# Maximal regularity for fractional difference equations with finite delay on
UMD spaces

Jichao Zhang, Shangquan Bu♯ Jichao Zhang: School of Science, Hubei University
of Technology, Wuhan 430068, China<EMAIL_ADDRESS>(Jichao Zhang)  Shangquan
Bu: Department of Mathematical Sciences, University of Tsinghua, Beijing
100084, China<EMAIL_ADDRESS>(Shangquan Bu)

###### Abstract.

In this paper, we study the $\ell^{p}$-maximal regularity for the fractional
difference equation with finite delay:

$\begin{cases}\Delta^{\alpha}u(n)=Au(n)+\gamma u(n-\lambda)+f(n),\
n\in\mathbb{N}_{0},\lambda\in\mathbb{N},\gamma\in\mathbb{R};\\\ u(i)=0,\ \
i=-\lambda,-\lambda+1,\cdots,1,2,\end{cases}$

where $A$ is a bounded linear operator defined on a Banach space $X$,
$f:\mathbb{N}_{0}\rightarrow X$ is an $X$-valued sequence and $2<\alpha<3$. We
introduce an operator theoretical method based on the notion of
$\alpha$-resolvent sequence of bounded linear operators, which gives an
explicit representation of solution. Further, using Blunck’s operator-valued
Fourier multipliers theorems on $\ell^{p}(\mathbb{Z};X)$, we completely
characterize the $\ell^{p}$-maximal regularity of solution when $1<p<\infty$
and $X$ is a UMD space.

###### Key words and phrases:

Fractional difference equations; Maximal regularity; UMD spaces;
R-boundedness; Finite delay

###### 2010 Mathematics Subject Classification:

Primary 34A08; Secondary 35R11, 39A06, 39A14, 43A22.

♯ The corresponding author.

1 This work was supported by the NSF of China (Grant No. 12171266)

##  1\. Introduction

The purpose of this paper is to study the existence and uniqueness of solution
and the $\ell^{p}$-maximal regularity for the fractional difference equation
with finite delay:

(1.1) $\Delta^{\alpha}u(n)=Au(n)+\gamma u(n-\lambda)+f(n),\
n\in\mathbb{N}_{0},\lambda\in\mathbb{N},\gamma\in\mathbb{R}$

with the initial conditions $u(i)=0$ when $i=-\lambda,-\lambda+1,\cdots,1,2$,
where $A$ is a bounded linear operator defined on a Banach space $X$,
$f:\mathbb{N}_{0}\rightarrow X$ is an $X$-valued sequence, $2<\alpha<3$ and
$1<p<\infty$. Here, we denote by $\mathbb{N}_{0}$ the set of all non negative
integers, $\Delta^{\alpha}$ denotes the discrete fractional operator of order
$\alpha>0$ in the Riemann-Liouville sense (see Definition 2.1 in the second
section).

The equation (1.1) has been the subject of research of many authors due to its
applications in many fields of sciences such as fractional nonlocal continuum
mechanics and physics [19, 20]. Much literature have been devoted to such
problem [8, 15, 16]. For instance, when $0<\alpha\leq 1$, Lizama and Murillo-
Arcila [15] studied the $\ell^{p}$-maximal regularity for (1.1) with the
initial conditions $u(i)=0$ when $i=-\lambda,-\lambda+1,\cdots,-1,0$, and they
proved that when the underlying Banach space $X$ is a UMD space, $1<p<\infty$,
$\sup_{n\in\mathbb{N}_{0}}\|S_{\alpha}(n)\|<\infty$ and
$\\{z^{1-\alpha}(z-1)^{\alpha}-\gamma
z^{-\lambda}:|z|=1,z\not=1\\}\subset\rho(A)$, then (1.1) has
$\ell^{p}$-maximal regularity if and only if the sets

$\big{\\{}z^{1-\alpha}(z-1)^{\alpha}[z^{1-\alpha}(z-1)^{\alpha}-\gamma
z^{-\lambda}-A]^{-1}:|z|=1,z\neq 1\big{\\}}$

and

$\big{\\{}z^{-\lambda}[z^{1-\alpha}(z-1)^{\alpha}-\gamma
z^{-\lambda}-A]^{-1}:|z|=1,z\neq 1\big{\\}}$

are $R$-bounded, where $S_{\alpha}(n)\in B(X)$ is an appropriate sequence
defined the parameters $\alpha,\ \gamma,\ \lambda$ and $A$. Later, when
$1<\alpha\leq 2$, Leal, Lizama and Murillo-Arcila [16] further considered the
$\ell^{p}$-maximal regularity for (1.1) with the initial conditions $u(i)=0$
when $i=-\lambda,-\lambda+1,\cdots,0,1$, they have shown that when $X$ is a
UMD space, $1<p<\infty$, $\sup_{n\in\mathbb{N}_{0}}\|S_{\alpha}(n)\|<\infty$
and $\\{z^{2-\alpha}(z-1)^{\alpha}-\gamma z^{-\lambda}:|z|=1,z\neq
1\\}\subset\rho(A)$, then (1.1) has the $\ell^{p}$-maximal regularity if and
only if the sets

$\big{\\{}z^{2-\alpha}(z-1)^{\alpha}[z^{2-\alpha}(z-1)^{\alpha}-\gamma
z^{-\lambda}-A]^{-1}:|z|=1,z\neq 1\big{\\}}$

and

$\big{\\{}z^{-\lambda}[z^{2-\alpha}(z-1)^{\alpha}-\gamma
z^{-\lambda}-A]^{-1}:|z|=1,z\neq 1\big{\\}}$

are $R$-bounded, where $S_{\alpha}(n)\in B(X)$ is also an appropriate sequence
defined $\alpha,\ \gamma,\ \lambda$ and $A$. However, the validity of such
characterization for the case $2<\alpha<3$ was left as an open problem. The
aim of this paper is to give a positive answer to this problem.

