# Shadowing and hyperbolicity for linear delay difference equations

Lucas Backes  Departamento de Matemática, Universidade Federal do Rio Grande
do Sul, Av. Bento Gonçalves 9500, CEP 91509-900, Porto Alegre, RS, Brazil.
<EMAIL_ADDRESS>,  Davor Dragičević  Faculty of Mathematics, University
of Rijeka, Radmile Matejčić 2, 51000 Rijeka, Croatia
<EMAIL_ADDRESS>and  Mihály Pituk  Department of Mathematics,
University of Pannonia, Egyetem út 10, 8200 Veszprém, Hungary; HUN–REN–ELTE
Numerical Analysis and Large Networks Research Group, Budapest, Hungary
<EMAIL_ADDRESS>
###### Abstract.

It is known that hyperbolic linear delay difference equations are shadowable
on the half-line. In this paper, we prove the converse and hence the
equivalence between hyperbolicity and the positive shadowing property for the
following two classes of linear delay difference equations: (a) for
nonautonomous equations with finite delays and uniformly bounded compact
coefficient operators in (possibly infinite-dimensional) Banach spaces, (b)
for Volterra difference equations with infinite delay in finite dimensional
spaces.

###### Key words and phrases:

Delay difference equation; shadowing; hyperbolicity; exponential dichotomy

###### 2020 Mathematics Subject Classification:

Primary: 37C50; Secondary: 37B99

##  1\. Introduction

Let $\mathbb{Z}$ and $\mathbb{C}$ denote the set of integers and the set of
complex numbers, respectively. For $k\in\mathbb{Z}$, define
$\mathbb{Z}^{+}_{k}=\\{\,n\in\mathbb{Z}:n\geq k\,\\}$ and
$\mathbb{Z}^{-}_{k}=\\{\,n\in\mathbb{Z}:n\leq k\,\\}$. Throughout the paper,
we shall assume as a standing assumption that $(X,|\cdot|)$ is a Banach space.
The symbol $\mathcal{L}(X)$ will denote the space of all bounded linear
operators $A\colon X\to X$ equipped with the operator norm,
$|A|=\sup_{|x|=1}|Ax|$ for $A\in\mathcal{L}(X)$.

Consider the linear autonomous difference equation

$x(n+1)=Ax(n),$ (1.1)

where $A\in\mathcal{L}(X)$. Given $\delta\geq 0$, by a _$\delta$
-pseudosolution_ of Eq. (1.1) on $\mathbb{Z}^{+}_{0}$, we mean a function
$y\colon\mathbb{Z}^{+}_{0}\to X$ such that

$\sup_{n\geq 0}|y(n+1)-Ay(n)|\leq\delta.$

Note that for $\delta=0$ the pseudosolution becomes a true solution of Eq.
(1.1) on $\mathbb{Z}^{+}_{0}$. We say that Eq. (1.1) is _shadowable_ on
$\mathbb{Z}^{+}_{0}$ or that it has the _positive shadowing property_ if for
every $\epsilon>0$ there exists $\delta>0$ such that for every
$\delta$-pseudosolution $y$ of (1.1) on $\mathbb{Z}^{+}_{0}$ there exists a
true solution $x$ of (1.1) on $\mathbb{Z}^{+}_{0}$ such that

$\sup_{n\geq 0}|x(n)-y(n)|\leq\epsilon.$

The shadowing of Eq. (1.1) is closely related to its hyperbolicity. Recall
that Eq. (1.1) is _hyperbolic_ if $\sigma(A)$ does not intersect the unit
circle $|\lambda|=1$ in $\mathbb{C}$, where $\sigma(A)$ denotes the spectrum
of $A$.

In a recent paper [4], Bernardes  _et al._ have studied various shadowing
properties of Eq. (1.1). Among others, they have shown that if Eq. (1.1) is
hyperbolic, then it is shadowable on $\mathbb{Z}^{+}_{0}$ (see [4, Theorem
13]). Moreover, if the coefficient $A\in\mathcal{L}(X)$ in Eq. (1.1) is a
compact operator, then the converse is also true. As usual, an operator
$A\colon X\to X$ is _compact_ if for every bounded set $S\subset X$ the image
$A(S)$ has compact closure in $X$. Thus, we have the following theorem.

######  Theorem 1.

[4, Theorem 15] Let $A\in\mathcal{L}(X)$ be a compact operator. Then the
following statements are equivalent. _(i)_ Eq. (1.1) is shadowable on
$\mathbb{Z}^{+}_{0}$;

_(ii)_ Eq. (1.1) is hyperbolic.

######  Remark 1.

As noted above, the implication (ii) $\Rightarrow$ (i) in Theorem 1 is true if
we assume merely that $A\in\mathcal{L}(X)$. The compactness of
$A\in\mathcal{L}(X)$ is important only for the validity of the converse
implication (i) $\Rightarrow$ (ii) (see [4, Remark 14]).

Now let us consider the finite dimensional case $X=\mathbb{C}^{d}$, where $d$
is a positive integer and $\mathbb{C}^{d}$ denotes the $d$-dimensional space
complex column vectors. Then, the space $\mathcal{L}(\mathbb{C}^{d})$ can be
identified with $\mathbb{C}^{d\times d}$, the space of $d\times d$ matrices
with complex entries. Since linear operators between finite dimensional spaces
are compact and the spectrum of a square matrix $A\in\mathbb{C}^{d\times d}$
consists of the roots of its characteristic equation

$\det(\lambda E-A)=0,$ (1.2)

where $E\in\mathbb{C}^{d\times d}$ is the unit matrix, in this case, Theorem 1
can be reformulated as follows.

######  Theorem 2.

Let $A\in\mathbb{C}^{d\times d}$. Then the following statements are
equivalent. _(i)_ Eq. (1.1) is shadowable on $\mathbb{Z}^{+}_{0}$;

_(ii)_ The characteristic equation (1.2) has no root on the unit circle
$|\lambda|=1$.

Our aim in this paper is to extend Theorems 1 and 2 to more general classes of
linear difference equations with delay.

In Sec. 2, we will generalize Theorem 1 to the nonautonomous linear difference
equation with finite delays

$x(n+1)=\sum_{j=0}^{r}A_{j}(n)x(n-j),$ (1.3)

where $r\in\mathbb{Z}^{+}_{0}$ is the maximum delay and the coefficients
$A_{j}(n)\in\mathcal{L}(X)$, $0\leq j\leq r$, $n\in\mathbb{Z}^{+}_{0}$, are
compact linear operators which are uniformly bounded, i.e., there exists
$K\geq 1$ such that

$|A_{j}(n)|\leq K,\qquad n\in\mathbb{Z}^{+}_{0},\quad 0\leq j\leq r.$ (1.4)

The main result of Sec. 2 is formulated in Theorem 3, which may be viewed as a
discrete analogue of our recent shadowing theorem for delay differential
equations in $\mathbb{R}^{d}$ [1, Theorem 2.2]. It says that, under the above
hypotheses, Eq. (1.3) is shadowable on $\mathbb{Z}^{+}_{0}$ if and only if it
has an exponential dichotomy which is a nonautonomous variant of
hyperbolicity. To the best of our knowledge, this result is new even for
ordinary difference equations ($r=0$).

In Sec. 3, we will extend Theorem 2 to the linear Volterra difference equation
with infinite delay

$x(n+1)=\sum_{j=-\infty}^{n}A(n-j)x(j),$ (1.5)

where $A\colon\mathbb{Z}^{+}_{0}\to\mathbb{C}^{d\times d}$ satisfies

$\sum_{j=0}^{\infty}|A(j)|e^{\gamma j}<\infty\qquad\text{for some
$\gamma>0$}.$ (1.6)

The _characteristic equation_ of Eq. (1.5) has the form

$\det\varDelta(\lambda)=0,\qquad|\lambda|>e^{-\gamma},$ (1.7)

where

$\varDelta(\lambda)=\lambda
E-\sum_{j=0}^{\infty}\lambda^{-j}A(j),\qquad|\lambda|>e^{-\gamma}.$ (1.8)

The main result of Sec. 3, Theorem 4, says that in the natural (infinite
dimensional) phase space $\mathcal{B}_{\gamma}$ defined below Eq. (1.5) is
shadowable on $\mathbb{Z}^{+}_{0}$ if and only if its characteristic equation
(1.7) has no root on the unit circle $|\lambda|=1$.

The fact that nonautonomous linear delay difference equations, including (1.3)
and (1.5), are shadowable whenever they are hyperbolic follows from [7,
Theorem 1]. Therefore, in both cases (1.3) and (1.5), we need to prove only
the converse result. In the case of the nonautonomous equation with finite
delays (1.3), the proof follows similar lines as the proof our continuous time
result [1, Theorem 2.2] with nontrivial modifications because, instead of
$\mathbb{R}^{d}$, we consider Eq. (1.3) in a general infinite dimensional
Banach space. It is based on the eventual compactness of the solution
operator, combined with an input-output technique [2] and Schäffer’s result
about the existence of regular covariant subspaces of linear difference
equation in a Banach space [24]. A similar argument for the Volterra equation
with infinite delay (1.5) does not apply since its solution operator is not
eventually compact. In this case the proof will be based on the duality
between Eq. (1.5) and its formal adjoint equation which has been established
by Matsunaga  _et. al._ [16].

##  2\. Shadowing of nonautonomous linear difference equation with finite
delays

In this section, we consider the shadowing of the nonautonomous linear
difference equation with finite delays (1.3), where
$A_{j}\colon\mathbb{Z}^{+}_{0}\to\mathcal{L}(X)$, $0\leq j\leq r$, satisfy
condition (1.4). The _phase space_ for Eq. (1.3) is
$(\mathcal{B}_{r},\|\cdot\|)$, where $\mathcal{B}_{r}$ is the set of all
functions $\phi\colon[-r,0]\cap\mathbb{Z}\to X$ and

$\|\phi\|=\max_{-r\leq\theta\leq
0}|\phi(\theta)|,\qquad\phi\in\mathcal{B}_{r}.$

Eq. (1.3) can be written equivalently in a form of a _functional difference
equation_

$x(n+1)=L_{n}(x_{n}),$ (2.1)

where the _solution segment_ $x_{n}\in\mathcal{B}_{r}$ is defined by

$x_{n}(\theta)=x(n+\theta),\qquad\theta\in[-r,0]\cap\mathbb{Z},$

and $L_{n}\colon\mathcal{B}_{r}\to X$ is a bounded linear functional defined
by

$L_{n}(\phi)=\sum_{j=0}^{r}A_{j}(n)\phi(-j),\qquad\phi\in\mathcal{B}_{r},\quad
n\in\mathbb{Z}^{+}_{0},\quad 0\leq j\leq r.$

In view of (1.4), we have that

$\|L_{n}\|\leq M:=(r+1)K,\qquad n\in\mathbb{Z}^{+}_{0}.$ (2.2)

Given $m\in\mathbb{Z}^{+}_{0}$ and $\phi\in\mathcal{B}_{r}$, there exists a
unique function $x\colon\mathbb{Z}^{+}_{m-r}\to X$ satisfying Eq. (1.3) and
the _initial condition_ $x(m+\theta)=\phi(\theta)$ for
$\theta\in[-r,0]\cap\mathbb{Z}$. We shall call $x$ the _solution of Eq. ( 1.3)
with initial value $x_{m}=\phi$_. By a _solution of Eq. ( 1.3) on
$\mathbb{Z}_{m}^{+}$_, we mean a solution $x$ with initial value $x_{m}=\phi$
for some $\phi\in\mathcal{B}_{r}$.

