††2020 Mathematics Subject Classification. 39A05, 39A06, 39A24, 42A75, 47D99.
Key words and phrases. Abstract non-scalar Volterra difference equations,
abstract multi-term fractional difference equations, Weyl fractional
derivatives, Poisson transform, almost periodic solutions.
This research is partially supported by grant 451-03-68/2020/14/200156 of
Ministry of Science and Technological Development, Republic of Serbia.

# Abstract non-scalar Volterra difference equations

Marko Kostić  Faculty of Technical Sciences, University of Novi Sad, Trg D.
Obradovića 6, 21125 Novi Sad, Serbia<EMAIL_ADDRESS>
###### Abstract.

In this paper, we investigate the abstract non-scalar Volterra difference
equations. We employ the Poisson like transforms to connect the solutions of
the abstract non-scalar Volterra integro-differential equations and the
abstract non-scalar Volterra difference equations. We also investigate the
existence, uniqueness and almost periodicity of solutions to the abstract
multi-term fractional difference equations with Weyl fractional derivatives.

##  1\. Introduction and preliminaries

The theory of abstract Volterra integro-differential equations is still a very
active field of research of many mathematicians. Fractional calculus and
fractional differential equations, which have emerged as the most important
and prominent areas of applied mathematics in the last three decades or so,
can be viewed as a special part of this theory (cf. the monographs [24, 25,
27, 37] and references cited therein for more details on the subject).

Discrete fractional calculus, discrete fractional equations and discrete
Volterra equations have received much attention recently, as well (cf. the
monographs [3] by S. Abbas et al., [5] by M. H. Annaby, Z. S. Mansour, [14] by
R. A. C. Ferreira, [17] by C. Goodrich, A. C. Peterson and the research
articles [1, 2, 4, 6, 7, 8, 10, 12], [13, 16, 18, 19, 20, 33, 35, 36] for some
recent results obtained in this direction). Discrete fractional calculus is
incredibly important in modeling of various real phenomena appearing in the
theories of neural networks, complex dynamic systems, frequency response
analysis, image processing and interval-valued systems, e.g..

On the other hand, the abstract higher-order difference equations have been
considered in Chapter 13 of the monograph [15] by M. I. Gil, where it has been
assumed that all operator coefficients are bounded linear operators. In a
series of our recent research papers, we have initiated the study of various
classes of the abstract difference equations with closed linear operators or
multivalued linear operators. For example, in a joint research study [33] with
H. C. Koyuncuoğlu and D. Velinov, we have considered the abstract fractional
difference inclusion

(1.1) $\displaystyle\Delta^{\alpha}_{W}u(k)\in{\mathcal{A}}u(k+1)+f(k),\quad
k\in{\mathbb{Z}},$

where ${\mathcal{A}}$ is a multivalued linear operator on a complex Banach
space $(X,\|\cdot\|)$, $0<\alpha\leq 1$ and $\Delta^{\alpha}_{W}u(k)$ denotes
the Weyl fractional difference operator of order $\alpha$; concerning the
existence and uniqueness of almost periodic solutions and almost automorphic
solutions of the abstract fractional difference equation (1.1) in which
${\mathcal{A}}=A$ is a bounded linear operator, we would like to mention here
the important research articles of C. Lizama and his co-authors [1, 2, 4, 33].
Further on, in our recent research article [31], we have analyzed the
existence and uniqueness of asymptotically almost periodic solutions of the
following abstract Volterra difference inclusion

$\displaystyle u(v)\in f(v)+{\mathcal{A}}\bigl{(}a\ast_{0}u\bigr{)}(v),\quad
v\in{\mathbb{N}}_{0},$

where ${\mathcal{A}}$ is a multivalued linear operator in $X,$
$(a(v))_{v\in{\mathbb{N}}_{0}}$ and $(f(v))_{v\in{\mathbb{N}}_{0}}$ are given
sequences in $X.$ In the same paper, we have considered the existence and
uniqueness of almost periodic type solutions and almost automorphic type
solutions to the following abstract Volterra difference inclusion

$\displaystyle
u(k+1)\in{\mathcal{A}}\sum_{j=-\infty}^{k}a(k-j)u(j+1)+\sum_{j=-\infty}^{k}b(k-j)f(k),\quad
k\in{\mathbb{Z}},$

where ${\mathcal{A}}$ is a multivalued linear operator in $X,$
$(a(k))_{k\in{\mathbb{N}}_{0}}$ and $(b(k))_{k\in{\mathbb{N}}_{0}}$ are given
sequences in $X;$ cf. also the research article [23] by V. Keyantuo et al. for
the first steps made in this direction. We have introduced the notion of a
discrete $(a,k)$-regularized $C$-resolvent family and provided several results
about the generation of discrete $(a,k)$-regularized $C$-resolvent families
therein; for more details about the existence and uniqueness of almost
periodic type solutions to the abstract Volterra functional-differential-
difference equations, we refer the reader to the forthcoming monograph [30].

Concerning the abstract non-scalar Volterra integral equations, mention should
be made of the important monograph [37] by J. Prüss and the monographs [24,
25, 27] by M. Kostić. The main aim of this paper is to examine various classes
of the abstract non-scalar Volterra difference equations. We use the Poisson
like transforms to make a connection between the solutions of the abstract
non-scalar Volterra integro-differential equations and the abstract non-scalar
Volterra difference equations. We consider here the abstract degenerate
Volterra difference equation

(1.2) $Bu(v)=f(v)+\sum_{j=0}^{v}A(v-j)u(j),\quad v\in{\mathbb{N}}_{0},$

where $f:{\mathbb{N}}_{0}\rightarrow X$, $B$ is a closed linear operator on
$X$ and $A:{\mathbb{N}}_{0}\rightarrow L(Y,X)$, where $Y$ is a Banach space
which is continuously embedded in $X,$ as well as the nonautonomous
generalization of (1.2). We thoroughly analyze the well-posedness of (1.2) in
the following particular case:

(1.3) $\displaystyle A(v)=a_{1}(v)A_{1}+...+a_{n}(v)A_{n},\quad
v\in{\mathbb{N}}_{0},$

where $n\in{\mathbb{N}},$ $A_{1},...,A_{n}$ are closed linear operators on
$X,$ and $(a_{1}(v))_{v\in{\mathbb{N}}_{0}},...,$
$(a_{n}(v))_{v\in{\mathbb{N}}_{0}}$ are given sequences in $X.$ The equation
(1.2) is a special case of the equation

(1.4) $Bu(v)=f(v)+\sum_{i=1}^{n}\sum_{j=0}^{v+v_{i}}A_{i}(v+v_{i}-j)u(j),\quad
v\in{\mathbb{N}}_{0},$

where $v_{1},...,v_{n}\in{\mathbb{N}}_{0}$ and
$A_{1}:{\mathbb{N}}_{0}\rightarrow L(Y,X),...,\
A_{n}:{\mathbb{N}}_{0}\rightarrow L(Y,X).$ We will not consider the well-
posedness of problem (1.4) in general case and we will only focus our
attention here to the case in which the kernel
$(A_{i}(v))_{v\in{\mathbb{N}}_{0}}$ can be expressed as
$A_{i}(\cdot):=a_{i}(\cdot)A_{i},$ $1\leq i\leq n.$

In [32], we have recently analyzed the abstract multi-term fractional
difference equations with Riemann-Liouville and Caputo fractional derivatives.
In this paper, we continue the above-mentioned research study by investigating
the abstract multi-term fractional difference equations with Weyl fractional
derivatives. In contrast to [32], the domain of solutions of the considered
fractional difference equations is the whole integer line ${\mathbb{Z}};$
moreover, we do not accompany the considered fractional difference equations
with any initial conditions. Here, we also want to emphasize that the use of
sequences $a_{1}(\cdot)=k^{m_{1}-\alpha_{1}}(\cdot),...,$
$a_{n}(\cdot)=k^{m_{n}-\alpha_{n}}(\cdot)$ in (1.3), where
$(k^{\alpha}(v))_{v\in{\mathbb{N}}_{0}}$ is the Cèsaro sequence, plays a
crucial role in our analysis; cf. Section 4 for the notion and more details.

The structure of this paper can be briefly described as follows. After
explaining the basic notation and terminology used in the paper, we recall the
basic facts about the abstract Volterra integral equations of non-scalar type
in Subsection 1.1. Discrete $(A,k,B)$-regularized $C$-resolvent families are
investigated in Section 2; in Section 3, we introduce and systematically
analyze various notions of the $(k,C,B,(A_{i})_{1\leq i\leq n})$-solution
operator families connected with the use of kernel
$(A(v))_{v\in{\mathbb{N}}_{0}}$ given by (1.3). The asymptotically almost
periodic type solutions of the abstract multi–term discrete abstract Cauchy
problem

$\displaystyle
Bu(v)=f(v)+\sum_{i=1}^{n}A_{i}\bigl{(}a_{i}\ast_{0}u\bigr{)}(v+v_{i}),\quad
v\in{\mathbb{N}}_{0},$

where $v_{1},...,v_{n}\in{\mathbb{N}}_{0},$ are analyzed in Subsection 3.1.

The abstract multi-term fractional difference equations with Weyl fractional
derivatives are investigated in Section 4; here, we also investigate the well-
posedness of the following abstract multi-term difference equation

$\displaystyle
Bu(v)=A_{1}\sum_{l=-\infty}^{v+v_{1}}a_{1}(v+v_{1}-l)u(l)+...+A_{n}\sum_{l=-\infty}^{v+v_{n}}a_{n}(v+v_{n}-l)u(l),\quad
v\in{\mathbb{Z}},$

where $B,\ A_{1},...,\ A_{n}$ are closed linear operator on $X.$ In
particular, we consider the existence of solutions to the problems

$\displaystyle\bigl{(}\Delta^{m}Bu\bigr{)}(v)$
$\displaystyle=A_{1}\Bigl{(}\Delta^{\alpha_{1}}_{W}u\Bigr{)}(v+v_{1})+...$
$\displaystyle+A_{n}\Bigl{(}\Delta^{\alpha_{n}}_{W}u\Bigr{)}(v+v_{n})+\Delta^{m}(k\circ
Cf)(v)+\Delta^{m}g(v),\ v\in{\mathbb{Z}}$

and

$\displaystyle B\bigl{(}\Delta^{m_{n}}h\bigr{)}(v)$
$\displaystyle=\sum_{j=1}^{n-1}A_{j}\Bigl{(}\Delta^{m_{n}-m_{j}}\Delta^{\alpha_{j}}_{W}h\Bigr{)}(v+v_{j})$
$\displaystyle+A_{n}\Bigl{(}\Delta^{\alpha_{n}}_{W}h\Bigr{)}(v+v_{n})+(k\circ
Cf)(v)+g(v),\ v\in{\mathbb{Z}};$

cf. Theorem 4.1 for the notation and more details. It is also worthwhile to
mention that we connect the solutions of the abstract multi-term fractional
differential equation

$A_{n}D_{t}^{\alpha_{n}}u(t)+A_{n-1}D_{t}^{\alpha_{n-1}}u(t)+...+A_{1}D_{t}^{\alpha_{1}}u(t)=0,\quad
t>0,$

where $0\leq\alpha_{1}<\alpha_{2}<...<\alpha_{n}$ and $D_{t}^{\alpha}u(t)$
denotes the Riemann-Liouville fractional derivative of function $u(t)$ of
order $\alpha>0,$ with the solutions of the abstract multi-term fractional
difference equation

$\displaystyle A_{n}\Bigl{[}\Delta^{\alpha_{n}}_{W}u\Bigr{]}(v)$
$\displaystyle+A_{n-1}\Bigl{[}\Delta^{\alpha_{n-1}}_{W}u\Bigr{]}\bigl{(}v+m_{n}-m_{n-1}\bigr{)}$
$\displaystyle+...+A_{1}\Bigl{[}\Delta^{\alpha_{1}}_{W}u\Bigr{]}\bigl{(}v+m_{n}-m_{1}\bigr{)}=-g(v),\quad
v\in{\mathbb{Z}};$

cf. Example 4.4 for more details. We also consider certain connections between
the abstract multi-term fractional differential equations with Caputo
derivatives and the abstract multi-term fractional difference equation of the
above form; we particularly analyze the existence of (almost periodic type)
solutions to the problem

$\Bigl{[}\Delta^{\alpha}_{W}u\Bigr{]}(v)+Au(v+m)=-g_{\omega}(v),\quad
v\in{\mathbb{Z}},$

where $\alpha>0$ and $m=\lceil\alpha\rceil;$ cf. Example 4.6 for more details.
All considered Volterra difference equations and fractional difference
equations can be unsolvable with respect to the highest derivative. The final
section of paper is reserved for the conclusions and final remark about the
topics under our consideration.

Notation and terminology. In the sequel, we will always assume that $X$ and
$Y$ are two complex Banach spaces such that $Y$ is continuously embedded in
$X$ and $C\in L(X).$ If $C$ is injective, then $[R(C)]$ denotes the Banach
space $R(C)$ equipped with the norm $\|x\|_{R(C)}:=\|C^{-1}x\|_{X}$, $x\in
R(C)$; the norm in $X$, resp. $Y$, is denoted by $\|\cdot\|_{X}$, resp.
$\|\cdot\|_{Y}.$ By the norm continuity we mean the continuity in $L(X)$ and,
in many places, we do not distinguish $S(\cdot)$ $(U(\cdot))$ and its
restriction to $Y$; $[D(B)]$ denotes the Banach space $D(B)$ equipped with the
graph norm. For the notion and more details about the multivalued linear
operators, we refer the rader to [27].

The Gamma function is denoted by $\Gamma(\cdot)$ and the principal branch is
always used to take the powers. Define
$g_{\zeta}(t):=t^{\zeta-1}/\Gamma(\zeta)$ and $0^{\zeta}:=0$ ($\zeta>0,$
$t>0$); $\lceil s\rceil:=\inf\\{k\in{\mathbb{Z}}:k\geq s\\}$
($s\in{\mathbb{R}}$). Set ${\mathbb{N}}_{k}:=\\{1,....,k\\}$ and
${\mathbb{N}}_{k}^{0}:=\\{0,1,....,k\\}$ for $k\in{\mathbb{N}}.$ If the
sequences $(a_{k})_{k\in{\mathbb{N}}_{0}}$ and
$(b_{k})_{k\in{\mathbb{N}}_{0}}$ are given, then we define
$(a\ast_{0}b)(\cdot)$ by $(a\ast_{0}b)(k):=\sum_{j=0}^{k}a_{k-j}b_{j},\
k\in{\mathbb{N}}_{0}.$ Let us recall that the convolution product $\ast_{0}$
is commutative and associative. Further on, if $\alpha>0,$ then the Cesàro
sequence $(k^{\alpha}(v))_{v\in{\mathbb{N}}_{0}}$ is defined by
$k^{\alpha}(v):=\Gamma(v+\alpha)/(\Gamma(\alpha)v!).$ It is well-known that,
for every $\alpha>0$ and $\beta>0$, we have $k^{\alpha}\ast_{0}k^{\beta}\equiv
k^{\alpha+\beta}$ and $|k^{\alpha}(v)-g_{\alpha}(v)|=O(g_{\alpha}(v)|1/v|)$,
$v\in{\mathbb{N}};$ in particular, $k^{\alpha}(v)\sim g_{\alpha}(v),$
$v\rightarrow+\infty.$ Define $k^{0}(0):=1$ and $k^{0}(v):=0,$
$v\in{\mathbb{N}};$ then $k^{\alpha}\ast_{0}k^{\beta}\equiv k^{\alpha+\beta}$
for all $\alpha,\ \beta\geq 0.$

If the sequences $(a_{k})_{k\in{\mathbb{N}}_{0}}$ and
$(b_{k})_{k\in{\mathbb{Z}}}$ are given, then we define the Weyl convolution
product $(a\circ b)(\cdot)$ by

$(a\circ b)(v):=\sum_{l=-\infty}^{v}a(v-l)b(l),\quad v\in{\mathbb{Z}}.$

Due to [23, Theorem 3.12(ii)-(iii)], some logical assumptions ensure that (cf.
also the proof of Theorem 4.1 below):

$\displaystyle\bigl{(}f\ast_{0}g\bigr{)}\circ h=g\circ(f\circ h)=f\circ(g\circ
h).$

Suppose that $\alpha>0$, $m=\lceil\alpha\rceil$ and $I=(0,\infty).$ The
Riemann-Liouville fractional integral of order $\alpha>0$ is defined for any
function $f\in L^{1}(I:X),$ by
$J_{t}^{\alpha}f(t):=\bigl{(}g_{\alpha}*f\bigr{)}(t),$ $t>0.$ The Riemann-
Liouville fractional derivative of order $\alpha>0$ is defined for any
function $f\in L^{1}(I:X)$ with $g_{m-\alpha}\ast f\in W^{m,1}(I:X),$ by

$\displaystyle
D_{t}^{\alpha}f(t):=\frac{d^{m}}{dt^{m}}\bigl{(}g_{m-\alpha}*f\bigr{)}(t)=D_{t}^{m}J_{t}^{m-\alpha}f(t),\quad
t>0,$

while the Caputo fractional derivative $\mathbf{D}_{t}^{\alpha}u(t)$ is
defined for any function $u\in C^{m-1}([0,\infty):X)$ for which
$g_{m-\alpha}*(u-\sum_{k=0}^{m-1}u_{k}g_{k+1})\in C^{m}([0,\infty):X)$, by

$\mathbf{D}_{t}^{\alpha}u(t):=\frac{d^{m}}{dt^{m}}\bigg{[}g_{m-\alpha}*\bigg{(}u-\sum_{k=0}^{m-1}u_{k}g_{k+1}\bigg{)}\bigg{]}.$

###  1.1. Abstract non-scalar Volterra integral equations

Let $A(t)$ be a locally integrable function from $[0,\tau)$ into $L(Y,X)$. We
need to recall the notion introduced in [27, Definition 2.9.1-Definition
2.9.3]:

######  Definition 1.1.

Let $k\in C([0,\tau))$ and $k\neq 0$, let $\tau\in(0,\infty]$, $f\in
C([0,\tau):X)$, and let $A\in L_{loc}^{1}([0,\tau):L(Y,X))$. Of concern is the
following degenerate Volterra equation:

(1.5) $Bu(t)=f(t)+\int^{t}_{0}A(t-s)u(s)\,ds,\quad t\in[0,\tau).$

Then a function $u\in C([0,\tau):[D(B)])$ is said to be a strong solution of
(1.5) if and only if $u\in L^{\infty}_{loc}([0,\tau):Y)$ and (1.5) holds on
$[0,\tau)$.