In order to obtain our main results. We first introduce a similar sequence of
bounded linear operators $S_{\alpha}(n)$ defined by the parameters $\alpha,\
\gamma,\ \lambda$ and $A$, called the $\alpha$-resolvent sequence
$(S_{\alpha}(n))_{n\in\mathbb{N}_{0}}$, which will give an explicit
representation of solution for (1.1). Precisely, let $2<\alpha<3$ and let
$S_{\alpha}(-\lambda)=S_{\alpha}(-\lambda+1)=\cdots=S_{\alpha}(-1)=0$,
$S_{\alpha}(0)=S_{\alpha}(1)=S_{\alpha}(2)=I$, and

$\displaystyle S_{\alpha}(n+3)$
$\displaystyle-2S_{\alpha}(n+2)+S_{\alpha}(n+1)=A(k^{\alpha-2}*S_{\alpha})(n)+\gamma(k^{\alpha-2}*S_{\alpha}^{\lambda})(n)$
$\displaystyle+k^{\alpha-2}(n+3)I+(1-\alpha)k^{\alpha-2}(n+2)I+\frac{(\alpha-1)(\alpha-2)}{2}k^{\alpha-2}(n+1)I$

when $n\in\mathbb{N}_{0}$, $\lambda\in\mathbb{N}$ and $\gamma\in\mathbb{R}$,
where $S_{\alpha}^{\lambda}(n)=S_{\alpha}(n-\lambda)$ and $k^{\alpha-2}$ is
defined by (2.4) below. We show that when $f:\mathbb{N}_{0}\to X$ is given,
there exists a unique solution $u:\mathbb{N}_{0}\to X$ of (1.1) which is given
by the formula

$\displaystyle u(n)=(h_{\alpha}*S_{\alpha}*f)(n-3)$

when $n\geq 3$. Here the function
$h_{\alpha}:\mathbb{N}_{0}\rightarrow\mathbb{R}$ is defined by
$h_{\alpha}(0)=1,h_{\alpha}(1)=\alpha-1$, $h_{\alpha}(2)=\alpha(\alpha-1)/2$
and

$h_{\alpha}(n+3)+(1-\alpha)h_{\alpha}(n+2)+(\alpha-1)(\alpha-2)h_{\alpha}(n+1)/2=0$

when $n\in\mathbb{N}_{0}$.

We show that when the underlying Banach space $X$ is a UMD space,
$1<p<\infty$, $\sup_{n\in\mathbb{N}_{0}}\|S_{\alpha}(n)\|<\infty$ and
$\\{z^{3-\alpha}(z-1)^{\alpha}-\gamma z^{-\lambda}:|z|=1,z\not=\pm
1\\}\subset\rho(A)$, then (1.1) has the $\ell^{p}$-maximal regularity if and
only if the sets

(1.2)
$\displaystyle\big{\\{}z^{3-\alpha}(z-1)^{\alpha}[z^{3-\alpha}(z-1)^{\alpha}-\gamma
z^{-\lambda}-A]^{-1}:|z|=1,z\neq\pm 1\big{\\}}$

and

(1.3) $\displaystyle\big{\\{}z^{-\lambda}[z^{3-\alpha}(z-1)^{\alpha}-\gamma
z^{-\lambda}-A]^{-1}:|z|=1,z\neq\pm 1\big{\\}}$

are $R$-bounded.

It is clear that the $R$-boundedness of the sets (1.2) and (1.3) do not depend
on the space parameter $p$. Thus when the underlying Banach space $X$ is a UMD
space, $\sup_{n\in\mathbb{N}_{0}}\|S_{\alpha}(n)\|<\infty$,
$\\{z^{3-\alpha}(z-1)^{\alpha}-\gamma z^{-\lambda}:|z|=1,z\not=\pm
1\\}\subset\rho(A)$, if (1.1) has the $\ell^{p}$-maximal regularity for some
$1<p<\infty$, then (1.1) has the $\ell^{p}$-maximal regularity for all
$1<p<\infty$. Since every norm bounded subset of $B(X)$ is actually
$R$-bounded when $X$ is a Hilbert space, we deduce that when the underlying
Banach space $X$ is a Hilbert space, $1<p<\infty$,
$\sup_{n\in\mathbb{N}_{0}}\|S_{\alpha}(n)\|<\infty$ and
$\\{z^{3-\alpha}(z-1)^{\alpha}-\gamma z^{-\lambda}:|z|=1,z\not=\pm
1\\}\subset\rho(A)$, then (1.1) has the $\ell^{p}$-maximal regularity if and
only if the sets (1.2) and (1.3) are norm bounded.

We notice that the fractional difference equation (1.1) in the case
$2<\alpha<3$ and $\gamma=0$ was previously studied in [21]. Our results
recover the explicit formula of solution and the characterization of
$\ell^{p}$-maximal regularity for (1.1) obtained in [21]. The main tool in
this paper is the operator-valued Fourier multipliers theorems on
$\ell^{p}(\mathbb{Z};X)$ obtained by Blunck [5], we will transform the
$\ell^{p}$-maximal regularity of (1.1) to an operator-valued Fourier
multiplier problem on $\ell^{p}(\mathbb{Z};X)$.

We notice that the research in this paper was motivated by recent studies on
discrete mathematical models proved that they can serve as a new
microstructural basis for fractional nonlocal continuum mechanics and physics
[18, 19]. Fractional difference equations can be also used to formulate
adequate models in nanomechanics [19, 20] and therefore further studies in
this class of fractional difference equations deserve to be investigated. Our
contribution in this paper provides a new qualitative advance in this line of
research, that incorporates tools from operator theory and allows the analysis
of maximal regularity for a very general but still simple model. The study of
more complex dynamical systems that include unbounded operators is still an
open problem. This task will be the objective of forthcoming works.

This paper is organized as follows: in section 2, we recall some basic
concepts and results in the existing literature related to fractional
difference operators, UMD spaces, $R$-boundedness, the discrete time Fourier
transform and Blunck’s Fourier multipliers theorems that will be later used;
in section 3, we study the $\alpha$-resolvent sequence defined by the
parameters $\alpha,\ \gamma,\ \lambda$ and $A$, and give an explicit
representation of solution for (1.1); in the last section, we give a
characterization of the $\ell^{p}$-maximal regularity for equation (1.1) when
$1<p<\infty$ and $X$ is a UMD space.