For each $n,m\in\mathbb{Z}^{+}_{0}$ with $n\geq m$, the _solution operator_
$T(n,m)\colon\mathcal{B}_{r}\to\mathcal{B}_{r}$ is defined by
$T(n,m)\phi=x_{n}$ for $\phi\in\mathcal{B}_{r}$, where $x$ is the unique
solution of Eq. (1.3) with initial value $x_{m}=\phi$. It is easily seen that
for all $n,k,m\in\mathbb{Z}^{+}_{0}$ with $n\geq k\geq m$,

$\displaystyle T(m,m)=I,$ (2.3) $\displaystyle T(n,m)=T(n,k)T(k,m),$ (2.4)
$\displaystyle\|T(n,m)\|\leq e^{\omega(n-m)},$ (2.5)

where $I$ denotes the identity operator on $\mathcal{B}_{r}$ and
$\omega=\log(M(1+r))$.

Now we can introduce the definitions of shadowing and exponential dichotomy
for Eq. (1.3) (equivalently, (2.1)).

######  Definition 1.

We say that Eq. (1.3) is _shadowable on $\mathbb{Z}^{+}_{0}$_ if for each
$\epsilon>0$ there exists $\delta>0$ such that for every function
$y\colon\mathbb{Z}^{+}_{-r}\to X$ satisfying

$\sup_{n\geq 0}|y(n+1)-L_{n}(y_{n})|\leq\delta,$

there exists a solution $x$ of (1.3) on $\mathbb{Z}^{+}_{0}$ such that

$\sup_{n\geq 0}\|x_{n}-y_{n}\|\leq\epsilon.$

######  Definition 2.

We say that Eq. (1.3) admits an _exponential dichotomy (on
$\mathbb{Z}^{+}_{0}$)_ if there exist a sequence of projections
$(P_{n})_{n\in\mathbb{Z}^{+}_{0}}$ on $\mathcal{B}_{r}$ and constants
$D,\lambda>0$ with the following properties:

  * •

for $n,m\in\mathbb{Z}^{+}_{0}$ with $n\geq m$,

$P_{n}T(n,m)=T(n,m)P_{m},$ (2.6)

and
$T(n,m)\rvert_{\operatorname{ker}P_{m}}\colon\operatorname{ker}P_{m}\to\operatorname{ker}P_{n}$
is onto and invertible;

  * •

for $n,m\in\mathbb{Z}^{+}_{0}$ with $n\geq m$,

$\|T(n,m)P_{m}\|\leq De^{-\lambda(n-m)};$ (2.7)

  * •

for $n,m\in\mathbb{Z}^{+}_{0}$ with $n\leq m$,

$\|T(n,m)Q_{m}\|\leq De^{-\lambda(m-n)},$ (2.8)

where $Q_{m}=I-P_{m}$ and
$T(n,m):=\left(T(m,n)\rvert_{\operatorname{ker}P_{n}}\right)^{-1}$.

The main result of this section is the following shadowing theorem for Eq.
(1.3).

######  Theorem 3.

Suppose that the coefficients $A_{j}(n)\in\mathcal{L}(X)$,
$n\in\mathbb{Z}^{+}_{0}$, $0\leq j\leq r$, of Eq. (1.3) are compact operators
satisfying condition (1.4). Then, the following statements are equivalent.
_(i)_ Eq. (1.3) is shadowable on $\mathbb{Z}^{+}_{0}$;

_(ii)_ Eq. (1.3) admits an exponential dichotomy.

######  Remark 2.

It is well-known that the autonomous linear equation (1.1) admits an
exponential dichotomy if and only if the spectrum $\sigma(A)$ does not
intersect the unit circle $|\lambda|=1$. Therefore, Theorem 3 is a
generalization of Theorem 1 to the nonautonomous delay difference equation
(1.3). Its conclusion is new even for ordinary difference equations ($r=0$).

######  Remark 3.

The implication (ii) $\Rightarrow$ (i) in Theorem 3 is a consequence of [7,
Theorem 1] with $f=0$, $c=0$ and $\mu=1$ which does not require the
compactness of the coefficients. Thus, this implication is true even without
the compactness assumption. However, for the validity of the converse
implication (i) $\Rightarrow$ (ii) the compactness of the coefficient
operators of Eq. (1.3) is essential (see Remark 1).

###### Proof of Theorem 3.

As noted in Remark 3, we need to prove only the implication (i) $\Rightarrow$
(ii). Suppose that Eq. (1.3) is shadowable on $\mathbb{Z}^{+}_{0}$. We will
show that it admits an exponential dichotomy. We split the proof into several
auxiliary results.

######  Claim 1.

Eq. (1.3) has the following _Perron type property_ : for each bounded function
$z\colon\mathbb{Z}^{+}_{0}\to X$, there exists a bounded function
$x\colon\mathbb{Z}^{+}_{-r}\to X$ which satisfies

$x(n+1)=L_{n}(x_{n})+z(n),\qquad n\in\mathbb{Z}^{+}_{0}.$ (2.9)

###### Proof of Claim 1.

Let $z\colon\mathbb{Z}^{+}_{0}\to X$ be an arbitrary bounded function. If
$z(n)=0$ for every $n\in\mathbb{Z}^{+}_{0}$, then (2.9) is trivially satisfied
with $x(n)=0$ for every $n\in\mathbb{Z}^{+}_{-r}$. Now suppose that $z(n)\neq
0$ for some $n\in\mathbb{Z}^{+}_{0}$ so that
$\|z\|_{\infty}:=\sup_{n\in\mathbb{Z}^{+}_{0}}|z(n)|>0$. Choose a constant
$\delta>0$ corresponding to the choice of $\epsilon=1$ in Definition 1. Take
an arbitrary solution $y\colon\mathbb{Z}^{+}_{-r}\to X$ of the nonhomogeneous
equation

$y(n+1)=L_{n}(y_{n})+\frac{\delta}{\|z\|_{\infty}}z(n),\qquad
n\in\mathbb{Z}^{+}_{0}.$

(The unique solution $y$ with initial value $y_{0}=0$ is sufficient for our
purposes.) Since

$\sup_{n\geq 0}|y(n+1)-L_{n}(y_{n})|\leq\delta,$

according to Definition 1, there exists a solution $\tilde{x}$ of Eq. (2.1) on
$\mathbb{Z}^{+}_{0}$ such that

$\sup_{n\geq{-r}}|\tilde{x}(n)-y(n)|=\sup_{n\geq 0}\|\tilde{x}_{n}-y_{n}\|\leq
1.$

Define a function $x\colon\mathbb{Z}^{+}_{-r}\to X$ by

$x(n):=\frac{\|z\|_{\infty}}{\delta}(\,y(n)-\tilde{x}(n)\,),\qquad
n\in\mathbb{Z}^{+}_{-r}.$

It can be easily verified that $x$ satisfies (2.9) and

$\sup_{n\in\mathbb{Z}^{+}_{-r}}|x(n)|\leq\frac{\|z\|_{\infty}}{\delta}<\infty.$

The proof of the claim is complete. ∎

For each $m\in\mathbb{Z}^{+}_{0}$, define

$\mathcal{S}(m)=\bigl{\\{}\phi\in\mathcal{B}_{r}:\sup_{n\geq
m}\|T(n,m)\phi\|<\infty\bigr{\\}}.$

Clearly, $\mathcal{S}(m)$ is a subspace of $\mathcal{B}_{r}$ which will be
called the _stable subspace of Eq. ( 2.1) at $m\in\mathbb{Z}^{+}_{0}$_.

######  Claim 2.

For each $n,m\in\mathbb{Z}^{+}_{0}$ with $n\geq m$, we have that

$[\,T(n,m)\,]^{-1}(\mathcal{S}(n))=\mathcal{S}(m).$

###### Proof of Claim 2.

Let $n$ and $m$ be as in the statement. If $\phi\in\mathcal{S}(m)$, then (see
(2.4))

$\sup_{k\geq n}\|T(k,n)T(n,m)\phi\|=\sup_{k\geq
n}\|T(k,m)\phi\|\leq\sup_{k\geq m}\|T(k,m)\phi\|<\infty.$

This shows that $T(n,m)\phi\in\mathcal{S}(n)$ and hence
$\phi\in[\,T(n,m)\,]^{-1}(\mathcal{S}(n))$.

Now suppose that $\phi\in[\,T(n,m)\,]^{-1}(\mathcal{S}(n))$. Then,
$T(n,m)\phi\in\mathcal{S}(n)$, which implies that

$\sup_{k\geq n}\|T(k,m)\phi\|=\sup_{k\geq n}\|T(k,n)T(n,m)\phi\|<\infty.$

Hence

$\sup_{k\geq m}\|T(k,m)\phi\|\leq\max_{m\leq k\leq
n-1}\|T(k,m)\phi\|+\sup_{k\geq n}\|T(k,m)\phi\|<\infty.$

Thus, $\phi\in\mathcal{S}(m)$. ∎

######  Claim 3.

For $n,m\in\mathbb{Z}^{+}_{0}$ with $n\geq m$, we have the algebraic sum
decomposition

$\mathcal{B}_{r}=T(n,m)\mathcal{B}_{r}+\mathcal{S}(n).$ (2.10)

###### Proof of Claim 3.

It is sufficient to prove the claim for $m=0$. Indeed, assuming that the
desired conclusion holds for $m=0$, we now fix an arbitrary $m\geq 1$. Then,
for every $n\geq m$ and $\phi\in\mathcal{B}_{r}$, there exist
$\phi_{1}\in\mathcal{B}_{r}$ and $\phi_{2}\in\mathcal{S}(n)$ such that
$\phi=T(n,0)\phi_{1}+\phi_{2}$. Hence,

$\phi=T(n,0)\phi_{1}+\phi_{2}=T(n,m)T(m,0)\phi_{1}+\phi_{2}\in
T(n,m)\mathcal{B}_{r}+\mathcal{S}(n).$

Thus, (2.10) holds. Therefore, from now on, we suppose that $m=0$. Evidently,
for every $\phi\in\mathcal{B}_{r}$,

$\phi=T(0,0)\phi+0\in T(0,0)\mathcal{B}_{r}+\mathcal{S}(0).$

Thus, (2.10) holds for $n=m=0$. Now suppose that $n>0$ and let
$\phi\in\mathcal{B}_{r}$ be arbitrary. Define $v\colon\mathbb{Z}^{+}_{n-r}\to
X$ by

$v(k)=\begin{cases}\phi(k-n)&\quad\text{for $n-r\leq k\leq n$},\\\
0&\quad\text{for $k\geq n+1$}\end{cases}$

so that $v_{n}=\phi$ and $z\colon\mathbb{Z}^{+}_{0}\to X$ by

$z(k)=\begin{cases}0&\quad\text{for $0\leq k\leq n-1$},\\\
v(k+1)-L_{k}(v_{k})&\quad\text{for $k\geq n$}.\end{cases}$

Clearly, $v$ is bounded on $\mathbb{Z}^{+}_{n-r}$. From this and (2.2), we
find that $\sup_{k\geq 0}|z(k)|<\infty$. By Claim 1, there exists a function
$x\colon\mathbb{Z}^{+}_{-r}\to X$ such that $\sup_{k\geq-r}|x(k)|<\infty$ and
(2.9) holds. Moreover, it follows from the definition of $z$ that

$v(k+1)=L_{k}(v_{k})+z(k),\qquad k\geq n.$

From this and (2.9), we conclude that $x-v$ is a solution of Eq. (2.1) on
$\mathbb{Z}^{+}_{n}$ and thus

$x_{k}-v_{k}=T(k,n)(x_{n}-v_{n})=T(k,n)(x_{n}-\phi),\qquad k\geq n.$

Since both $x$ and $v$ are bounded on $\mathbb{Z}^{+}_{n-r}$, this implies
that $x_{n}-\phi\in\mathcal{S}(n)$. On the other hand, since $z(k)=0$ for
$0\leq k<n$, we have that $x_{n}=T(n,0)x_{0}$. This implies that

$\phi=x_{n}+(\phi-x_{n})=T(n,0)x_{0}+(\phi-x_{n})\in
T(n,0)\mathcal{B}_{r}+\mathcal{S}(n).$

Since $\phi\in\mathcal{B}_{r}$ was arbitrary, we conclude that (2.10) holds
for $m=0$. ∎

######  Claim 4.

For each $m\in\mathbb{Z}^{+}_{0}$, $\mathcal{S}(m)$ is the image of a Banach
space under the action of a bounded linear operator.

