# Hyers-Ulam Stability of a System of First Order Linear Recurrences with Constant Coefficients

Authors: Bing Xu; Janusz Brzdęk
Journal: Discrete Dynamics in Nature and Society
 (2015)

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## Abstract

We study the Hyers-Ulam stability in a Banach spaceX of the system of first order linear difference equations of the form xn+1=Axn+dn for n∈N0 (nonnegative integers), where A is a given r×r matrix with real or complex coefficients, respectively,  and (dn)n∈N0 is a fixed sequence in Xr. That is, we investigate the sequences (yn)n∈N0 in Xr such that δ∶=supn∈N0yn+1-Ayn-dn<∞ (with the maximum norm in Xr) and show that, in the case where all the eigenvalues of A are not of modulus 1, there is a positive real constant c (dependent only on A) such that, for each such a sequence (yn)n∈N0, there is a solution (xn)n∈N0 of the system with supn∈N0yn-xn≤cδ.


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## Body

## 1. Introduction

The issue of stability of a functional equation can be expressed in the following way.When must a function satisfying an equation approximately (in some sense) be near an exact solution to the equation? It has been motivated by a question raised in 1940 by Ulam, concerning approximate homomorphisms of groups (see [1, 2]). The first partial answer to Ulam’s question (in the case of Cauchy’s functional equation in Banach spaces) was given by Hyers in [1]. After that result, a great number of papers on the subject have been published (see, e.g., monographs [3–5], survey articles [6–11], and the references given there), generalizing Ulam’s problem and Hyers’s theorem in various directions and to other equations (not necessarily functional) (see [12]). In particular, some results have been proved in [13], which concern the stability of linear difference equations of higher order of form (1). We describe them as follows.LetT be either N0 (the set of nonnegative integers) or Z (the set of integers), let K be either the field of reals R or the field of complex numbers C, let p∈N (the set of positive integers), let a1,…,ap∈K be fixed, and let (bn)n∈T be a given sequence in a Banach space X over K. The investigation of the Hyers-Ulam stability in X of the difference equation
(1)xn+p=a1xn+p-1+⋯+apxn+bn,n∈T,
actually means a study of the sequences (yn)n∈T in X, satisfying the condition
(2)δ:=sup⁡n∈Tyn+p-a1yn+p-1-⋯-apyn-bn<∞.LetS={z∈C:z=1} and t1,…,tp∈C denote the roots of the characteristic equation of (1), which has the following form: zp=a1zp-1+⋯+ap. The following two theorems have been proved in [13] (see also [14, 15]).Theorem 1.
Letδ>0 and t1,…,tp∈C∖S. Suppose that (yn)n∈T is a sequence in X such that (2) holds. Then, there exists a sequence (xn)n∈T in X satisfying (1) such that
(3)yn-xn≤δ1-t1·⋯·1-tp,n∈T.
Moreover,(a)
(xn)n∈T is unique if and only if ti>1 for i=1,…,p or T=Z;(b)
ifti>1 for i∈{1,…,p} or T=Z, then (xn)n∈T is the unique sequence in X such that (1) holds and sup⁡n∈Txn-yn<∞;(c)
ifT=N0 and ti<1 for some i∈{1,…,p}, then the cardinality of the set of all sequences (xn)n∈T in X, satisfying (1) and (3), equals the cardinality of X.Theorem 2.
Let|tj|=1 for some j∈{1,…,p}. Then, for any δ>0, there exists a sequence (yn)n∈T in X, satisfying inequality (2), such that, for every sequence (xn)n∈T in X, fulfilling recurrence (1),
(4)sup⁡n∈Tyn-xn=∞.
Moreover, if t1,…,tp∈K or there is a bounded sequence (xn)n∈T in X fulfilling (1), then (yn)n∈T can be chosen unbounded.We somehow complement those results in this paper by the study of the Hyers-Ulam stability of the following system of first order linear difference equations inX with constant coefficients aij∈K, i,j=1,…,r (r∈N is fixed):
(5)xn+11=a11xn1+a12xn2+⋯+a1rxnr+dn1,xn+12=a21xn1+a22xn2+⋯+a2rxnr+dn2,⋮xn+1r=ar1xn1+ar2xn2+⋯+arrxnr+dnr,
for all n∈N0, where dn1,…,dnr∈X for n∈N0 are given. If we write
(6)A=a11a12⋯a1ra21a22⋯a2r⋮⋮⋱⋮ar1ar2⋯arr,xn=xn1xn2⋮xnr,dn=dn1dn2⋮dnr,
then (5) can be expressed in the following simple form:
(7)xn+1=Axn+dn,n∈N0.
To simplify the notations, we consider xn and dn to be elements of Xr, when it is convenient (and when this makes no confusion); that is, we identify xn with (xn1,…,xnr) and dn with (dn1,…,dnr).Our results correspond, in particular, not only to the outcomes in [13], but also to those in [14, 15], where similar problems have been studied for r=1.