##  2\. Preliminaries

In this section, we briefly recall some necessary notions and results related
to fractional difference operators, UMD spaces, $R$-boundedness, the discrete
time Fourier transform and Blunck’s Fourier multipliers theorems, which will
be used in the sequel.

Let $X$ be a Banach space. We denote by $S(\mathbb{N}_{0};X)$ the vector space
consisting of all vector-valued sequences $u:\mathbb{N}_{0}\rightarrow X$.
Similarly we denote by $S(\mathbb{Z};X)$ the vector space consisting of all
vector-valued sequences $u:\mathbb{Z}\rightarrow X$. The forward Euler
operator $\Delta:S(\mathbb{N}_{0};X)\rightarrow S(\mathbb{N}_{0};X)$ is
defined as follows:

$\displaystyle\Delta u(n)=u(n+1)-u(n)$

when $n\in\mathbb{N}_{0}$. For every $m\in\mathbb{N}$, we define recursively
the m-th order forward difference operator
$\Delta^{m}:S(\mathbb{N}_{0};X)\rightarrow S(\mathbb{N}_{0};X)$ by
$\Delta^{m}=\Delta^{m-1}\Delta$.

Recall that the finite convolution $*$ between two sequences $f\in
S(\mathbb{N}_{0};\mathbb{C})$ and $g\in S(\mathbb{N}_{0};X)$ is defined by

(2.1) $(f*g)(n):=\sum_{j=0}^{n}f(n-j)g(j)$

when $n\in\mathbb{N}_{0}$. It is easy to verify that if $h\in
S(\mathbb{N}_{0};\mathbb{C})$, then

(2.2) $((f*h)*g)(n)=(f*(h*g))(n)=\sum_{i+j+k=n}f(i)h(j)g(k)$

when $n\in\mathbb{N}_{0}$. Hence we will use the expression $f*g*h$ without
any confusion.

The following definition of fractional sum, due to the previous works [1, 4],
was formally presented by Lizama in [12]. Let $0<\beta\leq 1$ and $u\in
S(\mathbb{N}_{0};X)$ be given. The fractional sum of $u$ of order $\beta$ is
defined by

(2.3) $\Delta^{-\beta}u(n):=(k^{\beta}*u)(n)=\sum_{j=0}^{n}k^{\beta}(n-j)u(j)$

when $n\in\mathbb{N}_{0}$, where

(2.4) $\displaystyle
k^{\beta}(j):=\frac{\Gamma(\beta+j)}{\Gamma(\beta)\Gamma(j+1)}$

when $j\in\mathbb{N}_{0}$, where $\Gamma$ is the Gamma function. It is easy to
verify that this sequence $(k^{\beta}(j))_{n\in\mathbb{N}_{0}}$ satisfies the
following semigroup property

(2.5) $\displaystyle(k^{\beta}*k^{\alpha})(n)=k^{\beta+\alpha}(n)$

when $n\in\mathbb{N}_{0},\beta,\alpha\in\mathbb{C}$.

The next definition corresponds to an analogous version of fractional
derivative in the sense of Riemann-Liouville, see [2, 17].

######  Definition 2.1.

Let $\alpha>0,\ \alpha\notin\mathbb{N}$ and $u\in S(\mathbb{N}_{0};X)$ be
given. The fractional difference operator of $u$ of order $\alpha$ is defined
by

(2.6) $\Delta^{\alpha}u(n):=\Delta^{m}\Delta^{-(m-\alpha)}u(n)$

when $n\in\mathbb{N}_{0}$, where $m\in\mathbb{N}$ is the unique integer $m$
satisfying $m-1<\alpha<m$. For more study of fractional difference operators,
we refer the readers to [7, 13, 14]

Let $u\in S(\mathbb{Z};X)$ be given, the discrete time Fourier transform of
$u$ is given by

$\hat{u}(z):=\sum_{j=-\infty}^{\infty}z^{-j}u(j)$

for all $|z|=1$, whenever it exists. We notice that the Fourier transform of
$u$ is sometimes also denoted by $\mathcal{F}(u)$. It is clear that when $f\in
S(\mathbb{N}_{0};\mathbb{C})$ and $g\in S(\mathbb{N}_{0};X)$, then

(2.7) $\widehat{(f*g)}(z)=\hat{f}(z)\hat{g}(z)$

when both sides are well defined for all $|z|=1$.

Let $k^{\alpha}$ be defined by (2.4) for $0<\alpha\leq 1$. It follows that the
Fourier transform of $k^{\alpha}$ is given by

(2.8) $\displaystyle\ \hat{k}^{\alpha}(z)=\frac{z^{\alpha}}{(z-1)^{\alpha}}$

when $|z|=1$ and $z\not=1$.

We say that a Banach space $X$ is a UMD (Unconditionality of Martingale
Differences) space if for all $1<p<\infty$, there exists a constant $C>0$
(depending only on $p$ and $X$) such that for any martingale $(g_{n})_{n\geq
0}\subset L^{p}(\Omega,\Sigma,\mu;X)$ and all scalars
$|\varepsilon_{n}|=1,n=1,2\cdots,N$, the following inequality holds:

$\Big{\|}g_{0}+\sum_{n=1}^{N}\varepsilon_{n}(g_{n}-g_{n-1})\Big{\|}_{L^{p}(\Omega,\Sigma,\mu;X)}\leq
C\big{\|}g_{N}\big{\|}_{L^{p}(\Omega,\Sigma,\mu;X)}.$

It is well known that $L_{p}$ spaces, Schatten class $S_{p}$ and Sobolev
spaces $W^{m,p}$ are UMD spaces when $1<p<\infty$. UMD spaces have played a
very important part in vector-valued harmonic analysis and probability theory,
see [6].

Let $X$ be a Banach space, we denote by $B(X)$ the space of all bounded linear
operators on $X$. Let $r_{j}$ be the $j$-th Rademacher function defined on
$[0,1]$ by $r_{j}(t)=\text{sgn(sin}(2^{j}t))$ whenever $j\geq 1$.

