R E S E A R C H
Open Access
Hyers-Ulam stability of the first-order
matrix difference equations
Soon-Mo Jung
**Correspondence:
[email protected] Mathematics Section, College of Science and Technology, Hongik University, Sejong, 339-701, Republic of Korea
Abstract
In this paper, we prove the Hyers-Ulam stability of the first-order linear homogeneous matrix difference equationsxi=Axi–1andxi–1=Axifor all integersi∈Z.
MSC: Primary 39A45; 39B82; secondary 39A06; 39B42
Keywords: difference equation; matrix difference equation; recurrence; Hyers-Ulam stability; approximation
1 Introduction
Throughout this paper, letnbe a fixed positive integer. Thenth order linear homogeneous difference equation with constant coefficients is of the form
ai=αai–+αai–+· · ·+αnai–n, ()
whereα,α, . . . ,αnare constants. For example, the second-order difference equation with
constant coefficients has the form
ai=αai–+βai–. ()
The solution of () is called the Fibonacci numbers whenα=β= ,a= , anda= ,
Lucas numbers whenα=β= ,a= , anda= , Pell numbers whenα= ,β= ,a= ,
anda= , Pell-Lucas numbers whenα= ,β= , anda=a= , and Jacobsthal numbers
ifα= ,β= ,a= , anda= .
The polynomial
p(x) =xn–αxn––αxn––· · ·–αn–x–αn
is called the characteristic polynomial of the difference equation ().
If the rootsr,r, . . . ,rnof the characteristic polynomial are distinct, then the solution of
the difference equation () is given by
ai=kri+kri+· · ·+knrin,
where the coefficientsk,k, . . . ,knare uniquely determined under the initial conditions of
the difference equation.
If the characteristic polynomial has rootsr,r, . . . ,rdwith multiplicitym,m, . . . ,md,
respectively, then the solution of the difference equation () is given by
ai= d
j= mj
k=
cjkik–rij,
where thecjk are constants andm+m+· · ·+md=n(see [, ]). For the Hyers-Ulam
stability of the linear difference equations, we may refer to [–]. Let (Cn, ·
n) be a complex normed space, each of whose elements is a column vector,
and let Cn×nbe a vector space consisting of all (n×n) complex matrices. We choose a
norm · n×non Cn×nwhich is compatible with · n,i.e., both norms obey
ABn×n≤ An×nBn×n and Axn≤ An×nxn ()
for all A, B∈Cn×nandx∈Cn.
A matrix difference equation is a difference equation with matrix coefficients in which the value of vector of variables at one point is dependent on the values of preceding (suc-ceeding) points.
In this paper, we prove the Hyers-Ulam stability of the first-order linear homogeneous matrix difference equationsxi= Axi–andxi–= Axifor all integersi∈Z, where the
tran-sition matrix A is nonsingular. More precisely, we prove that if a sequence{yi}i∈Zsatisfies
the inequalityyi– Ayi–n≤εfor alli∈Zresp.yi–– Ayin≤εfor alli∈Z, then there
exist a solution{xi}i∈Z⊂Cnof the first-order matrix difference equation () resp. () and
a constantK> such thatyi–xin≤Kεfor all integersi≥. (We refer the reader to
[–] for the exact definition of Hyers-Ulam stability.)
It should be remarked that many interesting theorems have been proved in [, ] con-cerning the linear (or nonlinear) recurrences. Especially in , the Hyers-Ulam stability of the first-order matrix difference equations has been proved in [] in a general setting. The substantial difference of this paper from [] lies in the fact that the stability problems for the ‘backward’ difference equations have been treated in Section of this paper.
2 Hyers-Ulam stability ofxi=Axi–1
In this section, we investigate the Hyers-Ulam stability of the first-order linear homoge-neous matrix difference equation
xi= Axi– ()
for all integersi∈Z, where
xi=
⎛ ⎜ ⎜ ⎜ ⎜ ⎝
xi
xi
.. .
xin
⎞ ⎟ ⎟ ⎟ ⎟ ⎠∈C
n and A=
⎛ ⎜ ⎜ ⎜ ⎜ ⎝
a a · · · an
a a · · · an
..
. ... . .. ...
an an · · · ann
⎞ ⎟ ⎟ ⎟ ⎟ ⎠∈C
n×n.
