We focus on the behavior of solutions of the difference equation xn=b1xn−1+b2xn−2+⋯+bnx0+yn, n=1,2,…, where (bk) is a fixed sequence of complex numbers, and (yk) is a fixed sequence in a complex Banach
space. We give the general solution of this difference equation. To examine the asymptotic behavior
of solutions, we compute the spectra of operators which correspond to such type of difference equations. These operators are represented by upper triangular or lower triangular infinite banded Toeplitz
matrices.

1. Introduction

The theory of difference equations is one of the most important representations of real world problems. The situation of an event at a fixed time usually depends on the situations of the event in the history. One of the ways to mathematically model such an event is to find a difference equation that directly or asymptotically describes the dependence of the situation at a time to the situations of the event in the history.

Let X be a complex Banach space. In this work, we are interested in a difference equation of the form xn=b1xn-1+b2xn-2+⋯+bnx0+yn,n=1,2,…, where (bk) and (yk) are fixed sequences in ℂ and X, respectively, and x0=y0. The difference equation (1.1) is a generalization of the difference equations investigated by Copson [1], Popa [2], and Stević [3]. We will give the general solution of the system of equations (1.1) and examine the different types of stability conditions by determining the spectra of related matrix operators. We will also examine the system of equations which correspond to the transpose of these matrices.

Let X and Y be Banach spaces and T:X→Y be a bounded linear operator. By ℛ(T), we denote the range of T, that is, ℛ(T)={y∈Y:y=Tx; x∈X}. By B(X), we denote the set of all bounded linear operators on X into itself. If X is any Banach space and T∈B(X), then the adjoint T* of T is a bounded linear operator on the dual X* of X defined by (T*ϕ)(x)=ϕ(Tx) for all ϕ∈X* and x∈X. Let X≠{θ} and T:𝒟(T)→X be a linear operator with domain 𝒟(T)⊂X. With T, we associate the operator Tλ=T-λI,
where λ is a complex number, and I is the identity operator on 𝒟(T). If Tλ has an inverse, which is linear, we denote it by Tλ-1, that is, Tλ-1=(T-λI)-1
and call it the resolvent operator of T. Many properties of Tλ and Tλ-1 depend on λ, and spectral theory is concerned with those properties. For instance, we will be interested in the set of all λ in the complex plane such that Tλ-1 exists. Boundedness of Tλ-1 is another property that will be essential. We will also ask for what λ's the domain of Tλ-1 is dense in X. For our investigation of T, Tλ, and Tλ-1, we need some basic concepts in spectral theory which are given as follows (see [4, page 370-371]).

Let X≠{θ} be a complex normed space, and let T:𝒟(T)→X be a linear operator with domain 𝒟(T)⊂X. A regular value λ of T is a complex number such that

Tλ-1 exists,

Tλ-1 is bounded,

Tλ-1 is defined on a set which is dense in X.

The resolvent set ρ(T) of T is the set of all regular values λ of T. Its complement σ(T)=ℂ∖ρ(T) in the complex plane ℂ is called the spectrum of T. Furthermore, the spectrum σ(T) is partitioned into three disjoint sets as follows. The point spectrum σp(T) is the set such that Tλ-1 does not exist. A λ∈σp(T) is called an eigenvalue of T. The continuous spectrum σc(T) is the set such that Tλ-1 exists and satisfies (R3) but not (R2). The residual spectrum σr(T) is the set such that Tλ-1 exists but does not satisfy (R3).

We will write ℓ∞, c and c0 for the spaces of all bounded, convergent, and null sequences, respectively. By ℓp, we denote the space of all p-absolutely summable sequences, where 1≤p<∞. Let μ and γ be two sequence spaces and A=(ank) be an infinite matrix of real or complex numbers ank, where n, k∈ℕ. Then, we say that A defines a matrix mapping from μ into γ, and we denote it by writing A:μ→γ, if for every sequence x=(xk)∈μ, the sequence Ax={(Ax)n}, the A-transform of x, is in γ, where (Ax)n=∑kankxk(n∈N).
By (μ,γ), we denote the class of all matrices A such that A:μ→γ. Thus, A∈(μ,γ) if and only if the series on the right side of (1.4) converges for each n∈ℕ and every x∈μ, and we have Ax={(Ax)n}n∈ℕ∈γ for all x∈μ.