######  Definition 2.2.

Let $X$ be a Banach spaces. A set $W\subset B(X)$ is said to be Rademacher
bounded ($R$-bounded, shortly) [9], if there exists $C\geq 0$ such that

$\displaystyle\Big{\|}\sum_{j=1}^{n}r_{j}T_{j}x_{j}\Big{\|}_{L^{1}([0,1];X)}\leq
C\Big{\|}\sum_{j=1}^{n}r_{j}x_{j}\Big{\|}_{L^{1}([0,1];X)}$

for all $T_{1},T_{2},\cdots,T_{n}\in W$, $x_{1},x_{2},\cdots,x_{n}\in X$ and
$n\in\mathbb{N}.$

######  Remark 2.3.

It is clear that when $\mathrm{W_{1}},\mathrm{W_{2}}\subset B(X)$ are
$R$-bounded sets, then the sets

$\mathrm{W_{1}W_{2}:=\big{\\{}ST:S\in W_{1},T\in W_{2}\big{\\}}},\
\mathrm{W_{1}+W_{2}}:=\big{\\{}S+T:S\in W_{1},T\in W_{2}\big{\\}}$

are still $R$-bounded. It is easy to see that if $\mathrm{W}$ is a bounded
subset of the complex plane $\mathbb{C}$, then the set $\\{\mu
I:\mu\in\mathrm{W}\\}$ is also $R$-bounded, where $I$ stands for the identity
operator on $X$. This follows easily from the Kahane’s contraction principle
[11].

In what follows we denote by $\mathbb{T}=(-\pi,0)\cup(0,\pi)$. Let $X$ be a
Banach space and let $G:\mathbb{T}\rightarrow B(X)$ be bounded and measurable.
Let $f\in S(\mathbb{Z};X)$ with finite support, i.e., the set
$\\{n\in\mathbb{Z}:f(n)\not=0\\}$ is finite. Then the function $t\to
G(t)(\mathcal{F}f)(e^{it})$ defined on $t\in\mathbb{T}$ is bounded and
measurable. Thus its inverse Fourier transform

$\mathcal{F}^{-1}\big{[}G(\cdot)(\mathcal{F}f)(e^{i\cdot})\big{]}(n)=\frac{1}{2\pi}\int_{0}^{2\pi}G(t)(\mathcal{F}f)(e^{it})e^{int}dt$

makes sense for all $n\in\mathbb{Z}$. Let $1\leq p<\infty$ be given. We say
that $G$ is an $\ell^{p}$-Fourier multiplier, if there exists a constant $C>0$
independent from $f$, such that

$\Big{(}\sum_{n\in\mathbb{Z}}\big{\|}\mathcal{F}^{-1}\big{[}G(\cdot)(\mathcal{F}f)(e^{i\cdot})\big{]}(n)\big{\|}^{p}\Big{)}^{1/p}\leq
C\Big{(}\sum_{n\in\mathbb{Z}}\|f(n)\|^{p}\Big{)}^{1/p}$

for all $f\in S(\mathbb{Z};X)$ with finite support. In this case, there exists
a unique bounded linear operator $T_{G}\in B(\ell^{p}(\mathbb{Z};X))$, such
that

(2.9) $G(t)(\mathcal{F}f)(e^{it})=\mathcal{F}(T_{G}f)(e^{it})$

when $t\in\mathbb{T}$ and $f\in S(\mathbb{Z};X)$ with finite support. Here we
used the easy fact that the set of all elements $f\in S(\mathbb{Z};X)$ with
finite support is dense in $\ell^{p}(\mathbb{Z};X)$.

We finish this section with the following $\ell^{p}$-Fourier multipliers
established by Blunck [5], which will be fundamental in our investigation.

######  Theorem 2.4.

Let $1<p<\infty$ and let $X$ be a UMD space. Assume that
$G:\mathbb{T}\rightarrow B(X)$ is differentiable and the sets

$\big{\\{}G(t):t\in\mathbb{T}\big{\\}}~{}~{}and\ \
\big{\\{}(e^{it}-1)(e^{it}+1)G^{\prime}(t):t\in\mathbb{T}\big{\\}}$

are $R$-bounded. Then $G$ is an $\ell^{p}$-Fourier multiplier.

######  Theorem 2.5.

Let $1\leq p<\infty$ and let $X$ be a Banach space. Let
$G:\mathbb{T}\rightarrow B(X)$ be continuous and bounded. Assume that $G$ is
an $\ell^{p}$-Fourier multiplier. Then the set $\\{G(t):t\in\mathbb{T}\\}$ is
$R$-bounded.

##  3\. The Representation of Solutions

In this section, we will introduce a concept of $\alpha$-resolvent sequence of
bounded operators, which will give an explicit representation of the solution
for the fractional difference equation (1.1) when $2<\alpha<3$. We observe
that the equation (1.1) in the case $0<\alpha\leq 1$ with the initial
conditions $u(i)=0$ when $i=-\lambda,-\lambda+1,\cdots,-1,0$ was previously
studied by Lizama and Murillo-Arcila [15]. Later, Leal, Lizama and Murillo-
Arcila [16] further studied the equation (1.1) in the case $1<\alpha\leq 2$
with the initial conditions $u(i)=0$ when $i=-\lambda,-\lambda+1,\cdots,0,1$.

###  3.1. The $\alpha$-Resolvent Sequence with Finite Delay

We first introduce an similar $\alpha$-resolvent sequence $S_{\alpha}(n)$ when
$n\geq-\lambda$ used in [15, 16].

######  Definition 3.1.

Let $A$ be a bounded linear operator defined on a Banach space $X$, and let
$\gamma\in\mathbb{R}$, $\lambda\in\mathbb{N}$ and $2<\alpha<3$. We define the
$\alpha$-resolvent sequence $S_{\alpha}(n)$ when $n\geq-\lambda$ generated by
$A,\alpha,\gamma$ and $\lambda$ by

(i) $S_{\alpha}(-i)=0$ when $i=1,\cdots,\lambda$;;

(ii) $S_{\alpha}(0)=S_{\alpha}(1)=S_{\alpha}(2)=I$;

(iii)
$S_{\alpha}(n+3)-2S_{\alpha}(n+2)+S_{\alpha}(n+1)=A(k^{\alpha-2}*S_{\alpha})(n)+\gamma(k^{\alpha-2}*S_{\alpha}^{\lambda})(n)+k^{\alpha-2}(n+3)I+(1-\alpha)k^{\alpha-2}(n+2)I+\frac{(\alpha-1)(\alpha-2)}{2}k^{\alpha-2}(n+1)I$
when $n\in\mathbb{N}_{0}$, where $S_{\alpha}^{\lambda}(n)$ is defined by
$S_{\alpha}^{\lambda}(n)=S_{\alpha}(n-\lambda)$.