## 2. Some Auxiliary Results

By an elementary induction onn, we obtain the following simple observation.Lemma 3.
If a sequence(xn)n∈N0 in Xr satisfies (7), then
(8)xn=Anx0+∑k=1n‍An-kdk-1,n∈N.In this paper,(9)Cjn:=nj=n!(n-j)!j!,n,j∈N0,n≥j,
denote the binomial coefficients and Cjn:=0 for j>n. The subsequent formula is well known:
(10)∑n=0∞‍Cnn+jwn=11-wj+1,w∈[0,1),j∈N0.
Also, replacing w by 1/x, we easily obtain that
(11)∑n=0∞‍Cnn+jx-(n+j+1)=1x-1j+1,x∈(1,∞),j∈N0.Further, write(12)A:=max⁡1≤i≤r∑j=1r‍aij,x:=max⁡1≤i≤rxi,x=x1,…,xr∈Xr.
Then, (Xr,·) is a Banach space and we have the following result, which will be useful in the proof of the main theorem.Theorem 4.
LetJλ,r be a Jordan matrix of the form
(13)Jλ,r=λ10⋯000λ1⋯00⋮⋮⋮⋱⋮⋮000⋯λ1000⋯0λr×r,
with some λ∈C∖S. If a sequence (yn)n∈N0 in Xr satisfies
(14)δ:=sup⁡n∈N0yn+1-Jλ,ryn-bn<∞,
then there exists a sequence (xn)n∈N0 in Xr such that
(15)xn+1=Jλ,rxn+bn,n∈N0,(16)sup⁡n∈N0yn-xn≤δ∑j=1r‍1-λ-j.Proof.
Letcn:=yn+1-Jλ,ryn-bn for n∈N0. Then, by (14), cn≤δ for n∈N0 and (see Lemma 3)
(17)yn=Jλ,rny0+∑k=1n‍Jλ,rn-k(bk-1+ck-1),n∈N.
Case 1 (
λ
<
1
). Define the sequence (xn)n∈N0 by (15) with x0=y0. Then,
(18)yn-xn=∑k=1n‍Jλ,rn-kck-1≤δ∑k=1n‍Jλ,rn-k,n∈N.
It is easy to show by induction on n that Jλ,rn is an upper (right) triangular matrix of the form
(19)Jλ,rn=λnC1nλn-1⋯Cr-1nλn-(r-1)0λn⋯Cr-2nλn-(r-2)⋮⋮⋱⋮00⋯C1nλn-100⋯λnr×r
(it is enough to use the well-known formula Ck+1n+1=Ckn+Ck+1n) whence we derive that Jλ,rn=∑j=0r-1‍Cjn|λ|n-j for n∈N0.
Since, in view of (10),
(20)∑n=0∞‍Cjnλn-j=11-λj+1,j=0,…,r,
we have
(21)∑n=0∞‍Jλ,rn=∑n=0∞‍∑j=0r-1‍Cjnλn-j=∑j=0r-1‍∑n=0∞‍Cjnλn-j=∑j=1r‍11-λj
and, by (18),
(22)yn-xn≤∑k=0∞‍Jλ,rkδ=δ∑j=1r‍11-λj,n∈N0.
That is, inequality (16) holds.
Case 2 (
|
λ
|
>
1
). Since Jλ,r-1 is an upper triangular matrix of the form
(23)Jλ,r-1=λ-1-1λ2⋯-1r-1λr0λ-1⋯-1r-2λr-1⋮⋮⋱⋮00⋯-1λ200⋯λ-1r×r,
it is easy to check that Jλ,r-n is also an upper triangular matrix for each n∈N and has the form
(24)Jλ,r-n=λ-n-Cn-1nλn+1⋯-1r-1Cn-1n+r-2λn+r-10λ-n⋯-1r-2Cn-1n+r-3λn+r-2⋮⋮⋱⋮00⋯-Cn-1nλn+100⋯λ-nr×r.
Hence,
(25)Jλ,r-n=∑j=0r-1‍Cn-1n-1+jλ-(n+j),n∈N.
Consequently, in view of (11),
(26)∑n=1∞‍Jλ,r-n=∑n=0∞‍∑j=0r-1‍Cnn+jλ-(n+j+1)=∑j=0r-1‍∑n=0∞‍Cnn+jλ-n+j+1=∑j=1r‍1λ-1j.
Taking into account that Jλ,r-kck-1≤Jλ,r-kck-1≤δJλ,r-k for k∈N, we deduce that the series ∑k=1∞‍Jλ,r-kck-1 is convergent. Take
(27)s:=∑k=1∞‍Jλ,r-kck-1
and define (xn)n∈N0 by (15) with x0=y0+s. Then, (see Lemma 3)
(28)xn=Jλ,rn(y0+s)+∑k=1n‍Jλ,rn-kbk-1,n∈N.
Next, by (26), for every n∈N, we have
(29)yn-xn=∑k=1n‍Jλ,rn-kck-1-Jλ,rns=∑k=1n‍Jλ,rn-kck-1-Jλ,rn∑k=1∞‍Jλ,r-kck-1=∑k=1n‍Jλ,rn-kck-1-∑k=1∞‍Jλ,rn-kck-1=∑k=n+1∞‍Jλ,rn-kck-1,
whence
(30)yn-xn≤∑k=1∞‍Jλ,r-kδ=δ∑j=1r‍1λ-1j,n∈N0.


