3 wheren ∈ Z, Sn N Sn, for some positive integer N.
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 matrixPn with period N such that Ψn PnC n−n0.
Theorem 3.3 The zero solution of 3.3 is
i stable if and only if the Floquet exponents have modulus less than or equal to one; those with
modulus of one are semisimple;
ii asymptotically 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 1.1 is stable if and only if there exists a k × k regular periodic
matrix Sn satisfying 2.8 such that
i the Floquet exponents of Sn have modulus less than or equal to one; those with modulus
of one are semisimple;
ii ρS−1n0An0 ≤ 1; those eigenvalues of S−1n0An0 of unit modulus are semisimple.
Theorem 3.5 The zero solution of 1.1 is asymptotically stable if and only if there exists a k × k
regular periodic matrix Sn satisfying 2.8 such that either
i all the Floquet exponents of Sn lie inside the unit disk and ρS−1n0An0 ≤ 1; those
eigenvalues of S−1n0An0 of unit modulus are semisimple; or
ii the Floquet exponents of Sn have modulus less than or equal to one; those with modulus
of one are semisimple; and ρS−1n0An0 < 1.
Remark 3.6 Let Sn be periodic with period N The Floquet exponents mentioned in
Theorem 3.3are the eigenvalues ofC, where C N SN − 1SN − 2 · · · S0.
Example 3.7 Consider the system
xn 1 −1
n β n1
β −n −1n
xn, 0 < β < 1. 3.4
Trang 6Note that the conditions ofExample 2.3are all satisfied It follows that
Sn −1
0 −1n1
Now,
C2 S1S0 −β
0 −1
for which the eigenvalues areλ1 −1, λ2 −β2.
On the other hand, for
B S−10A0
⎡
⎣1β 1
−1 −1
⎤
ρB < 1 if 2/3 < β < 1, and ρB 1 if β 2/3.
Applying Theorems3.4and3.5, we see that the zero solution of3.4 is asymptotically stable if 2/3 < β < 1, and is stable if β 2/3.
In fact, the unique solution of3.4 satisfying x0 x0is
xn HnB n x0 1
μ2− μ1
⎡
⎣ Q −1nn − 12 β n μ n2 − μ n
1
−1nn1/2 μ n
1− μ n
⎤
⎦ x0, 3.8
whereμ1 −γ −γ2− 4γ/2, μ2 −γ γ2− 4γ/2, γ 1−1/β, Q −1 nn−1/2 β n μ n
1μ2−
1/β − μ n
2μ1− 1/β, and M −1 nn1/2 μ n
2μ2 1 − μ n
1μ1 1
It is easy to see that limn → ∞ xn 0 if 2/3 < β < 1, and xn is bounded if β 2/3.
Remark 3.8 In the computation of HnB n,Hn is calculated by usingExample 2.3, andB n
is obtained by the method given in6,7
Acknowledgment
The authors would like to thank to Professor A ˘gacık Zafer for his valuable contributions to
Section 3
References
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3 S N Elaydi, An Introduction to Difference Equations, Undergraduate Texts in Mathematics, Springer,
New York, NY, USA, 1996
4 M Kipnis and D Komissarova, “Stability of a delay difference system,” Advances in Difference Equations, vol 2006, Article ID 31409, 9 pages, 2006.
5 V Lakshmikantham and D Trigiante, Theory of Difference Equations: Numerical Methods and Application, vol 181 of Mathematics in Science and Engineering, Academic Press, Boston, Mass, USA, 1988.
6 S N Elaydi and W A Harris Jr., “On the computation of A n ,” SIAM Review, vol 40, no 4, pp 965–971,
1998
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... k×k Trang 5Theorem 3.2 Consider system 1.1 with An A, a constant... < β < 1. 3.4
Trang 6Note that the conditions ofExample 2.3are all satisfied It follows that
Sn... 1 , 2.14
Trang 4then2.7 holds Moreover, from 2.8 we have
B