######  Remark 3.2.

Note that the above Definition 3.1 corresponds to the definition of
$\alpha$-resolvent sequence defined in [21] when $\gamma=0$. We need to define
$S_{\alpha}(-1)=\cdots=S_{\alpha}(-\lambda)=0$ as $S_{\alpha}^{\lambda}(n)$ is
defined by $S_{\alpha}(n-\lambda)$ when $n\in\mathbb{N}_{0}$.

######  Remark 3.3.

Let $\gamma\in\mathbb{R}$, $\lambda\in\mathbb{N}$ and $2<\alpha<3$. Assume
that $z^{3-\alpha}(z-1)^{\alpha}-\gamma z^{-\lambda}\in\rho(A)$ for all
$|z|=1,\ z\not=1$. Then the Fourier transform of $S_{\alpha}$ is given by

(3.1)
$\widehat{S}_{\alpha}(z)=\Big{[}z^{3}+(1-\alpha)z^{2}+\frac{(\alpha-1)(\alpha-2)}{2}z\Big{]}\big{[}z^{3-\alpha}(z-1)^{\alpha}-\gamma
z^{-\lambda}-A\big{]}^{-1}$

when $|z|=1,\ z\not=1$. Indeed, taking the Fourier transform on both sides of
(iii) in Definition 3.1 and using (2.4), (2.7) and (i), (ii) in Definition
3.1, we obtain

$\displaystyle z^{3}\widehat{S}_{\alpha}(z)$
$\displaystyle-z^{3}-z^{2}-z-2\big{[}z^{2}\widehat{S}_{\alpha}(z)-z^{2}-z\big{]}+z\widehat{S}_{\alpha}(z)-z=A\widehat{k}^{\alpha-2}(z)\widehat{S}_{\alpha}(z)$
$\displaystyle+\gamma
z^{-\lambda}\widehat{k}^{\alpha-2}(z)\hat{S}_{\alpha}^{\lambda}(z)+z^{3}\widehat{k}^{\alpha-2}(z)-z^{3}-z^{2}(\alpha-2)-\frac{(\alpha-1)(\alpha-2)}{2}z$
$\displaystyle+(1-\alpha)\big{[}z^{2}\widehat{k}^{\alpha-2}(z)-z^{2}-z(\alpha-2)\big{]}+\frac{(\alpha-1)(\alpha-2)}{2}\big{(}z\widehat{k}^{\alpha-2}(z)-z\big{)}$

when $|z|=1,\ z\not=1$. Here we have used the fact that the function
$\widehat{k}^{\alpha-2}(z)$ is only well-defined when $|z|=1,\ z\not=1$ by
(2.8). It follows that

$\big{[}z^{3}-2z^{2}+z-\gamma
z^{-\lambda}\widehat{k}^{\alpha-2}(z)-A\widehat{k}^{\alpha-2}(z)\big{]}\widehat{S}_{\alpha}(z)=\Big{[}z^{3}+(1-\alpha)z^{2}+\frac{(\alpha-1)(\alpha-2)}{2}z\Big{]}\widehat{k}^{\alpha-2}(z)$

when $|z|=1,\ z\not=1$. Thus, by (2.8), we have

$\widehat{S}_{\alpha}(z)=\Big{[}z^{3}+(1-\alpha)z^{2}+\frac{(\alpha-1)(\alpha-2)}{2}z\Big{]}\big{[}z^{3-\alpha}(z-1)^{\alpha}-\gamma
z^{-\lambda}-A\big{]}^{-1}$

when $|z|=1,\ z\not=1$.

We will need the following Lemma proved in [21].

######  Lemma 3.4.

([21, Lemma 3.1]) Let $2<\alpha<3$, $b:\mathbb{N}_{0}\rightarrow\mathbb{C}$
and $P:\mathbb{N}_{0}\rightarrow X$, where $X$ is a Banach space. Then

$\displaystyle\Delta^{\alpha}(b*P)(n)$
$\displaystyle=(b*\Delta^{\alpha}P)(n)+b(n+3)P(0)+b(n+2)\big{[}P(1)-\alpha
P(0)\big{]}$ $\displaystyle+b(n+1)\Big{[}P(2)-\alpha
P(1)+\frac{\alpha(\alpha-1)}{2}P(0)\Big{]}$

when $n\in\mathbb{N}_{0}$.

In order to prove our main result of this section, we need the following
lemmas.

######  Lemma 3.5.

Let $X$ be a Banach space, $A\in B(X)$, $2<\alpha<3$, $\gamma\in\mathbb{R}$,
$\lambda\in\mathbb{N}$ and let $(S_{\alpha}(n))_{n\geq-\lambda}$ be the
$\alpha$-resolvent sequence given by Definition 3.1. Then

$\Delta^{\alpha}S_{\alpha}(n)=AS_{\alpha}(n)+\gamma S_{\alpha}^{\lambda}(n)$

when $0\leq n\leq 2$.

