R E S E A R C H Open AccessHyers-Ulam stability of the first-order matrix difference equations * Correspondence: smjung@hongik.ac.kr Mathematics Section, College of Science and Technology
Trang 1R E S E A R C H Open Access
Hyers-Ulam stability of the first-order
matrix difference equations
* Correspondence:
smjung@hongik.ac.kr
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 equationsx i= Axi–1andx i–1= Axi for all integers i∈ Z.
MSC: Primary 39A45; 39B82; secondary 39A06; 39B42 Keywords: difference equation; matrix difference equation; recurrence; Hyers-Ulam
stability; approximation
1 Introduction
Throughout this paper, let n be a fixed positive integer The nth order linear homogeneous
difference equation with constant coefficients is of the form
where α, α, , α nare constants For example, the second-order difference equation with constant coefficients has the form
a i = αa i–+ βa i– ()
The solution of () is called the Fibonacci numbers when α = β = , a= , and a= ,
Lucas numbers when α = β = , a= , and a= , Pell numbers when α = , β = , a= ,
and a= , Pell-Lucas numbers when α = , β = , and a= a= , and Jacobsthal numbers
if α = , β = , a= , and a=
The polynomial
p (x) = x n – αx n–– αx n––· · · – α n–x – α n
is called the characteristic polynomial of the difference equation ()
If the roots r, r, , r nof the characteristic polynomial are distinct, then the solution of the difference equation () is given by
a i = kri + kri+· · · + k n r i n,
where the coefficients k, k, , k nare uniquely determined under the initial conditions of the difference equation
© 2015 Jung This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, pro-vided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and
Trang 2If the characteristic polynomial has roots r, r, , r d with multiplicity m, m, , m d, respectively, then the solution of the difference equation () is given by
a i=
d
j=
m j
k=
c jk i k–r i j,
where the c jk are constants and m+ m+· · · + m d = 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 ×n be 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 Ax n≤ An ×n x n ()
for all A, B ∈ Cn ×nandx ∈ C n
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 equationsx i= Axi–andx i–= Axi for all integers i∈ Z, where the
tran-sition matrix A is nonsingular More precisely, we prove that if a sequence{y i}i∈Zsatisfies
the inequalityy i– Ayi–n ≤ ε for all i ∈ Z resp y i–– Ayin ≤ ε for all i ∈ Z, then there
exist a solution{x i}i∈Z ⊂ Cnof the first-order matrix difference equation () resp () and
a constant K > such that y i–x in ≤ Kε for all integers i ≥ (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 ofx i= Ax i–1
In this section, we investigate the Hyers-Ulam stability of the first-order linear
homoge-neous matrix difference equation
for all integers i∈ Z, where
x i=
⎛
⎜
⎜
⎝
x i
x i
x in
⎞
⎟
⎟
⎠∈ C
n and A=
⎛
⎜
⎜
⎝
a a · · · a n
a a · · · a n
. .
a n a n · · · a nn
⎞
⎟
⎟
⎠∈ C
n ×n
Theorem . Given a fixed positive integer n , let (C n,·n ) and (C n ×n,·n ×n ) be complex
normed spaces , whose elements are column vectors resp (n × n) complex matrices, with the
Trang 3property () Assume that the transition matrix A∈ Cn ×n is nonsingular and {ε i}i∈Zis a
sequence of nonnegative real numbers If a sequence {y i}i∈Z ⊂ Cn satisfies the inequality
for all i ∈ Z, then there exists a solution {x i}i∈Z ⊂ Cn of the first-order matrix difference
equation () such that
y i–x in≤
i
k=ε kAi –k
n ×n+Ai
n ×n y–xn (for i≥ ),
–i
k=ε k +iA–k
n ×n+A––i
n ×n y–xn (for i < ).