###### Proof of Claim 4.

Fix $m\in\mathbb{Z}^{+}_{0}$ and let $\mathcal{X}$ denote the Banach space of
all bounded functions $x\colon\mathbb{Z}^{+}_{m-r}\to X$ equipped with the
supremum norm,

$\|x\|_{\mathcal{X}}:=\sup_{k\geq m-r}|x(k)|<\infty,\qquad x\in\mathcal{X}.$

Let $\mathcal{X}^{\prime}$ denote the set of all $x\in\mathcal{X}$ which are
solutions of (2.1) on $\mathbb{Z}^{+}_{0}$. We will show that
$\mathcal{X}^{\prime}$ is a closed subspace of $\mathcal{X}$. To this end, let
$(x^{j})_{j\in\mathbb{Z}^{+}_{0}}$ be a sequence in $\mathcal{X}^{\prime}$
such that $x^{j}\to y$ in $\mathcal{X}$ as $j\to\infty$ for some
$y\in\mathcal{X}$. Then, $x^{j}(k)\to y(k)$ as $j\to\infty$ for every $k\geq
m-r$. Moreover, for each $n\geq m$, we have that $x^{j}_{n}\to y_{n}$ in
$\mathcal{B}_{r}$ as $j\to\infty$. From this, by letting $j\to\infty$ in the
equation

$x^{j}(n+1)=L_{n}(x^{j}_{n}),\qquad n\geq m,$

and using the continuity of the coefficients $L_{n}$, we conclude that

$y(n+1)=L_{n}(y_{n}),\qquad n\geq m.$

Thus, $y$ is a solution of Eq. (2.1) on $\mathbb{Z}^{+}_{m}$ and hence
$y\in\mathcal{X}^{\prime}$. This shows that $\mathcal{X}^{\prime}$ is a closed
subspace of $\mathcal{X}$ and hence it is a Banach space. Now define
$\Phi\colon\mathcal{X}^{\prime}\to\mathcal{B}_{r}$ by $\Phi(x)=x_{m}$ for
$x\in\mathcal{X}^{\prime}$. Clearly, $\Phi$ is a bounded linear operator with
$\|\Phi\|\leq 1$ and $\Phi(\mathcal{X}^{\prime})=\mathcal{S}(m)$. ∎

######  Claim 5.

For $n,m\in\mathbb{Z}^{+}_{0}$ with $n\geq m+r+1$, the solution operator
$T(n,m)\colon\mathcal{B}_{r}\to\mathcal{B}_{r}$ is compact.

###### Proof of Claim 5.

Suppose that $m\in\mathbb{Z}^{+}_{0}$ and $\phi\in\mathcal{B}_{r}$. Let $x$
denote the unique solution of (2.1) with initial value $x_{m}=\phi$. From
(2.1) an (2.2), we obtain for $n\geq m$,

$|x(n+1)|\leq\|L_{n}\|\|x_{n}\|\leq M\|x_{n}\|,$

and hence

$\|x_{n+1}\|\leq|x(n+1)|+\|x_{n}\|\leq(1+M)\|x_{n}\|.$

Since $\|x_{m}\|=\|\phi\|$, this implies by induction on $n$ that

$\|x_{n}\|\leq(1+M)^{n-m}\|\phi\|\qquad\text{for $n\geq m$}.$ (2.11)

Clearly, $\iota(\phi)=(\phi(-r),\phi(-r+1),\dots,\phi(0))$ is an isometric
isomorphism between the phase space $\mathcal{B}_{r}$ and the $(r+1)$-fold
product space $X^{r+1}$ endowed with the maximum norm, $\|x\|=\max_{1\leq
j\leq r+1}|x_{j}|$ for $x=(x_{1},x_{2},\dots,x_{r+1})\in X^{r+1}$. With this
identification, we have that

$T(m+r+1,m)\phi=x_{m+r+1}=(x(m+1),x(m+2),\dots,x(m+r+1)),$

which, together with Eq. (2.1), implies that

$T(m+r+1,m)\phi=\bigl{(}L_{m}(x_{m}),L_{m+1}(x_{m+1}),\dots,L_{m+r}(x_{m+r})\bigr{)}.$
(2.12)

Let $S$ be an arbitrary bounded subset of $\mathcal{B}_{r}$. Then, there
exists $\rho>0$ such that $\|\phi\|\leq\rho$ for all $\phi\in S$. From this
and (2.11), we obtain that

$\|x_{m+j}\|\leq(1+M)^{r}\rho\qquad\text{whenever $\phi\in S$ and $0\leq j\leq
r$}.$

Therefore, if $\phi\in S$, then the segments $x_{m},x_{m+1},\dots,x_{m+r}$ of
the corresponding solution $x$ of (2.1) with initial value $x_{m}=\phi$ belong
to the closed ball of radius $(1+M)^{r}\rho$ around zero in $\mathcal{B}_{r}$,
which will be denoted by $D$. From this and (2.12), we conclude that

$T(m+r+1,m)(S)\subset
C:=\overline{L_{n}(D)}\times\overline{L_{m+1}(D)}\times\dots\times\overline{L_{m+r}(D)}.$
(2.13)

Since $L_{m},L_{m+1},\dots,L_{m+r}$ are compact operators and $D$ is a bounded
set, the closures of the image sets $L_{m}(D),L_{m+1}(D),\dots,L_{m+r}(D)$ are
compact subsets of $X$. Therefore, $C$ is a product of $r+1$ compact subsets
of $X$ which is a compact subset of the product space $X^{r+1}$. In view of
(2.13), $T(m+r+1,m)(S)$ is a subset of the compact set $C\subset X^{r+1}$ and
hence it is relatively compact in $X^{r+1}$. Since $S$ was an arbitrary
bounded subset of $\mathcal{B}_{r}$, this proves that
$T(m+r+1,m)\colon\mathcal{B}_{r}\to\mathcal{B}_{r}$ is a compact operator.
Finally, if $n>m+r+1$, then

$T(n,m)=T(n,m+r+1)T(m+r+1,m)$

is a product of the bounded linear operator $T(n,m+r+1)$ and the compact
operator $T(m+r+1,m)$ and hence it is compact (see, e.g., [25]). ∎

######  Claim 6.

The stable subspace $\mathcal{S}(0)$ of Eq. (2.1) is closed and has finite
codimension in $\mathcal{B}_{r}$.

Before giving a proof of Claim 6, let us recall the following notions [24].
Let $B$ be a Banach space. A subspace $S$ of $B$ is called _subcomplete in
$B$_ if there exist a Banach space $Z$ and a bounded linear operator
$\Phi\colon Z\to B$ such that $\Phi(Z)=S$

Let $\mathcal{A}\colon\mathbb{Z}^{+}_{0}\to\mathcal{L}(B)$ be an operator-
valued map. For $n,m\in\mathbb{Z}^{+}_{0}$ with $n\geq m$, define the
corresponding _transition operator_ $U(n,m)\colon B\to B$ is defined by

$U(n,m)=\mathcal{A}(n-1)\mathcal{A}(n-2)\cdots\mathcal{A}(m)\qquad\text{for
$n>m$}$

and $U(n,n)=I$ for $n\in\mathbb{Z}^{+}_{0}$, where $I$ denotes the identity
operator on $B$. A sequence $Y=(Y(n))_{n\in\mathbb{Z}^{+}_{0}}$ of subspaces
in $B$ is called a _covariant sequence for $\mathcal{A}$_ if

$[\,\mathcal{A}(n)\,]^{-1}(Y(n+1))=Y(n)\qquad\text{for all
$n\in\mathbb{Z}^{+}_{0}$}.$

A covariant sequence $Y=(Y(n))_{n\in\mathbb{Z}^{+}_{0}}$ for $\mathcal{A}$ is
called _algebraically regular_ if

$U(n,0)B+Y(n)=B\qquad\text{for each $n\in\mathbb{Z}^{+}_{0}$}.$

Finally, a covariant sequence $Y=(Y(n))_{n\in\mathbb{Z}^{+}_{0}}$ for
$\mathcal{A}$ is called _subcomplete_ if the subspace $Y(n)$ is subcomplete in
$B$ for all $n\in\mathbb{Z}^{+}_{0}$.

The proof of Claim 6 will be based on the following result due to Schäffer
[24].

######  Lemma 1.

([24, Lemma 3.4]) Let $B$ be a Banach space and
$\mathcal{A}\colon\mathbb{Z}^{+}_{0}\to\mathcal{L}(B)$. Suppose that
$Y=(Y(n))_{n\in\mathbb{Z}^{+}}$ is a subcomplete algebraically regular
covariant sequence for $\mathcal{A}$. If the transition operator $U(n,m)\colon
B\to B$ is compact for some $n,m\in\mathbb{Z}^{+}_{0}$ with $n>m$, then the
subspaces $Y(n)$, $n\in\mathbb{Z}^{+}_{0}$, are closed and have constant
finite codimension in $B$.

Now we can give a proof of Claim 6.

###### Proof of Claim 6.

Claims 2, 3 and 4 guarantee that the stable subspaces
$Y(n):=\mathcal{S}(n)\subset\mathcal{B}_{r}$ of Eq. (2.1) form a subcomplete
algebraically regular covariant sequence for
$\mathcal{A}\colon\mathbb{Z}^{+}\to\mathcal{L}(\mathcal{B}_{r})$ defined by

$\mathcal{A}(n):=T(n+r+1,n),\qquad n\in\mathbb{Z}^{+}_{0}.$

According to Claim 5, the associated transition operator
$U(n+1,n)=\mathcal{A}(n)$ is compact. By the application of Lemma 1, we
conclude that $Y(0)=\mathcal{S}(0)$ is closed and has finite codimension in
$\mathcal{B}_{r}$. ∎

By Claim 6, the stable subspace $\mathcal{S}(0)$ is closed and has finite
codimension in $\mathcal{B}_{r}$. This implies that $\mathcal{S}(0)$ is
complemented in $\mathcal{B}_{r}$ (see, e.g., [20, Lemma 4.21, p. 106]). More
precisely, there exists a subspace $\mathcal{U}$ of $\mathcal{B}_{r}$ such
that $\dim\mathcal{U}=\operatorname{codim}\mathcal{S}(0)<\infty$ and

$\mathcal{B}_{r}=\mathcal{S}(0)\oplus\mathcal{U}.$ (2.14)

######  Claim 7.

For each bounded function $z\colon\mathbb{Z}^{+}_{0}\to X$, there exists a
unique bounded function $x\colon\mathbb{Z}^{+}_{-r}\to X$ with
$x_{0}\in\mathcal{U}$ which satisfies (2.9). Moreover, there exists a constant
$C>0$, independent of $z$, such that

$\sup_{n\geq-r}|x(n)|\leq C\sup_{n\geq 0}|z(n)|.$ (2.15)

###### Proof of Claim 7.

By Claim 1, there exists a bounded function
$\tilde{x}\colon\mathbb{Z}^{+}_{-r}\to X$ that satisfies

$\tilde{x}(n+1)=L_{n}(\tilde{x}_{n})+z(n),\qquad n\in\mathbb{Z}^{+}_{0}.$

On the other hand, (2.14) implies the existence of $\phi_{1}\in\mathcal{S}(0)$
and $\phi_{2}\in\mathcal{U}$ such that

$\tilde{x}_{0}=\phi_{1}+\phi_{2}.$

Define $x\colon\mathbb{Z}^{+}_{-r}\to X$ by

$x(n)=\tilde{x}(n)-y(n),\qquad n\geq-r,$

where $y$ is a solution of Eq. (2.1) with initial value $y_{0}=\phi_{1}$.
Since $y_{0}=\phi_{1}\in\mathcal{S}(0)$, we have that
$\sup_{n\geq-r}|y(n)|<\infty$. Then, $x$ satisfies (2.9),
$x_{0}=\tilde{x}_{0}-\phi_{1}=\phi_{2}\in\mathcal{U}$ and
$\sup_{n\geq-r}|x(n)|<\infty$. We claim that $x$ with the desired properties
is unique. Indeed, if $\bar{x}$ is an arbitrary function with the desired
properties, then $x_{0}-\bar{x}_{0}\in\mathcal{U}\cap\mathcal{S}(0)=\\{0\\}$.
Thus, $x_{0}=\bar{x}_{0}$ and hence $x=\bar{x}$ identically on
$\mathbb{Z}^{+}_{-r}$.