###### Proof.

By using (2.1), (2.3) and (2.6), we have

(3.2) $\displaystyle\Delta^{\alpha}S_{\alpha}(0)$
$\displaystyle=\Delta^{3}\Delta^{-(3-\alpha)}S_{\alpha}(0)$
$\displaystyle=\Delta^{-(3-\alpha)}S_{\alpha}(3)-3\Delta^{-(3-\alpha)}S_{\alpha}(2)$
$\displaystyle\ \ \ \
+3\Delta^{-(3-\alpha)}S_{\alpha}(1)-\Delta^{-(3-\alpha)}S_{\alpha}(0)$
$\displaystyle=\sum_{j=0}^{3}k^{3-\alpha}(3-j)S_{\alpha}(j)-3\sum_{j=0}^{2}k^{3-\alpha}(2-j)S_{\alpha}(j)$
$\displaystyle\ \ \ \
+3\sum_{j=0}^{1}k^{3-\alpha}(1-j)S_{\alpha}(j)-k^{3-\alpha}(0)S_{\alpha}(0)$
$\displaystyle=S_{\alpha}(3)+k^{3-\alpha}(1)I-2k^{3-\alpha}(2)I+k^{3-\alpha}(3)I-I.$

It follows from Definition 3.1 that

(3.3) $\displaystyle S_{\alpha}(3)=$ $\displaystyle
I+A(k^{\alpha-2}*S_{\alpha})(0)+\gamma(k^{\alpha-2}*S_{\alpha}^{\lambda})(0)$
$\displaystyle+k^{\alpha-2}(3)I+(1-\alpha)k^{\alpha-2}(2)I+\frac{(\alpha-1)(\alpha-2)}{2}k^{\alpha-2}(1)I.$

Thus, (3.2)and (3.3) together imply that

$\displaystyle\Delta^{\alpha}S_{\alpha}(0)$
$\displaystyle=I+A(k^{\alpha-2}*S_{\alpha})(0)+\gamma(k^{\alpha-2}*S_{\alpha}^{\lambda})(0)+k^{\alpha-2}(3)I+(1-\alpha)k^{\alpha-2}(2)I$
$\displaystyle\ \ \ \
+\frac{(\alpha-1)(\alpha-2)}{2}k^{\alpha-2}(1)I+k^{(3-\alpha)}(1)I-2k^{(3-\alpha)}(2)I+k^{(3-\alpha)}(3)I-I$
$\displaystyle=A(k^{\alpha-2}*S_{\alpha})(0)+\gamma(k^{\alpha-2}*S_{\alpha}^{\lambda})(0)$
$\displaystyle=A(k^{\alpha-2}*S_{\alpha})(0)+\gamma S_{\alpha}^{\lambda}(0)$
$\displaystyle=AS_{\alpha}(0)+\gamma S_{\alpha}^{\lambda}(0).$

From Definition 3.1 again, we have

(3.4) $\displaystyle S_{\alpha}(4)=$ $\displaystyle
2S_{\alpha}(3)-I+A(k^{\alpha-2}*S_{\alpha})(1)+\gamma(k^{\alpha-2}*S_{\alpha}^{\lambda})(1)$
$\displaystyle+k^{\alpha-2}(4)I+(1-\alpha)k^{\alpha-2}(3)I+\frac{(\alpha-1)(\alpha-2)}{2}k^{\alpha-2}(2)I$

and

(3.5) $\displaystyle S_{\alpha}(5)=$ $\displaystyle
2S_{\alpha}(4)-S_{\alpha}(3)+A(k^{\alpha-2}*S_{\alpha})(2)+\gamma(k^{\alpha-2}*S_{\alpha}^{\lambda})(2)$
$\displaystyle+k^{\alpha-2}(5)I+(1-\alpha)k^{\alpha-2}(4)I+\frac{(\alpha-1)(\alpha-2)}{2}k^{\alpha-2}(3)I.$

Again by (2.1), (2.3) and (2.6), we obtain

(3.6) $\displaystyle\Delta^{\alpha}S_{\alpha}(1)$
$\displaystyle=\Delta^{3}\Delta^{-(3-\alpha)}S_{\alpha}(1)$
$\displaystyle=\Delta^{-(3-\alpha)}S_{\alpha}(4)-3\Delta^{-(3-\alpha)}S_{\alpha}(3)$
$\displaystyle\ \ \ \
+3\Delta^{-(3-\alpha)}S_{\alpha}(2)-\Delta^{-(3-\alpha)}S_{\alpha}(1)$
$\displaystyle=\sum_{j=0}^{4}k^{3-\alpha}(4-j)S_{\alpha}(j)-3\sum_{j=0}^{3}k^{3-\alpha}(3-j)S_{\alpha}(j)$
$\displaystyle\ \ \ \
+3\sum_{j=0}^{2}k^{3-\alpha}(2-j)S_{\alpha}(j)-\sum_{j=0}^{1}k^{3-\alpha}(1-j)S_{\alpha}(j)$
$\displaystyle=S_{\alpha}(4)-\alpha
S_{\alpha}(3)+k^{3-\alpha}(2)I-2k^{3-\alpha}(3)I$ $\displaystyle\ \ \ \
+k^{3-\alpha}(4)I+(\alpha-1)I.$

Therefore, by using (3.3), (3.4) and (3.6) we deduce that

$\displaystyle\Delta^{\alpha}S_{\alpha}(1)$
$\displaystyle=(2-\alpha)S_{\alpha}(3)-I+A(k^{\alpha-2}*S_{\alpha})(1)+\gamma(k^{\alpha-2}*S_{\alpha}^{\lambda})(1)I$
$\displaystyle\ \ \ \
+k^{\alpha-2}(4)+(1-\alpha)k^{\alpha-2}(3)I+\frac{(\alpha-1)(\alpha-2)}{2}k^{\alpha-2}(2)I$
$\displaystyle\ \ \ \
+k^{3-\alpha}(2)I-2k^{3-\alpha}(3)I+k^{3-\alpha}(4)I+(\alpha-1)I$
$\displaystyle=AS_{\alpha}(1)+k^{\alpha-2}(4)I+(3-2\alpha)k^{\alpha-2}(3)I+\frac{3(\alpha-1)(\alpha-2)}{2}k^{\alpha-2}(2)I$
$\displaystyle\ \ \ \
-\frac{(\alpha-1)(\alpha-2)^{2}}{2}k^{\alpha-2}(1)I+k^{3-\alpha}(2)I-2k^{3-\alpha}(3)I$
$\displaystyle\ \ \ \
+k^{3-\alpha}(4)I+(2-\alpha)\gamma(k^{\alpha-2}*S_{\alpha}^{\lambda})(0)+\gamma(k^{\alpha-2}*S_{\alpha}^{\lambda})(1)$
$\displaystyle=AS_{\alpha}(1)+\gamma S_{\alpha}^{\lambda}(1).$