Theorem . Given a fixed positive integer n,let(Cn,·n)and(Cn×n,·n×n)be complex
property().Assume that the transition matrixA∈Cn×nis nonsingular and{ε
i}i∈Zis a
sequence of nonnegative real numbers.If a sequence{yi}i∈Z⊂Cnsatisfies the inequality
yi– Ayi–n≤εi ()
for all i∈Z,then there exists a solution{xi}i∈Z⊂Cnof the first-order matrix difference
equation()such that
yi–xin≤ i
k=εkAin–×kn+Ain×ny–xn (for i≥),
–i
k=εk+iA–kn×n+A––ni×ny–xn (for i< ).
Proof Assume that a sequence{yi}i∈Z⊂Cnsatisfies the inequality () for alli∈Z. First,
we assume thatiis a nonnegative integer. It then follows from () and () that
yi– Aiyn≤ yi– Ayi–n+Ayi–– Ayi–n
+Ayi–– Ayi–n+· · ·+Ai–y– Aiyn ≤ yi– Ayi–n+An×nyi–– Ayi–n
+An×nyi–– Ayi–n+· · ·+Ain–×ny– Ayn ≤εi+An×nεi–+An×nεi–+· · ·+Ain–×nε
=Ain×n
i
k=
εkA–n×kn. ()
It is obvious that a sequence{xi}i∈Z⊂Cnsatisfies the first-order matrix difference
equa-tion () if and only if
xi= Aix ()
for eachi∈Z, where we set Ai= (A–)–ifor all negative integersi. Hence, by () and (),
we have
yi–xin≤yi– Aiyn+Aiy– Aixn+Aix–xin
≤ i
k=
εkAin–×kn+Ain×ny–xn
for any integeri≥.
On the other hand, we supposeiis a negative integer. For this case, it follows from () and () that
yi– Aiyn
=yi–
A––iyn
≤yi– A–yi+n+A–yi+–
A–yi+n
+A–yi+–
A–yi+n+· · ·+A–
–i–
y––
≤A–
n×nAyi–yi+n+A –
n×nAyi+–yi+n
+A–
n×nAyi+–yi+n+· · ·+A ––i
n×nAy––yn ≤A–n×nεi++A–
n×nεi++A –
n×nεi++· · ·+A ––i
n×nε
=
–i
k=
εk+iA–kn×n. ()
Moreover, by () and (), we have
yi–xin≤yi–
A––iyn+A–
–i
y–
A––ixn
+A––ix–xin ≤
–i
k=
εk+iA– k n×n+A
––i
n×ny–xn
for all integersi< .
In view of (), if we assume the initial condition in the previous theorem, we can easily prove the uniqueness of the sequence{xi}i∈Zas we see in the following corollary.
Corollary . Given a fixed positive integer n,let(Cn,·
n)and(Cn×n,·n×n)be complex
normed spaces,whose elements are column vectors resp. (n×n)complex matrices,with the property().Assume that the transition matrixA∈Cn×nis nonsingular and{ε
i}i∈Zis
a sequence of nonnegative real numbers.If a sequence{yi}i∈Z⊂Cnsatisfies the inequality
()for all i∈Z,then there exists a unique solution{xi}i∈Z⊂Cnof the first-order matrix
difference equation()with the initial conditionx=ysuch that
yi–xin≤ i
k=εkAni–×kn (for i≥),
–i
k=εk+iA–kn×n (for i< ).
Some of the most important matrix norms are induced byp-norms. For ≤p≤ ∞, the matrix norm induced by thep-norm,
Ap:=sup x=
Axp xp
,
is called the matrixp-norm. For example, we get
A=max
≤j≤n n
i=
|aij| and A∞=max ≤i≤n
n
j= |aij|.
It is well known that the matrixp-norm, together with thep-norm, satisfies the conditions in (), where
x=
n
j=
|xj| and x∞=max ≤j≤n|xj|
In the following corollary, we prove the Hyers-Ulam stability of the second-order linear homogeneous difference equation with constant coefficients.
Corollary . Let(C, ·
∞)and(C×, · ∞)be complex normed spaces and letα,β,
γ be complex numbers satisfying the conditions
α+ β= , β= , γ = . ()
Assume thatε> is an arbitrary constant.If a sequence{ai}i∈Zof complex numbers
satis-fies the inequality
|ai–αai––βai–| ≤ε ()
for all i∈Z,then there exists a sequence{ci}i∈Zof complex numbers such that c–=a–,
c=a,ci=αci–+βci–,and
|ai–ci| ≤ i
k=εAi∞–k (for i≥),
–i
k=εA–k∞ (for i< ),
whereA∞=max{|α|+|β/γ|,|γ|}andA–
∞=max{|/γ|,|α/β|+|γ/β|}.