Finally, we show the existence of a constant $C>0$ such that (2.15) holds. Let
$\mathcal{X}_{0}$ and $\mathcal{X}_{-r}$ denote the Banach space of all
bounded $X$-valued functions defined on $\mathbb{Z}^{+}_{0}$ and
$\mathbb{Z}^{+}_{-r}$, respectively, equipped with the supremum norm. For
$z\in\mathcal{X}_{0}$, define $\mathcal{F}(z)=x$, where $x$ is the unique
bounded solution of the nonhomogeneous equation (2.9) with
$x_{0}\in\mathcal{U}$. (The existence and uniqueness of $x$ is guaranteed by
the first part of the proof.) Evidently, $\mathcal{F}(z)=x\in\mathcal{X}_{-r}$
for $z\in\mathcal{X}_{0}$ and
$\mathcal{F}\colon\mathcal{X}_{0}\to\mathcal{X}_{-r}$ is a linear operator. We
will now observe that $\mathcal{F}$ is a closed operator. Indeed, let
$(z^{k})_{k\in\mathbb{Z}^{+}}$ be a sequence in $\mathcal{X}_{0}$ such that
$z^{k}\to z$ for some $z\in\mathcal{X}_{0}$ and $x^{k}:=\mathcal{F}(z^{k})\to
x$ for some $x\in\mathcal{X}_{-r}$. Then, letting $k\to+\infty$ in

$x^{k}(n+1)=L_{n}(x_{n}^{k})+z^{k}(n)$

for each fixed $n$, we get that

$x(n+1)=L_{n}(x_{n})+z(n),\qquad n\in\mathbb{Z}^{+}_{0}.$

That is, $x$ satisfies (2.9). Now, since $x^{k}_{0}\in\mathcal{U}$ for
$k\in\mathbb{Z}^{+}_{0}$ and $\mathcal{U}$ is a finite-dimensional and hence a
closed subset of $\mathcal{B}_{r}$, we have that
$x_{0}=\lim_{k\to\infty}x^{k}_{0}\in\mathcal{U}$. Therefore,
$x\in\mathcal{X}_{-r}$ is a bounded function satisfying (2.9) with
$x_{0}\in\mathcal{U}$. Hence $\mathcal{F}(z)=x$ which shows that
$\mathcal{F}\colon\mathcal{X}_{0}\to\mathcal{X}_{-r}$ is a closed operator.
According to the Closed Graph Theorem (see, e.g., [25, Theorem 4.2-I, p.
181]), $\mathcal{F}$ is bounded, which implies that (2.15) holds with
$C=\|\mathcal{F}\|$, the operator norm of $\mathcal{F}$. ∎

For $n\in\mathbb{Z}^{+}$, define

$\mathcal{U}(n)=T(n,0)\mathcal{U}$

so that $\mathcal{U}(0)=\mathcal{U}$. It is easily seen that

$T(n,m)\mathcal{S}(m)\subset\mathcal{S}(n)\quad\text{and}\quad
T(n,m)\mathcal{U}(m)=\mathcal{U}(n)$ (2.16)

whenever $n,m\in\mathbb{Z}^{+}_{0}$ with $n\geq m$.

######  Claim 8.

For $n,m\in\mathbb{Z}^{+}_{0}$ with $n\geq m$,
$T(n,m)\rvert_{\mathcal{U}(m)}\colon\mathcal{U}(m)\to\mathcal{U}(n)$ is
invertible.

###### Proof of Claim 8.

In view of (2.16), we only need to show that the operator above is injective.
Let $n,m\in\mathbb{Z}^{+}_{0}$ with $n\geq m$ and $\phi\in\mathcal{U}(m)$ be
such that $T(n,m)\phi=0$. Since $\phi\in\mathcal{U}(m)$, there exists
$\bar{\phi}\in\mathcal{U}(0)=\mathcal{U}$ such that $\phi=T(m,0)\bar{\phi}$.
Let $x\colon\mathbb{Z}^{+}_{-r}\to X$ be the solution of (2.1) with initial
value $x_{0}=\bar{\phi}$. Since $x(k)=0$ for all sufficiently large
$k\in\mathbb{Z}^{+}$, we have that $\sup_{k\geq-r}|x(k)|<\infty$. It follows
from the uniqueness in Claim 7, applied for $z\equiv 0$, that $x\equiv 0$.
This implies that $\bar{\phi}=\phi=0$. ∎

######  Claim 9.

For each $n\in\mathbb{Z}^{+}_{0}$, $\mathcal{B}_{r}$ can be decomposed into
the direct sum

$\mathcal{B}_{r}=\mathcal{S}(n)\oplus\mathcal{U}(n).$ (2.17)

###### Proof of Claim 9.

Since $\mathcal{U}(0)=\mathcal{U}$, for $n=0$, the decomposition (2.17)
follows immediately from (2.14). Now suppose that $n\geq 1$ and let
$\phi\in\mathcal{B}_{r}$ be arbitrary. Let $v\colon\mathbb{Z}^{+}_{n-r}\to X$
and $z\colon\mathbb{Z}^{+}_{0}\to X$ be as in the proof of Claim 3. Since
$\sup_{k\geq 0}|z(k)|<\infty$, by Claim 7 there exists a unique function
$x\colon\mathbb{Z}^{+}_{-r}\to X$ such that $x_{0}\in\mathcal{U}$,
$\sup_{k\geq{-r}}|x(k)|<\infty$ and (2.9) holds. By the same reasoning as in
the proof of Claim 3, we have that $x_{n}-\phi\in\mathcal{S}(n)$. Moreover,
$x_{n}=T(n,0)x_{0}\in\mathcal{U}(n)$. Consequently,

$\phi=(\phi-x_{n})+x_{n}\in\mathcal{S}(n)+\mathcal{U}(n).$

Suppose now that $\phi\in\mathcal{S}(n)\cap\mathcal{U}(n)$. Then, there exists
$\bar{\phi}\in\mathcal{U}$ such that $\phi=T(n,0)\bar{\phi}$. Consider the
unique solution $x\colon\mathbb{Z}^{+}_{-r}\to X$ of Eq. (2.1) with
$x_{0}=\bar{\phi}$. Then, $x$ satisfies (2.9) with $z\equiv 0$,
$x_{0}=\bar{\phi}\in\mathcal{U}$ and $\sup_{k\geq{-r}}|x(k)|<\infty$. By the
uniqueness in Claim 7, we conclude that $x\equiv 0$. Therefore, $\bar{\phi}=0$
which implies that $\phi=0$. The proof of the Claim is completed. ∎

######  Claim 10.

There exists $Q>0$ such that

$\|T(n,m)\phi\|\leq Q\|\phi\|,$

for every $n,m\in\mathbb{Z}^{+}_{0}$ with $n\geq m$ and
$\phi\in\mathcal{S}(m)$.

###### Proof of Claim 10.

Fix $m\in\mathbb{Z}^{+}_{0}$ and $\phi\in\mathcal{S}(m)$. Let
$u\colon\mathbb{Z}^{+}_{m-r}\to X$ be the solution of Eq. (2.1) with initial
value $u_{m}=\phi$. Define $x\colon\mathbb{Z}^{+}_{-r}\to X$ and
$z\colon\mathbb{Z}^{+}_{0}\to X$ by

$x(k)=\begin{cases}u(k)&\text{ for $k\geq{m+1}$;}\\\ 0&\text{ for $-r\leq
k\leq m$}\end{cases}$

and

$z(k)=x(k+1)-L_{k}(x_{k}),\qquad k\geq 0,$

respectively. Evidently, (2.9) is satisfied, and since
$\phi\in\mathcal{S}(m)$, we have that $\sup_{k\geq{-r}}|x(k)|<\infty$.
Moreover, $x_{0}=0\in\mathcal{U}$. Furthermore, $z(k)=0$ for $0\leq k\leq m-1$
and $k\geq m+r+1$. Thus, using (2.2), (2.5) and the fact that
$\|x_{k}\|\leq\|u_{k}\|$, we find that

$\begin{split}\sup_{k\geq 0}|z(k)|&=\sup_{m\leq k\leq m+r}|z(k)|\\\
&\leq\sup_{m\leq k\leq m+r}|x(k+1)|+\sup_{m\leq k\leq m+r}|L_{k}x_{k}|\\\
&\leq\sup_{m\leq k\leq m+r}|u(k+1)|+M\sup_{m\leq k\leq m+r}\|x_{k}\|\\\
&\leq\sup_{m\leq k\leq m+r}|L_{k}(u_{k})|+M\sup_{m\leq k\leq m+r}\|u_{k}\|\\\
&\leq 2M\sup_{m\leq k\leq m+r}\|u_{k}\|\\\ &=2M\sup_{m\leq k\leq
m+r}\|T(k,m)\phi\|\\\ &\leq 2Me^{\omega r}\|\phi\|.\end{split}$

From the last inequality and conclusion (2.15) of Claim 7, we conclude that

$\sup_{k\geq{m+1}}|u(k)|\leq\sup_{k\geq{-r}}|x(k)|\leq C\sup_{k\geq
0}|z(k)|\leq 2CMe^{\omega r}\|\phi\|.$

Hence,

$\|T(n,m)\phi\|=\|u_{n}\|\leq 2CMe^{\omega r}\|\phi\|,\qquad n\geq m+r+1.$

Since (2.2) implies that

$\|T(n,m)\phi\|\leq e^{\omega r}\|\phi\|,\qquad m\leq n\leq m+r,$

the conclusion of the claim holds with

$Q:=\max\\{\,e^{\omega r},2CMe^{\omega r}\,\\}>0.$

∎

######  Claim 11.

There exist $D,\lambda>0$ such that

$\|T(n,m)\phi\|\leq De^{-\lambda(n-m)}\|\phi\|,$

for every $n,m\in\mathbb{Z}^{+}_{0}$ with $n\geq m$ and
$\phi\in\mathcal{S}(m)$.