## 3. The Main Result

Letλ1,…,λm be the eigenvalues of A with multiplicities r1,…,rm, respectively. There exists a nonsingular matrix Q in Cr×r with A=QJQ-1, where
(31)J=Jλ1,r1⊕⋯⊕Jλm,rm,Jλj,rj=λj10⋯000λj1⋯00⋮⋮⋮⋱⋮⋮000⋯λj1000⋯0λjrj×rj,j=1,…,m.The next theorem is the main result of this paper.Theorem 5.
Assume thatλj∈C∖S for j=1,…,m. For any sequence (zn)n∈N0 in Xr, satisfying
(32)δ:=sup⁡n∈N0zn+1-Azn-dn<∞,
there exists a sequence (xn)n∈N0 in Xr such that
(33)xn+1=Axn+dn,n∈N0,(34)sup⁡n∈N0zn-xn≤δQQ-1max⁡j=1,…,m∑k=1rj‍11-λjk.Proof.
Let(zn)n∈N0 be a sequence in Xr such that (32) holds.
First, consider the case whereK=C. Write yn:=Q-1zn and bn:=Q-1dn for n∈N0. Then,
(35)yn+1-Jyn-bn≤Q-1zn+1-Azn-dn≤Q-1δ=:δ0,n∈N0.
Define projectionspj:Xr→Xrj (for j=1,…,m) by
(36)p1w1,…,wr:=(w1,…,wr1),p2(w1,…,wr):=(wr1+1,…,wr1+r2),⋮pm-1(w1,…,wr):=(wr1+⋯+rm-2+1,…,wr1+⋯+rm-1),pm(w1,…,wr):=(wr1+⋯+rm-1+1,…,wr1+⋯+rm)
for (w1,…,wr)∈Xr. For simplicity, we write yn=(p1(yn),…,pm(yn)) and bn=(p1(bn),…,pm(bn)) for n∈N0. It is easily seen that (in analogous notation)
(37)Jyn=Jλ1,r1p1yn,…,Jλm,rmpmyn,n∈N0,pj(yn+1)-Jλj,rjpj(yn)-pj(bn)≤yn+1-Jyn-bn≤δ0,n∈N0,j=1,…,m.
According to Theorem 4 (applied for each j∈{1,…,m}, separately), there exists a sequence (un)n∈N0 in Xr such that
(38)pj(un+1)=Jλj,rjpj(un)+pj(bn),n∈N0,pj(un)-pj(yn)≤δ0∑k=1rj‍11-λjk,n∈N0,j=1,…,m.
Clearly,
(39)un+1=Jun+bn,n∈N0,yn-un≤δ0max⁡∑k=1rj‍11-λjk:j=1,…,m,n∈N0.
Letxn:=Qun for n∈N0. Then,
(40)xn+1=Qun+1=QJun+Qbn=QJQ-1xn+dn,n∈N0,zn-xn=Qyn-Qun≤Qyn-un≤Qδ0max⁡∑k=1rj‍11-λjk:j=1,…,m,n∈N0.
Now, consider the caseK=R. Define the linear structure in X^:=X2 by (x,y)+(z,w):=(x+z,y+w) and (α+iβ)(x,y):=(αx-βy,βx+αy) for x,y,z,w∈X,α,β∈R. Then, X^ is a complex Banach space (see, e.g., [16, page 39], [17], or [18, 1.9.6, page 66]), when endowed with the Taylor norm ∥·∥T given by
(41)(x,y)T:=sup⁡0≤θ≤2π(cos⁡θ)x+(sinθ)y,x,y∈X.
Note that
(42)max⁡{x,y}≤(x,y)T≤x+y,x,y∈X.
Definep^1:X2→X by p^1(w1,w2):=w1 for w1,w2∈X. Let z^n:=(zn,0) and d^n:=(dn,0) for n∈N0. Then, (z^n)n∈N0 is a sequence in X^ and
(43)z^n+1-Az^n-d^nT=zn+1-Azn-dn≤δ,n∈N0.
So, by the first part of the proof, there is a sequence (x^n)n∈N0 in X^ such that
(44)x^n+1=Ax^n+d^n,n∈N0,sup⁡n∈N0z^n-x^nT≤δQQ-1max⁡j=1,…,m∑k=1rj‍11-λjk.
Write xn:=p^1(x^n) for n∈N0. Then, it is easily seen that (33) and (34) are valid (in view of (42)).Remark 6.
The assumption thatλj≠1 for j=1,…,m cannot be omitted in the general case (at least when r=1), in view of Theorem 2.
It seems that our method cannot be easily applied to the systems of linear difference equations of higher orders, because it is difficult in such cases to obtain a formula analogous as (8).Open Problems. There arises a natural question if some results similar to Theorem 2 and statements (a)–(c) of Theorem 1 can be obtained for difference equation (33) with r>1 (also with N0 replaced by Z).



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