Using a similar argument in (3.6), we have

(3.7) $\displaystyle\Delta^{\alpha}S_{\alpha}(2)=$ $\displaystyle
S_{\alpha}(5)-\alpha
S_{\alpha}(4)+\frac{\alpha(\alpha-1)}{2}S_{\alpha}(3)+k^{3-\alpha}(5)I-2k^{3-\alpha}(4)I$
$\displaystyle+k^{3-\alpha}(3)I-k^{3-\alpha}(2)I+2k^{3-\alpha}(1)I-k^{\alpha-2}(0)I.$

Therefore, applying (3.3), (3.4), (3.5) and (3.7), we deduce that

$\displaystyle\Delta^{\alpha}S_{\alpha}(2)$
$\displaystyle=AS_{\alpha}(2)+\frac{\alpha(\alpha-3)}{2}I+k^{\alpha-2}(5)I+(3-2\alpha)k^{\alpha-2}(4)I$
$\displaystyle\ \ \ \
+(\alpha-2)(2\alpha-3)k^{\alpha-2}(3)I+\frac{(2-\alpha)(1-\alpha)(5-2\alpha)}{2}k^{\alpha-2}(2)I$
$\displaystyle\ \ \ \
-\frac{(\alpha-1)(3-\alpha)(\alpha-2)^{2}}{4}k^{\alpha-2}(1)I+k^{3-\alpha}(5)I-2k^{3-\alpha}(4)I$
$\displaystyle\ \ \ \
+k^{3-\alpha}(3)I-k^{3-\alpha}(2)I+2k^{3-\alpha}(1)I+\frac{(\alpha-2)(\alpha-3)}{2}\gamma
S_{\alpha}^{\lambda}(0)$ $\displaystyle\ \ \ \
+(2-\alpha)\gamma(k^{\alpha-2}*S_{\alpha}^{\lambda})(1)+\gamma(k^{\alpha-2}*S_{\alpha}^{\lambda})(2)$
$\displaystyle=AS_{\alpha}(2)+\gamma S_{\alpha}^{\lambda}(2).$

The proof is completed. ∎

######  Lemma 3.6.

Let $X$ be a Banach space, $A\in B(X)$, $2<\alpha<3$, $\gamma\in\mathbb{R}$,
$\lambda\in\mathbb{N}$ and let $(S_{\alpha}(n))_{n\geq-\lambda}$ be the
$\alpha$-resolvent sequence given by Definition 3.1. Then

$\Delta^{\alpha}S_{\alpha}(n)=AS_{\alpha}(n)+\gamma S_{\alpha}^{\lambda}(n)$

for all $n\in\mathbb{N}_{0}$.

###### Proof.

First note that

(3.8)
$\Delta^{\alpha}k^{\alpha-2}(n)=\Delta^{3}\Delta^{-(3-\alpha)}k^{\alpha-2}(n)=\Delta^{3}(k^{(3-\alpha)}*k^{\alpha-2})(n)=\Delta^{3}k^{1}(n)=0$

when $n\in\mathbb{N}_{0}$ as $k^{1}(n)=1$ for all $n\in\mathbb{N}_{0}$. By
(iii) in Definition 3.1, we have

$\displaystyle\sum_{j=0}^{n}k^{3-\alpha}(j)S$
${}_{\alpha}(n+3-j)-2\sum_{j=0}^{n}k^{3-\alpha}(j)S_{\alpha}(n+2-j)+\sum_{j=0}^{n}k^{3-\alpha}(j)S_{\alpha}(n+1-j)$
$\displaystyle=A\sum_{j=0}^{n}k^{3-\alpha}(j)(k^{\alpha-2}*S_{\alpha})(n-j)+\gamma\sum_{j=0}^{n}k^{3-\alpha}(j)(k^{\alpha-2}*S_{\alpha}^{\lambda})(n-j)$
$\displaystyle\ \ \ \
+\sum_{j=0}^{n}k^{3-\alpha}(j)k^{\alpha-2}(n+3-j)I+(1-\alpha)\sum_{j=0}^{n}k^{3-\alpha}(j)k^{\alpha-2}(n+2-j)I$
$\displaystyle\ \ \ \
+\frac{(\alpha-1)(\alpha-2)}{2}\sum_{j=0}^{n}k^{3-\alpha}(j)k^{\alpha-2}(n+1-j)I$

when $n\in\mathbb{N}_{0}$. This implies that

$\displaystyle\Delta^{-(3-\alpha)}$ $\displaystyle
S_{\alpha}(n+3)-2\Delta^{-(3-\alpha)}S_{\alpha}(n+2)+\Delta^{-(3-\alpha)}S_{\alpha}(n+1)$
$\displaystyle\ \ \ \ \
-k^{3-\alpha}(n+1)S_{\alpha}(2)-k^{3-\alpha}(n+2)S_{\alpha}(1)-S_{\alpha}(0)k^{3-\alpha}(n+3)$
$\displaystyle\ \ \ \ \
+2k^{3-\alpha}(n+1)S_{\alpha}(1)+2k^{3-\alpha}(n+2)S_{\alpha}(0)-k^{3-\alpha}(n+1)S_{\alpha}(0)$
$\displaystyle=A\Delta^{-(3-\alpha)}(k^{\alpha-2}*S_{\alpha})(n)+\gamma\Delta^{-(3-\alpha)}(k^{\alpha-2}*S_{\alpha}^{\lambda})(n)+\Delta^{-(3-\alpha)}k^{\alpha-2}(n+3)I$
$\displaystyle\ \ \ \
+(1-\alpha)\Delta^{-(3-\alpha)}k^{\alpha-2}(n+2)I+\frac{(\alpha-1)(\alpha-2)}{2}\Delta^{-(3-\alpha)}k^{\alpha-2}(n+1)I$
$\displaystyle\ \ \ \
-k^{3-\alpha}(n+3)k^{\alpha-2}(0)I-k^{3-\alpha}(n+2)k^{\alpha-2}(1)I-k^{3-\alpha}(n+1)k^{\alpha-2}(2)I$
$\displaystyle\ \ \ \
-(1-\alpha)k^{3-\alpha}(n+2)k^{\alpha-2}(0)I-(1-\alpha)k^{3-\alpha}(n+1)k^{\alpha-2}(1)I$
$\displaystyle\ \ \ \
-\frac{(\alpha-1)(\alpha-2)}{2}k^{3-\alpha}(n+1)k^{\alpha-2}(0)I$