###### Proof of Claim 11.

We claim that if

$N>eCMQ^{2}(r+1)+r+1$ (2.18)

with $Q$ as in Claim 10, then for every $m\in\mathbb{Z}^{+}_{0}$ and
$\phi\in\mathcal{S}(m)$,

$\|T(n,m)\phi\|\leq\frac{1}{e}\|\phi\|\qquad\text{for $n\geq m+N$}.$ (2.19)

Suppose, for the sake of contradiction, that (2.18) holds and there exist
$m\in\mathbb{Z}^{+}_{0}$ and $\phi\in\mathcal{S}(m)$ such that

$\|T(n,m)\phi\|>\frac{1}{e}\|\phi\|\qquad\text{for some $n\geq m+N$}.$ (2.20)

Fix $n\geq m+N$ such that the first inequality in (2.20) holds and let $u$
denote the solution of the homogeneous equation Eq. (2.1) with initial value
$u_{m}=\phi$ so that $u_{n}=T(n,m)\phi$. Therefore, the first inequality in
(2.20) can be written as

$\|u_{n}\|>\frac{1}{e}\|\phi\|.$ (2.21)

In view of (2.16), $\phi\in\mathcal{S}(m)$ implies that
$u_{j}=T(j,m)\phi\in\mathcal{S}(j)$ for $j\geq m$. Therefore, by Claim 10, we
have that

$\|u_{n}\|=\|T(n,j)u_{j}\|\leq Q\|u_{j}\|\qquad\text{whenever $m\leq j\leq
n$}.$

From the last inequality and (2.21), we find that $\|u_{j}\|>0$ whenever
$m\leq j\leq n$. This, together with the fact that $n\geq m+N>m+r+1$ and hence
$n-r-1>m$, implies that we can define a function
$x\colon\mathbb{Z}^{+}_{-r}\to X$ by

$x(k)=\begin{cases}\chi(k)u(k)&\text{for $k\geq m$,}\\\ 0&\text{for $-r\leq
k\leq m-1$,}\end{cases}$

where

$\chi(k)=\begin{cases}0\qquad&\text{for $-r\leq k\leq m$},\\\\[2.0pt]
\displaystyle\sum_{j=m}^{k-1}\|u_{j}\|^{-1}\qquad&\text{for $m+1\leq k\leq
n-r-1$},\\\\[10.0pt]
\displaystyle\sum_{j=m}^{n-r-1}\|u_{j}\|^{-1}\qquad&\text{for $k\geq
n-r$}.\end{cases}$

Evidently, $x$ satisfies the nonhomogeneous equation

$x(k+1)=L_{k}(x_{k})+z(k),\qquad k\in\mathbb{Z}^{+}_{0},$ (2.22)

where $z\colon\mathbb{Z}^{+}_{0}\rightarrow X$ is defined by

$z(k)=x(k+1)-L_{k}(x_{k}),\qquad k\in\mathbb{Z}^{+}_{0}.$

Since $u_{m}=\phi\in\mathcal{S}(m)$ implies that $u$ is bounded on
$\mathbb{Z}^{+}_{m-r}$ and $0\leq\chi(k)\leq\chi(n-r)$ for $k\geq{-r}$, it
follows that $x$ is bounded on $\mathbb{Z}^{+}_{-r}$. From this and (2.2), we
obtain that $z$ is also bounded on $\mathbb{Z}^{+}_{0}$. Since
$x_{0}=0\in\mathcal{U}$, by Claim 7, we have that

$\sup_{k\geq{-r}}|x(k)|\leq C\sup_{k\geq 0}|z(k)|.$ (2.23)

Our objective now is to estimate the norm of $z(k)$. Since $x(k)=0$ for
$-r\leq k\leq m$, we have that $z(k)=0$ for $0\leq k\leq m-1$. The function
$\chi$ is constant on $\mathbb{Z}^{+}_{n-r}$, therefore $x$ is a constant
multiple of the solution $u$ of the homogeneous equation (2.1) on
$\mathbb{Z}^{+}_{n-r}$. Thus, $x$ also satisfies the homogeneous equation
(2.1) for $k\geq n$ and hence $z(k)=0$ for $k\geq n$. It remains to consider
the case when $m\leq k\leq n-1$. Let such a $k$ be fixed. By definition, we
have

$\begin{split}z(k)&=x(k+1)-L_{k}(x_{k})\\\
&=\chi(k+1)u(k+1)-L_{k}(\chi_{k}u_{k})\\\
&=\chi(k+1)L_{k}(u_{k})-L_{k}(\chi_{k}u_{k})\\\
&=L_{k}(\chi(k+1)u_{k}-\chi_{k}u_{k}).\end{split}$

Therefore, using (2.2), it follows that

$|z(k)|\leq M\|\chi(k+1)u_{k}-\chi_{k}u_{k}\|.$

Let $\theta\in[-r,0]\cap\mathbb{Z}$. Then

$\begin{split}|\left(\chi(k+1)u_{k}-\chi_{k}u_{k}\right)(\theta)|&=|\left(\chi(k+1)-\chi(k+\theta)\right)u(k+\theta)|,\end{split}$

and hence

$\begin{split}|\left(\chi(k+1)u_{k}-\chi_{k}u_{k}\right)(\theta)|&\leq\sum_{j=m}^{k}\|u_{j}\|^{-1}|u(k+\theta)|\qquad\text{whenever
}k+\theta\leq m\end{split}$

and

$\begin{split}|\left(\chi(k+1)u_{k}-\chi_{k}u_{k}\right)(\theta)|&\leq\sum_{j=k+\theta}^{k}\|u_{j}\|^{-1}|u(k+\theta)|\qquad\text{whenever
}k+\theta>m.\end{split}$

In view of (2.16), $u_{m}=\phi\in\mathcal{S}(m)$ implies that
$u_{j}=T(j,m)\phi\in\mathcal{S}(j)$ for $j\geq m$. Therefore, by Claim 10, for
$m\leq j\leq k$, we have

$|u(k+\theta)|\leq\|u_{k}\|=\|T(k,j)u_{j}\|\leq Q\|u_{j}\|$

so that $\|u_{j}\|^{-1}|u(k+\theta)|\leq Q$. From this, we conclude that if
$k+\theta\leq m$ so that $k-m\leq-\theta\leq r$, then

$\sum_{j=m}^{k}\|u_{j}\|^{-1}|u(k+\theta)|\leq Q(k-m+1)\leq Q(r+1),$

while in case $k+\theta>m$, we have

$\sum_{j=k+\theta}^{k}\|u_{j}\|^{-1}|u(k+\theta)|\leq Q(-\theta+1)\leq
Q(r+1).$

Therefore, in both cases $k+\theta\leq m$ and $k+\theta>m$, we have that

$\begin{split}|\left(\chi(k+1)u_{k}-\chi_{k}u_{k}\right)(\theta)|&\leq
Q(r+1).\end{split}$

Since $\theta\in[-r,0]\cap\mathbb{Z}$ was arbitrary, this implies that

$\|\chi(k+1)u_{k}-\chi_{k}u_{k}\|\leq Q(r+1),$

which, combined with (2), yields

$|z(k)|\leq MQ(r+1).$

We have shown that the last inequality is valid whenever $m\leq k\leq n-1$ and
$z(k)=0$ otherwise. Hence

$\sup_{k\geq 0}|z(k)|\leq MQ(r+1).$

This, together with (2.23), implies that

$\sup_{k\geq-r}|x(k)|\leq CMQ(r+1).$ (2.24)

Since $n-r\geq n-N\geq m$ and $\chi$ is nondecreasing on
$\mathbb{Z}^{+}_{-r}$, we have for $\theta\in[-r,0]\cap\mathbb{Z}$,

$|x(n+\theta)|=\chi(n+\theta)|u(n+\theta)|\geq\chi(n-r)|u(n+\theta)|=|u(n+\theta)|\sum_{j=m}^{n-r-1}\|u_{j}\|^{-1}.$

Hence

$\|x_{n}\|\geq\|u_{n}\|\sum_{j=m}^{n-r-1}\|u_{j}\|^{-1}.$

According to Claim 10, $u_{m}=\phi\in\mathcal{S}(m)$ implies that
$\|u_{j}\|=\|T(j,m)\phi\|\leq Q\|\phi\|$ for $j\geq m$. This, together with
the previous inequality, yields

$\|x_{n}\|\geq\|u_{n}\|\sum_{j=m}^{n-r-1}\|u_{j}\|^{-1}\geq\|u_{n}\|\frac{n-m-r}{Q\|\phi\|}.$

The last inequality, combined with (2.21) and (2.24), implies that

$\begin{split}CMQ(r+1)\geq\|x_{n}\|\geq\|u_{n}\|\frac{n-m-r}{Q\|\varphi\|}>\frac{n-m-r}{eQ}.\end{split}$

However, this contradicts the fact that $n\geq m+N$ with $N$ satisfying
(2.18). Thus, (2.19) holds whenever $m\in\mathbb{Z}^{+}_{0}$ and
$\phi\in\mathcal{S}(m)$.

Using (2.19), we can easily complete the proof. Choose an integer $N$
satisfying (2.18). Let $n\geq m$. Then, $n-m=kN+h$ for some
$k\in\mathbb{Z}^{+}_{0}$ and $0\leq h\leq N-1$. From (2.19) and Claim 10, we
obtain for $\phi\in\mathcal{S}(m)$,

$\begin{split}\|T(n,m)\phi\|&=\|T(m+kN+h,m)\phi\|\\\
&=\|T(m+kN+h,m+kN)T(m+kN,m)\phi\|\\\ &\leq Q\|T(m+kN,m)\phi\|\\\ &\leq
Qe^{-k}\|\phi\|\\\ &\leq eQe^{-\frac{n-m}{N}}\|\phi\|.\end{split}$

Hence, the conclusion of the claim holds with $D=eQ$ and $\lambda=1/N$. ∎

######  Claim 12.

There exists $Q^{\prime}>0$ such that

$\|T(n,m)\phi\|\leq Q^{\prime}\|\phi\|,$

for every $n,m\in\mathbb{Z}^{+}_{0}$ with $n\leq m$ and
$\phi\in\mathcal{U}(m)$.

###### Proof of Claim 12.

Given $m\in\mathbb{Z}^{+}_{0}$ and $\phi\in\mathcal{U}(m)$, there exists
$\bar{\phi}\in\mathcal{U}$ such that $\phi=T(m,0)\bar{\phi}$. Let
$u\colon\mathbb{Z}^{+}_{-r}\to X$ be the solution of Eq. (2.1) such that
$u_{0}=\bar{\phi}$. Consider $x\colon\mathbb{Z}^{+}_{-r}\to X$ and
$z\colon\mathbb{Z}^{+}_{0}\to X$ given by

$x(k)=\begin{cases}u(k)&\text{for $-r\leq k\leq m$}\\\\[10.0pt] 0&\text{for
$k\geq m+1$,}\end{cases}$

and

$z(k)=x(k+1)-L_{k}(x_{k}),\qquad k\in\mathbb{Z}^{+}_{0},$

so that (2.9) is satisfied. Moreover, since $x(k)=0$ for $k\geq{m+1}$, it
follows that $\sup_{k\geq{-r}}|x(k)|<\infty$. Furthermore,
$x_{0}=u_{0}\in\mathcal{U}$ and $z(k)=0$ for $0\leq k\leq m-1$ and $k\geq
m+r+1$. Proceeding as in the proof of Claim 10, it can be shown that

$\sup_{k\geq 0}|z(k)|\leq 2Me^{\omega r}\|\phi\|.$

From conclusion (2.15) of Claim 7, we conclude that

$\sup_{-r\leq k\leq m}|u(k)|\leq\sup_{k\geq{-r}}|x(k)|\leq C\sup_{k\geq
0}|z(k)|\leq 2CMe^{\omega r}\|\phi\|.$

This implies that the conclusion of the claim holds with

$Q^{\prime}:=2CMe^{\omega r}>0.$

∎

######  Claim 13.

There exist $D^{\prime},\lambda^{\prime}>0$ such that

$\|T(n,m)\phi\|\leq D^{\prime}e^{-\lambda^{\prime}(m-n)}\|\phi\|$

for every $n,m\in\mathbb{Z}^{+}_{0}$ with $n\leq m$ and
$\phi\in\mathcal{U}(m)$.