Proof Assume that a sequence{y i}i∈Z ⊂ Cn satisfies the inequality () for all i∈ Z First,
we assume that i is a nonnegative integer It then follows from () and () that
y i– Ai y
n ≤ y i– Ay i–n+ Ay i–– Ay i–
n
+ Ay i–– Ay i–
n+· · · + Ai–y– Ai y
n
≤ y i– Ayi–n+An ×n y i–– Ayi–n
+A
n ×n y i–– Ayi–n+· · · + Ai–
n ×n y– Ayn
≤ ε i+An ×n ε i–+A
n ×n ε i–+· · · + Ai–
n ×n ε
=Ai
n ×n
i
k=
ε kA–k
It is obvious that a sequence{x i}i∈Z ⊂ Cnsatisfies the first-order matrix difference equa-tion () if and only if
for each i∈ Z, where we set Ai= (A–)–i for all negative integers i Hence, by () and (),
we have
y i–x in≤ y i– Ai y
n+ Ai y– Ai x
n+ Ai x–x i
n
≤
i
k=
ε kAi –k
n ×n+Ai
n ×n y–xn
for any integer i≥
On the other hand, we suppose i is a negative integer For this case, it follows from ()
and () that
y i– Ai y
n
= y i–
A– –i y
n
≤ y i– A–y i+
n+ A–y i+–
A– y i+
n
+ A– y i+–
A– y i+ +· · · + A– –i– y––
A– –i y
Trang 4n ×n Ay i–y i+n+ A–
n ×n Ay i+–y i+n
+ A–
n ×n Ay i+–y i+n+· · · + A– –i
n ×n Ay––yn
≤ A–
n ×n ε i++ A–
n ×n ε i++ A–
n ×n ε i++· · · + A– –i
n ×n ε
=
–i
k=
ε k +i A– k
Moreover, by () and (), we have
y i–x in≤ y i–
A– –i y
n+ A– –i y–
A– –i x
n
+ A– –i
x–x i
n
≤
–i
k=
ε k +i A– k
n ×n+ A– –i
n ×n y–xn
In view of (), if we assume the initial condition in the previous theorem, we can easily prove the uniqueness of the sequence{x i}i∈Zas we see in the following corollary
Corollary . Given a fixed positive integer n , let (C n,·n ) and (C n ×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 ×n is nonsingular and {ε i}i∈Zis
a sequence of nonnegative real numbers If a sequence {y i}i∈Z ⊂ Cn satisfies the inequality
() for all i ∈ Z, then there exists a unique solution {x i}i∈Z ⊂ Cn of the first-order matrix
difference equation () with the initial condition x=ysuch that
y i–x in≤ i k=ε kAi –k
n ×n (for i≥ ),
–i
k=ε k +iA–k
n ×n (for i < ).
Some of the most important matrix norms are induced by p-norms For ≤ p ≤ ∞, the matrix norm induced by the p-norm,
Ap:= sup
x =
Ax p
x p
,
is called the matrix p-norm For example, we get
A= max
≤j≤n
n
i=
|a ij| and A∞= max
≤i≤n
n
j=
|a ij|
It is well known that the matrix p-norm, together with the p-norm, satisfies the conditions
in (), where
x=
n
j=
|x j | and x∞= max
≤j≤n|x j| for anyx ∈ C n
Trang 5In 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 let α, β,
γ be complex numbers satisfying the conditions
Assume that ε > is an arbitrary constant If a sequence {a i}i∈Zof complex numbers
satis-fies the inequality
for all i ∈ Z, then there exists a sequence {c i}i∈Z of complex numbers such that c–= a–,
c= a, c i = αc i–+ βc i–, and
|a i – c i| ≤ i k=εAi –k
∞ (for i≥ ),
–i
k=εA–k
∞ (for i < ),
whereA∞= max{|α| + |β/γ |, |γ |} and A–∞= max{|/γ |, |α/β| + |γ /β|}
Proof If we define a sequence{b i}i∈Z of complex numbers by b i = γ a i–, it then follows
from () that
|a i – αa i––β γ b i–| ≤ ε,
|b i – γ a i–| =
for any i∈ Z If we set
y i:=
a i
b i
and A:=
α β γ
γ
,
then we get
y i– Ay i–∞≤ ε
for each i∈ Z.
According to Corollary ., there exists a unique solution{x i}i∈Z ⊂ Cof the first-order matrix difference equation () with the initial conditionx= a
γ a– such that
y i–x i∞≤
i
k=εAi –k
∞ (for i≥ ),
–i
k=εA–k
∞ (for i < ).