Several authors have studied the spectrum and fine spectrum of linear operators defined by matrices over some sequence spaces. Rhoades [5] examined the fine spectra of the weighted mean operators. Reade [6] worked on the spectrum of the Cesàro operator over the sequence space c0. Gonzáles [7] studied the fine spectrum of the Cesàro operator over the sequence space ℓp. Yıldırım [8] examined the fine spectra of the Rhally operators over the sequence spaces c0 and c. Akhmedov and Başar [9] have determined the fine spectrum of the difference operator Δ over ℓp. Later, Bilgiç et al. [10] worked on the spectrum of the operator B(r,s,t), defined by a triple-band lower triangular matrix, over the sequence spaces c0 and c. Recently, Altun and Karakaya [11] determined the fine spectra of Lacunary operators.

Let L=[01000⋯00100⋯00010⋯⋮⋮⋮⋮⋮⋱],
which is the left shift operator, and let the transpose of L beR=Lt=[00000⋯10000⋯01000⋯⋮⋮⋮⋮⋮⋱],
which is the right shift operator. Let D be the unit disc {z∈ℂ:|z|≤1}.

Let a=(a0,a1,a2,…). A lower triangular Toeplitz matrix corresponding to a is in the form Ra=[a00000⋯a1a0000⋯a2a1a000⋯⋮⋮⋮⋮⋮⋱].
And an upper triangular Toeplitz matrix corresponding to a is in the form La=[Ra]t.

Lemma 1.1.

Let μ∈{ℓ1,c0,c,ℓ∞}. Then Ra∈B(μ) if and only if a∈ℓ1. Moreover ∥Ra∥μ=∥a∥ℓ1.

Proof.

Let us do the proof for μ=c. The proof for μ=c0 or ℓ∞ is similar. Let ∥·∥ denote the norm of c. Firstly, we have
‖Ra‖c=sup∥x∥=1‖Ra(x)‖=sup∥x∥=1(supn|∑k=0nakxn-k|)≤sup∥x∥=1(supn∑k=0n|ak||xn-k|)≤supn∑k=0n|ak|=∑k=0∞|ak|.
Now, fix n∈ℕ, and let a′=(ak′) be a sequence such that
ak′={|an-k|an-kifan-k≠0,k≤n,0otherwise,
then we have
‖Ra‖c≥‖Ra(a′)‖≥|(Ra(a′))n|=∑k=0n|ak|.
Hence, ∥Ra∥c=∥a∥ℓ1.

Now, let μ=ℓ1, and let ∥·∥ denote the norm of ℓ1. We have‖Ra‖l1=sup∥x∥=1‖Ra(x)‖=sup∥x∥=1(∑n=0∞|∑k=0nakxn-k|)≤sup∥x∥=1(∑n=0∞∑k=0n|ak||xn-k|)≤sup∥x∥=1(limn→∞∑k=0n∑j=0n|ak||xj|)≤(limn→∞∑k=0n|ak|)=∑k=0∞|ak|.
On the other hand,
‖Ra‖l1≥‖Ra(1,0,0,…)‖=∑k=0∞|ak|.
So, we have the same norm ∥Ra∥ℓ1=∥a∥ℓ1.

Remark 1.2.

We have B(μ)=(μ,μ) for the sequence spaces in Lemma 1.1 since these spaces are BK spaces.

We also have an La version of the last lemma, for which we leave the proof to the reader.

Lemma 1.3.

Let μ∈{ℓ1,c0,c,ℓ∞}. La∈B(μ) if and only if a∈ℓ1. Moreover ∥La∥μ=∥a∥ℓ1.

For any sequence a, let us associate the function fa(z)=∑k=0∞akzk.

2. Spectra of the OperatorsTheorem 2.1.

Let a∈ℓ1. Then Ra=fa(R) and La=fa(L).

Proof.

Let us do the proof for Ra. The proof for La can be done similarly. Let a(n)=a-(a0,a1,…,an,0,0,…). Then
∑k=0nakRk=[a00000⋯a1a0000⋯a2a1a000⋯⋮⋮⋮⋮⋮⋱anan-1an-2⋯a0⋯0anan-1an-2⋯⋯00anan-1an-2⋯⋮⋮⋮⋮⋮⋱],Ra-∑k=0nakRk=Ra(n).
So by Lemma 1.1,
‖Ra-∑k=0nakRk‖c0=‖Ra(n)‖c0=∑k=n+1∞|ak|.
Hence,
limn→∞‖Ra-∑k=0nakRk‖c0=0,
and so Ra=fa(R).

Theorem 2.2.