###### Proof of Claim 13.

We claim that if

$N^{\prime}>eCM(Q^{\prime})^{2}(r+1)$ (2.25)

with $Q^{\prime}$ as in Claim 12, then for every $m\geq N^{\prime}$ and
$\phi\in\mathcal{U}(m)$,

$\|T(n,m)\phi\|\leq\frac{1}{e}\|\phi\|\qquad\text{whenever $0\leq n\leq
m-N^{\prime}$}.$ (2.26)

Suppose, for the sake of contradiction, that (2.25) holds and there exist
$m\geq N^{\prime}$ and $\phi\in\mathcal{U}(m)$ such that

$\|T(n,m)\phi\|>\frac{1}{e}\|\phi\|\qquad\text{for some~{}$n$ with $0\leq
n\leq m-N^{\prime}$}.$ (2.27)

Fix $n$ with $0\leq n\leq m-N^{\prime}$ such that the first inequality in
(2.27) holds. Evidently, (2.27) implies that $\phi\in\mathcal{U}(m)$ is
nonzero. Therefore, there exists a nonzero $\bar{\phi}\in\mathcal{U}$ such
that $\phi=T(m,0)\bar{\phi}$. Let $u$ denote the unique solution of Eq. (2.1)
with $u_{0}=\bar{\phi}$ so that $u_{m}=T(m,0)u_{0}=T(m,0)\bar{\phi}=\phi$.
Since $u_{n}=T(n,m)u_{m}=T(n,m)\phi$, the first inequality in (2.27) can be
written as

$\|u_{n}\|>\frac{1}{e}\|\phi\|.$ (2.28)

Choose a sequence $\psi\colon\mathbb{Z}^{+}_{0}\to[0,1]$ such that

$\psi(j)=1\quad\text{for $0\leq j\leq
m$}\qquad\text{and}\qquad\psi(j)=0\quad\text{for $j\geq m+1$}.$ (2.29)

By Claim 8, $0\neq\bar{\phi}\in\mathcal{U}$ implies that
$u_{j}=T(j,0)\bar{\phi}\neq 0$ for $j\geq 0$. Therefore, we can define a
function $x\colon\mathbb{Z}^{+}_{-r}\to X$ by

$x(k)=\chi(k)u(k)\qquad\text{for $k\geq{-r}$},$

where

$\chi(k)=\begin{cases}\displaystyle\sum_{j=0}^{\infty}\psi(j)\|u_{j}\|^{-1}\qquad&\text{for
$-r\leq k\leq 0$},\\\\[15.0pt]
\displaystyle\sum_{j=k}^{\infty}\psi(j)\|u_{j}\|^{-1}\qquad&\text{for $k\geq
1$}.\end{cases}$

Note that

$x_{0}=cu_{0}=c\bar{\phi}\in\mathcal{U},\qquad\text{where
$c=\chi(0)=\sum_{j=0}^{m}\psi(j)\|u_{j}\|^{-1}$.}$

Since $\psi(k)=0$ and hence $x(k)=0$ for $k\geq m+1$, we have that
$\sup_{k\geq{-r}}|x(k)|<\infty$. Moreover, $x$ satisfies (2.9) with
$z\colon\mathbb{Z}^{+}_{0}\to X$ defined by

$z(k)=x(k+1)-L_{k}(x_{k}),\qquad k\geq 0.$ (2.30)

Since $x_{k}=0$ for $k\geq m+r+1$, it follows that $z(k)=0$ for
$k\geq{m+r+1}$. In particular, $z$ is bounded on $\mathbb{Z}^{+}_{0}$. From
(2.30) and Claim 7, we conclude that

$\sup_{k\geq{-r}}|x(k)|\leq C\sup_{k\geq{0}}|z(k)|.$ (2.31)

Let $k\geq 0$ be arbitrary. By the same calculations as in the proof of Claim
10, we have

$\begin{split}z(k)&=L_{k}(\chi(k+1)u_{k}-\chi_{k}u_{k}).\end{split}$

From this and (2.2), we find that

$|z(k)|\leq M\|\chi(k+1)u_{k}-\chi_{k}u_{k}\|.$ (2.32)

Let $\theta\in[-r,0]\cap\mathbb{Z}$. Then

$\begin{split}|\left(\chi(k+1)u_{k}-\chi_{k}u_{k}\right)(\theta)|&=|\left(\chi(k+1)-\chi(k+\theta)\right)u(k+\theta)|.\end{split}$

From this and the definition of $\chi$, taking into account that
$0\leq\psi\leq 1$ on $\mathbb{Z}^{+}_{0}$, we conclude that

$|\left(\chi(k+1)u_{k}-\chi_{k}u_{k}\right)(\theta)|\leq\sum_{j=0}^{k}\|u_{j}\|^{-1}|u(k+\theta)|\qquad\text{whenever
$k+\theta\leq 0$}.$ (2.33)

and

$|\left(\chi(k+1)u_{k}-\chi_{k}u_{k}\right)(\theta)|\leq\sum_{j=k+\theta}^{k}\|u_{j}\|^{-1}|u(k+\theta)|\qquad\text{whenever
$k+\theta>0$}.$ (2.34)

By Claim 8, $u_{0}=\bar{\phi}\in\mathcal{U}$ implies that
$u_{j}=T(j,0)u_{0}\in\mathcal{U}(j)$ for $j\geq 0$. Therefore, according to
Claim 12, if $k+\theta<0$, then

$|u(k+\theta)|\leq\|u_{0}\|=\|T(0,j)u_{j}\|\leq Q^{\prime}\|u_{j}\|$

and hence $\|u_{j}\|^{-1}|u(k+\theta)|\leq Q^{\prime}$ for $j\geq 0$. If
$k+\theta\leq 0$ so that $k\leq-\theta\leq r$, then from the last ineaquality
and (2.33), we obtain that

$|\left(\chi(k+1)u_{k}-\chi_{k}u_{k}\right)(\theta)|\leq
Q^{\prime}(r+1)\qquad\text{whenever $k+\theta\leq 0$}.$ (2.35)

It follows by similar arguments that if $k+\theta>0$, then

$|u(k+\theta)|\leq\|u_{k+\theta}\|=\|T(k+\theta,j)u_{j}\|\leq
Q^{\prime}\|u_{j}\|$

and hence $\|u_{j}\|^{-1}|u(k+\theta)|\leq Q^{\prime}$ for $j\geq k+\theta$.
This, together with (2.34), yields

$|\left(\chi(k+1)u_{k}-\chi_{k}u_{k}\right)(\theta)|\leq
Q^{\prime}(r+1)\qquad\text{whenever $k+\theta>0$}.$ (2.36)

Since $\theta\in[-r,0]\cap\mathbb{Z}$ was arbitrary, (2.35) and (2.36) imply
that

$\begin{split}\|\chi(k+1)u_{k}-\chi_{k}u_{k}\|\leq
Q^{\prime}(r+1).\end{split}$

Since $k\geq 0$ was arbitrary, the last inequality, combined with (2.32),
implies that

$\sup_{k\geq 0}|z(k)|\leq MQ^{\prime}(r+1).$

This, together with (2.31), yields

$\sup_{k\geq-r}|x(k)|\leq CMQ^{\prime}(r+1).$ (2.37)

Since $\chi$ is nonincreasing on $\mathbb{Z}^{+}_{-r}$, we have for
$\theta\in[-r,0]\cap\mathbb{Z}$,

$|x(n+\theta)|=\chi(n+\theta)|u(n+\theta)|\geq\chi(n)|u(n+\theta)|=|u(n+\theta)|\sum_{j=n}^{m}\|u_{j}\|^{-1},$

the last equality being a consequence of (2.29). Hence

$\|x_{n}\|\geq\|u_{n}\|\sum_{j=n}^{m}\|u_{j}\|^{-1}.$

Since $u_{m}=\phi\in\mathcal{U}(m)$, by Claim 12, we have that
$\|u_{j}\|=\|T(j,m)\phi\|\leq Q^{\prime}\|\phi\|$ for $0\leq j\leq m$. This,
together with the previous inequality, gives

$\|x_{n}\|\geq\|u_{n}\|\frac{m-n+1}{Q^{\prime}\|\phi\|}.$

This, combined with (2.28) and (2.37), yields

$\begin{split}CMQ^{\prime}(r+1)\geq\|x_{n}\|\geq\|u_{n}\|\frac{m-n+1}{Q^{\prime}\|\phi\|}>\frac{m-n+1}{eQ^{\prime}}.\end{split}$

The last inequality contradicts the fact that $n-m\geq N^{\prime}$ with
$N^{\prime}$ satisfying (2.25). Thus, (2.26) holds whenever $m\geq N^{\prime}$
and $\phi\in\mathcal{U}(m)$.

Now, using (2.26), we can easily complete the proof. Let $0\leq n\leq m$ and
$\phi\in\mathcal{U}(m)$. Choose an integer $N^{\prime}$ satisfying (2.25).
Then, $m-n=kN^{\prime}+h$ for some $k\in\mathbb{Z}^{+}_{0}$ and $0\leq h\leq
N^{\prime}-1$. From (2.26) and Claim 12, we obtain

$\begin{split}\|T(n,m)\phi\|&=\|T(n,n+kN^{\prime}+h)\phi\|\\\
&=\|T(n,n+kN^{\prime})T(n+kN^{\prime},n+kN^{\prime}+h)\phi\|\\\ &\leq
e^{-k}\|T(n+kN^{\prime},n+kN^{\prime}+h)\phi\|\\\ &\leq
Q^{\prime}e^{-k}\|\phi\|\\\ &\leq
eQ^{\prime}e^{-\frac{m-n}{N^{\prime}}}\|\phi\|.\end{split}$

Thus, the conclusion of the claim holds with $D^{\prime}=eQ^{\prime}$ and
$\lambda^{\prime}=1/N^{\prime}$. ∎

For each $n\in\mathbb{Z}^{+}_{0}$, let $P_{n}$ denote the projection of
$\mathcal{B}_{r}$ onto $\mathcal{S}(n)$ along $\mathcal{U}(n)$ associated with
the decomposition (2.17).

######  Claim 14.

The projections $P_{n}$, $n\in\mathbb{Z}^{+}_{0}$, are uniformly bounded,
i.e.,

$\sup_{n\geq 0}\|P_{n}\|<\infty.$

###### Proof of Claim 14.

Since $\operatorname{ker}P_{n}=\mathcal{U}(n)$ and
$\operatorname{im}P_{n}=\mathcal{S}(n)$ for $n\in\mathbb{Z}^{+}_{0}$, Claims
11 and 13 show that the hypotheses of Lemma 3.1 of Huy and Van Minh [10] are
satisfied with $X=\mathcal{B}_{r}$ and $A_{n}=T(n+1,n)$. Therefore, the
desired conclusion follows from [10, Lemma 3.1]. ∎

Now we can complete the proof of Theorem 3. Let $\phi\in\mathcal{B}_{r}$ and
$n,m\in\mathbb{Z}^{+}_{0}$ with $n\geq m$ be fixed. From
$P_{m}\phi\in\mathcal{S}(m)$ and (2.16), we have that
$T(n,m)P_{m}\phi\in\mathcal{S}(n)$. Hence

$P_{n}T(n,m)P_{m}\phi=T(n,m)P_{m}\phi.$

Similarly, considering $Q_{m}=I-P_{m}$, $Q_{m}\phi\in\mathcal{U}(m)$ and
(2.16) imply that $T(n,m)Q_{m}\phi\in\mathcal{U}(n)$. Hence

$P_{n}T(n,m)Q_{m}\phi=0.$

From the above relations, taking into account that $\phi=P_{m}\phi+Q_{m}\phi$,
we conclude that

$P_{n}T(n,m)\phi=P_{n}T(n,m)P_{m}\phi+P_{n}T(n,m)Q_{m}\phi=T(n,m)P_{m}\phi.$

Since $\phi\in\mathcal{B}_{r}$ was arbitrary, this proves (2.6).

Evidently, $\operatorname{ker}P_{m}=\mathcal{U}(m)$ for $m\in\mathbb{Z}^{+}$.
Therefore, from Claim 8 and (2.16), it follows that the restriction
$T(n,m)\rvert_{\operatorname{ker}P(m)}\colon\operatorname{ker}P(m)\to\operatorname{ker}P(n)$
is invertible and onto. Furthermore, by Claim 14, the projections $P_{n}$,
$n\in\mathbb{Z}^{+}_{0}$, are uniformly bounded. Combining this fact with
Claims 11 and 13, we conclude that the exponential estimates (2.7) and (2.8)
are also satisfied. Thus, (2.1) admits an exponential dichotomy. ∎

######  Remark 4.

In the proof of Theorem 3, we have shown that the Perron property (see Claim
1) implies the existence of an exponential dichotomy for Eq. (1.3). Results of
this type have a long history that goes back to the pioneering works of Perron
[19] for ordinary differential equations and Li [12] for difference equations.
Subsequent important contributions are due to Massera and Schäffer [13, 14],
Daleckiĭ and Kreĭn [6], Coppel [5] and Henry [8], who was the first to
consider the case of noninvertible dynamics. For more recent contributions, we
refer to [9, 10, 11, 15, 21, 22, 23] and the references therein. A
comprehensive overview of the relationship between hyperbolicity and the
Perron property is given in [2].