In view of (), this last inequality implies that
a i
γ a i–
– Ai
a
γ a–
∞
≤
i
k=εAi –k
∞ (for i≥ ),
–i
k=εA–k
Trang 6Since the transition matrix A has two distinct eigenvalues λ = α–
√
α+β
α+√
α+β
, which are the roots of the characteristic equation λ– αλ – β = , the matrix A
can be expressed as
with
C=
λ λ
γ γ
λ
λ
γ (λ– λ)
γ –λ
–γ λ
By (), we obtain
Ai= CDiC–
γ (λ– λ)
λ λ
γ γ
λ i
λ i
γ –λ
–γ λ
γ (λ– λ)
γ (λ i+
– λ i+
) –λλ(λ i
– λ i
)
γ(λ i– λ i) –γ λλ(λ i–– λ i–)
for every integer i≥ Using this equality, it follows from () that
a i–a–a–λ
λ–λ λ i++a–a–λ
λ–λ λ i+
γ a i–– γ a–a–λ
λ–λ λ i+ γ a–a–λ
λ–λ λ i
∞
≤
i
k=
εAi –k
for all integers i≥
On the other hand, the inverse matrix A–has two distinct eigenvalues ω=–α–
√
α+β
β = –βλand ω=–α+
√
α+β
β = –βλ, which are roots of the characteristic equation ω+α β ω–
β = Hence, the matrix A–may be expressed as
A–=
γ
γ
β –α β
=
γ ω γ ω
ω
ω
γ ω γ ω
–
Using (), we have
Ai=
A– –i
=
γ ω γ ω
ω –i
ω –i
γ ω γ ω
–
γ (ω– ω)
γ ωω(ω –i– – ω –i– ) ω –i – ω –i
γωω(ω–i – ω –i) γ (ω –i – ω –i )
for all integers i < Thus, the inequality () yields
a i–a––aω
ω–ω ω –i +a––aω
ω–ω ω–i
γ a i–– γ a––aω
ω–ω ω –i + γ a––aω
ω–ω ω–i
∞≤
–i
k=
ε A– k
for any integer i < .
Trang 7Finally, considering (), (), and [], Theorem ., if we set
c i:=
a–a–λ
λ–λ λ i+–a–a–λ
λ–λ λ i+ (for i≥ ),
a––aω
ω–ω ω –i
–a––aω
ω–ω ω –i
(for i < ), then we get c–= a–, c= a, and it follows from () and () that
|a i – c i| ≤ i k=εAi –k
∞ (for i≥ ),
–i
k=εA–k
∞ (for i < ).
Furthermore, it is not difficult to show that the sequence{c i}i∈Zsatisfies the second-order
linear difference equation
c i = αc i–+ βc i–
If we set γ = ±α±
√
α+β
in Corollary ., then we get
lim
β→∞A∞· A– ∞= .
For example, if we set γ = α+
√
α+β
and β > , then we have
⎧
⎨
⎩
A∞= maxα+√
α+β
,–α+
√
α+β
,
A–∞= maxα+√
α+β
β ,–α+
√
α+β
β
,
()
and hence
lim
β→∞A∞· A– ∞= lim
β→∞
β·√
β =
For the case when γ = α–
√
α+β
, γ = –α+
√
α+β
, or γ = –α–
√
α+β
, we analogously obtain limβ→∞A∞· A–∞=
If α and β are simultaneously small in absolute value, then the second-order difference
equation () has the Hyers-Ulam stability as we see in the following example
Example . Given an ε > , assume that a sequence {a i}i∈Zof complex numbers satisfies
the inequality
a i–
a i––
a i–
≤ ε
for all i ∈ Z With α =
and β =, it follows from () thatA∞=+
√
andA–∞=
+ √
Using these values, Corollary . implies that there exists a sequence{c i}i∈Z of
complex numbers such that c–= a–, c= a, c i=c i–+c i–, and
|a i – c i| ≤
√
+
– √
+
i
ε (for i≥ ),
√
+ √
+ –i
– ε (for i < ).