In the following definition, we introduce the main solution concepts for
dealing with (1.5):

######  Definition 1.2.

Let $\tau\in(0,\infty]$, $k\in C([0,\tau))$, $k\neq 0$ and $A\in
L_{loc}^{1}([0,\tau):L(Y,X))$. A family $(S(t))_{t\in[0,\tau)}$ in
$L(X,[D(B)])$ is said to be an $(A,k,B)$-regularized $C$-pseudoresolvent
family if and only if the following holds:

  * (S1)c

The mappings $t\mapsto S(t)x$, $t\in[0,\tau)$ and $t\mapsto BS(t)x$,
$t\in[0,\tau)$ are continuous in $X$ for every $x\in X$, $BS(0)=k(0)C$ and
$S(t)C=CS(t)$, $t\in[0,\tau)$.

  * (S2)

Set $U(t)x:=\int^{t}_{0}S(s)x\,ds$, $x\in X$, $t\in[0,\tau)$. Then (S2) means
$U(t)Y\subseteq Y$, $U(t)_{\mid Y}\in L(Y)$, $t\in[0,\tau)$ and $(U(t)_{\mid
Y})_{t\in[0,\tau)}$ is locally Lipschitz continuous in $L(Y)$.

  * (S3)

The resolvent equations

(1.6) $BS(t)y=k(t)Cy+\int^{t}_{0}A(t-s)dU(s)y,\quad t\in[0,\tau),\ y\in Y,$
(1.7) $BS(t)y=k(t)Cy+\int^{t}_{0}S(t-s)A(s)y\,ds,\quad t\in[0,\tau),\ y\in Y,$

hold; (1.6), resp. (1.7), is called the first resolvent equation, resp. the
second resolvent equation.

An $(A,k,B)$-regularized $C$-pseudoresolvent family $(S(t))_{t\in[0,\tau)}$ is
said to be an $(A,k,B)$-regularized $C$-resolvent family if additionally:

  * (S4)

For every $y\in Y$, $S(\cdot)y\in L^{\infty}_{loc}([0,\tau):Y)$.

An operator family $(S(t))_{t\in[0,\tau)}$ in $L(X,[D(B)])$ is called a weak
$(A,k,B)$-regularized $C$-pseudoresolvent family if and only if (S1)c and
(1.7) hold.

The condition (S3) can be rewritten in the following equivalent form:

$\displaystyle\begin{split}\text{(S3)'}\qquad\qquad&BU(t)y=\Theta(t)Cy+\int^{t}_{0}A(t-s)U(s)y\,ds,\quad
t\in[0,\tau),\ y\in Y,\\\
&BU(t)y=\Theta(t)Cy+\int^{t}_{0}U(t-s)A(s)y\,ds,\quad t\in[0,\tau),\ y\in
Y.\end{split}$

We also need the following notion:

######  Definition 1.3.

Let $\tau\in(0,\infty]$, $k\in C([0,\tau))$, $k\neq 0$ and $A\in
L_{loc}^{1}([0,\tau):L(Y,X))$. A strongly continuous operator family
$(V(t))_{t\in[0,\tau)}\subseteq L(X)$ is said to be an $(A,k,B)$-regularized
$C$-uniqueness family if and only if

$V(t)By=k(t)Cy+\int^{t}_{0}V(t-s)A(s)y\,ds,\quad t\in[0,\tau),\ y\in Y\cap
D(B).$

##  2\. Discrete $(A,k,B)$-regularized $C$-resolvent families

Suppose that $B(k)$ is a closed linear operator acting in $X$
($k\in{\mathbb{N}}_{0}$), and $A:{\mathbb{N}}_{0}\rightarrow L(Y,X)$. We would
like to introduce the following notion:

######  Definition 2.1.

  * (i)

Let $f:{\mathbb{N}}_{0}\rightarrow X$. Then it is said that a sequence
$(u_{k})_{k\in{\mathbb{N}}_{0}}$ is a solution of the abstract degenerate
Volterra difference equation

(2.1) $B(v)u(v)=f(v)+\sum_{j=0}^{v}A(v-j)u(j),\quad v\in{\mathbb{N}}_{0}$

if and only if $u(j)\in D(B(j))\cap Y$ for all $j,\ v\in{\mathbb{N}}_{0}$, and
(2.1) holds.

  * (ii)

Let $k:{\mathbb{N}}_{0}\rightarrow{\mathbb{C}}$ and $k\neq 0.$ Then a family
$(S(v))_{v\in{\mathbb{N}}_{0}}$ in $L(X)$ is said to be a discrete
$(A,k,B)$-regularized $C$-resolvent family if and only if $S(v)C=CS(v)$,
$v\in{\mathbb{N}}_{0}$ and the following holds:

    * (S1)

$S(v)Y\subseteq Y$, $S(v)_{\mid Y}\in L(Y)$, $v\in{\mathbb{N}}_{0}$,

(2.2) $B(v)S(v)y=k(v)Cy+\sum_{j=0}^{v}A(v-j)S(j)y,\quad v\in{\mathbb{N}}_{0},\
y\in Y,$

and

(2.3) $B(v)S(v)y=k(v)Cy+\sum_{j=0}^{v}S(v-j)A(j)y,\quad v\in{\mathbb{N}}_{0},\
y\in Y;$

(2.2), resp. (2.3), is called the first resolvent equation, resp. the second
resolvent equation.

An operator family $(S(v))_{v\in{\mathbb{N}}_{0}}$ in $L(X)$ is called a
discrete weak $(A,k,B)$-regularized $C$-resolvent family if and only if
$S(v)C=CS(v)$, $v\in{\mathbb{N}}_{0}$, $S(v)Y\subseteq Y$, $S(v)_{\mid Y}\in
L(Y)$, $v\in{\mathbb{N}}_{0}$ and (2.3) hold.

  * (iii)

Let $k:{\mathbb{N}}_{0}\rightarrow{\mathbb{C}}$ and $k\neq 0.$ Then a strongly
continuous operator family $(W(v))_{v\in{\mathbb{N}}_{0}}\subseteq L(X)$ is
said to be a discrete $(A,k,B)$-regularized $C$-uniqueness family if and only
if, for every $v\in{\mathbb{N}}_{0},$ we have

$W(v)B(v)y=k(v)Cy+\sum_{j=0}^{v}W(v-j)A(j)y,\quad y\in Y\cap D(B(v)).$

The notion introduced in Definition 2.1(ii)-(iii) has not been considered
elsewhere even in the case that $B={\rm I}.$ Further on, it is clear that the
assumptions $S(v)Y\subseteq Y$, $S(v)_{\mid Y}\in L(Y)$,
$v\in{\mathbb{N}}_{0}$ and $A(j)S(v)y=S(v)A(j)y$ for all $j,\
v\in{\mathbb{N}}_{0}$ and $y\in Y$ ensure that (2.3) implies (2.2). We feel it
is our duty to say that the equation (2.2) is only an unsatisfactory attempt
to provide a discrete analogue of the first resolvent equation (1.6) as well
as that the last mentioned equation can be also considered in the
nonautonomous setting (wth certain difficulties).

In the following proposition, we analyze the uniqueness of solutions to (2.1):

######  Proposition 2.2.

Suppose that a sequence $(u_{k})_{k\in{\mathbb{N}}_{0}}$ is a solution of
(2.1) and the operator $B(v)-A(0)$ is injective for all
$v\in{\mathbb{N}}_{0}.$ Then $u(\cdot)$ is uniquely determined.

###### Proof.

Suppose that $f\equiv 0$ in (2.1). Then it is clear that (2.1) with $v=0$
gives $B(0)u(0)=A(0)u(0),$ so that $u(0)$ is uniquely determined. If $v=1,$
then we get $B(1)u(1)=A(1)u(0)+A(0)u(1),$ so that $u(1)$ is uniquely
determined, as well. Proceeding in this way and using the injectiveness of the
operator $B(v)-A(0)$ for all $v\in{\mathbb{N}}_{0},$ we simply obtain the
required statement. ∎

Further on, if $B(v)\equiv B,$ $g:{\mathbb{N}}_{0}\rightarrow{\mathbb{C}}$ and
$g\neq 0,$ then we define $S_{g}(v)x:=(g\ast_{0}S)(v)x,$
$v\in{\mathbb{N}}_{0}$ and $x\in X.$ It can be simply shown that, if
$(S(v))_{v\in{\mathbb{N}}_{0}}\subseteq L(X)$ is a discrete (weak)
$(A,k,B)$-regularized $C$-resolvent family
[$(W(v))_{v\in{\mathbb{N}}_{0}}\subseteq L(X)$ is a discrete
$(A,k,B)$-regularized $C$-uniqueness family], then
$(S_{g}(v))_{v\in{\mathbb{N}}_{0}}\subseteq L(X)$
[$(W_{g}(v))_{v\in{\mathbb{N}}_{0}}\subseteq L(X)$] is a discrete (weak)
$(A,k\ast_{0}g,B)$-regularized $C$-resolvent family [discrete
$(A,k\ast_{0}g,B)$-regularized $C$-uniqueness family].

Concerning the notion of a discrete $(A,k,B)$-regularized $C$-uniqueness
family, we would like to state the following result (cf. also [27, Proposition
2.9.5(i)]; the interested reader may try to transfer the statement of [27,
Proposition 2.9.4(i)] to the discrete solution operator families, as well):

######  Proposition 2.3.

Suppose that $k:{\mathbb{N}}_{0}\rightarrow{\mathbb{C}}$, $k\neq 0$, a
sequence $(u_{k})_{k\in{\mathbb{N}}_{0}}$ is a solution of the abstract
degenerate Volterra difference equation (2.1) and a strongly continuous
operator family $(W(v))_{v\in{\mathbb{N}}_{0}}\subseteq L(X)$ is a discrete
$(A,k,B)$-regularized $C$-uniqueness family.

  * (i)

Then $W(0)f(0)=k(0)Cu(0).$

  * (ii)

If $B(v)\equiv B,$ then we have $(kC\ast_{0}u)(v)=(W\ast_{0}f)(v)$ for all
$v\in{\mathbb{N}}_{0}.$ In particular, if $k(0)\neq 0$ and $C$ is injective,
then there exists at most one solution of (2.1).

###### Proof.

The given assumptions imply

$W(0)B(0)u(0)=k(0)Cu(0)+W(0)A(0)u(0)\ \ \mbox{ and }\ \
B(0)f(0)=f(0)+A(0)u(0).$

This simply yields $W(0)f(0)=k(0)Cu(0).$ To prove (ii) with $B(v)\equiv B,$
observe first that

$W(1)Bu(0)=k(1)Cu(0)+W(1)A(0)u(0)+W(0)A(1)u(0)\mbox{ and
}Bu(0)=f(0)+A(0)u(0).$

Hence,

(2.4) $\displaystyle W(1)f(0)=k(1)Cu(0)+W(0)A(1)u(0).$

Similarly, we have

$W(0)Bu(1)=k(0)Cu(1)+W(0)A(0)u(1)\mbox{ and }Bu(1)=f(1)+A(1)u(0)+A(0)u(1).$

Hence,

(2.5) $\displaystyle W(0)f(1)=k(0)Cu(1)-W(0)A(1)u(0).$

Keeping in mind (2.4)-(2.5), we get that $(kC\ast_{0}u)(1)=(W\ast_{0}f)(1).$
Considering the terms $W(v)Bu(0),$ $W(v-1)Bu(1),...,$ and $W(0)Bu(v),$ a
similar line of reasoning gives that $(kC\ast_{0}u)(v)=(W\ast_{0}f)(v)$ for
all $v\in{\mathbb{N}}.$ It is clear that, if $f\equiv 0$, $k(0)\neq 0$ and $C$
is injective, then $(kC\ast_{0}u)(v)=0$ for all $v\in{\mathbb{N}}_{0};$
inductively, it readily follows that $u\equiv 0.$ ∎

We continue by stating the following result (cf. also Theorem 4.1 with
$v_{1}=...=v_{n}=0$):

######  Theorem 2.4.

Suppose that $k:{\mathbb{N}}_{0}\rightarrow{\mathbb{C}},$ $k\neq 0$ and
$(S(v))_{v\in{\mathbb{N}}_{0}}\subseteq L(X)$ is an operator family such that
$S(v)Y\subseteq Y$, $S(v)_{\mid Y}\in L(Y)$, $v\in{\mathbb{N}}_{0}$, (2.2)
holds (this, in particular, holds if $(S(v))_{v\in{\mathbb{N}}_{0}}$ is a
discrete $(A,k,B)$-regularized $C$-resolvent family),
$\sum_{v=0}^{+\infty}\|S(v)_{\mid Y}\|_{L(Y)}<+\infty$ and
$\sum_{v=0}^{+\infty}\|[A\ast_{0}S](v)\|_{L(Y,X)}<+\infty.$ Suppose, further,
that _(i)_ or _(ii)_ holds, where:

  * (i)

$f:{\mathbb{Z}}\rightarrow Y$ is a bounded sequence, $k\in
l^{1}({\mathbb{Z}}:Y)$ and $\sum_{v=0}^{+\infty}\|A(v)\|_{L(Y,X)}<+\infty;$

  * (ii)

$f\in l^{1}({\mathbb{Z}}:Y),$ $k:{\mathbb{Z}}\rightarrow Y$ is a bounded
sequence and $\sup_{v\geq 0}\|A(v)\|_{L(Y,X)}<+\infty.$

Define $u(v):=\sum_{l=-\infty}^{v}S(v-l)f(l),$ $v\in{\mathbb{Z}}.$ Then
$u(\cdot)$ is bounded, $u\in l^{1}({\mathbb{Z}}:Y)$ provided that _(ii)_
holds, and

(2.6) $\displaystyle B(v)u(v)=(k\circ
Cf)(v)+\sum_{l=-\infty}^{v}A(v-l)u(l),\quad v\in{\mathbb{Z}}.$

###### Proof.

It is clear that the sequence $(k\circ Cf)(\cdot)$ is well-defined, $u(\cdot)$
is well-defined and bounded if (i) holds as well as that $u\in
l^{1}({\mathbb{Z}}:X)$ if (b) holds. The functional equality (S1) and the
closedness of the linear operator $B(v)$ in combination with the assumptions
$\sum_{v=0}^{+\infty}\|S(v)_{\mid Y}\|_{L(Y)}<+\infty$ and
$\sum_{v=0}^{+\infty}\|[A\ast_{0}S](v)\|_{L(Y,X)}<+\infty$ simply imply the
following:

$\displaystyle Bu(v)-(k\circ Cf)(v)=\sum_{l=-\infty}^{v}BS(v-l)f(l)-(k\circ
Cf)(v)$ $\displaystyle=-(k\circ
Cf)(v)+\sum_{l=-\infty}^{v}\Bigl{[}k(v-l)Cf(l)+\bigl{[}A\ast_{0}S\bigr{]}(v-l)f(l)\Bigr{]}$
$\displaystyle=\sum_{l=-\infty}^{v}\bigl{[}A\ast_{0}S\bigr{]}(v-l)f(l)=\sum_{s=0}^{+\infty}\bigl{[}A\ast_{0}S\bigr{]}(s)f(v-s)$
$\displaystyle=\sum_{s=0}^{+\infty}A(s)\sum_{r=0}^{+\infty}S(r)f(v-s-r)=\sum_{s=0}^{+\infty}A(s)u(v-s)=\sum_{l=-\infty}^{v}A(v-l)u(l),\
v\in{\mathbb{Z}},$

where we have also used the Fubini theorem in the last line of computation. ∎

The main result of this section reads as follows:

######  Theorem 2.5.

Suppose that $a\in{\mathbb{R}},$ $\omega\in{\mathbb{R}}\setminus\\{0\\},$
$k:{\mathbb{N}}_{0}\rightarrow{\mathbb{C}}$ and $k\neq 0,$ $(S(t))_{t\geq 0}$
in $L(X,[D(B)])$ is a weak $(A,k,B)$-regularized $C$-pseudoresolvent family
and $(W(t))_{t\geq 0}\subseteq L(X)$ is an $(A,k,B)$-regularized
$C$-uniqueness family. Suppose further that there exist finite real constants
$M\geq 1$ and $\epsilon>0$ such that
$\|A(t)\|_{L(Y,X)}+\|S(t)\|_{L(X,[D(B)])}+\|W(t)\|_{L(X)}\leq
M\exp(t(a-\epsilon)),$ $t\geq 0.$ Define

$\displaystyle k_{a,w}(v):=\int^{+\infty}_{0}e^{-at}\frac{(\omega
t)^{v}}{v!}k(t)\,dt,\quad v\in{\mathbb{N}}_{0},$

,

$\displaystyle A_{a,w}(v)y:=\int^{+\infty}_{0}e^{-at}\frac{(\omega
t)^{v}}{v!}A(t)y\,dt,\quad v\in{\mathbb{N}}_{0},\ y\in Y,$ $\displaystyle
S_{a,w}(v)x:=\int^{+\infty}_{0}e^{-at}\frac{(\omega t)^{v}}{v!}S(t)x\,dt,\quad
v\in{\mathbb{N}}_{0},\ x\in X$

and

$\displaystyle W_{a,w}(v)x:=\int^{+\infty}_{0}e^{-at}\frac{(\omega
t)^{v}}{v!}W(t)x\,dt,\quad v\in{\mathbb{N}}_{0},\ x\in X.$