Let μ be one of the sequence spaces c0,c,ℓ∞, or ℓp with 1≤p<∞. Then L is a bounded linear operator over μ with ∥L∥μ=1 and σ(L,μ)=D.

Proof.

Let us do the proof first for μ=ℓp for 1≤p<∞. Let x=(x1,x2,…),
‖Lx‖p=‖(x2,x3,…)‖p=(∑i=2∞|xi|p)1/p≤‖x‖p
and ∥(0,1,0,0,…)∥p=1=∥L(0,1,0,0,…)∥p, hence ∥L∥p=1. In a similar way, we can show that ∥L∥μ=1 also for μ∈{c0,c,ℓ∞}. This means the spectral radius is less or equal to 1 and so
σ(L,μ)⊂D,
for μ∈{c0,c,ℓp,ℓ∞}.

Now, let us examine the eigenvalues for L. If Lx=λx, thenx2=λx1,x3=λx2,⋮
If x1=0, then xk=0 for all k. So, let x1≠0, then
xk=λk-1x1.
Hence, for any λ with |λ|<1, the sequence x=(1,λ,λ2,…)∈μ is an eigenvector for μ∈{c0,c,ℓp,ℓ∞}. Hence, σp(L,μ)⊃{λ∈ℂ:|λ|<1}. Combining this with (2.5), we have
σ(L,μ)=D,
for μ∈{c0,c,ℓp,ℓ∞}, since the spectrum is a closed set.

Theorem 2.3.

Let μ be one of the sequence spaces c0,c,ℓ∞ or ℓp with 1≤p<∞. Then R is a bounded linear operator over μ with ∥R∥μ=1 and σ(R,μ)=D.

Proof.

The boundedness of the operator can be proved as in the proof of Theorem 2.2. Now, we will use the fact that the spectrum of a bounded operator over a Banach space is equal to the spectrum of the adjoint operator. The adjoint operator is the transpose of the matrix for c0 and ℓp with 1≤p<∞. Hence,
σ(R,l1)=σ(L,c0)=Dσ(R,c0)=σ(L,l1)=Dσ(R,lp)=σ(L,lq)=Dσ(R,l∞)=σ(L,l1)=D1<p<∞.
It is known by Cartlidge [12] that if a matrix operator A is bounded on c, then σ(A,c)=σ(A,ℓ∞). Then we also have
σ(R,c)=σ(R,l∞)=D.

Theorem 2.4.

Let a be a sequence such that fa is holomorphic in a region containing D. Then σ(Ra,μ)=σ(La,μ)=fa(D).

Proof.

By the spectral mapping theorem for holomorphic functions (see, e.g., [13, page 569]) we have
σ(Ra,μ)=σ(fa(R),μ)=fa(σ(R,μ))=fa(D),σ(La,μ)=σ(fa(L),μ)=fa(σ(L,μ))=fa(D).

Theorem 2.5.

Let μ be a sequence space. If Ra is not a multiple of the identity mapping, then
σp(Ra,μ)=∅.

Proof.

Suppose λ is an eigenvalue and x=(x0,x1,…)∈μ is an eigenvector which corresponds to λ. Then Rax=λx and so we have the following linear system of equations.
a0x0=λx0,a1x0+a0x1=λx1,a2x0+a1x1+a0x2=λx2,⋮
Let xk be the first nonzero entry of x. Then the system of equations reduces to
a0xk=λxk,a1xk+a0xk+1=λxk+1,a2xk+a1xk+1+a0xk+2=λxk+2,⋮
From the first equation we get λ=a0, and using the other equations in the given order we get ak=0 for k>0. This means there is no solution if there exists any k>0 with ak≠0.

3. Applications to the System of Equations

Let us consider the evolutionary difference equation yn=anx0+an-1x1+⋯+a0xn(a0≠0),n=0,1,2,…. Here (an) is a sequence of complex numbers. The condition a0≠0 is needed to make the system of equations (3.1) solvable. By solvability of a system of equations we mean that for any given sequence (yn) of complex numbers there exists a unique sequence (xn) of complex numbers satisfying the equations. Equation (3.1) may be written in the form xn=b1xn-1+b2xn-2+⋯+bnx0+un,
for n=1,2,…, and x0=u0, by the change of variables bk=-ak/a0 for k∈{1,2,…} and where un is a function of n variables with un=un(x0,x1,…,xn-1)=yn/a0.

Let 00=1 for the following theorem.

Theorem 3.1.