Proof If we define a sequence{bi}i∈Zof complex numbers bybi=γai–, it then follows
from () that
|ai–αai––βγbi–| ≤ε,
|bi–γai–|=
for anyi∈Z. If we set
yi:=
ai
bi
and A:=
α βγ γ
,
then we get
yi– Ayi–∞≤ε
for eachi∈Z.
According to Corollary ., there exists a unique solution{xi}i∈Z⊂Cof the first-order
matrix difference equation () with the initial conditionx=
a
γa–
such that
yi–xi∞≤ i
k=εAi∞–k (fori≥),
–i
k=εA–k∞ (fori< ).
In view of (), this last inequality implies that
ai γai–
– Ai
a γa–
∞ ≤
i
k=εAi∞–k (fori≥),
–i
k=εA–k∞ (fori< ).
Since the transition matrix A has two distinct eigenvalues λ =
α–√α+β
andλ =
α+√α+β
, which are the roots of the characteristic equationλ
–αλ–β= , the matrix A
can be expressed as
A= CDC– ()
for every integeri≥. Using this equality, it follows from () that
βλ, which are roots of the characteristic equationω
+α
βω–
β = . Hence, the matrix A
–may be expressed as
A–=
for all integersi< . Thus, the inequality () yields
Finally, considering (), (), and [], Theorem ., if we set
Furthermore, it is not difficult to show that the sequence{ci}i∈Zsatisfies the second-order
linear difference equation
, we analogously obtain
limβ→∞A∞· A–∞= .
Ifαandβare simultaneously small in absolute value, then the second-order difference equation () has the Hyers-Ulam stability as we see in the following example.
3 Hyers-Ulam stability ofxi–1=Axi
In practical applications, we sometimes consider the first-order linear homogeneous ma-trix difference equation
xi–= Axi ()
instead of (), where the transition matrix A is a nonsingular matrix of Cn×n.
We now investigate the Hyers-Ulam stability of the matrix difference equation ().
Theorem . Given a fixed positive integer n,let(Cn,·n)and(Cn×n,·n×n)be complex
normed spaces,whose elements are column vectors resp. (n×n)complex matrices,with the property().Assume that the transition matrixA∈Cn×nis nonsingular and{εi}i∈Zis a
sequence of nonnegative real numbers.If a sequence{yi}i∈Z⊂Cnsatisfies the inequality
yi–– Ayin≤εi ()
for all i∈Z,then there exists a solution{xi}i∈Z⊂Cnof the first-order matrix difference
equation()such that
yi–xin≤ i
k=εkA–ni+–×nk+A–in×ny–xn (for i≥),
–i
k=εk+iAkn–×n+A–n×iny–xn (for i< ).
()
Proof Assume that a sequence{yi}i∈Z⊂Cnsatisfies the inequality () for alli∈Z. First,
we assume thatiis a nonnegative integer. Then, by () and (), we have
yi– A–iyn≤yi– A–yi–n+A–yi–– A–yi–n
+A–yi–– A–yi–n+· · ·+A–i+y– A–iyn ≤εiA–n×n+εi–A–
n×n+εi–A –
n×n+· · ·+εA –i
n×n
=
i
k=
εkA– i+–k n×n .
Obviously, a sequence{xi}i∈Z⊂Cnsatisfies the first-order matrix difference equation
() if and only if
xi= A–ix ()
for alli∈Z, where we set A–i= (A–)ifor each integeri≥. Hence, we get
yi–xin≤yi– A–iyn+A–iy– A–ixn+A–ix–xin
≤ i
k=
εkA– i+–k n×n +A
–i
n×ny–xn
On the other hand, ifiis a negative integer, then it follows from () and () that
yi– A–iyn=yi– Ayi+n+Ayi+– Ayi+n
+Ayi+– Ayi+n+· · ·+A–i–y–– A–iyn ≤εi++εi+An×n+εi+An×n+· · ·+εA–ni×–n
=
–i
k=
εk+iAkn–×n.
Thus, by () and the last inequality, we obtain
yi–xin≤yi– A–iyn+A–iy– A–ixn+A–ix–xin
≤ –i
k=
εk+iAkn–×n+A–n×iny–xn
for any integeri< .
We now remark that if we apply Theorem . in place of the proof of Theorem ., then we would obtain an inequality () below, which seems not to be better than the inequality () given in Theorem ., as we see in the following remark, whose proof we omit.