Then $(S_{a,w}(v))_{v\in{\mathbb{N}}_{0}}$ is a discrete weak
$(A,k_{a,w},B)$-regularized $C$-resolvent family and
$(W_{a,w}(v))_{v\in{\mathbb{N}}_{0}}$ is a discrete
$(A,k_{a,w},B)$-regularized $C$-uniqueness family. Furthermore, if
$(S(t))_{t\geq 0}$ is an $(A,k,B)$-regularized $C$-pseudoresolvent family,
then for any $v\in{\mathbb{N}}$ and $y\in Y,$ we have

(2.7) $\displaystyle
BS_{a,w}(v)y=k_{a,\omega}(v)Cy+a\omega^{v}\bigl{(}A_{a,w}\ast_{0}U\bigr{)}(v)y-\omega^{v}(-1)^{v}\bigl{(}A_{a,w}\ast_{0}U\bigr{)}(v-1)y.$

###### Proof.

Suppose that $v\in{\mathbb{N}}_{0}$ and $y\in Y$ are fixed. Applying the
second resovent equation, the basic operational properties of the vector-
valued Laplace transform and the Leibniz rule, it readily follows that (cf.
also the proof of [36, Theorem 3.4]):

$\displaystyle BS_{a,w}(v)y=\int^{+\infty}_{0}e^{-at}\frac{(\omega
t)^{v}}{v!}S(t)y\,dt=\int^{+\infty}_{0}e^{-at}\frac{(\omega
t)^{v}}{v!}BS(t)y\,dt$ $\displaystyle=\int^{+\infty}_{0}e^{-at}\frac{(\omega
t)^{v}}{v!}\Biggl{[}k(t)Cy+\int^{t}_{0}S(t-s)A(s)y\,ds\Biggr{]}\,dt$
$\displaystyle=k_{a,w}(v)Cy+(-1)^{v}\frac{\omega^{v}}{v!}\Biggl{[}\hat{S}(\lambda)\hat{A}(\lambda)\Biggr{]}^{(v)}_{\lambda=a}$
$\displaystyle=k_{a,w}(v)Cy+(-1)^{v}\frac{\omega^{v-j}\omega^{j}}{v!}\sum_{j=0}^{v}\binom{v}{j}\hat{S}^{(v-j)}(a)\hat{A}^{(j)}(a)$
$\displaystyle=k_{a,w}(v)Cy+\frac{\omega^{v-j}\omega^{j}}{v!}\sum_{j=0}^{v}\binom{v}{j}\int^{+\infty}_{0}e^{-at}t^{v-j}S(t)\Biggl{[}\int^{+\infty}_{0}e^{-ar}r^{v-j}A(r)y\,dr\Biggr{]}\,dt$
$\displaystyle=k_{a,w}(v)Cy+\bigl{(}S_{a,w}\ast_{0}A_{a,w}\bigr{)}(v)y,$

as required. The corresponding statement for
$(W_{a,w}(v))_{v\in{\mathbb{N}}_{0}}$ follows similarly; suppose now that
$(S(t))_{t\geq 0}$ is an $(A,k,B)$-regularized $C$-pseudoresolvent family. Fix
$v\in{\mathbb{N}}$ and $y\in Y.$ Applying the partial integration and (S3)’,
we get:

$\displaystyle BS_{a,w}(v)y=\int^{+\infty}_{0}e^{-at}\Biggl{[}a\frac{(\omega
t)^{v}}{v!}-\omega\frac{(\omega t)^{v-1}}{(v-1)!}\Biggr{]}BU(t)y\,dt$
$\displaystyle=\int^{+\infty}_{0}e^{-at}\Biggl{[}a\frac{(\omega
t)^{v}}{v!}-\omega\frac{(\omega
t)^{v-1}}{(v-1)!}\Biggr{]}\cdot\Biggl{\\{}\int^{t}_{0}k(s)\,ds\,Cy+(A\ast
U)(t)y\Biggr{\\}}\,dt$
$\displaystyle=k_{a,w}(v)Cy+\int^{+\infty}_{0}e^{-at}\Biggl{[}a\frac{(\omega
t)^{v}}{v!}-\omega\frac{(\omega t)^{v-1}}{(v-1)!}\Biggr{]}(A\ast U)(t)y\,dt.$

Then we can argue as in the first part of the proof to complete it. ∎

It is worth mentioning that Theorem 2.5 can be applied to the abstract non-
scalar Volterra integral equations considered by J. Prüss in [37, Chapter
7–Chapter 9] (see, especially, the analysis of viscoelastic Timoshenko beam on
p. 240) as well as to the abstract degenerate non-scalar Volterra integral
equations considered in [27, Theorem 2.9.7] (see, especially, the applications
with the abstract differential operators in $L^{p}$-spaces given on pp.
240–241). In such a way, we can consider the well-posedness for certain
classes of the abstract semidiscrete non-scalar Volterra difference equations.

Further on, the integrability of $(A,k,B)$-regularized $C$-resolvent families
has been considered by J. Prüss in [37, Section 10.5, pp. 277–281], provided
that $k\equiv 1$ and $B=C={\rm I}.$ It is worth noting that the condition
$\sum_{v=0}^{+\infty}\|S(v)_{\mid Y}\|_{L(Y)}<+\infty$ in Theorem 2.4 holds
for the resolvents $(S(v)\equiv S_{1,1}(v))_{v\in{\mathbb{N}}_{0}}$ obtained
by applying Theorem 2.5 to the resolvent operator families used in the
analysis of Volterra non-scalar equations of variational type (cf. [37,
Corollary 10.7, Corollary 10.8; pp. 280–281]). In general case of
exponentially bounded $(A,k,B)$-regularized $C$-resolvent families, we must
use different values of $a,$ which should be sufficiently large, and $w>0$ in
order to enure the validity of condition $\sum_{v=0}^{+\infty}\|S(v)_{\mid
Y}\|_{L(Y)}<+\infty$ in Theorem 2.4. If this is the case, then we can consider
the existence and uniqueness of almost periodic type solutions to (2.6); the
usual almost periodic solutions can be considered if the condition (i) in
Theorem 2.4 holds, while the (metrically) Weyl almost periodic solutions of
(2.6) can be considered if the condition (ii) in Theorem 2.4 holds (cf. [30]
for more details).

We close this section by providing some useful observations:

######  Remark 2.6.

Define also

$\displaystyle\Theta_{a,w}(v):=\int^{+\infty}_{0}e^{-at}\frac{(\omega
t)^{v}}{v!}\int^{t}_{0}k(s)\,ds\,dt,\quad v\in{\mathbb{N}}_{0}$

and

$\displaystyle U_{a,w}(v)y:=\int^{+\infty}_{0}e^{-at}\frac{(\omega
t)^{v}}{v!}U(t)y\,dt,\quad v\in{\mathbb{N}}_{0},\ y\in Y.$

Then we have

$BU_{a,w}(v)y=\Theta_{a,w}(v)Cy+\bigl{(}U_{a,w}\ast_{0}A_{a,w}\bigr{)}(v)y,\quad
v\in{\mathbb{N}}_{0},\ y\in Y$

and

$BU_{a,w}(v)y=\Theta_{a,w}(v)Cy+\bigl{(}A_{a,w}\ast_{0}U_{a,w}\bigr{)}(v)y,\quad
v\in{\mathbb{N}}_{0},\ y\in Y,$

so that, for every $y\in Y$, the sequence $(u(v)\equiv
U_{a,w}(v)y)_{v\in{\mathbb{N}}_{0}}$ is a strong solution of problem (2.1)
with $B(v)\equiv B$ and $f(v)\equiv\Theta_{a,w}(v)Cy.$

######  Remark 2.7.

In our recent research article [32], we have analyzed the following Poisson
type transform

$\displaystyle v\mapsto
S_{a,w,j}(v)x:=\int^{+\infty}_{0}e^{-(at)^{j}}\frac{(\omega
t)^{v}}{v!}S(t)x\,dt,\quad v\in{\mathbb{N}}_{0},\ x\in X,$

where $j\in{\mathbb{N}}$ and $j\geq 2.$ It would be very difficult to state an
analogue of Theorem 2.5 for this transform because we cannot so simply compute
the term

$\int^{+\infty}_{0}e^{-(at)^{j}}\frac{(\omega
t)^{v}}{v!}\Biggl{[}\int^{t}_{0}S(t-s)A(s)y\Biggr{]}\,dt,\quad
v\in{\mathbb{N}}_{0},\ x\in X$

in the newly arisen situation.

##  3\. $(k,C,B,(A_{i})_{1\leq i\leq n},(v_{i})_{1\leq i\leq n})$-solution
operator families

In this section, we analyze various classes of $(k,C,B,(A_{i})_{1\leq i\leq
n},(v_{i})_{1\leq i\leq n})$-solution operator families connected with the use
of kernel $(A_{i}(v)\equiv a_{i}(v)A_{i})_{v\in{\mathbb{N}}_{0}}$ in (1.4). If
$v_{1},...,v_{n}\in{\mathbb{N}}_{0},$ then we define
$v_{max}:=\max(v_{1},...,v_{n})$ and
$M:=\\{i\in{\mathbb{N}}_{n}:v_{i}=v_{max}\\}.$ We will always assume
henceforth that the following condition holds:

(3.1) $\displaystyle a_{i}:{\mathbb{N}}_{0}\rightarrow{\mathbb{C}}\ \ \mbox{
for all }\ \ i\in{\mathbb{N}}_{n}\ \ \mbox{ and }\ \ a_{i}(0)\neq 0,\quad i\in
M.$

We continue by introducing the following notion:

######  Definition 3.1.

Suppose that $B$, $A_{1},...,A_{n}$ are closed linear operators on $X$, $C\in
L(X)$, $v_{1},...,v_{n}\in{\mathbb{N}}_{0},$
${\mathcal{I}}\subseteq{\mathbb{N}}_{n},$
$k:{\mathbb{N}}_{0}\rightarrow{\mathbb{C}},$ $k\neq 0$ and (3.1) holds. Then
we say that the operator family $(S(v))_{v\in{\mathbb{N}}_{0}}\subseteq L(X)$
is a discrete:

  * (i)

$(k,C,B,(A_{i})_{1\leq i\leq n},(v_{i})_{1\leq i\leq n})$-existence family if
and only if the mapping $x\mapsto
A_{i}\bigl{(}a_{i}\ast_{0}S\bigr{)}(v+v_{i})x,$ $x\in X$ belongs to $L(X)$ for
$v\in{\mathbb{N}}_{0}$, $1\leq i\leq n$ and

(3.2) $\displaystyle
BS(v)x=k(v)Cx+\sum_{i=1}^{n}A_{i}\bigl{(}a_{i}\ast_{0}S\bigr{)}(v+v_{i})x,\quad
v\in{\mathbb{N}}_{0},\ x\in X.$

  * (ii)

$(k,C,B,(A_{i})_{1\leq i\leq n},(v_{i})_{1\leq i\leq
n},{\mathcal{I}})$-existence family if and only if
$(S(v))_{v\in{\mathbb{N}}_{0}}$ is $(k,C,B,(A_{i})_{1\leq i\leq
n},(v_{i})_{1\leq i\leq n})$-existence family and $S(v)A_{i}\subseteq
A_{i}S(v)$ for all $v\in{\mathbb{N}}_{0}$ and
$i\in{\mathbb{N}}_{n}\setminus{\mathcal{I}}.$

Any discrete $(k,C,B,(A_{i})_{1\leq i\leq n},(v_{i})_{1\leq i\leq
n})$-existence family $(S(v))_{v\in{\mathbb{N}}_{0}}$ satisfies
$(S(v))_{v\in{\mathbb{N}}_{0}}\subseteq L(X,[D(B)]).$ In the case that
$v_{1}=v_{2}=...=v_{n}=0,$ we omit the term “$(v_{i})_{1\leq i\leq n}$” from
the notation. If this is not the case, then we cannot expect the uniqueness of
$(k,C,B,(A_{i})_{1\leq i\leq n},(v_{i})_{1\leq i\leq n})$-existence family
when all terms of this tuple are given in advance. For example, if $n=1,$
$B=C={\rm I},$ $A_{1}=2{\rm I}$, $a_{1}\equiv k\equiv 1$ and $v_{1}=1,$ then
we have $S(0)x+2S(1)x+Cx=0,$ so that neither $S(0)$ nor $S(1)$ can be uniquely
determined. The notion of a discrete $(k,C,B,(A_{i})_{1\leq i\leq
n})$-existence family looks similar but it is not exactly the same as the
notion of a weak discrete $(A,k,B)$-regularized $C$-resolvent family with
$B(\cdot)\equiv B$, $Y:=X$ and $A(v)y:=\sum_{i=1}^{n}a_{i}(v)A_{i}y$ for all
$y\in Y$ and $v\in{\mathbb{N}}_{0}.$

We continue by stating the following result:

######  Proposition 3.2.

Suppose that $B$, $A_{1},...,A_{n}$ are closed linear operators on $X$, $C\in
L(X)$, $v_{1},...,v_{n}\in{\mathbb{N}}_{0},$
$k:{\mathbb{N}}_{0}\rightarrow{\mathbb{C}},$ $k\neq 0$, (3.1) holds and
$(S(v))_{v\in{\mathbb{N}}_{0}}\subseteq L(X)$ is a discrete
$(k,C,B,(A_{i})_{1\leq i\leq n},(v_{i})_{1\leq i\leq n})$-existence family. If
$x\in X,$ $i\in{\mathbb{N}}_{n}$ and $v_{i}=0$, then $S(v)x\in D(A_{i})$ for
all $v\in{\mathbb{N}}_{0}$ and $i\in{\mathbb{N}}_{n};$ the same holds for each
$i\in{\mathbb{N}}_{n}$ with $v_{i}>0,$ provided that $S(0)x\in
D(A_{i}),...,S(v_{i}-1)x\in D(A_{i}).$

###### Proof.

We will prove the statement in case $i\in{\mathbb{N}}_{n}$ and $v_{i}>0$. Then
we have $(a_{i}\ast_{0}S)(v_{i})x\in D(A_{i}).$ Since we have assumed
$a_{i}(0)\neq 0$ and $S(0)x\in D(A_{i}),...,S(v_{i}-1)x\in D(A_{i}),$ this
simply implies $S(v_{i})x\in D(A_{i}).$ Since $(a_{i}\ast_{0}S)(1+v_{i})x\in
D(A_{i}),$ we similarly obtain $S(v_{i}+1)x\in D(A_{i}).$ Proceeding by
induction, we get $S(v)x\in D(A_{i})$ for all $v\in{\mathbb{N}}_{0}.$ ∎

Now we will state and prove the following result (cf. also [31, Theorem 2.5]):

######  Theorem 3.3.

Suppose that $B$, $A_{1},...,A_{n}$ are closed linear operators on $X$, $C\in
L(X)$ is injective, $k:{\mathbb{N}}_{0}\rightarrow{\mathbb{C}},$ $k(0)\neq 0$
and (3.1) holds.

  * (i)

Suppose, further, that $(S(v))_{v\in{\mathbb{N}}_{0}}\subseteq L(X)$ is a
discrete $(k,C,B,(A_{i})_{1\leq i\leq n})$-existence family such that
$S(0)Bx=BS(0)x$ and $S(0)A_{i}x=A_{i}S(0)x$ for all $x\in D(B)\cap
D(A_{1})\cap...\cap D(A_{n}).$ Then $(B-\sum_{i=0}^{n}a_{i}(0)A_{i})^{-1}C\in
L(X),$ $S(0)=k(0)(B-\sum_{i=0}^{n}a_{i}(0)A_{i})^{-1}C$,

$\displaystyle S(v)x$
$\displaystyle=\Biggl{(}B-\sum_{i=0}^{n}a_{i}(0)A_{i}\Biggr{)}^{-1}$ (3.3)
$\displaystyle\times\Biggl{[}k(v)Cx+\sum_{i=1}^{n}A_{i}\sum_{j=0}^{v-1}a_{i}(v-j)S(j)x\Biggr{]},\quad
v\in{\mathbb{N}},\ x\in X,$

and $A_{i}S(v)\in L(X)$ for all $i\in{\mathbb{N}}_{n}$ and
$v\in{\mathbb{N}}_{0}$.