The solution of the difference equation (3.2) is
xn=un+c1un-1+c2un-2+⋯+cnu0,n=1,2,…, with
ck=∑m1,m2,…,mk(m1+m2+⋯+mkm1,m2,…,mk)b1m1b2m2⋯bkmk,k=1,2,…, where the summation is over nonnegative integers satisfying m1+2m2+⋯+kmk=k.

Proof.

We have b1=c1. For k≥2, let us show that
bk+bk-1c1+bk-2c2+⋯+b1ck-1=ck,k=2,3,…. We can see that the left side of (3.5) can be written in the form
∑m1,m2,…,mka(m1,m2,…,mk)b1m1b2m2⋯bkmk,
where the summation is over nonnegative integers satisfying m1+2m2+⋯+kmk=k. To show (3.5), let us fix k≥2 and the numbers m1,m2,…,mk such that m1+2m2+⋯+kmk=k. Then for the coefficient a(m1,m2,…,mk), we have two cases.Case 1 (<inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M268"><mml:msub><mml:mrow><mml:mi>m</mml:mi></mml:mrow><mml:mrow><mml:mi>k</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:math></inline-formula>).

m1=m2=⋯=mk-1=0, and then
a(m1,m2,…,mk)=a(0,0,…,0,1)=1=(10,0,…,0,1)=(m1+m2+⋯+mkm1,m2,…,mk).

Case 2 (<inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M271"><mml:msub><mml:mrow><mml:mi>m</mml:mi></mml:mrow><mml:mrow><mml:mi>k</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>).

By induction we have
a(m1,m2,…,mk)=a(m1,m2,…,mk-1,0)=∑mj≥1,1≤j≤k-1(m1+m2+⋯+mk-1-1m1,m2,…,mj-1,mj-1,mj+1,mj+2,…,mk-1)=(m1+m2+⋯+mk-1m1,m2,…,mk-1)=(m1+m2+⋯+mkm1,m2,…,mk).

So, by Cases 1 and 2, we havebk+bk-1c1+⋯+b1ck-1=∑m1,m2,…,mka(m1,m2,…,mk)b1m1b2m2⋯bkmk=∑m1,m2,…,mk(m1+m2+⋯+mkm1,m2,…,mk)b1m1b2m2⋯bkmk=ck.

Now, we will prove the theorem by induction over n. x1=u1+b1u0=u1+c1u0 and so (3.3) is true for n=1. Suppose (3.3) is true for n≤s for a positive integer s. We have xs+1=us+1+b1xs+b2xs-1+⋯+bs+1x0=us+1+b1(us+c1us-1+c2us-2+⋯+csu0)+b2(us-1+c1us-2+c2us-3+⋯+cs-1u0)+⋯+bs+1u0=us+1+b1us+(b2+b1c1)us-1+(b3+b2c1+b1c2)us-2+⋯+(bs+1+bsc1+bs-1c2+⋯+b1cs)u0=us+1+c1us+c2us-1⋯+cs+1u0.
Hence, (3.3) is true for n=s+1.

A special case of (3.1) is the one where (an) consists of finitely many nonzero terms. So there exists a fixed k∈ℕ such that the equations turn into the form
yn={akxn-k+ak-1xn-k+1+⋯+a0xnifn>k,anx0+an-1x1+⋯+a0xnifn≤k,(a0≠0),n=0,1,2,….

We give the following theorem, which is a direct consequence of Lemma 1.1, to compare the results of it with the results of the next theorem.

Theorem 3.2.

Let (an) be a sequence of complex numbers such that the system of difference equations (3.1) hold for the complex number sequences x=(xn) and y=(yn). Then the following are equivalent:

boundedness of (xn) always implies boundedness of (yn),

convergence of (xn) always implies convergence of (yn),

xn→0 always implies yn→0,

∑|xn|<∞ always implies ∑|yn|<∞,

∑|an|<∞.

Theorem 3.3.

Suppose fa is a nonconstant holomorphic function on a region containing D. Let (an) be a sequence of complex numbers such that the system of difference equations (3.1) hold for the complex number sequences x=(xn) and y=(yn). Then the following are equivalent:

boundedness of (yn) always implies boundedness of (xn),

convergence of (yn) always implies convergence of (xn),

yn→0 always implies xn→0,

∑|yn|p<∞ always implies ∑|xn|p<∞,

fa(z) has no zero in the unit disc D.

Proof.