##  3\. Shadowing of linear Volterra difference equations with infinite delay

In this section, we are interested in the shadowing of the Volterra difference
equation with infinite delay (1.5), where the kernel $A$ satisfies condition
(1.6). The phase space for Eq. (1.5) is the Banach space
$(\mathcal{B}_{\gamma},\|\cdot\|)$ given by

$\mathcal{B}_{\gamma}=\biggl{\\{}\,\phi\colon\mathbb{Z}^{-}_{0}\to\mathbb{C}^{d}:\sup_{\theta\in\mathbb{Z}^{-}_{0}}|\phi(\theta)|e^{\gamma\theta}<\infty\,\biggr{\\}},\qquad\|\phi\|=\sup_{\theta\in\mathbb{Z}^{-}_{0}}|\phi(\theta)|e^{\gamma\theta},\qquad\phi\in\mathcal{B}_{\gamma}.$

Under condition (1.6), Eq. (1.5) can be written equivalently in the form

$x(n+1)=L(x_{n}),$ (3.1)

where $x_{n}\in\mathcal{B}_{\gamma}$ is defined by $x_{n}(\theta)=x(n+\theta)$
for $\theta\in\mathbb{Z}^{-}_{0}$ and
$L\colon\mathcal{B}_{\gamma}\to\mathbb{C}^{d}$ is a bounded linear functional
defined by

$L(\phi)=\sum_{j=0}^{\infty}A(j)\phi(-j),\qquad\phi\in\mathcal{B}_{\gamma}.$

It is known (see, e.g., [16], [18]) that if (1.6) holds, then for every
$\phi\in\mathcal{B}_{\gamma}$, there exists a unique function
$x\colon\mathbb{Z}\to\mathbb{C}^{d}$ satisfying Eq. (1.5) (equivalently, Eq.
(3.1)) such that $x(\theta)=\phi(\theta)$ for $\theta\in\mathbb{Z}^{-}_{0}$.
We shall call $x$ the _solution of Eq. ( 1.5) (or (3.1)) on
$\mathbb{Z}^{+}_{0}$ with initial value $x_{0}=\phi$_. By a _solution of Eq. (
1.5) on $\mathbb{Z}^{+}_{0}$_, we mean a solution $x$ with initial value
$x_{0}=\phi$ for some $\phi\in\mathcal{B}_{\gamma}$.

For Eq. (1.5), the definition of shadowing can be modified as follows.

######  Definition 3.

We say that Eq. (1.5) is _shadowable on $\mathbb{Z}^{+}_{0}$_ if for each
$\epsilon>0$ there exists $\delta>0$ such that for every function
$y\colon\mathbb{Z}\to\mathbb{C}^{d}$ satisfying

$\sup_{n\geq 0}|y(n+1)-L(y_{n})|\leq\delta,$

there exists a solution $x$ of (1.5) on $\mathbb{Z}^{+}_{0}$ such that

$\sup_{n\geq 0}\|x_{n}-y_{n}\|\leq\epsilon.$

The main result of this section is the following theorem which shows that,
under condition (1.6), Eq. (1.5) is shadowable on $\mathbb{Z}^{+}_{0}$ if and
only if it is hyperbolic.

######  Theorem 4.

Suppose that (1.6) holds. Then, the following statements are equivalent. _(i)_
Eq. (1.5) is shadowable on $\mathbb{Z}^{+}_{0}$;

_(ii)_ The characteristic equation (1.7) has no root on the unit circle
$|\lambda|=1$.

Before we give a proof of Theorem 4, we summarize some facts from the spectral
theory of linear Volterra difference equations with infinite delay ([16],
[17], [18]).

For each $n\in\mathbb{Z}^{+}_{0}$, define
$T(n)\colon\mathcal{B}_{\gamma}\to\mathcal{B}_{\gamma}$ by
$T(n)\phi=x_{n}(\phi)$ for $\phi\in\mathcal{B}_{\gamma}$, where $x(\phi)$ is
the unique solution of Eq. (1.5) on $\mathbb{Z}^{+}_{0}$ with initial value
$x_{0}(\phi)=\phi$. It is well-known (see [16, 17]) that $T(n)$ is a bounded
linear operator in $\mathcal{B}_{\gamma}$ which has the semigroup property
$T(0)=I$, the identity on $\mathcal{B}_{\gamma}$, and $T(n+m)=T(n)T(m)$ for
$n,m\in\mathbb{Z}^{+}_{0}$. As a consequence, we have that

$T(n)=T^{n},\qquad n\in\mathbb{Z}^{+}_{0},\qquad\text{where\quad$T:=T(1)$}.$

From the definition of the _solution operator_ $T=T(1)$ and Eq. (1.5), we have
that

$[T(\phi)](\theta)=\begin{cases}\displaystyle\sum_{j=0}^{\infty}A(j)\phi(-j)&\qquad\text{for
$\theta=0$,}\\\ \phi(\theta+1)&\qquad\text{for $\theta\leq-1$}.\end{cases}$
(3.2)

If (1.6) holds, then the characteristic function $\varDelta$ defined by (1.8)
is an analytic function of the complex variable $\lambda$ in the region
$|\lambda|>e^{-\gamma}$. Denote by $\varSigma$ the set of characteristic roots
of Eq. (1.5),

$\varSigma=\\{\,\lambda\in\mathbb{C}:|\lambda|>e^{-\gamma},\,\det\varDelta(\lambda)=0\,\\},$

and define

$\varSigma^{cu}=\\{\,\lambda\in\varSigma:|\lambda|\geq 1\\}.$

It follows from the analycity of $\varDelta$ that $\varSigma^{cu}$ is a finite
spectral set for $T$. The corresponding spectral projection $\varPi^{cu}$ on
$\mathcal{B}_{\gamma}$ is defined by

$\varPi^{cu}=\frac{1}{2\pi i}\int_{C}(\lambda I-T)^{-1}\,d\lambda,$

where $C$ is any rectifiable Jordan curve which is disjoint with $\varSigma$
and contains $\varSigma^{cu}$ in its interior, but no point of
$\varSigma^{s}:=\varSigma\setminus\varSigma^{cu}$. The phase space
$\mathcal{B}_{\gamma}$ can be decomposed into the direct sum

$\mathcal{B}_{\gamma}=\mathcal{B}_{\gamma}^{cu}\oplus\mathcal{B}_{\gamma}^{s}$

with $\mathcal{B}_{\gamma}^{cu}=\varPi^{cu}(\mathcal{B}_{\gamma})$ and
$\mathcal{B}_{\gamma}^{s}=\varPi^{s}(\mathcal{B}_{\gamma})$, where
$\varPi^{s}=I-\varPi^{cu}$. The subspaces $\mathcal{B}_{\gamma}^{cu}$ and
$\mathcal{B}_{\gamma}^{s}$ are called the _center-unstable subspace_ and the
_stable subspace_ of $\mathcal{B}_{\gamma}$, respectively. The spectra of the
restrictions $T^{cu}:=T|_{\mathcal{B}_{\gamma}^{cu}}$ and
$T^{s}:=T|_{\mathcal{B}_{\gamma}^{s}}$ satisfy

$\sigma(T^{cu})=\varSigma^{cu}\quad\text{ and
}\quad\sigma(T^{s})=\sigma(T)\setminus\varSigma^{cu}=\\{\,\lambda\in\sigma(T):|\lambda|<1\,\\}.$

If $\varSigma^{cu}$ is nonempty, then it consists of finitely many eigenvalues
of $T$,

$\varSigma^{cu}=\\{\,\lambda_{1},\dots,\lambda_{r}\,\\},$

and $\mathcal{B}_{\gamma}^{cu}$ can be written as a direct sum of the
nullspaces

$\mathcal{B}_{\gamma}^{cu}=\operatorname{ker}((T-\lambda_{1}I)^{p_{1}})\oplus\cdots\oplus\operatorname{ker}((T-\lambda_{r}I)^{p_{r}}),$

where $p_{j}$ is the index (ascent) of $\lambda_{j}$, $j=1,\dots,r$. (see [16,
Remark 2.1, p. 62]).

An explicit representation of the spectral projection $\varPi^{cu}$ can be
given using the duality between Eq. (1.5) and its _formal adjoint equation_

$y(n-1)=\sum_{j=0}^{\infty}y(n+j)A(j),\qquad n\in\mathbb{Z}^{+}_{0},$ (3.3)

where $y(n)\in\mathbb{C}^{d*}$. Here $\mathbb{C}^{d*}$ denotes the
$d$-dimensional space of complex row vectors with a norm $|\cdot|$ which is
compatible with the given norm on $\mathbb{C}^{d}$, i.e.,
$|x^{*}x|\leq|x^{*}||x|$ for all $x\in\mathbb{C}^{d}$. The superscript ∗
indicates the conjugate transpose. The phase space for the formal adjoint
equation (3.3) is the Banach space
$(\mathcal{B}_{\tilde{\gamma}}^{\sharp},\|\cdot\|)$ defined by

$\mathcal{B}_{\tilde{\gamma}}^{\sharp}=\biggl{\\{}\,\psi\colon\mathbb{Z}^{+}_{0}\to\mathbb{C}^{d*}:\sup_{\zeta\in\mathbb{Z}^{+}_{0}}|\psi(\zeta)|e^{-\tilde{\gamma}\zeta}<\infty\,\biggr{\\}},\quad\|\psi\|=\sup_{\zeta\in\mathbb{Z}^{+}_{0}}|\psi(\zeta)|e^{-\tilde{\gamma}\zeta},\quad\psi\in\mathcal{B}_{\tilde{\gamma}}^{\sharp},$

where $\tilde{\gamma}$ is a fixed number such that $0<\tilde{\gamma}<\gamma$.
The solution operator
$T^{\sharp}\colon\mathcal{B}_{\tilde{\gamma}}^{\sharp}\to\mathcal{B}_{\tilde{\gamma}}^{\sharp}$
of Eq. (3.3) is given by

$[T^{\sharp}(\psi)](\zeta)=\begin{cases}\displaystyle\sum_{j=0}^{\infty}\psi(j)A(j)&\qquad\text{if
$\zeta=0$,}\\\ \psi(\zeta-1)&\qquad\text{if $\zeta\geq 1$}.\end{cases}$ (3.4)

Define a _bilinear form_
$\langle\cdot,\cdot\rangle\colon\mathcal{B}_{\tilde{\gamma}}^{\sharp}\times\mathcal{B}_{\gamma}\to\mathbb{C}$
by

$\langle\psi,\phi\rangle=\psi(0)\phi(0)+\sum_{j=1}^{\infty}\sum_{\zeta=0}^{j-1}\psi(\zeta+1)A(j)\phi(\zeta-j),\qquad\phi\in\mathcal{B}_{\gamma},\quad\psi\in\mathcal{B}_{\tilde{\gamma}}^{\sharp}.$

As shown in [16, Ineq. (3.3), p. 64], this bilinear form is bounded, i.e.,
there exists $K>0$ such that

$|\langle\psi,\phi\rangle|\leq
K\|\psi\|\|\phi\|,\qquad\phi\in\mathcal{B}_{\gamma},\quad\psi\in\mathcal{B}_{\tilde{\gamma}}^{\sharp}.$
(3.5)

Moreover, between Eqs. (1.5) and (3.3), we have the following _duality
relation_ (see [16, Lemma 3.1])

$\langle\psi,T\phi\rangle=\langle
T^{\sharp}\psi,\phi\rangle,\qquad\phi\in\mathcal{B}_{\gamma},\quad\psi\in\mathcal{B}_{\tilde{\gamma}}^{\sharp}.$
(3.6)

It is known that $T$ and $T^{\sharp}$ have the same spectrum and the dimension
of the subspace

$\mathcal{N}^{\sharp}:=\operatorname{ker}((T^{\sharp}-\lambda_{1}I)^{p_{1}})\oplus\cdots\oplus\operatorname{ker}((T^{\sharp}-\lambda_{r}I)^{p_{r}})$

of $\mathcal{B}_{\tilde{\gamma}}^{\sharp}$ is the same as the (finite)
dimension of the center-unstable subspace $\mathcal{B}_{\gamma}^{cu}$, which
will be denoted by $s$. Let $\\{\,\phi_{1},\dots,\phi_{s}\,\\}$ and
$\\{\,\psi_{1},\dots,\psi_{s}\,\\}$ be bases for $\mathcal{B}_{\gamma}^{cu}$
and $\mathcal{N}^{\sharp}$, respectively. Define
$\varPhi=(\phi_{1},\dots,\phi_{s})$ and
$\varPsi=\operatorname{col}(\psi_{1},\dots,\psi_{s})$. Then, the $s\times s$
matrix $\langle\varPsi,\varPhi\rangle$ given by
$\langle\varPsi,\varPhi\rangle=(\langle\psi_{i},\phi_{j}\rangle)_{i,j=1,\dots,s}$
is nonsingular, therefore, by replacing $\varPsi$ with
$\langle\varPsi,\varPhi\rangle^{-1}\varPsi$, we may (and do) assume that
$\langle\varPsi,\varPhi\rangle=E$, the $s\times s$ unit matrix. The projection
$\varPi^{cu}\colon\mathcal{B}_{\gamma}\to\mathcal{B}_{\gamma}^{cu}$ can be
given explicitly by (see [16, Theorem 3.1])

$\varPi^{cu}\phi=\varPhi\langle\varPsi,\phi\rangle,\qquad\phi\in\mathcal{B}_{\gamma},$
(3.7)

where $\langle\varPsi,\phi\rangle$ denotes the column vector
$\operatorname{col}(\langle\psi_{1},\phi\rangle,\dots,\langle\psi_{s},\phi\rangle)$.