  * (ii)

Suppose that $C\in L(X)$ is injective,
$(B-\sum_{i=0}^{n}a_{i}(0)A_{i})^{-1}C\in L(X)$ and, for every
$l\in{\mathbb{N}}$ and for every choice of integers $a_{j}\in{\mathbb{N}}_{n}$
for $1\leq j\leq l,$ we have

(3.4)
$\displaystyle\Biggl{[}\prod_{j=1}^{l}\Biggl{(}B-\sum_{i=0}^{n}a_{i}(0)A_{i}\Biggr{)}^{-1}A_{a_{j}}\Biggr{]}\cdot\Biggl{(}B-\sum_{i=0}^{n}a_{i}(0)A_{i}\Biggr{)}^{-1}C\in
L(X).$

Define $S(0):=k(0)(B-\sum_{i=0}^{n}a_{i}(0)A_{i})^{-1}C$ and $S(v),$
$v\in{\mathbb{N}},$ recursively by (3.3). Then
$(S(v))_{v\in{\mathbb{N}}_{0}}\subseteq L(X)$ is well-defined, $A_{i}S(v)\in
L(X)$ for all $i\in{\mathbb{N}}_{n}$, $v\in{\mathbb{N}}_{0}$ and
$(S(v))_{v\in{\mathbb{N}}_{0}}$ is a unique discrete $(k,C,B,(A_{i})_{1\leq
i\leq n})$-existence family. Furthermore, if
${\mathcal{I}}\subseteq{\mathbb{N}}_{n}$ and

(3.5) $\displaystyle\begin{split}&CB\subseteq BC,\ CA_{i}\subseteq
A_{i}C\mbox{ for all }i\in{\mathbb{N}}_{n}\setminus{\mathcal{I}},\\\
&\bigl{(}\forall
i\in{\mathbb{N}}_{n}\setminus{\mathcal{I}}\bigr{)}\,\bigl{(}\forall x\in
D(A_{i})\cap D(B)\bigr{)}\,Bx\in D(A_{i}),\ A_{i}x\in D(B)\mbox{ and
}A_{i}Bx=BA_{i}x,\\\ &\bigl{(}\forall
i\in{\mathbb{N}}_{n}\setminus{\mathcal{I}}\bigr{)}\bigl{(}\forall
j\in{\mathbb{N}}_{n}\bigr{)}\,\bigl{(}\forall x\in D(A_{i})\cap
D(A_{j})\bigr{)}\\\ &A_{j}x\in D(A_{i}),\ A_{i}x\in D(A_{j})\mbox{ and
}A_{i}A_{j}x=A_{j}A_{i}x,\end{split}$

resp. there exist a closed linear operator $A$ and the complex polynomials
$P_{B}(\cdot),$ $P_{1}(\cdot),...,\ P_{n}(\cdot)$ such that $CA\subseteq AC$
and $B=P_{B}(A),$ $A_{1}=P_{1}(A),...,\ A_{n}=P_{n}(A),$ then
$(S(v))_{v\in{\mathbb{N}}_{0}}$ is a discrete $(k,C,B,(A_{i})_{1\leq i\leq
n},{\mathcal{I}})$-existence family, resp. $(S(v))_{v\in{\mathbb{N}}_{0}}$ is
a discrete
$(k,C,B,(A_{i})_{1\leq i\leq n},\emptyset)$-existence family.

  * (iii)

Suppose that $C={\rm I}$, $(B-\sum_{j=0}^{n}a_{j}(0)A_{j})^{-1}\in L(X),$
$\sum_{v=0}^{+\infty}|a_{i}(v)|<+\infty$ for $1\leq i\leq n$,
$\sum_{v=0}^{\infty}|k(v)|<+\infty,$ and _(a)_ or _(b)_ holds, where:

    * (a)

$A_{i}\in L(X)$ for $1\leq i\leq n$ and

(3.6) $\displaystyle
1>\sum_{i=1}^{n}\sum_{v=1}^{+\infty}\bigl{|}a_{i}(v)\bigr{|}\cdot\Biggl{\|}\Biggl{(}B-\sum_{j=0}^{n}a_{j}(0)A_{j}\Biggr{)}^{-1}A_{i}\Biggr{\|}.$

    * (b)

Suppose that $C={\rm I}$, (LABEL:univer) holds or there exist a closed linear
operator $A$ and the complex polynomials $P_{B}(\cdot),$ $P_{1}(\cdot),...,\
P_{n}(\cdot)$ such that $B=P_{B}(A),$ $A_{1}=P_{1}(A),...,\ A_{n}=P_{n}(A),$
and

(3.7) $\displaystyle
1>\sum_{i=1}^{n}\sum_{v=1}^{+\infty}\bigl{|}a_{i}(v)\bigr{|}\cdot\Biggl{\|}A_{i}\Biggl{(}B-\sum_{j=0}^{n}a_{j}(0)A_{j}\Biggr{)}^{-1}\Biggr{\|}.$

Then the requirements in _(ii)_ hold and we have

(3.8) $\displaystyle\sum_{v=0}^{+\infty}\|S(v)\|<+\infty\ \ \mbox{ and }\ \
\sum_{v=0}^{+\infty}\Bigl{\|}A_{i}\bigl{(}a_{i}\ast_{0}S\bigr{)}(v)\Bigr{\|}<+\infty\
\ (1\leq i\leq n),$

provided that _(a)_ holds, resp. we have (3.8) and

(3.9) $\displaystyle\sum_{v=0}^{+\infty}\|A_{i}S(v)\|<+\infty\ \ (1\leq i\leq
n),$

provided that _(b)_ holds.

###### Proof.

Suppose that the assumptions in (i) hold and $x\in X.$ Then (3.2) with $v=0$
implies $BS(0)x=k(0)Cx+\sum_{i=1}^{n}a_{i}(0)A_{i}S(0)x$ so that
$(B-\sum_{i=0}^{n}a_{i}(0)A_{i})S(0)x=k(0)Cx$ and $S(0)x\in
k(0)(B-\sum_{i=0}^{n}a_{i}(0)A_{i})^{-1}Cx.$ Now we will prove that the
multivalued linear operator $(B-\sum_{i=0}^{n}a_{i}(0)A_{i})^{-1}C$ is single-
valued so that $(B-\sum_{i=0}^{n}a_{i}(0)A_{i})C\in L(X)$ and
$S(0)=k(0)(B-\sum_{i=0}^{n}a_{i}(0)A_{i})^{-1}C.$ In actual fact, let us
assume that $y\in(B-\sum_{i=0}^{n}a_{i}(0)A_{i})^{-1}C0$ for some $y\in X.$
This implies $(B-\sum_{i=0}^{n}a_{i}(0)A_{i})y=0$ and $y\in D(B)\cap
D(A_{1})\cap...\cap D(A_{n})$ since $a_{1}(0)\neq 0,...,a_{n}(0)\neq 0;$ now
the prescribed assumption implies that
$0=S(0)(B-\sum_{i=0}^{n}a_{i}(0)A_{i})y=(B-\sum_{i=0}^{n}a_{i}(0)A_{i})S(0)y=k(0)Cy.$
Since $k(0)\neq 0$ and $C$ is injective, this simply implies $y=0$, the
required conclusions and $A_{i}S(0)\in L(X)$ for all $i\in{\mathbb{N}}_{n}$ by
the closed graph theorem.

Since $BS(v)\in L(X)$, $A_{i}\bigl{(}a_{i}\ast_{0}S\bigr{)}(v)\in L(X)$ for
$v\in{\mathbb{N}}_{0}$, $1\leq i\leq n,$

(3.10) $\displaystyle
BS(v)x=k(v)Cx+\sum_{i=1}^{n}A_{i}\bigl{[}a_{i}(0)S(v)x+...+a_{i}(v)S(0)x\Bigr{]},\quad
v\in{\mathbb{N}}_{0},\ x\in X$

and $a_{1}(0)\neq 0,...,a_{n}(0)\neq 0,$ we can inductively prove that
$A_{i}S(v)\in L(X)$ for all $i\in{\mathbb{N}}_{n}$ and $v\in{\mathbb{N}}_{0}$.
Moreover, (3.10) easily implies (3.3) after a simple calculation.

In order to prove (ii), let us observe first that the mappings $S(0)$, $BS(0)$
and $x\mapsto A_{i}\bigl{(}a_{i}\ast_{0}S\bigr{)}(0)x,$ $x\in X$ belong to
$L(X)$ for $v=0$, $1\leq i\leq n$ as well as that (3.2) holds with $v=0$ and
$A_{j}S(0)\in L(X)$ all $j\in{\mathbb{N}}_{0},$ since we have assumed that
$a_{i}\neq 0$ for all $i\in{\mathbb{N}}_{n}.$ Now we proceed by induction.
Assume that the mappings $S(v^{\prime}),$ $BS(v^{\prime})$ and $x\mapsto
A_{i}\bigl{(}a_{i}\ast_{0}S\bigr{)}(v^{\prime})x,$ $x\in X$ belong to $L(X)$
for all $v^{\prime}<v\in{\mathbb{N}}$, $1\leq i\leq n$ as well as that (3.2)
holds for all $v^{\prime}<v$ and $A_{j}S(v^{\prime})\in L(X)$ for all
$j\in{\mathbb{N}}_{0}.$ The induction hypothesis together with the
representation formula (3.3), the assumption (3.4) and the closed graph
theorem imply that $S(v)\in L(X)$ and $BS(v)\in L(X).$ Since

$A_{i}\bigl{(}a_{1}\ast_{0}S\bigr{)}(v)x=A_{i}\Biggl{[}a_{i}(0)S(v)x+\sum_{j=1}^{v-1}a_{i}(v-j)S(j)x\Biggr{]},\quad
x\in X,\ 1\leq i\leq n,$

the representation formula (3.3), the closed graph theorem and the assumption
$a_{i}\neq 0$ for all $i\in{\mathbb{N}}_{0}$ imply that $A_{i}S(v)\in L(X)$
and the mapping $x\mapsto A_{i}\bigl{(}a_{i}\ast_{0}S\bigr{)}(v)x,$ $x\in X$
belongs to $L(X)$ for $1\leq i\leq n.$ Keeping in mind (3.3), the above simply
implies (3.2), so that the conclusions stated in the first part of theorem
hold true. Suppose now that ${\mathcal{I}}\subseteq{\mathbb{N}}_{n}$ and the
commutation relations in (LABEL:univer) hold. We will prove that
$(S(v))_{v\in{\mathbb{N}}_{0}}$ is a discrete $(k,C,B,(A_{i})_{1\leq i\leq
n},{\mathcal{I}})$-existence family, i.e., that for each fixed $x\in D(A_{i})$
we have $A_{i}x\in D(S(v))$ and $A_{i}S(v)x=S(v)A_{i}x$ for all
$v\in{\mathbb{N}}_{0}$ and $i\in{\mathbb{N}}_{n}\setminus{\mathcal{I}}.$ If
$v=0,$ then we need to prove that

$A_{i}\Biggl{(}B-\sum_{j=0}^{n}a_{j}(0)A_{j}\Biggr{)}^{-1}Cx=\Biggl{(}B-\sum_{j=0}^{n}a_{j}(0)A_{j}\Biggr{)}^{-1}CA_{i}x,$

i.e.,

$\Biggl{(}B-\sum_{i=0}^{n}a_{j}(0)A_{j}\Biggr{)}A_{i}\Biggl{(}B-\sum_{j=0}^{n}a_{j}(0)A_{j}\Biggr{)}^{-1}Cx=CA_{i}x.$

Using the equalities in the second line and the fourth line of (LABEL:univer),
the above is equivalent to

$A_{i}\Biggl{(}B-\sum_{i=0}^{n}a_{j}(0)A_{j}\Biggr{)}\Biggl{(}B-\sum_{j=0}^{n}a_{j}(0)A_{j}\Biggr{)}^{-1}Cx=CA_{i}x,$

i.e., with $A_{i}Cx=CA_{i}x$, which it has been assumed in the first line of
(LABEL:univer). Now we proceed by induction. Suppose that for each
$v^{\prime}<v$, $i\in{\mathbb{N}}_{n}\setminus{\mathcal{I}}$ and $x\in
D(A_{i})$ we have $A_{i}x\in D(S(v^{\prime}))$ and
$A_{i}S(v^{\prime})x=S(v^{\prime})A_{i}x$. Let us prove that the last equality
holds with $v^{\prime}=v.$ Using (3.3), it suffices to show that:

$\displaystyle A_{i}$
$\displaystyle\Biggl{(}B-\sum_{j=0}^{n}a_{j}(0)A_{j}\Biggr{)}^{-1}\Biggl{[}k(v)Cx+\sum_{j=1}^{n}A_{j}\sum_{l=0}^{v-1}a_{j}(v-l)S(l)x\Biggr{]}$
$\displaystyle=\Biggl{(}B-\sum_{j=0}^{n}a_{j}(0)A_{j}\Biggr{)}^{-1}\Biggl{[}k(v)C+\sum_{j=1}^{n}A_{j}\sum_{l=0}^{v-1}a_{j}(v-l)S(l)\Biggr{]}A_{i}x.$

The equality of terms containing $k(v)Cx$ is clear, while the equality of the
remaining parts follows from the induction hypothesis and the assumption made
in the third line and the fourth line od (LABEL:univer), since
$\sum_{l=0}^{v-1}a_{j}(v-l)S(l)x\in D(A_{j})$ for all $j\in{\mathbb{N}}_{n},$
$v\in{\mathbb{N}}$ and $x\in X.$ It is quite easy to show that, if there exist
a closed linear operator $A$ and the complex polynomials $P_{B}(\cdot),$
$P_{1}(\cdot),...,\ P_{n}(\cdot)$ such that $CA\subseteq AC$ and $B=P_{B}(A),$
$A_{1}=P_{1}(A),...,\ A_{n}=P_{n}(A),$ then $(S(v))_{v\in{\mathbb{N}}_{0}}$ is
a discrete $(k,C,B,(A_{i})_{1\leq i\leq n},\emptyset)$-existence family.

If $C={\rm I}$ and $(B-\sum_{j=0}^{n}a_{j}(0)A_{j})^{-1}\in L(X),$ then the
equation (3.4) automatically holds by the closed graph theorem. Suppose now
that the assumptions in (iii)(a) hold. Then we have

$\displaystyle\sum_{v=0}^{+\infty}\|S(v)\|$
$\displaystyle\leq\Biggl{\|}\Biggl{(}B-\sum_{i=0}^{n}a_{i}(0)A_{i}\Biggr{)}^{-1}\Biggr{\|}\cdot\sum_{v=0}^{+\infty}|k(v)|$
$\displaystyle+\sum_{i=1}^{n}\Biggl{\|}\Biggl{(}B-\sum_{j=0}^{n}a_{j}(0)A_{j}\Biggr{)}^{-1}A_{i}\Biggr{\|}\cdot\sum_{v=0}^{+\infty}\sum_{j=0}^{v-1}\bigl{|}a_{i}(v-j)\bigr{|}\cdot\|S(j)\|$
$\displaystyle=\Biggl{\|}\Biggl{(}B-\sum_{i=0}^{n}a_{i}(0)A_{i}\Biggr{)}^{-1}\Biggr{\|}\cdot\sum_{v=0}^{+\infty}|k(v)|$
$\displaystyle+\sum_{i=1}^{n}\sum_{v=1}^{+\infty}\bigl{|}a_{i}(v)\bigr{|}\cdot\Biggl{\|}\Biggl{(}B-\sum_{j=0}^{n}a_{j}(0)A_{j}\Biggr{)}^{-1}A_{i}\Biggr{\|}\cdot\sum_{v=0}^{+\infty}\|S(v)\|.$

Keeping in mind (3.6), we simply get the first estimate in (3.8). The second
estimate in (3.8) holds since $A_{i}\in L(X)$ for $1\leq i\leq n$ and

$\sum_{v=0}^{+\infty}\Bigl{\|}A_{i}\bigl{(}a_{i}\ast_{0}S\bigr{)}(v)\Bigr{\|}\leq\bigl{\|}A_{i}\bigr{\|}\cdot\sum_{v=0}^{+\infty}\bigl{|}a_{i}(v)\bigr{|}\cdot\sum_{v=0}^{+\infty}\bigl{\|}S(v)\bigr{\|}\
\ (1\leq i\leq n).$

Suppose now that the assumptions in (iii)(b) hold. Then we similarly obtain

$\displaystyle\sum_{v=0}^{+\infty}\|S(v)\|$
$\displaystyle\leq\Biggl{\|}\Biggl{(}B-\sum_{i=0}^{n}a_{i}(0)A_{i}\Biggr{)}^{-1}\Biggr{\|}\cdot\sum_{v=0}^{+\infty}|k(v)|$
$\displaystyle+\sum_{i=1}^{n}\Biggl{\|}A_{i}\Biggl{(}B-\sum_{j=0}^{n}a_{j}(0)A_{j}\Biggr{)}^{-1}\Biggr{\|}\cdot\sum_{v=0}^{+\infty}\sum_{j=0}^{v-1}\bigl{|}a_{i}(v-j)\bigr{|}\cdot\|S(j)\|$
$\displaystyle=\Biggl{\|}\Biggl{(}B-\sum_{i=0}^{n}a_{i}(0)A_{i}\Biggr{)}^{-1}\Biggr{\|}\cdot\sum_{v=0}^{+\infty}|k(v)|$
$\displaystyle+\sum_{i=1}^{n}\sum_{v=1}^{+\infty}\bigl{|}a_{i}(v)\bigr{|}\cdot\Biggl{\|}A_{i}\Biggl{(}B-\sum_{j=0}^{n}a_{j}(0)A_{j}\Biggr{)}^{-1}\Biggr{\|}\cdot\sum_{v=0}^{+\infty}\|S(v)\|.$

Now we can use (3.7) to deduce the first estimate in (3.8). Let $1\leq i\leq
n;$ then the second estimate in (3.8) can be proved as follows: Clearly, we
have

$\sum_{v=0}^{\infty}\Biggr{\|}A_{i}\sum_{j=0}^{v}a_{i}(v-j)S(j)\Biggl{\|}\leq\sum_{v=0}^{+\infty}|a_{i}(v)|\cdot\sum_{v=0}^{+\infty}\|A_{i}S(v)\|$

Therefore, it suffices to show (3.9). But, arguing as above and using our
commuting assumptions (LABEL:univer), it follows that

$\displaystyle\sum_{v=0}^{+\infty}\|A_{i}S(v)\|$
$\displaystyle\leq\Biggl{\|}A_{i}\Biggl{(}B-\sum_{j=0}^{n}a_{j}(0)A_{j}\Biggr{)}^{-1}\Biggr{\|}\cdot\sum_{v=0}^{+\infty}|k(v)|$
$\displaystyle+\sum_{v=0}^{+\infty}\sum_{j=1}^{n}\Biggl{\|}A_{i}\Biggl{(}B-\sum_{j=0}^{n}a_{j}(0)A_{j}\Biggr{)}^{-1}A_{j}\sum_{l=0}^{v-1}a_{j}(v-l)S(l)\Biggr{\|}$
$\displaystyle=\Biggl{\|}A_{i}\Biggl{(}B-\sum_{j=0}^{n}a_{j}(0)A_{j}\Biggr{)}^{-1}\Biggr{\|}\cdot\sum_{v=0}^{+\infty}|k(v)|$
$\displaystyle+\sum_{v=0}^{+\infty}\sum_{j=1}^{n}\Biggl{\|}A_{j}\Biggl{(}B-\sum_{j=0}^{n}a_{j}(0)A_{j}\Biggr{)}^{-1}\sum_{l=0}^{v-1}a_{j}(v-l)A_{i}S(l)\Biggr{\|}$
$\displaystyle\leq\Biggl{\|}A_{i}\Biggl{(}B-\sum_{j=0}^{n}a_{j}(0)A_{j}\Biggr{)}^{-1}\Biggr{\|}\cdot\sum_{v=0}^{+\infty}|k(v)|$
$\displaystyle+\sum_{j=1}^{n}\sum_{v=1}^{+\infty}\bigl{|}a_{j}(v)\bigr{|}\cdot\Biggl{\|}A_{j}\Biggl{(}B-\sum_{j=0}^{n}a_{j}(0)A_{j}\Biggr{)}^{-1}\Biggr{\|}\cdot\sum_{v=0}^{+\infty}\|A_{i}S(v)\|.$

Now the required conclusion simply follows from (3.7). ∎

We proceed with some useful observations:

######  Remark 3.4.

  * (i)

As already clarified, the equation (3.4) automatically holds if $C={\rm I};$
moreover, in general case, we have $C^{l}D_{l}\in L(X),$ where $D_{l}$ denotes
the operator appearing on the left-hand side of (3.4).