Let us prove only (i)⇔(v). We will omit the proofs of (ii)⇔(v), (iii)⇔(v), (iv)⇔(v) since they are similarly proved. Since fa is a holomorphic function on a region containing D, we have a∈ℓ1 which means Ra is bounded by Lemma 1.1. Suppose boundedness of yn implies boundedness of xn. Then the operator Ra∈(ℓ∞,ℓ∞) is onto. We have Rax=y and since fa is not constant Ra∈(ℓ∞,ℓ∞) is one to one by Theorem 2.5. Hence, Ra is bijective and by the open mapping theorem Ra-1 is continuous. This means that λ=0 is not in the spectrum σ(Ra,ℓ∞), so 0∉fa(D).

For the inverse implication, suppose fa(z) has no zero on the unit disc D. So, λ=0 is in the resolvent set ρ(Ra,ℓ∞). Hence by Lemma 7.2-3 of [4] Ra-1 is defined on the whole space ℓ∞, which means that the boundedness of (yn) implies the boundedness of (xn).

Corollary 3.4.

Let a polynomial P(z)=a0+a1z+⋯+akzk be given such that the system of difference equations (3.11) hold for the complex number sequences x=(xn) and y=(yn). Then the following are equivalent:

boundedness of (yn) always implies boundedness of (xn),

convergence of (yn) always implies convergence of (xn),

yn→0 always implies xn→0,

∑|yn|p<∞ always implies ∑|xn|p<∞,

all zeros of P(z) are outside the unit disc D.

Now consider the system of equations yn=∑j=0∞ajxn+j,n=0,1,2,…. Here (an) is a sequence of complex numbers.

A special case of (3.12) is the one where (an) consists of finitely many nonzero terms. So there exists a fixed k∈ℕ such that the equations turn into the form yn=∑j=0kajxn+jn=0,1,2,….

Now, we again give a theorem, which is a direct consequence of Lemma 1.3, to compare the results of it with the results of the next theorem.

Theorem 3.5.

Let (an) be a sequence of complex numbers such that the system of difference equations (3.12) hold for the complex number sequences x=(xn) and y=(yn). Then the following are equivalent:

boundedness of (xn) always implies boundedness of (yn),

convergence of (xn) always implies convergence of (yn),

xn→0 always implies yn→0,

∑|xn|<∞ always implies ∑|yn|<∞,

∑|an|<∞.

Theorem 3.6.

Suppose that fa is a holomorphic function on a region containing D. Let (an) be a sequence of complex numbers such that the system of difference equations (3.12) hold for the complex number sequences x=(xn) and y=(yn). Then the following are equivalent:

boundedness of (yn) always implies a unique bounded solution (xn),

convergence of (yn) always implies a unique convergent solution (xn),

yn→0 always implies a unique solution (xn) with xn→0,

∑|yn|p<∞ always implies a unique solution (xn) with ∑|xn|p<∞,

fa(z) has no zero on the unit disc D.

Proof.

Let us prove only (i)⇔(v). We will omit the proofs of (ii)⇔(v), (iii)⇔(v), (iv)⇔(v) since they are similarly proved. Suppose boundedness of yn implies a unique bounded solution xn. Then the operator La∈(ℓ∞,ℓ∞) is bijective. Since fa is a holomorphic function on a region containing D, we have a∈ℓ1 which means La is bounded by Lemma 1.3. By the open mapping theorem La-1 is continuous. This means that λ=0 is not in the spectrum σ(La,ℓ∞), so 0∉fa(D).

For the inverse implication, suppose that fa(z) has no zero on the unit disc D. So, λ=0 is in the resolvent set ρ(La,ℓ∞). Hence by Lemma 7.2-3 of [4] La-1 is defined on the whole space ℓ∞, which means that the boundedness of (yn) implies a bounded unique solution (xn).

Corollary 3.7.

Let a polynomial P(z)=a0+a1z+⋯+akzk be given such that the system of difference equations (3.13) hold for the complex number sequences x=(xn) and y=(yn). Then the following are equivalent:

boundedness of (yn) always implies a bounded unique solution (xn),

convergence of (yn) always implies a convergent unique solution (xn),

yn→0 always implies a unique solution (xn) with xn→0,

∑|yn|p<∞ always implies a unique solution (xn) with ∑|xn|p<∞,

all zeros of P(z) are outside the unit disc D.

Remark 3.8.

We see that the unit disc also has an important role in examining the different types of stability conditions of difference equations or system of equations related with holomorphic functions.

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