The subspace $\mathcal{B}_{\gamma}^{cu}$ is invariant under the solution
operator $T$. If $B$ denotes the representation matrix of the linear
transformation $T|_{\mathcal{B}_{\gamma}^{cu}}$ with respect to the basis
$\varPhi$ of $\mathcal{B}_{\gamma}^{cu}$, then

$T\varPhi=\varPhi B\qquad\text{and}\qquad\sigma(B)=\varSigma^{cu}.$ (3.8)

A similar argument yields the existence of a square matrix $C$ such that

$T^{\sharp}\varPsi=C\varPsi\qquad\text{and}\qquad\sigma(C)=\varSigma^{cu}.$
(3.9)

Now suppose that $x$ is a solution of the nonhomogeneous equation

$x(n+1)=L(x_{n})+p(n),\qquad n\in\mathbb{Z}^{+}_{0}.$ (3.10)

with initial value $x_{0}=\phi$ for some $\phi\in\mathcal{B}_{\gamma}$. Then,
$x$ satisfies the following representation formula in $\mathcal{B}_{\gamma}$
(see [17, Theorem 2.1]), which is called the _variation of constants formula_
for Eq. (3.10) in the phase space,

$x_{n}=T(n)\phi+\sum_{j=0}^{n-1}T(n-1-j)\varGamma p(j),\qquad
n\in\mathbb{Z}^{+}_{0},$ (3.11)

where the operator $\varGamma\colon\mathbb{C}^{d}\to\mathcal{B}_{\gamma}$ is
defined by

$[\varGamma x](\theta)=\begin{cases}x&\qquad\text{if $\theta=0$,}\\\
0&\qquad\text{if $\theta\leq-1$}.\end{cases}$ (3.12)

Evidently,

$\|\varGamma x\|=|x|,\qquad x\in\mathbb{C}^{d}.$ (3.13)

Finally, let $z(n)$ be the coordinate of the projection $\varPi^{cu}x_{n}$
with respect to the basis $\varPhi$, i.e., $\varPi^{cu}x_{n}=\varPhi z(n)$ for
$n\in\mathbb{Z}^{+}_{0}$. In view of (3.7), $z(n)$ is given explicitly by

$z(n)=\langle\varPsi,x_{n}\rangle,\qquad n\in\mathbb{Z}^{+}_{0}.$ (3.14)

Moreover, it is known (see [18, Theorem 3]) that $z$ satisfies the following
first order difference equation in $\mathbb{C}^{s}$,

$z(n+1)=Bz(n)+\langle\varPsi,\varGamma p(n)\rangle,\qquad
n\in\mathbb{Z}^{+}_{0},$ (3.15)

with $B$ as in (3.8).

Now we are in a position to give a proof of Theorem 3.

###### Proof of Theorem 3.

$(i)\Rightarrow(ii)$. Suppose, for the sake of contradiction, that Eq. (1.5)
is shadowable, but (ii) does not hold. The shadowing property of Eq. (1.5)
implies the following type Perron type property.

######  Claim 15.

For every bounded function $p\colon\mathbb{Z}^{+}_{0}\to\mathbb{C}^{d}$, there
exists a function $x\colon\mathbb{Z}\to\mathbb{C}^{d}$ satisfying the
nonhomogeneous equation (3.10) with

$\sup_{n\geq 0}|x(n)|<\infty.$ (3.16)

The proof of Claim 15 is almost identical with that of Claim 1 in the of
Theorem 3, therefore we omit it. Since (ii) does not hold, there exists a
characteristic root $\lambda\in\varSigma$ with $|\lambda|=1$. Evidently,
$\lambda\in\varSigma^{cu}$, therefore the second relation in (3.8) implies the
existence of a nonzero vector $v\in\mathbb{C}^{s*}$ such that

$vB=\lambda v.$ (3.17)

Define $p\colon\mathbb{Z}^{+}_{0}\to\mathbb{C}^{d}$ by

$p(n)=\lambda^{n+1}(v\varPsi(0))^{*},\qquad n\in\mathbb{Z}^{+}_{0}.$ (3.18)

Since $\sup_{n\geq 0}|p(n)|=|(v\varPsi(0))^{*}|<\infty$, by Claim 15, the
nonhomogeneous equation (3.10) has at least one solution $x$ on
$\mathbb{Z}^{+}_{0}$ such that (3.16) holds. The corresponding coordinate
function $z$ defined by (3.14) satisfies the difference equation (3.15). From
(3.5), (3.13), (3.14) and (3.16), we obtain that $z$ and hence the function
$u\colon\mathbb{Z}^{+}_{0}\to\mathbb{C}$ defined by

$u(n)=vz(n),\qquad n\in\mathbb{Z}^{+}_{0},$ (3.19)

is bounded on $\mathbb{Z}^{+}_{0}$. Multiplying Eq. (3.15) from the left by
$v$ and using (3.17), we obtain that

$u(n+1)=\lambda u(n)+v\langle\varPsi,\varGamma p(n)\rangle,\qquad
n\in\mathbb{Z}^{+}_{0}.$ (3.20)

It follows from (3.12), (3.18) and the definition of the bilinear form
$\langle\cdot,\cdot\rangle$ that

$\langle\varPsi,\varGamma
p(n)\rangle=\varPsi(0)p(n)=\lambda^{n+1}\varPsi(0)(v\varPsi(0))^{*}.$

This, together with (3.20), implies that

$u(n+1)=\lambda u(n)+c\lambda^{n+1},\qquad n\in\mathbb{Z}^{+}_{0},$ (3.21)

with

$c=(v\varPsi(0))(v\varPsi(0))^{*}=|(v\varPsi(0))^{*}|_{2}^{2},$ (3.22)

where $|\cdot|_{2}$ denotes the $l_{2}$-norm on $\mathbb{C}^{d}$. From Eq.
(3.21), it follows by the variation of constants formula that

$u(n)=\lambda^{n}(u(0)+cn),\qquad n\in\mathbb{Z}^{+}_{0}.$ (3.23)

It follows from the duality (3.6) that $B=C$, where $B$ and $C$ have the
meaning from (3.8) and (3.9), respectively. Indeed, (3.6) implies that

$B=\langle\varPsi,\varPhi\rangle B=\langle\varPsi,\varPhi
B\rangle=\langle\varPsi,T\varPhi\rangle=\langle
T^{\sharp}\varPsi,\varPhi\rangle=\langle
C\varPsi,\varPhi\rangle=C\langle\varPsi,\varPhi\rangle=C.$

From (3.2) and the relation $T\varPhi=\varPhi B$ (see (3.8)), we find that

$\varPhi(\theta)=\varPhi(0)B^{\theta},\qquad\theta\in\mathbb{Z}^{-}_{0}.$
(3.24)

Similarly, from (3.4) and the relation $T^{\sharp}\varPsi=B\varPsi$ (see
(3.9)), we have that

$\varPsi(\zeta)=B^{-\zeta}\varPsi(0),\qquad\zeta\in\mathbb{Z}^{+}_{0}.$ (3.25)

Next we will show that

$v\varPsi(0)\neq 0.$ (3.26)

Suppose, for the sake of contradiction, that $v\varPsi(0)=0$. Then, by (3.17)
and (3.25), we have

$v\varPsi(\zeta)=vB^{-\zeta}\varPsi(0)=\lambda^{-\zeta}v\varPsi(0)=0\qquad\text{for
all $\zeta\in\mathbb{Z}^{+}_{0}$}.$

Thus, $v\varPsi$ is identically zero on $\mathbb{Z}^{+}_{0}$. On the other
hand, $v\neq 0$ implies that $v\varPsi$ is a nontrivial linear combination of
the basis elements $\psi_{1},\dots,\psi_{s}$ of $\mathcal{N}^{\sharp}$ and
hence it cannot be identically zero on $\mathbb{Z}^{+}_{0}$. This
contradiction proves that (3.26) holds.

From (3.22) and (3.26), we find that $c>0$. From this and (3.23), taking into
account that $|\lambda|=1$, we obtain that

$|u(n)|=|u(0)+cn|\rightarrow\infty,\qquad n\to\infty,$

which contradicts the boundedness of $u$.

$(ii)\Rightarrow(i)$. Suppose that the characteristic equation (1.7) has no
root with $|\lambda|=1$. Then, the exponential estimates of the solution
operator on the stable and unstable subspaces of $\mathcal{B}_{\gamma}$ (see,
e.g., [18, Theorem 1]) imply that Eq. (1.5) admits an exponential dichotomy on
$\mathbb{Z}^{+}_{0}$ as defined in [7] with projections $P_{n}=\varPi^{s}$ for
$n\in\mathbb{Z}^{+}_{0}$. By the application of [7, Theorem 1] with $f_{n}=0$,
$c=0$ and $\mu=1$, we conclude that Eq. (1.5) is shadowable on
$\mathbb{Z}^{+}_{0}$. ∎

######  Remark 5.

It is known that certain solutions of Eq. (1.5) can be continued backward in
the sense that they satisfy Eq. (1.5) for all $n\in\mathbb{Z}$. Such solutions
are sometimes called _global solutions_ or _entire solutions_. In a recent
paper [3], Barreira and Valls have considered the Ulam–Hyers stability, a
special case of shadowing, of the global solutions for difference equations
with finite delays. In [3, Theorem 5], they have proved the analogue of a
recent shadowing theorem [7, Theorem 1] for global solutions. Moreover, in the
special case of linear autonomous equations in finite dimensional spaces they
have shown that the Ulam–Hyers stability of the global solutions is equivalent
to the existence of an exponential dichotomy whenever the coefficients are
scalar (see [3, Theorem 8]) or the Jordan blocks associated with the central
directions are diagonal (see [3, Theorem 9]). Using a similar argument as in
the proof of Theorem 4, it can be shown that the assumption about the Jordan
blocks in [3, Theorem 9] is superfluous. In this sense, Theorem 4 may be
viewed as an improvement of [3, Theorem 9].

## Acknowledgements

L. Backes was partially supported by a CNPq-Brazil PQ fellowship under Grant
No. 307633/2021-7. D. Dragičević was supported in part by Croatian Science
Foundation under the Project IP-2019-04-1239 and by the University of Rijeka
under the Projects uniri-prirod-18-9 and uniri-prprirod-19-16. M. Pituk was
supported by the Hungarian National Research, Development and Innovation
Office grant no. K139346.

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