  * (ii)

The assumptions made in (LABEL:univer) do not imply that the operators
$A_{1},...,A_{n}$ are bounded, in general. A simple counterexample can be
given with $B=C={\rm I},$ the operators $A_{i}$ being bounded for
$i\in{\mathbb{N}}_{n}\setminus{\mathcal{I}},$ by assuming also that
$R(A_{i})\subseteq D(A_{j})$ for all $j\in{\mathbb{N}}_{n}$ and
$A_{i}A_{j}x=A_{j}A_{i}x$ for all $j\in{\mathbb{N}}_{n}$ and $x\in D(A_{j}).$

  * (iii)

It would be very tempting to deduce the estimates (3.8) and (3.9) using the
argumentation contained in the proofs of parts (ii)-(iii) of [31, Theorem
2.5].

Now we will consider the situation in which $v_{max}>0.$ Set
$M:=\\{i\in{\mathbb{N}}_{n}:v_{i}=v_{max}\\};$ then we have the following
analogue of Theorem 3.3:

######  Theorem 3.5.

  * (i)

Suppose that $B$, $A_{1},...,A_{n}$ are closed linear operators on $X$, $C\in
L(X)$, $k:{\mathbb{N}}_{0}\rightarrow{\mathbb{C}},$ $k(0)\neq 0,$ (3.1) holds,
$(S(v))_{v\in{\mathbb{N}}_{0}}\subseteq L(X)$ is a discrete
$(k,C,B,(A_{i})_{1\leq i\leq n},(v_{i})_{1\leq i\leq n})$-existence family,
and for every $i\in M$ and $x\in X,$ we have $S(0)x\in
D(A_{i}),...,S(v_{max}-1)x\in D(A_{i})$. Then $S(v)x\in D(A_{i})$ for all
$v\in{\mathbb{N}}_{0},$ $x\in X$, $i\in M,$

(3.11) $\displaystyle
BS(0)x=k(0)Cx+\sum_{i=1}^{n}\bigl{(}a_{i}\ast_{0}S\bigr{)}(v_{i})$

and $(S(v)$ is a uniquely determined for $v>v_{max}$, provided that the
operator $\sum_{i\in M}a_{i}(0)A_{i}$ is injective.

Furthermore, let us assume that _(a)_ -_(c)_ hold, where:

    * (a)

$[\sum_{i\in M}a_{i}(0)A_{i}]^{-1}B\in L(X)$ and $[\sum_{i\in
M}a_{i}(0)A_{i}]^{-1}C\in L(X);$

    * (b)

$[\sum_{i\in M}a_{i}(0)A_{i}]^{-1}A_{j}(a_{i}\ast_{0}S)(v+v_{i})\in L(X)$ for
all $v\in{\mathbb{N}}_{0}$ and $j\in{\mathbb{N}}_{n}\setminus M;$

    * (c)

$[\sum_{i\in M}a_{i}(0)A_{i}]^{-1}A_{j}S(v)\in L(X)$ for all
$v\in{\mathbb{N}}_{0}$ and $j\in M.$

Then, for every $v\in{\mathbb{N}}$ and $x\in X$, we have

$\displaystyle S\bigl{(}v+v_{max}\bigr{)}x=\Biggl{[}\sum_{i\in
M}a_{i}(0)A_{i}\Biggr{]}^{-1}\Biggl{\\{}BS(v)x-k(v)Cx$ (3.12)
$\displaystyle-\sum_{i\in{\mathbb{N}}_{n}\setminus
M}A_{i}\bigl{(}a_{i}\ast_{0}S\bigr{)}(v+v_{i})x-\sum_{i\in
M}A_{i}\Bigl{[}\bigl{(}a_{i}\ast_{0}S\bigr{)}(v+v_{i})x-a_{i}(0)S(v+v_{max})x\Bigr{]}\Biggr{\\}},$

and $A_{i}S(v)\in L(X)$ for all $i\in M$ and $v\in{\mathbb{N}}_{0}$.

  * (ii)

Suppose that $B$, $A_{1},...,A_{n}$ are closed linear operators on $X$, $C\in
L(X)$, $k:{\mathbb{N}}_{0}\rightarrow{\mathbb{C}},$ $k(0)\neq 0,$ (3.1) holds,
and the operators $S(0)\in L(X),...,S(v_{max})\in L(X)$ satisfy (3.11).
Suppose, further, that:

    * (a)

$[\sum_{i\in M}a_{i}(0)A_{i}]^{-1}\in L(X)$ and $[\sum_{i\in
M}a_{i}(0)A_{i}]^{-1}B\in L(X);$

    * (b)

$A_{j}\in L(X)$ for all $j\in{\mathbb{N}}_{n}\setminus M;$

    * (c)

$[\sum_{i\in M}a_{i}(0)A_{i}]^{-1}A_{j}S(v)\in L(X)$ for all
$v\in{\mathbb{N}}_{v_{max}}^{0}$ and $j\in M.$

Define, for every $v\in{\mathbb{N}}$, the operator $S\bigl{(}v+v_{max})$
through (3.12). Then $(S(v))_{v\in{\mathbb{N}}_{0}}\subseteq L(X)$ is a
discrete $(k,C,B,(A_{i})_{1\leq i\leq n},(v_{i})_{1\leq i\leq n})$-existence
family and $A_{i}S(v)\in L(X)$ for all $i\in M$ and $v\in{\mathbb{N}}_{0}$.
Furthermore, if there exist a closed linear operator $A$ and the complex
polynomials $P_{B}(\cdot),$ $P_{1}(\cdot),...,\ P_{n}(\cdot)$ such that
$CA\subseteq AC,$ $B=P_{B}(A),$ $A_{1}=P_{1}(A),...,\ A_{n}=P_{n}(A),$ and
conditions _(a)_ -_(c)_ hold, then $(S(v))_{v\in{\mathbb{N}}_{0}}$ is a
discrete
$(k,C,B,(A_{i})_{1\leq i\leq n},\emptyset,(v_{i})_{1\leq i\leq n})$-existence
family provided that
$S(v)A\subseteq AS(v)$ for $0\leq v\leq v_{max}.$

  * (iii)

Suppose that the assumptions in _(ii)_ hold,
$\sum_{v=0}^{+\infty}|a_{i}(v)|<+\infty$ for $1\leq i\leq n$,
$\sum_{v=0}^{\infty}|k(v)|<+\infty,$ and _(a)_ or _(b)_ holds, where:

    * (a)

$A_{i}\in L(X)$ for all $i\in{\mathbb{N}}_{n},$ and

$\displaystyle 1>\Biggl{\|}\Biggl{[}\sum_{j\in
M}a_{j}(0)A_{j}\Biggr{]}^{-1}B\Biggr{\|}$
$\displaystyle+\sum_{v=0}^{+\infty}\bigl{|}a_{i}(v)\bigr{|}\cdot\sum_{i\in{\mathbb{N}}_{n}\setminus
M}\Biggl{\|}\Biggl{[}\sum_{j\in M}a_{j}(0)A_{j}\Biggr{]}^{-1}A_{i}\Biggr{\|}$
$\displaystyle+\sum_{v=1}^{+\infty}\bigl{|}a_{i}(v)\bigr{|}\cdot\sum_{i\in
M}\Biggl{\|}\Biggl{[}\sum_{j\in M}a_{j}(0)A_{j}\Biggr{]}^{-1}A_{i}\Biggr{\|}.$

    * (b)

Suppose that there exist a closed linear operator $A$ and the complex
polynomials $P_{B}(\cdot),$ $P_{i}(\cdot)$ ($i\in M$) such that $B=P_{B}(A),$
$A_{i}=P_{i}(A)$ for $i\in M$, $A_{j}\in L(X)$ for all
$j\in{\mathbb{N}}\setminus M,$ and

$\displaystyle 1>\Biggl{\|}\Biggl{[}\sum_{j\in
M}a_{j}(0)A_{j}\Biggr{]}^{-1}B\Biggr{\|}$
$\displaystyle+\sum_{v=0}^{+\infty}\bigl{|}a_{i}(v)\bigr{|}\cdot\sum_{i\in{\mathbb{N}}_{n}\setminus
M}\Biggl{\|}\Biggl{[}\sum_{j\in M}a_{j}(0)A_{j}\Biggr{]}^{-1}A_{i}\Biggr{\|}$
$\displaystyle+\sum_{v=1}^{+\infty}\bigl{|}a_{i}(v)\bigr{|}\cdot\sum_{i\in
M}\Biggl{\|}A_{i}\Biggl{[}\sum_{j\in M}a_{j}(0)A_{j}\Biggr{]}^{-1}\Biggr{\|}.$

Then we have

(3.13) $\displaystyle\sum_{v=0}^{+\infty}\|S(v)\|<+\infty\ \ \mbox{ and }\ \
\sum_{v=0}^{+\infty}\Bigl{\|}A_{i}\bigl{(}a_{i}\ast_{0}S\bigr{)}\bigl{(}v+v_{i}\bigr{)}\Bigr{\|}<+\infty\
\ (1\leq i\leq n),$

provided that _(a)_ holds, resp. we have (3.13) and

(3.14)
$\displaystyle\sum_{v=0}^{+\infty}\Bigl{\|}A_{i}S\bigl{(}v\bigr{)}\Bigr{\|}<+\infty\
\ (i\in M),$

provided that _(b)_ holds and $A_{j}A_{i}\subseteq A_{i}A_{j}$ for every $i\in
M$ and $j\in{\mathbb{N}}_{n}\setminus M.$

###### Proof.

It is clear that (3.11) holds (cf. (3.2) with $v=0$) as well as that
Proposition 3.2 implies $S(v)x\in D(A_{i})$ for all $v\in{\mathbb{N}}_{0},$
$x\in X$, $i\in M.$ Plugging $v=1$ in (3.2) and using this fact, we get

$\displaystyle\Biggl{[}\sum_{i\in
M}a_{i}(0)A_{i}\Biggr{]}S\bigl{(}v_{max}+1\bigr{)}x$
$\displaystyle=BS(1)x-k(1)Cx-\sum_{i\in{\mathbb{N}}_{n}\setminus
M}A_{i}\bigl{(}a_{i}\ast_{0}S\bigr{)}(1+v_{i})x$ $\displaystyle-\sum_{i\in
M}A_{i}\Bigl{[}\bigl{(}a_{i}\ast_{0}S\bigr{)}(1+v_{i})x-a_{i}(0)S(1+v_{max})x\Bigr{]}.$

If the operator $\sum_{i\in M}a_{i}(0)A_{i}$ is injective, the above implies
that the value of $S(v_{max}+1)$ is uniquely determined; moreover, if (a)-(c)
hold, then we obtain that the formula (3.12) is valid with $v=1$ as well as
that $A_{i}S(v_{max}+1)\in L(X)$ for all $i\in M.$ Repeating this argumetation
with $v=2,$ $v=3,...,$ we obtain the part (i). The first conclusion in (ii)
follows similarly, while the remaining parts can be deduced as in the proof of
Theorem 3.3. ∎

It is also worth noting that, in Definition 3.1, we can consider arbitrary
sequence $(h(v;x))_{v\in{\mathbb{N}}_{0}}$ in $X$ as a substitute of the
already considered sequence $(k(v)Cx)_{v\in{\mathbb{N}}_{0}},$ especially in
the case that $v_{max}>0.$

###  3.1. The well-posedness and asymptotically almost periodic type solutions
of the corresponding discrete abstract Cauchy problems

In this subsection, we will briefly consider the following discrete abstract
Cauchy problem:

(3.15) $\displaystyle
Bu(v)=f(v)+\sum_{i=1}^{n}A_{i}\bigl{(}a_{i}\ast_{0}u\bigr{)}(v+v_{i}),\quad
v\in{\mathbb{N}}_{0},$

where $v_{1},...,v_{n}\in{\mathbb{N}}_{0}.$ We say that a mapping
$u:{\mathbb{N}}_{0}\rightarrow X$ is:

  * (i)

a mild solution of (3.15) if and only if $u(v)\in D(B)$ and
$(a_{i}\ast_{0}u)(v+v_{i})\in D(A_{i})$ for all $v\in{\mathbb{N}}_{0},$
$i\in{\mathbb{N}}_{n}$ and (3.1) holds;

  * (ii)

a strong solution of (3.15) if and only if $u(v)\in D(A_{i})$ for all
$v\in{\mathbb{N}}_{0},$ $i\in{\mathbb{N}}_{n}$ and

$\displaystyle
Bu(v)=f(v)+\sum_{i=1}^{n}\bigl{(}a_{i}\ast_{0}A_{i}u\bigr{)}(v+v_{i}),\quad
v\in{\mathbb{N}}_{0}.$

It is obvious that any strong solution of (3.15) is a mild solution of (3.15).
If $(S(v))_{v\in{\mathbb{N}}_{0}}$ is a discrete $(k,C,B,(A_{i})_{1\leq i\leq
n},(v_{i})_{1\leq i\leq n})$-existence family, then it is also clear that the
function $u(v):=S(v)x,$ $v\in{\mathbb{N}}_{0},$ where $x\in X,$ is a mild
solution of (3.15). Since we have assumed (3.1), it readily follows that any
mild solution of (3.15) is also a strong solution of (3.15), if for each
$i\in{\mathbb{N}}_{n}$ with $v_{i}>0$ we have $u(0)\in
D(A_{i}),...,u(v_{i}-1)\in D(A_{i}).$ Furthermore, the problem (3.15) has at
most one mild (strong) solution provided that for each $i\in{\mathbb{N}}_{n}$
with $v_{i}>0$ we have $u(0)\in D(A_{i}),...,u(v_{i}-1)\in D(A_{i})$ as well
as that the operator $B-\sum_{i=1}^{n}a_{i}(0)A_{i}$ is injective; cf. also
the proof of Proposition 3.2. The interested reader may try to prove some
discrete variation of parameters formula for the problem (3.15); cf. also [31,
Proposition 3.3].

In the remainder of this subsection, we will consider the case
$v_{1}=v_{2}=...=v_{n}=0$, only. The proof of following result is simple and
therefore omitted:

######  Proposition 3.6.

Suppose that $B$, $A_{1},...,A_{n}$ are closed linear operators on $X$, $C\in
L(X)$, $k:{\mathbb{N}}_{0}\rightarrow{\mathbb{C}},$ $k\neq 0$,
$g:{\mathbb{N}}_{0}\rightarrow{\mathbb{C}},$ $g\neq 0$, and
$(S(v))_{v\in{\mathbb{N}}_{0}}\subseteq L(X)$ is a discrete
$(k,C,B,(A_{i})_{1\leq i\leq n})$-existence family. If $x\in X,$ then the
sequence $u(v):=(g\ast_{0}S)(v)x,$ $v\in{\mathbb{N}}_{0}$ is a strong solution
of (3.15) with $f(v)=(k\ast_{0}g)(v),$ $v\in{\mathbb{N}}_{0}.$

Now we will explain how Proposition 3.6 can be applied in the analysis of the
existence and uniqueness of asymptotically almost periodic type solutions of
the discrete abstract Cauchy problem (3.15), with $v_{1}=v_{2}=...=v_{n}=0;$
concerning the almost periodic functions (sequences), the almost automorphic
functions (sequences) and their applications, one may refer, e.g., to the
monographs [11, 26, 28, 29, 34].

Suppose that there exist a bounded sequence $h:{\mathbb{Z}}\rightarrow X$ and
a sequence $q:{\mathbb{N}}_{0}\rightarrow X$ vanishing at plus infinity such
that $g(v)=h(v)+q(v)$ for all $v\in{\mathbb{N}}_{0}.$ If the requirements of
Proposition 3.6 hold, then we know that the function $u(v):=(g\ast_{0}S)(v)x,$
$x\in X,$ $v\in{\mathbb{N}}_{0}$ is a strong solution of (3.15) with
$f(v)=(k\ast_{0}g)(v),$ $v\in{\mathbb{N}}_{0}.$ If, additionally,
$\sum_{v=0}^{+\infty}\|S(v)\|<+\infty,$ then the argumentation given at the
end of [31, Subsection 3.1] shows that there exist a bounded sequence
$H:{\mathbb{Z}}\rightarrow X$ and a sequence $Q:{\mathbb{N}}_{0}\rightarrow X$
vanishing at plus infinity such that $u(v)=H(v)+Q(v)$ for all
$v\in{\mathbb{N}}_{0};$ moreover, $H(\cdot)$ is bounded and $T$-almost
periodic, where $T\in L(X),$ or bounded and ${\mathrm{R}}$-almost automorphic,
where ${\mathrm{R}}$ is any collection of sequences in ${\mathbb{Z}},$
provided that the forcing term $h(\cdot)$ has the same property (cf. [31] for
the notion).

