Abstract
We first give a theorem on the reducibility of linear system of difference equations of the form 
. Next, by the means of Floquet theory, we obtain some stability results. Moreover, some examples are given to illustrate the importance of the results.
Similar content being viewed by others
1. Introduction
Consider the homogeneous linear system of difference equations
where 
is a 
nonsingular matrix with real entries and 
If for some 
is specified, then (1.1) is called an initial value problem (IVP). The solution of this IVP is given by
where 
is the fundamental matrix defined by
However, (1.1) is called reducible to equation
if there is a nonsingular matrix 
with real entries such that
Let 
be a 
matrix function whose entries are real-valued functions defined for 
. Consider the system
Let 
be a fundamental matrix of (1.7) satisfying 
. This 
can be used to transform (1.1) 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 matrix 
, and provide an example for illustration. The results in this section are discrete analogues of the ones given in [1].
Theorem 2.1.
The homogeneous linear difference system (1.1) is reducible to (1.5) under the transformation (1.6) if and only if there exists a 
regular real matrix 
such that
hold.
Proof.
Let 
and 
be defined as above. Under the transformation (1.6), (1.1) becomes
and after reorganizing, we get
Thus, (1.1) is reducible to (1.5) with
Clearly, 
is the unique solution of the IVP:
where 
This problem is equivalent to solving (2.1). □
Corollary 2.2.
The homogeneous linear system of difference equation (1.1) is reducible to
with a constant matrix 
under transformation (1.6) if and only if there exists a 
regular real matrix 
defined for 
such that
hold.
Below, we give an example for Corollary 2.2 in the special case 
. To obtain the matrix 
, we choose a suitable form of the matrix 
.
Example 2.3.
Consider the system
where
-
(i)
are real-valued functions defined for 
such that 
for all 
-
(ii)
for all 
-
(iii)
are real-valued functions defined for 
such that 
for all 
for all 
We also assume that for all 
,
It is easy to verify that if we take
where
then (2.7) holds. Moreover, from (2.8) we have
In case 
for every 
, that is,
the relations (2.10), (2.12), and (2.13) take the form
where 
is a real constant and 
, 
are arbitrary real constants such that 
Corollary 2.4.
If there exists a 
regular constant matrix 
such that
then (1.1) reduces to (2.6) with 
It should be noted that in case the constant matrices 
and 
commute, that is, 
, then 
must be a constant matrix as well.
3. Stability of Linear Systems
It turns out that to obtain a stability result, one needs take 
, a periodic matrix [2]. Indeed, this allows using the Floquet theory for linear periodic system (1.7).
We need the following three well-known theorems [3–5].
Theorem 3.1.
Let 
be the fundamental matrix of (1.1) with 
The zero solution of (1.1) is
-
(i)
stable if and only if there exists a positive constant M such that
(3.1) -
(ii)
asymptotically stable if and only if
(3.2)where
is a norm in
.
is a norm in
.Theorem 3.2.
Consider system (1.1) with 
a constant regular matrix. Then its zero solution is
-
(i)
stable if and only if
and the eigenvalues of unit modulus are semisimple; -
(ii)
asymptotically stable if and only if
, where 
is an eigenvalue of
is the spectral radius of 
and the eigenvalues of unit modulus are semisimple;
, where 
is an eigenvalue of
is the spectral radius of 
Consider the linear periodic system
where 
, 
for some positive integer N.
From the literature, we know that if 
with 
is a fundamental matrix of (3.3), then there exists a constant 
matrix, whose eigenvalues are called the Floquet exponents, and periodic matrix 
with period N such that 
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 Theorems 3.1, 3.2, and 3.3, we obtain from Corollary 2.2 the following new stability criteria for (1.1).
Theorem 3.4.
The zero solution of (1.1) is stable if and only if there exists a 
regular periodic matrix 
satisfying (2.8) such that
-
(i)
the Floquet exponents of
have modulus less than or equal to one; those with modulus of one are semisimple; -
(ii)
; those eigenvalues of 
of unit modulus are semisimple.
have modulus less than or equal to one; those with modulus of one are semisimple;
; those eigenvalues of 
of unit modulus are semisimple.Theorem 3.5.
The zero solution of (1.1) is asymptotically stable if and only if there exists a 
regular periodic matrix 
satisfying (2.8) such that either
-
(i)
all the Floquet exponents of
lie inside the unit disk and 
; those eigenvalues of 
of unit modulus are semisimple; or -
(ii)
the Floquet exponents of
have modulus less than or equal to one; those with modulus of one are semisimple; and 
lie inside the unit disk and 
; those eigenvalues of 
of unit modulus are semisimple; or
have modulus less than or equal to one; those with modulus of one are semisimple; and 
Remark 3.6.
Let 
be periodic with period N. The Floquet exponents mentioned in Theorem 3.3 are the eigenvalues of 
where 
Example 3.7.
Consider the system
Note that the conditions of Example 2.3 are all satisfied. It follows that
Now,
for which the eigenvalues are 
On the other hand, for
if 
and 
if 
.
Applying Theorems 3.4 and 3.5, we see that the zero solution of (3.4) is asymptotically stable if 
and is stable if 
In fact, the unique solution of (3.4) satisfying 
is
where 
, 
, 
, 
, and 
.
It is easy to see that 
if 
and 
is bounded if 
Remark 3.8.
In the computation of 
, 
is calculated by using Example 2.3, and 
is obtained by the method given in [6, 7].
References
Tiryaki A:On the equation
. Mathematica Japonica 1988, 33(3):469-473.Rodriguez J, Etheridge DL: Periodic solutions of nonlinear second-order difference equations. Advances in Difference Equations 2005, 2005(2):173-192. 10.1155/ADE.2005.173
Elaydi SN: An Introduction to Difference Equations, Undergraduate Texts in Mathematics. Springer, New York, NY, USA; 1996:xiv+389.
Kipnis M, Komissarova D: Stability of a delay difference system. Advances in Difference Equations 2006, 2006:-9.
Lakshmikantham V, Trigiante D: Theory of Difference Equations: Numerical Methods and Application, Mathematics in Science and Engineering. Volume 181. Academic Press, Boston, Mass, USA; 1988:x+242.
Elaydi SN, Harris WA Jr.:On the computation of
. SIAM Review 1998, 40(4):965-971. 10.1137/S0036144597319235Zafer A: Calculating the matrix exponential of a constant matrix on time scales. Applied Mathematics Letters 2008, 21(6):612-616. 10.1016/j.aml.2007.06.006
. Mathematica Japonica 1988, 33(3):469-473.
. SIAM Review 1998, 40(4):965-971. 10.1137/S0036144597319235Acknowledgment
The authors would like to thank to Professor Ağacık Zafer for his valuable contributions to Section 3.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Tiryaki, A., Misir, A. Reducibility and Stability Results for Linear System of Difference Equations. Adv Differ Equ 2008, 867635 (2008). https://doi.org/10.1155/2008/867635
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1155/2008/867635