Trang 83 Hyers-Ulam stability ofx i–1= Ax i
In practical applications, we sometimes consider the first-order linear homogeneous
ma-trix difference equation
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 (C n,·n ) and (C n ×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 ×n is nonsingular and {ε i}i∈Zis a
sequence of nonnegative real numbers If a sequence {y i}i∈Z ⊂ Cn satisfies the inequality
for all i ∈ Z, then there exists a solution {x i}i∈Z ⊂ Cn of the first-order matrix difference
equation () such that
y i–x in≤ i k=ε kA–i +–k
n ×n +A–i
n ×n y–xn (for i≥ ),
–i
k=ε k +iAk–
n ×n+A–i
n ×n y–xn (for i < ). ()
Proof Assume that a sequence{y i}i∈Z ⊂ Cn satisfies the inequality () for all i∈ Z First,
we assume that i is a nonnegative integer Then, by () and (), we have
y i– A–i y
n≤ y i– A–y i–
n+ A–y i–– A–y i–
n
+ A–y i–– A–y i–
n+· · · + A–i+ y– A–i y
n
≤ ε i A–
n ×n + ε i– A–
n ×n + ε i– A–
n ×n+· · · + ε A– i
n ×n
=
i
k=
ε k A– i +–k
n ×n
Obviously, a sequence{x i}i∈Z ⊂ Cnsatisfies the first-order matrix difference equation () if and only if
for all i∈ Z, where we set A–i= (A–)i for each integer i≥ Hence, we get
y i–x in≤ y i– A–i y
n+ A–i y– A–i x
n+ A–i x–x i
n
≤
i
k=
ε k A– i +–k
n ×n + A– i
n ×n y–xn
for all integers i≥
Trang 9On the other hand, if i is a negative integer, then it follows from () and () that
y i– A–i y
n=y i– Ayi+n+ Ay i+– Ay i+
n
+ Ay i+– Ay i+
n+· · · + A–i– y–– A–i y
n
≤ ε i++ ε i+An ×n + ε i+A
n ×n+· · · + εA–i–
n ×n
=
–i
k=
ε k +iAk–
n ×n
Thus, by () and the last inequality, we obtain
y i–x in≤ y i– A–i y
n+ A–i y– A–i x
n+ A–i x–x i
n
≤
–i
k=
ε k +iAk–
n ×n+A–i
n ×n y–xn
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 integer n, let (C n, · 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{y i}i∈Z ⊂ Cnsatisfies the inequality
() for all i ∈ Z, then there exists a solution {x i}i∈Z ⊂ Cnof the first-order matrix difference
equation () such that
y i–x in
≤
ε i++ i k+=ε kA–i +–k
n ×n +A–i+
n ×n Ay–x–n (for i≥ ),
ε i++ –i– k= ε k +i+Ak
n ×n+A–i–
n ×n Ay–x–n (for i < ). ()
In view of (), assuming the initial condition in the previous theorem leads to the uniqueness of the sequence{x i}i∈Z, as we see in the following corollary
Corollary . Given a fixed positive integer n , let (C n,·n ) and (C n ×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 ×n is nonsingular and {ε i}i∈Zis
a sequence of nonnegative real numbers If a sequence {y i}i∈Z ⊂ Cn satisfies the inequality
() for all i ∈ Z, then there exists a solution {x i}i∈Z ⊂ Cn of the first-order matrix difference
equation () with the initial condition x=ysuch that
y i–x in≤ i k=ε kA–i +–k
n ×n (for i≥ ),
–i ε k +iAk–
×n (for i < ).
Trang 10In the next corollary, we investigate the Hyers-Ulam stability of the second-order linear homogeneous difference equation with constant coefficients
a i = αa i++ βa i+ ()
Corollary . Let(C, · ∞) and (C×, · ∞) be complex normed spaces and let α, β,
γ be complex numbers satisfying the conditions
Assume that ε > is an arbitrary constant If a sequence {a i}i∈Zof complex numbers
satis-fies the inequality
for all i ∈ Z, then there exists a sequence {c i}i∈Z of complex numbers such that c= a,
c= a, c i = αc i++ βc i+, and
|a i – c i| ≤ i k=εA–i +–k
∞ (for i≥ ),
–i
k=εAk–
∞ (for i < ),
whereA∞= max{|α| + |β/γ |, |γ |} and A–∞= max{|/γ |, |α/β| + |γ /β|}
Proof If we define a sequence{b i}i∈Z of complex numbers by b i = γ a i+, it then follows
from () that
|a i – αa i+–β γ b i+| ≤ ε,
|b i – γ a i+| =
for every i∈ Z Hence, if we set
y i:=
a i
b i
and A:=
α β γ
γ
,
then we get
y i– Ayi+∞≤ ε for all i∈ Z.
According to Corollary ., there exists a unique solution{x i}i∈Zof the first-order matrix difference equation () with the initial conditionx= a
γ a such that
y i–x i∞≤ i k=εA–i +–k
∞ (for i≥ ),
–i
k=εAk–
∞ (for i < ).
In view of () and the last inequality, we have
a i
γ a i+
– A–i
a
γ a
∞
≤
i
k=εA–i +–k
∞ (for i≥ ),
–i
k=εAk–
...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... 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 .,... class="text_page_counter">Trang 5
In the following corollary, we prove the Hyers- Ulam stability of the second -order linear homogeneous difference equation with constant coefficients