##  4\. The abstract multi-term fractional difference equations with Weyl
fractional derivatives

Given a one-dimensional sequence $(u_{k})$ in $X$, the Euler forward
difference operator $\Delta$ is defined by $\Delta u_{k}:=u_{k+1}-u_{k}.$ The
operator $\Delta^{m}$ is defined inductively; then, for every integer $m\geq
1,$ we have:

$\displaystyle\Delta^{m}u_{k}=\sum_{j=0}^{m}(-1)^{m-j}\binom{m}{j}u_{k+j}.$

If $u:{\mathbb{N}}_{0}\rightarrow X$ is a given sequence and
$\alpha\in(0,\infty)\setminus{\mathbb{N}}_{0},$ then we define the fractional
integral $\Delta^{-\alpha}u:{\mathbb{N}}_{0}\rightarrow X$ by

$\Delta^{-\alpha}u(v):=\sum_{j=0}^{v}k^{\alpha}(v-j)u(j),\quad
v\in{\mathbb{N}}_{0}.$

Set $m:=\lceil\alpha\rceil.$ Then the Riemann-Liouville fractional difference
operator of order $\alpha>0,$ $\Delta^{\alpha}$ shortly, is defined by

$\Delta^{\alpha}u(v):=\Bigl{[}\Delta^{m}\bigl{(}\Delta^{-(m-\alpha)}u\bigr{)}\Bigr{]}(v),\quad
v\in{\mathbb{N}}_{0}.$

Further on, if $\alpha>0,$ $m=\lceil\alpha\rceil$ and
$u:{\mathbb{Z}}\rightarrow X$ satisfies
$\sum_{v=-\infty}^{\infty}\|u(v)\|\cdot(1+|v|)^{m-\alpha-1}<+\infty,$ then we
define the Weyl fractional derivative

$\displaystyle\Bigl{[}\Delta^{\alpha}_{W}u\Bigr{]}(v):=\Bigl{[}\Delta^{m}\Bigl{(}\Delta^{-(m-\alpha)}_{W}u\Bigr{)}\Bigr{]}(v),\quad
v\in{\mathbb{Z}},$

where

$\Bigl{(}\Delta^{-(m-\alpha)}_{W}u\Bigr{)}(v):=\sum_{l=-\infty}^{v}k^{m-\alpha}(v-l)u(l),\quad
v\in{\mathbb{Z}}.$

Hence,

(4.1)
$\displaystyle\Bigl{[}\Delta^{\alpha}_{W}u\Bigr{]}(v)=\sum_{j=0}^{m}\sum_{l=-\infty}^{v+j}(-1)^{m-j}\binom{m}{j}k^{m-\alpha}(v+j-l)u(l),\quad
v\in{\mathbb{Z}}.$

Due to [1, Remark 2.4], the assumption
$\sum_{v=-\infty}^{\infty}\|u(v)\|\cdot(1+|v|)^{m-\alpha-1}<+\infty$ implies

(4.2)
$\displaystyle\Bigl{[}\Delta^{\alpha}_{W}u\Bigr{]}(v)=\Bigl{[}\Delta^{m}\Bigl{(}\Delta^{-(m-\alpha)}_{W}u\Bigr{)}\Bigr{]}(v)=\Bigl{[}\Delta^{-(m-\alpha)}_{W}\Bigl{(}\Delta^{m}u\Bigr{)}\Bigr{]}(v),\quad
v\in{\mathbb{Z}};$

furthermore, it can be simply proved that, for every real number $a,$ the
sequence $h(\cdot)=u(\cdot+a)$ satisfies
$\sum_{v=-\infty}^{\infty}\|h(v)\|\cdot(1+|v|)^{m-\alpha-1}<+\infty$ as well
as that (cf. (4.1)):

(4.3)
$\displaystyle\Bigl{[}\Delta^{\alpha}_{W}h\Bigr{]}(v)=\Bigl{[}\Delta^{\alpha}_{W}u\Bigr{]}(v+a),\quad
v\in{\mathbb{Z}}.$

In the sequel, we also write $\Delta^{\alpha}_{W}u(v)$ in place of
$[\Delta^{\alpha}_{W}u](v).$

If $A$ is a closed linear operator, then we must distinguish the terms
$A(\Delta^{\alpha}_{W}u)$ and $\Delta^{\alpha}_{W}(Au)$ for $\alpha>0.$ For
example, if $u(v)\equiv x\notin D(A),$ then we have

$\displaystyle\Delta^{\alpha}_{W}u(v)$
$\displaystyle=\sum_{j=0}^{m}\sum_{l=-\infty}^{v+j}(-1)^{m-j}\binom{m}{j}k^{m-\alpha}(v+j-l)x$
(4.4)
$\displaystyle=\sum_{j=0}^{m}(-1)^{m-j}\binom{m}{j}\Biggl{[}\sum_{s=0}^{\infty}k^{m-\alpha}(s)x\Biggr{]}=0,\quad
v\in{\mathbb{Z}},$

but the term $[\Delta^{\alpha}_{W}(Au)](v)$ is not well-defined for any
$v\in{\mathbb{Z}}.$

Define now $T_{i,L}^{W}u(v):=A_{i}\Delta_{W}^{\alpha_{i}}u(v)$, if
$v\in{\mathbb{Z}}$ and $i\in\mathbb{N}_{n}$, and
$T_{i,R}^{W}u(v):=\Delta_{C}^{\alpha_{i}}A_{i}u(v)$, if $v\in{\mathbb{Z}}$ and
$i\in\mathbb{N}_{n}$. We assume that, for every $v\in{\mathbb{N}}_{0}$ and
$i\in\mathbb{N}_{n}$, $T_{i}^{W}u(v)$ denotes either $T_{i,L}^{W}u(v)$ or
$T_{i,R}^{W}u(v)$. Of concern is the following abstract multi-term fractional
difference equation without initial conditions:

(4.5) $\sum_{i=1}^{n}T_{i}^{W}u(v+v_{i})=f(v),\quad v\in{\mathbb{Z}},$

where $v_{1},\dots,v_{n}\in{\mathbb{Z}}$ and $f:{\mathbb{Z}}\rightarrow X$ is
a given sequence. A strong solution of problem (4.5) is any sequence
$u:{\mathbb{Z}}\rightarrow X$ such that the term $T_{i}^{W}u(\cdot+v_{i})$ is
well-defined for any $i\in{\mathbb{N}}_{n}$ as well as that (4.5) identically
holds for any $v\in{\mathbb{Z}}$.

Set now $v_{min}:=\min\\{v_{1},...,v_{n}\\}$ and $y(v):=u(v+v_{min}),$
$v\in{\mathbb{Z}}.$ Then it is clear that the problem (4.5) is equivalent with
the probem (cf. (4.3)):

$\sum_{i=1}^{n}T_{i}^{W}y\bigl{(}v+[v_{i}-v_{min}]\bigr{)}=f(v),\quad
v\in{\mathbb{Z}},$

as well as that $v_{i}-v_{min}\geq 0$ for all $i\in{\mathbb{N}}_{n}.$
Therefore, in the analysis of the existence and uniqueness of solutions of the
problem (4.5), we may assume without loss of generality that $v_{1}\geq 0,\
...,\ v_{n}\geq 0$. This will be our standing assumption henceforth.

Due to (4.1), the problem (4.5) is equivalent with

$\displaystyle\sum_{i\in{\mathcal{I}}}A_{i}\Biggl{[}\sum_{j=0}^{m_{i}}\sum_{l=-\infty}^{v+j}(-1)^{m_{i}-j}\binom{m_{i}}{j}k^{m_{i}-\alpha}(v+j-l)u(l)\Biggr{]}$
$\displaystyle+\sum_{i\notin{\mathcal{I}}}\Biggl{[}\sum_{j=0}^{m_{i}}\sum_{l=-\infty}^{v+j}(-1)^{m_{i}-j}\binom{m_{i}}{j}k^{m_{i}-\alpha}(v+j-l)A_{i}u(l)\Biggr{]}=f(v),\quad
v\in{\mathbb{Z}},$

where ${\mathcal{I}}:=\\{i\in\mathbb{N}_{n}:\alpha_{i}>0\text{ and
}T_{i,L}^{W}u(v)\text{ appears on the left hand side of }\eqref{DFDEabzw}\\}$
We cannot expect the uniqueness of (periodic/almost periodic) strong solutions
of (4.5) in any sense: if $u(\cdot)$ is a strong solution of (4.5), then the
sequence $u_{x}(\cdot):=u(\cdot)+x,$ where $x\in D(A_{i})$ for all
$i\notin{\mathcal{I}}$, is likewise a strong solution of (4.5); cf. (4.4).

We continue our exposition with the observation that [31, Theorem 3.4] can be
helpful in the analysis of the existence of strong solutions to some classes
of the abstract multi-term fractional difference equations. This result is far
from being satisfactory in the general analysis of problem (4.5), when we need
the notion introduced in Definition 3.1.

The following result seems to be extremely important in the study of the
abstract multi-term fractional difference equations with Weyl fractional
derivatives:

######  Theorem 4.1.

Suppose that $v_{1}\geq 0,\ ...,\ v_{n}\geq 0$,
$(S(v))_{v\in{\mathbb{N}}_{0}}\subseteq L(X)$ is a discrete
$(k,C,B,(A_{i})_{1\leq i\leq n},(v_{i})_{1\leq i\leq n})$-existence family,
(3.8) and the following hold:

  * (a)

$f:{\mathbb{Z}}\rightarrow X$ is a bounded sequence, $k\in
l^{1}({\mathbb{N}}_{0}:Y)$ and $\sum_{v=0}^{+\infty}|a_{i}(v)|<+\infty$ for
$1\leq i\leq n,$ or

  * (b)

$f\in l^{1}({\mathbb{Z}}:X),$ $k:{\mathbb{N}}_{0}\rightarrow X$ is a bounded
sequence and $a_{i}:{\mathbb{Z}}\rightarrow{\mathbb{C}}$ is a bounded sequence
for $1\leq i\leq n.$

Define

$u(v):=\sum_{l=-\infty}^{v}S(v-l)f(l),\quad v\in{\mathbb{Z}}$

and

$\displaystyle g(v)$
$\displaystyle:=A_{1}\sum_{l=v+1}^{v+v_{1}}\bigl{(}a_{1}\ast_{0}S\bigr{)}(v+v_{1}-l)f(l)+...$
(4.6)
$\displaystyle+A_{n}\sum_{l=v+1}^{v+v_{n}}\bigl{(}a_{n}\ast_{0}S\bigr{)}(v+v_{n}-l)f(l),\quad
v\in{\mathbb{Z}}.$

Then $u(\cdot)$ is bounded if _(a)_ holds, $u\in l^{1}({\mathbb{Z}}:X)$ if
_(b)_ holds, and we have

$\displaystyle Bu(v)$
$\displaystyle=A_{1}\sum_{l=-\infty}^{v+v_{1}}a_{1}(v+v_{1}-l)u(l)+...$ (4.7)
$\displaystyle+A_{n}\sum_{l=-\infty}^{v+v_{n}}a_{n}(v+v_{n}-l)u(l)+(k\circ
Cf)(v)+g(v),\ v\in{\mathbb{Z}}.$

Especially, the following holds:

  * (i)

Suppose that $f\in l^{1}({\mathbb{Z}}:X)$ and $g(v)$ is given by (4.6), with
$a_{i}(v)=k^{m_{i}-\alpha_{i}}(v)$ for all $v\in{\mathbb{N}}_{0}$ and
$i\in{\mathbb{N}},$ where $m=m_{i}\in{\mathbb{N}}$ for all
$i\in{\mathbb{N}}_{n}.$ Then we have:

$\displaystyle\bigl{(}\Delta^{m}Bu\bigr{)}(v)$
$\displaystyle=A_{1}\Bigl{(}\Delta^{\alpha_{1}}_{W}u\Bigr{)}(v+v_{1})+...$
$\displaystyle+A_{n}\Bigl{(}\Delta^{\alpha_{n}}_{W}u\Bigr{)}(v+v_{n})+\Delta^{m}(k\circ
Cf)(v)+\Delta^{m}g(v),\ v\in{\mathbb{Z}}.$

  * (ii)

Suppose that $f\in l^{1}({\mathbb{Z}}:X)$ and $u=\Delta^{m_{n}}h$ for a
certain sequence $h:{\mathbb{Z}}\rightarrow X.$ Then we have:

$\displaystyle B\bigl{(}\Delta^{m_{n}}h\bigr{)}(v)$
$\displaystyle=\sum_{j=1}^{n-1}A_{j}\Bigl{(}\Delta^{m_{n}-m_{j}}\Delta^{\alpha_{j}}_{W}h\Bigr{)}(v+v_{j})$
$\displaystyle+A_{n}\Bigl{(}\Delta^{\alpha_{n}}_{W}h\Bigr{)}(v+v_{n})+(k\circ
Cf)(v)+g(v),\ v\in{\mathbb{Z}}.$

###### Proof.

It is clear that $u(\cdot)$ is well-defined and bounded if (a) holds as well
as $u\in l^{1}({\mathbb{Z}}:X)$ if (b) holds; furthermore, the sequence
$(k\circ Cf)(\cdot)$ is well-defined. The functional equality (3.2) and the
closedness of the linear operator $B$ together imply with the prescribed
assumptions and the Fubini theorem that:

$\displaystyle Bu(v)-(k\circ Cf)(v)=\sum_{l=-\infty}^{v}BS(v-l)f(l)-(k\circ
Cf)(v)=-(k\circ Cf)(v)$
$\displaystyle+\sum_{l=-\infty}^{v}\Biggl{[}k(v-l)Cf(l)+A_{1}\bigl{(}a_{1}\ast_{0}S\bigr{)}(v+v_{1}-l)f(l)+...+A_{n}\bigl{(}a_{n}\ast_{0}S\bigr{)}(v+v_{n}-l)f(l)\Biggr{]}$
$\displaystyle=\sum_{l=-\infty}^{v}\Biggl{[}A_{1}\bigl{(}a_{1}\ast_{0}S\bigr{)}(v+v_{1}-l)f(l)+...+A_{n}\bigl{(}a_{n}\ast_{0}S\bigr{)}(v+v_{n}-l)f(l)\Biggr{]}$
(4.8)
$\displaystyle=A_{1}\sum_{l=-\infty}^{v}\bigl{(}a_{1}\ast_{0}S\bigr{)}(v+v_{1}-l)f(l)+...+A_{n}\sum_{l=-\infty}^{v}\bigl{(}a_{n}\ast_{0}S\bigr{)}(v+v_{n}-l)f(l)$
$\displaystyle=A_{1}\sum_{l=-\infty}^{v+v_{1}}\bigl{(}a_{1}\ast_{0}S\bigr{)}(v+v_{1}-l)f(l)+...+A_{n}\sum_{l=-\infty}^{v+v_{n}}\bigl{(}a_{n}\ast_{0}S\bigr{)}(v+v_{n}-l)f(l)-g(v)$
$\displaystyle=\sum_{j=1}^{n}A_{j}\sum_{s=0}^{+\infty}\Biggl{[}a_{j}(s)S(0)+...+a_{j}(0)S(s)\Biggr{]}f(v+v_{j}-s)-g(v)$
(4.9)
$\displaystyle=\sum_{j=1}^{n}A_{j}\sum_{s=0}^{+\infty}a_{j}(s)\sum_{r=0}^{+\infty}S(r)f(v+v_{j}-s-r)-g(v)$
$\displaystyle=\sum_{j=1}^{n}A_{j}\sum_{l=-\infty}^{v+v_{j}}a_{j}(v+v_{j}-l)u(l)-g(v),\quad
v\in{\mathbb{Z}},$

so that (4.7) holds true; here, (4.8) follows from the assumption
$\sum_{v=0}^{+\infty}\|A_{i}\bigl{(}a_{i}\ast_{0}S\bigr{)}(v+v_{i})\|<+\infty$
for $1\leq i\leq n$ and the closedness of the linear operators
$A_{1},...,A_{n}$, while (4.9) follows from an application of the discrete
Fubini theorem, the assumption $\sum_{v=0}^{+\infty}\|S(v)\|<+\infty$ and the
assumption (a) or (b). The proofs of (i) and (ii) follow simply from this
general result with condition (b) being satisfied, the definitions of Weyl
fractional derivatives and the equation (4.2). ∎

We continue be providing certain observations:

######  Remark 4.2.

  * (i)

It is clear that $g(\cdot)\equiv 0,$ if $v_{1}=v_{2}=...=v_{n}=0$.

  * (ii)

The assumption $u(v)=\Delta^{m_{n}}h(v),$ $v\in{\mathbb{Z}}$ in (ii) implies
that $h(v)=u_{1}(v)+c_{0}+c_{1}v+...+c_{m_{n}-1}v^{m_{n}-1}$ for all
$v\in{\mathbb{Z}}$, for some sequence $(u_{1}(\cdot))$ depending on
$(u(\cdot))$ and arbitrary complex constants $c_{0},...,c_{m_{n}-1}.$ For more
details about the calculus of finite differences and its applications, we
refer the reader to the monographs [21] and [38].