Remark . Given a fixed positive integern, let (Cn, ·
n) and (Cn×n, · n×n) be complex
normed spaces, whose elements are column vectors resp. (n×n) complex matrices, with the property (). Assume that the transition matrix A∈Cn×nis nonsingular and{ε
i}i∈Zis
a sequence of nonnegative real numbers. If a sequence{yi}i∈Z⊂Cnsatisfies the inequality
() for alli∈Z, then there exists a solution{xi}i∈Z⊂Cnof the first-order matrix difference
equation () such that
yi–xin ≤ εi++
i+
k=εkA–ni+–×nk+A–ni+×nAy–x–n (fori≥), εi++
–i–
k= εk+i+Ank×n+An–×i–nAy–x–n (fori< ).
()
In view of (), assuming the initial condition in the previous theorem leads to the uniqueness of the sequence{xi}i∈Z, as we see in the following corollary.
Corollary . Given a fixed positive integer n,let(Cn,·
n)and(Cn×n,·n×n)be complex
normed spaces,whose elements are column vectors resp. (n×n)complex matrices,with the property().Assume that the transition matrixA∈Cn×nis nonsingular and{ε
i}i∈Zis
a sequence of nonnegative real numbers.If a sequence{yi}i∈Z⊂Cnsatisfies the inequality
()for all i∈Z,then there exists a solution{xi}i∈Z⊂Cnof the first-order matrix difference
equation()with the initial conditionx=ysuch that
yi–xin≤ i
k=εkA–ni+–×nk (for i≥),
–i
In the next corollary, we investigate the Hyers-Ulam stability of the second-order linear homogeneous difference equation with constant coefficients
ai=αai++βai+. ()
Corollary . Let(C, ·
∞)and(C×, · ∞)be complex normed spaces and letα,β, γ be complex numbers satisfying the conditions
α+ β= , β= , γ = . ()
Assume thatε> is an arbitrary constant.If a sequence{ai}i∈Zof complex numbers
satis-fies the inequality
|ai–αai+–βai+| ≤ε ()
for all i∈Z, then there exists a sequence{ci}i∈Z of complex numbers such that c=a,
c=a,ci=αci++βci+,and
|ai–ci| ≤ i
k=εA–i∞+–k (for i≥),
–i
k=εAk∞– (for i< ),
whereA∞=max{|α|+|β/γ|,|γ|}andA–∞=max{|/γ|,|α/β|+|γ/β|}.
Proof If we define a sequence{bi}i∈Zof complex numbers bybi=γai+, it then follows
from () that
|ai–αai+–βγbi+| ≤ε, |bi–γai+|=
for everyi∈Z. Hence, if we set
yi:=
ai
bi
and A:=
α βγ γ
,
then we get
yi– Ayi+∞≤ε
for alli∈Z.
According to Corollary ., there exists a unique solution{xi}i∈Zof the first-order matrix
difference equation () with the initial conditionx=
a
γa
such that
yi–xi∞≤ i
k=εA–i∞+–k (fori≥),
–i
k=εAk∞– (fori< ).
In view of () and the last inequality, we have
ai γai+
– A–i
a γa
∞ ≤
i
k=εA–i∞+–k (fori≥),
–i
k=εAk∞– (fori< ).
Since the matrix A has two distinct eigenvaluesλ=
α–√α+β
andλ=
α+√α+β , as we
did in the proof of Corollary ., ifiis a negative integer, then we get
A–i=
for all integersi< . By using () and this equality, we have
On the other hand, the inverse matrix A–has two distinct eigenvaluesω =–
nonnegative integer, then we obtain
A–i=A–i= –
for all integersi≥. Thus, it follows from () and the last equality that
Finally, considering () and (), we define
ci:=
Furthermore, it is easy to verify that the sequence{ci}i∈Zsatisfies
Thus, we get
Ifβ is large in absolute value, then the second-order difference equation () has the Hyers-Ulam stability as we see in the next example.
Example . Given anε> , assume that a sequence{ai}i∈Zof complex numbers satisfies
. Using these values, Corollary . implies that there exists a sequence {ci}i∈Zof complex numbers such thatc=a,c=a,ci=ci++ ci+, and If we apply Corollary . with the inequality
The inequalities () and () are equivalent. In this case with inequality () or (), there is more efficiency with Corollary . than Corollary . for any integeri.
Competing interests
The author declares that he has no competing interests.
Author’s contributions
The author read and approved the final manuscript.
Acknowledgements
This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (No. 2013R1A1A2005557).
Received: 26 February 2015 Accepted: 17 May 2015 References
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