  * (iii)

Suppose that $c_{i}\in{\mathbb{C}}$, $v_{1}=v_{2}=...=v_{n}=0$ and
$h_{i}\in{\mathbb{Z}}$ for $1\leq i\leq m.$ Define
$y(v):=\sum_{i=1}^{m}c_{i}u(v+h_{i}),$ $v\in{\mathbb{Z}};$ then (4.7) implies
after the substitution $s=v-l\geq 0$ that

$\displaystyle By(v)$
$\displaystyle=A_{1}\sum_{l=-\infty}^{v}a_{1}(v-l)y(l)+...+A_{n}\sum_{l=-\infty}^{v}a_{n}(v-l)y(l)$
$\displaystyle+\sum_{i=i}^{m}c_{i}\bigl{(}k\circ
Cf\bigr{)}\bigl{(}v+h_{i}\bigr{)},\ v\in{\mathbb{Z}}.$

We can similarly prove the following general result (the strong solutions of
problems under our consideration are obtained by plugging
${\mathcal{I}}=\emptyset$):

######  Theorem 4.3.

Suppose that $v_{1}\geq 0,\ ...,\ v_{n}\geq 0$,
${\mathcal{I}}\subseteq{\mathbb{N}}_{n},$
$(S(v))_{v\in{\mathbb{N}}_{0}}\subseteq L(X)$ is a discrete
$(k,C,B,(A_{i})_{1\leq i\leq n},(v_{i})_{1\leq i\leq
n},{\mathcal{I}})$-existence family, $\sum_{v=0}^{+\infty}\|S(v)\|<+\infty$,
$\sum_{v=0}^{+\infty}\|A_{i}\bigl{(}a_{i}\ast_{0}S\bigr{)}(v+v_{i})\|<+\infty$
for $i\in{\mathcal{I}}$ and the following holds:

  * (a)

$f:{\mathbb{Z}}\rightarrow X$ is a bounded sequence, $k\in
l^{1}({\mathbb{N}}_{0}:X)$ and $\sum_{v=0}^{+\infty}|a_{i}(v)|<+\infty$ for
$i\in{\mathcal{I}}$ or

  * (b)

$f\in l^{1}({\mathbb{Z}}:X),$ $k:{\mathbb{N}}_{0}\rightarrow X$ is a bounded
sequence and $a_{i}:{\mathbb{Z}}\rightarrow{\mathbb{C}}\setminus\\{0\\}$ is a
bounded sequence for $i\in{\mathcal{I}}$

as well as

  * (c)

$A_{i}f:{\mathbb{Z}}\rightarrow X$ is a bounded sequence,
$\sum_{v=0}^{+\infty}|a_{i}(v)|<+\infty$ for
$i\in{\mathbb{N}}_{n}\setminus{\mathcal{I}}$ and
$(S(v))_{v\in{\mathbb{N}}_{0}}\subseteq L(X)$ is a discrete
$(k,C,B,(A_{i})_{1\leq i\leq n},(v_{i})_{1\leq i\leq
n},{\mathcal{I}})$-existence family, or

  * (d)

$f\in l^{1}({\mathbb{Z}}:X)$, $\sum_{v=0}^{+\infty}\|A_{i}S(v)\|<+\infty$ for
all $i\in{\mathbb{N}}_{n}\setminus{\mathcal{I}}$ and
$a_{i}:{\mathbb{N}}_{0}\rightarrow{\mathbb{C}}\setminus\\{0\\}$ is a bounded
sequence for $i\in{\mathbb{N}}_{n}\setminus{\mathcal{I}},$ or

  * (e)

$f\in l^{1}({\mathbb{Z}}:[D(A_{i})])$ for all
$i\in{\mathbb{N}}_{n}\setminus{\mathcal{I}},$
$a_{i}:{\mathbb{N}}_{0}\rightarrow{\mathbb{C}}\setminus\\{0\\}$ is a bounded
sequence for $i\in{\mathbb{N}}_{n}\setminus{\mathcal{I}}$ and
$(S(v))_{v\in{\mathbb{N}}_{0}}\subseteq L(X)$ is a discrete
$(k,C,B,(A_{i})_{1\leq i\leq n},(v_{i})_{1\leq i\leq
n},{\mathcal{I}})$-existence family.

Define $u(\cdot)$ and $g(\cdot)$ in the same way as in the proof of _Theorem
4.1_. Then we have:

$\displaystyle Bu(v)$
$\displaystyle=\sum_{i\in{\mathcal{I}}}A_{i}\sum_{l=-\infty}^{v+v_{i}}a_{i}(v+v_{i}-l)u(l)$
$\displaystyle+\sum_{i\in{\mathbb{N}}_{n}\setminus{\mathcal{I}}}\sum_{l=-\infty}^{v+v_{i}}a_{i}(v+v_{i}-l)A_{i}u(l)+(k\circ
Cf)(v)+g(v),\ v\in{\mathbb{Z}}.$

Especially, the following holds for a function $f\in l^{1}({\mathbb{Z}}:X):$

  * (i)

Suppose that $a_{i}(v)=k^{m_{i}-\alpha_{i}}(v)$ for all $v\in{\mathbb{N}}_{0}$
and $i\in{\mathbb{N}},$ where $m=m_{i}\in{\mathbb{N}}$ for all
$i\in{\mathbb{N}}_{n}.$ If $\sum_{v=0}^{+\infty}\|A_{i}S(v)\|<+\infty$ for all
$i\in{\mathbb{N}}_{n}\setminus{\mathcal{I}}$ or $f\in
l^{1}({\mathbb{Z}}:[D(A_{i})])$ for all
$i\in{\mathbb{N}}_{n}\setminus{\mathcal{I}}$ and
$(S(v))_{v\in{\mathbb{N}}_{0}}\subseteq L(X)$ is a discrete
$(k,C,B,(A_{i})_{1\leq i\leq n},(v_{i})_{1\leq i\leq
n},{\mathcal{I}})$-existence family, then we have:

$\displaystyle\bigl{(}\Delta^{m}Bu\bigr{)}(v)$
$\displaystyle=\sum_{i\in{\mathcal{I}}}A_{i}\Bigl{(}\Delta^{\alpha_{i}}_{W}u\Bigr{)}(v+v_{i})$
$\displaystyle+\sum_{i\in{\mathbb{N}}_{n}\setminus{\mathcal{I}}}\Bigl{(}\Delta^{\alpha_{i}}_{W}A_{i}u\Bigr{)}(v+v_{i})+\Delta^{m}(k\circ
Cf)(v)+\Delta^{m}g(v),\quad v\in{\mathbb{Z}}.$

  * (ii)

Suppose that $\sum_{v=0}^{+\infty}\|A_{i}S(v)\|<+\infty$ for all
$i\in{\mathbb{N}}_{n}\setminus{\mathcal{I}}$ or $f\in
l^{1}({\mathbb{Z}}:[D(A_{i})])$ for all
$i\in{\mathbb{N}}_{n}\setminus{\mathcal{I}}$ and
$(S(v))_{v\in{\mathbb{N}}_{0}}\subseteq L(X)$ is a discrete
$(k,C,B,(A_{i})_{1\leq i\leq n},(v_{i})_{1\leq i\leq
n},{\mathcal{I}})$-existence family as well as $u=\Delta^{m_{n}}h$ for a
certain sequence
$h:{\mathbb{Z}}\rightarrow\bigcap_{i\in{\mathbb{N}}_{n}\setminus{\mathcal{I}}}D(A_{i}).$
Then we have:

$\displaystyle B\bigl{(}\Delta^{m_{n}}h\bigr{)}(v)$
$\displaystyle=\sum_{i\in{\mathcal{I}}}A_{i}\Bigl{(}\Delta^{m_{n}-m_{i}}\Delta^{\alpha_{i}}_{W}h\Bigr{)}(v+v_{i})$
$\displaystyle+\sum_{i\in{\mathbb{N}}_{n}\setminus{\mathcal{I}}}\Bigl{(}\Delta^{m_{n}-m_{i}}\Delta^{\alpha_{j}}_{W}A_{i}h\Bigr{)}(v+v_{i})+(k\circ
Cf)(v)+g(v),\ v\in{\mathbb{Z}}.$

It is clear that Theorem 4.1 and Theorem 4.3 can be illustrated with many
concrete examples in which the corresponding discrete $(k,0,0,(A_{i})_{1\leq
i\leq n},(v_{i})_{1\leq i\leq n},{\mathcal{I}})$-existence families are
integrable on account of Theorem 3.3(iii) or Theorem 3.5(iii). Now we will
provide one more important application of Theorem 4.3:

######  Example 4.4.

Suppose that $(T(t))_{t\geq 0}\subseteq L(X)$ is a strongly continuous
operator family such that $\int^{+\infty}_{0}\|T(t)\|\,dt<+\infty$ and, for
every $x\in X,$ the mapping $t\mapsto T(t)x,$ $t\geq 0$ is a mild solution of
the abstract Cauchy problem

$A_{n}D_{t}^{\alpha_{n}}T(t)x+A_{n-1}D_{t}^{\alpha_{n-1}}T(t)x+...+A_{1}D_{t}^{\alpha_{1}}T(t)x=0,\quad
t>0,$

accompanied with certain initial conditions (cf. [27, pp. 161–163] for more
details in this direction), where $0\leq\alpha_{1}<\alpha_{2}<...<\alpha_{n}$
and the operators $A_{j}$ are polynomials of a certain closed linear operator
$A$; the interested reader may consult [25]-[27] for more details and many
interesting examples of differential operators which can be used to provide
certain applications of the conclusions established here (we can also consider
the exponentially bounded operator families $(T(t))_{t\geq 0}$ using the
Poisson like transform from Theorem 2.5 since the statement of [2, Theorem
5.5] admits an extension in this framework; details can be left to the
interested readers). Set

$S(v)x:=\int^{+\infty}_{0}e^{-t}(t^{v}/v!)T(t)x\,dt,\quad
v\in{\mathbb{N}}_{0},\ x\in X.$

Then we have $\sum_{v=0}^{+\infty}\|S(v)\|<+\infty$ and

$\displaystyle A_{n}\bigl{(}\Delta^{\alpha_{n}}S(\cdot)x\bigr{)}(v)$
$\displaystyle+\sum_{s=1}^{n-1}A_{s}\bigl{(}\Delta^{\alpha_{s}}S(\cdot)x\bigr{)}(v+m_{n}-m_{s})=0,\
v\in{\mathbb{N}}_{0},\ x\in X;$

cf. [32]. In other words,

$\displaystyle
A_{n}\Biggl{[}\sum_{j=0}^{m_{n}}\sum_{l=0}^{v+j}(-1)^{m_{n}-j}\binom{m_{n}}{j}k^{m_{n}-\alpha_{n}}(v+j-l)S(l)x\Biggr{]}$
$\displaystyle+\sum_{s=1}^{n-1}A_{s}\Biggl{[}\sum_{j=0}^{m_{s}}\sum_{l=0}^{v+m_{n}-m_{s}+j}(-1)^{m_{s}-j}\binom{m_{s}}{j}k^{m_{s}-\alpha_{s}}(v+m_{n}-m_{s}+j-l)S(l)x\Biggr{]}=0,$

for any $v\in{\mathbb{N}}_{0}$ and $x\in X.$ Now it can be easily shown that
Theorem 4.3 is applicable with ${\mathcal{I}}=\emptyset,$ $B=C=0$ and
arbitrary non-trivial sequence $f\in l^{1}({\mathbb{Z}}:[D(A_{i})])$ for all
$i\in{\mathbb{N}}_{n};$ in our concrete situation, we have that
$(S(v))_{v\in{\mathbb{N}}_{0}}\subseteq L(X)$ is a discrete
$(k,0,0,(A_{i})_{1\leq i\leq n},(v_{i})_{1\leq i\leq
n},{\mathcal{I}})$-existence family with $v_{i}=m_{i}$ for $1\leq i\leq n$ and

$a_{s}(v):=\sum_{j=0}^{m_{s}}(-1)^{m_{s}-j}\binom{m_{s}}{j}k^{m_{s}-\alpha_{s}}\bigl{(}v+m_{n}+j-2m_{s}\bigr{)},\quad
v\in{\mathbb{N}}_{0}\ \ (1\leq s\leq n),$

where we have put $k^{m_{s}-\alpha_{s}}(v):=0$ if $v>0$ ($1\leq s\leq n$).
Next, let us observe that we have $a_{s}(0)=1,$ $i\in M$ and
$\sum_{v=0}^{+\infty}|a_{s}(v)|<+\infty$ for $s\in{\mathbb{N}}_{n}$ since
$a_{s}(v)\sim c_{s}v^{-2+m_{s}-\alpha_{s}}$ as $v\rightarrow+\infty;$ this
follows form a relatively simple analysis involving the identity
$\sum_{j=0}^{m_{s}}(-1)^{m_{s}-j}\binom{m_{s}}{j}=0$, the Lagrange mean value
theorem and the asymptotic formula $k^{\alpha}(v)\sim g_{\alpha}(v)$ as
$v\rightarrow+\infty$ ($\alpha>0$). Consequently, we have that there do not
exist an integer $m_{s}\in{\mathbb{N}},$ the complex constants
$\beta_{1;s},...,\beta_{m;s}$ and the translation numbers
$v_{1;s}^{\prime}\in{\mathbb{N}}_{0},...,v_{m;s}^{\prime}\in{\mathbb{N}}_{0}$
such that
$k^{m_{s}-\alpha_{s}}(v)=\beta_{1;s}a_{s}(v+v_{1;s}^{\prime})+...+\beta_{m;s}a_{s}(v+v_{m;s}^{\prime}).$

In our concrete situation, we are solving the equation

(4.10)
$\displaystyle\sum_{l=-\infty}^{v+v_{n}}a_{n}(v+v_{n}-l)A_{n}u(l)+...+\sum_{l=-\infty}^{v+v_{1}}a_{1}(v+v_{1}-l)A_{1}u(l)=-g(v),\
v\in{\mathbb{Z}},$

where $g(\cdot)$ is given by (4.6). Keeping in mind the representation of
$a_{s}(\cdot)$ for $1\leq s\leq n,$ we get

$\displaystyle\sum_{l=-\infty}^{v+m_{n}}\sum_{j=0}^{m_{n}}(-1)^{m_{n}-j}\binom{m_{n}}{j}k^{m_{n}-\alpha_{n}}\bigl{(}v+j-l\bigr{)}A_{n}u(l)$
$\displaystyle+\sum_{s=1}^{n-1}\sum_{l=-\infty}^{v+m_{s}}\sum_{j=0}^{m_{s}}(-1)^{m_{s}-j}\binom{m_{s}}{j}k^{m_{s}-\alpha_{s}}\bigl{(}v+m_{n}-m_{s}+j-l\bigr{)}A_{s}u(l)=-g(v),$

for any $v\in{\mathbb{Z}},$ i.e.,

$\displaystyle\sum_{j=0}^{m_{n}}(-1)^{m_{n}-j}\binom{m_{n}}{j}\sum_{l=-\infty}^{v+m_{n}}k^{m_{n}-\alpha_{n}}\bigl{(}v+j-l\bigr{)}A_{n}u(l)$
$\displaystyle+\sum_{s=1}^{n-1}\sum_{j=0}^{m_{s}}(-1)^{m_{s}-j}\binom{m_{s}}{j}\sum_{l=-\infty}^{v+m_{s}}k^{m_{s}-\alpha_{s}}\bigl{(}v+m_{n}-m_{s}+j-l\bigr{)}A_{s}u(l)=-g(v),$

for any $v\in{\mathbb{Z}}.$ Taking into account the formula (4.1), we finally
get

$\displaystyle\Bigl{[}\Delta^{\alpha_{n}}_{W}A_{n}u\Bigr{]}(v)$
$\displaystyle+\Bigl{[}\Delta^{\alpha_{n-1}}_{W}A_{n-1}u\Bigr{]}\bigl{(}v+m_{n}-m_{n-1}\bigr{)}$
$\displaystyle+...+\Bigl{[}\Delta^{\alpha_{1}}_{W}A_{1}u\Bigr{]}\bigl{(}v+m_{n}-m_{1}\bigr{)}=-g(v),\quad
v\in{\mathbb{Z}}.$

The term $g(\cdot)$ can be directly computed for small values of $\alpha_{j}$
($1\leq j\leq n$); for example, if $\alpha_{n}\leq 1,$ then we have

$g(v)=\sum_{i\in M}A_{i}\int_{0}^{+\infty}e^{-t}T(t)f(v+1)\,dt,\quad
v\in{\mathbb{Z}}.$

The kernels $a(v)=k^{m-\alpha}(v),$ $v\in{\mathbb{N}}_{0}$ are not integrable
provided that $\alpha\in(0,\infty)\setminus{\mathbb{N}}$ and
$m=\lceil\alpha\rceil.$ Therefore, we cannot use Theorem 3.3(iii) or Theorem
3.5(iii) to deduce the integrability of the corresponding discrete
$(k,B,C,(A_{i})_{1\leq i\leq n},(v_{i})_{1\leq i\leq
n},{\mathcal{I}})$-existence families. We can overcome this difficulty by
applying the following trick:

######  Theorem 4.5.

Suppose that $v_{1}\geq 0,\ ...,\ v_{n}\geq 0$,
${\mathcal{I}}\subseteq{\mathbb{N}}_{n},$
$(S(v))_{v\in{\mathbb{N}}_{0}}\subseteq L(X)$ is a discrete
$(k,C,B,(A_{i})_{1\leq i\leq n},(v_{i})_{1\leq i\leq
n},{\mathcal{I}})$-existence family, $\omega>0,$
$\sum_{v=0}^{+\infty}\|e^{-\omega v}S(v)\|<+\infty$,
$\sum_{v=0}^{+\infty}\|A_{i}\bigl{(}a_{i}\ast_{0}e^{-\omega\cdot}S(\cdot)\bigr{)}(v+v_{i})\|<+\infty$
for $i\in{\mathcal{I}}$ and the following holds:

  * (a)

$e^{-\omega\cdot}f:{\mathbb{Z}}\rightarrow X$ is a bounded sequence,
$e^{-\omega\cdot}k\in l^{1}({\mathbb{N}}_{0}:X)$ and
$\sum_{v=0}^{+\infty}|a_{i}(v)e^{-\omega(\cdot-v_{i})}|<+\infty$ for
$i\in{\mathcal{I}}$ or

  * (b)

$e^{-\omega\cdot}f\in l^{1}({\mathbb{Z}}:X),$
$e^{-\omega\cdot}k:{\mathbb{N}}_{0}\rightarrow X$ is a bounded sequence and
$e^{-\omega(\cdot-
v_{i})}a_{i}:{\mathbb{Z}}\rightarrow{\mathbb{C}}\setminus\\{0\\}$ is a bounded
sequence for $i\in{\mathcal{I}}$

as well as

  * (c)

$e^{-\omega\cdot}A_{i}f:{\mathbb{Z}}\rightarrow X$ is a bounded sequence,
$\sum_{v=0}^{+\infty}|a_{i}(v)e^{-\omega(\cdot-v_{i})}|<+\infty$ for
$i\in{\mathbb{N}}_{n}\setminus{\mathcal{I}}$ and and
$(S(v))_{v\in{\mathbb{N}}_{0}}\subseteq L(X)$ is a discrete
$(k,C,B,(A_{i})_{1\leq i\leq n},(v_{i})_{1\leq i\leq
n},{\mathcal{I}})$-existence family, or

  * (d)

$e^{-\omega\cdot}f\in l^{1}({\mathbb{Z}}:X)$,
$\sum_{v=0}^{+\infty}\|A_{i}e^{-\omega v}S(v)\|<+\infty$ for all
$i\in{\mathbb{N}}_{n}\setminus{\mathcal{I}}$ and $e^{-\omega(\cdot-
v_{i})}a_{i}:{\mathbb{N}}_{0}\rightarrow{\mathbb{C}}\setminus\\{0\\}$ is a
bounded sequence for $i\in{\mathbb{N}}_{n}\setminus{\mathcal{I}},$ or

  * (e)

$e^{-\omega\cdot}f\in l^{1}({\mathbb{Z}}:[D(A_{i})])$ for all
$i\in{\mathbb{N}}_{n}\setminus{\mathcal{I}},$ $e^{-\omega(\cdot-
v_{i})}a_{i}:{\mathbb{N}}_{0}\rightarrow{\mathbb{C}}\setminus\\{0\\}$ is a
bounded sequence for $i\in{\mathbb{N}}_{n}\setminus{\mathcal{I}}$ and
$(S(v))_{v\in{\mathbb{N}}_{0}}\subseteq L(X)$ is a discrete
$(k,C,B,(A_{i})_{1\leq i\leq n},(v_{i})_{1\leq i\leq
n},{\mathcal{I}})$-existence family.

Define

$u(v):=e^{\omega
v}\sum_{l=-\infty}^{v}\Bigl{[}e^{-\omega(v-l)}S(v-l)\Bigr{]}\cdot\Bigl{[}e^{-\omega
l}f(l)\Bigr{]},\quad v\in{\mathbb{Z}}$

and $g_{\omega}(\cdot)$ in the same way as in the proof of _Theorem 4.1_ with
the terms $g(\cdot),$ $a_{i}(\cdot),$ $S(\cdot)$ and $f(\cdot)$ replaced
therein with the terms $g_{\omega}(\cdot)$, $e^{-\omega(\cdot-
v_{i})}a_{i}(\cdot),$ $e^{-\omega\cdot}S(\cdot)$ and
$e^{-\omega\cdot}f(\cdot)$, respectively. Then we have:

$\displaystyle Bu(v)$
$\displaystyle=\sum_{i\in{\mathcal{I}}}A_{i}\sum_{l=-\infty}^{v+v_{i}}a_{i}(v+v_{i}-l)u(l)$
(4.11)
$\displaystyle+\sum_{i\in{\mathbb{N}}_{n}\setminus{\mathcal{I}}}\sum_{l=-\infty}^{v+v_{i}}a_{i}(v+v_{i}-l)A_{i}u(l)+e^{-\omega
v}(k\circ Cf)(v)+g_{\omega}(v),\ v\in{\mathbb{Z}}.$

Especially, the following holds for a function $e^{-\omega\cdot}f\in
l^{1}({\mathbb{Z}}:X):$

  * (i)

Suppose that $a_{i}(v)=k^{m_{i}-\alpha_{i}}(v)$ for all $v\in{\mathbb{N}}_{0}$
and $i\in{\mathbb{N}},$ where $m=m_{i}\in{\mathbb{N}}$ for all
$i\in{\mathbb{N}}_{n}.$ If $\sum_{v=0}^{+\infty}\|A_{i}e^{-\omega
v}S(v)\|<+\infty$ for all $i\in{\mathbb{N}}_{n}\setminus{\mathcal{I}}$ or
$e^{-\omega\cdot}f\in l^{1}({\mathbb{Z}}:[D(A_{i})])$ for all
$i\in{\mathbb{N}}_{n}\setminus{\mathcal{I}}$ and
$(S(v))_{v\in{\mathbb{N}}_{0}}\subseteq L(X)$ is a discrete
$(k,C,B,(A_{i})_{1\leq i\leq n},(v_{i})_{1\leq i\leq
n},{\mathbb{N}}_{n}\setminus{\mathcal{I}})$-existence family, then we have:

$\displaystyle\bigl{(}\Delta^{m}Bu\bigr{)}(v)$
$\displaystyle=\sum_{i\in{\mathcal{I}}}A_{i}\Bigl{(}\Delta^{\alpha_{i}}_{W}u\Bigr{)}(v+v_{i})$
$\displaystyle+\sum_{i\in{\mathbb{N}}_{n}\setminus{\mathcal{I}}}\Bigl{(}\Delta^{\alpha_{i}}_{W}A_{i}u\Bigr{)}(v+v_{i})+\Delta^{m}e^{-\omega
v}(k\circ Cf)(v)+\Delta^{m}g_{\omega}(v),\quad v\in{\mathbb{Z}}.$

  * (ii)

Suppose that $\sum_{v=0}^{+\infty}\|A_{i}e^{-\omega v}S(v)\|<+\infty$ for all
$i\in{\mathbb{N}}_{n}\setminus{\mathcal{I}}$ or $e^{-\omega\cdot}f\in
l^{1}({\mathbb{Z}}:[D(A_{i})])$ for all
$i\in{\mathbb{N}}_{n}\setminus{\mathcal{I}}$ and
$(S(v))_{v\in{\mathbb{N}}_{0}}\subseteq L(X)$ is a discrete
$(k,C,B,(A_{i})_{1\leq i\leq n},(v_{i})_{1\leq i\leq
n},{\mathbb{N}}_{n}\setminus{\mathcal{I}})$-existence family as well as
$u=\Delta^{m_{n}}h$ for a certain sequence
$h:{\mathbb{Z}}\rightarrow\bigcap_{i\in{\mathbb{N}}_{n}\setminus{\mathcal{I}}}D(A_{i}).$
Then we have:

$\displaystyle B\bigl{(}\Delta^{m_{n}}h\bigr{)}(v)$
$\displaystyle=\sum_{i\in{\mathcal{I}}}A_{i}\Bigl{(}\Delta^{m_{n}-m_{i}}\Delta^{\alpha_{i}}_{W}h\Bigr{)}(v+v_{i})$
$\displaystyle+\sum_{i\in{\mathbb{N}}_{n}\setminus{\mathcal{I}}}\Bigl{(}\Delta^{m_{n}-m_{i}}\Delta^{\alpha_{j}}_{W}A_{i}h\Bigr{)}(v+v_{i})+e^{-\omega
v}(k\circ Cf)(v)+g_{\omega}(v),\ v\in{\mathbb{Z}}.$

###### Proof.

The proof simply follows by applying Theorem 4.3, after observing that
$(e^{-\omega v}S(v))_{v\in{\mathbb{N}}_{0}}$ is a discrete
$(e^{-\omega\cdot}k(\cdot),C,B,(A_{i})_{1\leq i\leq n},(v_{i})_{1\leq i\leq
n},{\mathcal{I}})$-existence family with the kernels $a_{i}(\cdot)$ replaced
therein with the kernels $a_{i}(\cdot)e^{-\omega(\cdot-v_{i})}$ for $1\leq
i\leq n.$ ∎

As in many research articles published by now, this enables us to consider the
almost periodic features of the function $e^{-\omega\cdot}u(\cdot),$ where
$u(\cdot)$ solves (4.11); cf. [28] for more details on the subject. The
exponential boundedness of discrete $(k,B,C,(A_{i})_{1\leq i\leq
n},(v_{i})_{1\leq i\leq n},{\mathcal{I}})$-existence family can be proved in
many concrete situations; for example, if $v_{1}=...=v_{n}=0$, $k(\cdot)$ is
exponentially bounded, the kernels $a_{i}(\cdot)$ are bounded for $1\leq i\leq
n,$ $C={\rm I}$ and there exists a closed linear operator $A$ and the complex
polynomials $P_{B}(\cdot),$ $P_{i}(\cdot)$ ($1\leq i\leq n$) such that
$B=P_{B}(A)$ and $A_{i}=P_{i}(A)$ for $1\leq i\leq n$.

Finally, we will present the following illustrative applications of Theorem
4.5:

######  Example 4.6.

  * (i)

Suppose, for simplicity, that $\alpha>0,$ $m=\lceil\alpha\rceil$ and a closed
linear operator $-A$ is a subgenerator of
$(g_{\alpha},g_{\alpha})$-regularized $C$-resolvent family $(T(t))_{t\geq 0}$
such that there exist two real constants $M\geq 1$ and $c\in(0,1)$ such that
$\|T(t)\|\leq Me^{(1-c)t},$ $t\geq 0;$ see [27] for the notion and more
details. Then it is not difficult to show that, for every $x\in X,$ the
function $t\mapsto T(t)x,$ $t\geq 0$ is a mild solution of the abstract Cauchy
problem (cf. [22] for the case $1<\alpha\leq 2$; if $0<\alpha\leq 1,$ then we
recover some known results about the problem (1.1) with ${\mathcal{A}}=-A$):

$D_{t}^{\alpha}u(t)+Au(t)=0,\quad t>0.$

We define the sequence $(S(v))_{v\in{\mathbb{N}}_{0}}$ as in Example 4.4. Then
we have $\|S(v)\|\leq c^{-1}(c^{-1})^{v},$ $v\in{\mathbb{N}}_{0}$ so that
$\sum_{v=0}^{+\infty}\|e^{-\omega v}S(v)\|<+\infty$ for $\omega>\ln(c^{-1}).$
Let us assume that the sequence $f:{\mathbb{Z}}\rightarrow X$ satisfies the
assumptions (a) and (c) from the formulation of Theorem 4.5. If we define
$u(\cdot)$ as in Example 4.4, then the equation (4.10) holds with the sequence
$g(\cdot)$ replaced with the sequence $g_{\omega}(\cdot),$ obtained by
replacing the forcing term $f(\cdot)$ in (4.6) by $e^{-\omega\cdot}f(\cdot).$
Arguing in the same way as in Example 4.4, we get that

$\Bigl{[}\Delta^{\alpha}_{W}u\Bigr{]}(v)+Au(v+m)=-g_{\omega}(v),\quad
v\in{\mathbb{Z}}.$

  * (ii)

In this part, we will transfer the conclusions established in (i) for the
equations with the Caputo fractional derivatives. Suppose that $\alpha>0,$
$m=\lceil\alpha\rceil$ and a closed linear operator $-A$ is a subgenerator of
a global $(g_{\alpha},C)$-regularized resolvent family $(T(t))_{t\geq 0}$ such
that there exist two real constants $M\geq 1$ and $c\in(0,1)$ with
$\|T(t)\|\leq Me^{(1-c)t},$ $t\geq 0;$ see [9], [25] and [27] for the notion
and the corresponding examples. Then we know that, for every $x\in X,$ the
function $t\mapsto T(t)x,$ $t\geq 0$ is a mild solution of the abstract Cauchy
problem

$\mathbf{D}_{t}^{\alpha}u(t)+Au(t)=0,\quad t>0;\ u(0)=Cx,\ u^{(j)}(0)=0,\
1\leq j\leq m-1.$

Using the formula established in [32]:

$\displaystyle\int^{+\infty}_{0}$ $\displaystyle
e^{-t}\frac{t^{v+m_{n}}}{(v+m_{n})!}{\mathbf{D}}_{t}^{\alpha}u(t)\,dt=\bigl{(}\Delta^{\alpha}y\bigr{)}(v)$
$\displaystyle+\frac{(-1)^{v+m+1}}{(v+m)!}\sum_{k=0}^{m-1}u^{(k)}(0)(\alpha-1-k)\cdot...\cdot(\alpha-
k-v-m),\ v\in{\mathbb{N}}_{0},$

we can similarly prove that

$0=\bigl{(}\Delta^{\alpha}S\bigr{)}(v)x+AS(v+m)x+k(v)Cx,\quad
v\in{\mathbb{N}}_{0},\ x\in X,$

where

$k(v)=\frac{(-1)^{v+m+1}}{(v+m)!}(\alpha-1)\cdot...\cdot(\alpha-v-m),\quad
v\in{\mathbb{N}}_{0}.$

Define the sequence $(S(v))_{v\in{\mathbb{N}}_{0}}$ as in (i). Then
$(S(v))_{v\in{\mathbb{N}}_{0}}$ is a discrete $(k,0,C,({\rm
I},A),(0,m),\emptyset)$-existence family, with

$a_{2}(v)=\sum_{j=0}^{m}(-1)^{m-j}\binom{m}{j}k^{m-\alpha}\bigl{(}v+j-m\bigr{)}\
\ \mbox{ and }\ \ a_{1}(v)=k^{0}(v),\quad v\in{\mathbb{N}}_{0}.$

Arguing as in Example 4.4, with the same notion of the term
$g_{\omega}(\cdot),$ we obtain that

$\Bigl{[}\Delta^{\alpha}_{W}u\Bigr{]}(v)+Au(v+m)=-g_{\omega}(v)-e^{\omega
v}(k\circ Cf)(v),\quad v\in{\mathbb{Z}},$

provided that the sequence $f:{\mathbb{Z}}\rightarrow X$ satisfies the
assumptions (a) and (c) from the formulation of Theorem 4.5.

  * (iii)

Suppose, finally, that $0\leq\alpha_{1}<\alpha_{2}<....<\alpha_{n},$
$k\in{\mathbb{N}}_{m_{n}-1}^{0}$ and the resolvent operator family
$(T(t))_{t\geq 0}$ satisfies that there exist two real constants $M\geq 1$ and
$c\in(0,1)$ with $\|T(t)\|\leq Me^{(1-c)t},$ $t\geq 0$ as well as that, for
every $x\in X,$ the function $t\mapsto T(t)x,$ $t\geq 0$ is a mild solution of
the abstract Cauchy problem

$A_{n}\mathbf{D}_{t}^{\alpha_{n}}u(t)+...+A_{1}\mathbf{D}_{t}^{\alpha_{1}}u(t)=0,\
t>0;\ u^{(j)}(0)=\delta_{jk}Cx,\ 0\leq j\leq m_{n}-1;$

see [25] and [27] for the notion and the corresponding examples. Set
$A_{k}:=\\{j\in{\mathbb{N}}_{n-1}^{0}:m_{j}-1\geq k\\}$ and

$\displaystyle k(v):=\sum_{j\in
A_{k}}\frac{(-1)^{v+m_{j}+1}}{(v+m_{j})!}(\alpha-1-k)\cdot....\cdot(\alpha-1-k-m_{j}).$

We define the sequence $(S(v))_{v\in{\mathbb{N}}_{0}}$ as in Example 4.4. Then
we have

$\displaystyle A_{n}\bigl{(}\Delta^{\alpha_{n}}S(\cdot)x\bigr{)}(v)$
$\displaystyle+\sum_{s=1}^{n-1}A_{s}\bigl{(}\Delta^{\alpha_{s}}S(\cdot)x\bigr{)}(v+m_{n}-m_{s})=-k(v)Cx,\
v\in{\mathbb{N}}_{0},\ x\in X,$

so that $(S(v))_{v\geq 0}$ is a discrete
$(k,0,C,(A_{i})_{1\leq i\leq
n},(m_{n}-m_{1},...,m_{n}-m_{n-1},0),\emptyset)$-existence family. If the
sequence $f:{\mathbb{Z}}\rightarrow X$ satisfies the assumptions (a) and (c)
from the formulation of Theorem 4.5, then the foregoing argumentation shows
that

$\displaystyle\Bigl{[}\Delta^{\alpha_{n}}_{W}A_{n}u\Bigr{]}(v)$
$\displaystyle+\Bigl{[}\Delta^{\alpha_{n-1}}_{W}A_{n-1}u\Bigr{]}\bigl{(}v+m_{n}-m_{n-1}\bigr{)}$
$\displaystyle+...+\Bigl{[}\Delta^{\alpha_{1}}_{W}A_{1}u\Bigr{]}\bigl{(}v+m_{n}-m_{1}\bigr{)}=-g_{\omega}(v)-e^{\omega
v}(k\circ Cf)(v),\ v\in{\mathbb{Z}},$

where we define $g_{\omega}(\cdot)$ as in (ii).

##  5\. Conclusions and final remarks

In this paper, we have examined the abstract non-scalar Volterra difference
equations. We have employed the Poisson like transforms and, as a special case
of our analysis, we have investigated the existence, uniqueness and almost
periodicity to the abstract multi-term fractional difference equations with
Weyl fractional derivatives.

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