when $n\in\mathbb{N}_{0}$. It follows that

$\displaystyle\Delta^{-(3-\alpha)}$ $\displaystyle
S_{\alpha}(n+3)-2\Delta^{-(3-\alpha)}S_{\alpha}(n+2)+\Delta^{-(3-\alpha)}S_{\alpha}(n+1)$
$\displaystyle=A\Delta^{-(3-\alpha)}(k^{\alpha-2}*S_{\alpha})(n)+\gamma\Delta^{-(3-\alpha)}(k^{\alpha-2}*S_{\alpha}^{\lambda})(n)+\Delta^{-(3-\alpha)}k^{\alpha-2}(n+3)$
$\displaystyle\ \ \ \
+\frac{(\alpha-1)(\alpha-2)}{2}\Delta^{-(3-\alpha)}k^{\alpha-2}(n+1)+(1-\alpha)\Delta^{-(3-\alpha)}k^{\alpha-2}(n+2)$

when $n\in\mathbb{N}_{0}$. Therefore,

(3.9)
$\displaystyle\Delta^{\alpha}S_{\alpha}(n+3)-2\Delta^{\alpha}S_{\alpha}(n+2)+\Delta^{\alpha}S_{\alpha}(n+1)=$
$\displaystyle A\Delta^{\alpha}(k^{\alpha-2}*S_{\alpha})(n)$
$\displaystyle+\gamma\Delta^{\alpha}(k^{\alpha-2}*S_{\alpha}^{\lambda})(n).$

by (3.8) when $n\in\mathbb{N}_{0}$.

Applying Lemma 3.4 and (3.8), we obtain

(3.10) $\displaystyle\Delta^{\alpha}(k^{\alpha-2}*S_{\alpha})(n)=$
$\displaystyle(\Delta^{\alpha}k^{\alpha-2}*S_{\alpha})(n)+k^{\alpha-2}(0)S_{\alpha}(n+3)$
$\displaystyle\ \ \ \ +[k^{\alpha-2}(1)-\alpha
k^{\alpha-2}(0)]S_{\alpha}(n+2)+\Big{[}k^{\alpha-2}(2)$ $\displaystyle\ \ \ \
-\alpha
k^{\alpha-2}(1)+\frac{\alpha(\alpha-1)}{2}k^{\alpha-2}(0)\Big{]}S_{\alpha}(n+1)$
$\displaystyle=S_{\alpha}(n+3)-2S_{\alpha}(n+2)+S_{\alpha}(n+1)$

and

(3.11) $\displaystyle\Delta^{\alpha}(k^{\alpha-2}*S_{\alpha}^{\lambda})(n)$
$\displaystyle=(\Delta^{\alpha}k^{\alpha-2}*S_{\alpha}^{\lambda})(n)+k^{\alpha-2}(0)S_{\alpha}^{\lambda}(n+3)$
$\displaystyle\ \ \ \ +[k^{\alpha-2}(1)-\alpha
k^{\alpha-2}(0)]S_{\alpha}^{\lambda}(n+2)+\Big{[}k^{\alpha-2}(2)$
$\displaystyle\ \ \ \ -\alpha
k^{\alpha-2}(1)+\frac{\alpha(\alpha-1)}{2}k^{\alpha-2}(0)\Big{]}S_{\alpha}^{\lambda}(n+1)$
$\displaystyle=S_{\alpha}^{\lambda}(n+3)-2S_{\alpha}^{\lambda}(n+2)+S_{\alpha}^{\lambda}(n+1)$

when $n\in\mathbb{N}_{0}$.

Thus, replacing (3.10) and (3.11) in (3.9), we deduce that

$\Delta^{2}\Delta^{\alpha}S_{\alpha}(n+1)=\Delta^{2}AS_{\alpha}(n+1)+\gamma\Delta^{2}S_{\alpha}^{\lambda}(n+1)$

when $n\in\mathbb{N}_{0}$. The conclusion follows easily from Lemma 3.5 and
the proof is completed. ∎

###  3.2. The Existence and Uniqueness of Solution

We recall the following definition introduced in [21], which is necessary to
establish the main result of this section.

######  Definition 3.7.

([21, Lemma 3.2]) Let $2<\alpha<3$ be given, The function
$h_{\alpha}:\mathbb{N}_{0}\rightarrow\mathbb{R}$ is defined by
$h_{\alpha}(0)=1,~{}~{}h_{\alpha}(1)=\alpha-1,~{}~{}h_{\alpha}(2)=\frac{\alpha(\alpha-1)}{2}$,
and

(3.12)
$h_{\alpha}(n+3)+(1-\alpha)h_{\alpha}(n+2)+\frac{(\alpha-1)(\alpha-2)}{2}h_{\alpha}(n+1)=0$

when $n\geq 0$.

We observe that the Fourier transform of $h_{\alpha}$ is as follows:

(3.13)
$\hat{h}_{\alpha}(z)=\frac{z^{3}}{z^{2}+(1-\alpha)z+\frac{(\alpha-1)(\alpha-2)}{2}}.$

by Remark 3.3 in [21].

Now we are ready to prove the main result of this section.

######  Theorem 3.8.

Let $X$ be a Banach space, $A\in B(X)$, $2<\alpha<3$, $\gamma\in\mathbb{R}$,
$\lambda\in\mathbb{N}$ and let $f\in S(\mathbb{N}_{0};X)$ be given. Then $u\in
S(\mathbb{N}_{0};X)$ defined by $u(0)=u(1)=u(2)=0$, and

(3.14) $\displaystyle u(n)=(h_{\alpha}*S_{\alpha}*f)(n-3)$

when $n\geq 3$, is a solution of (1.1), and this is the unique solution of
(1.1).

###### Proof.

From Lemma 3.4, Lemma 3.6 and (3.12), we have

(3.15) $\displaystyle\Delta^{\alpha}\big{(}h_{\alpha}*S_{\alpha}\big{)}(n)$
$\displaystyle=(h_{\alpha}*\Delta^{\alpha}S_{\alpha})(n)+h_{\alpha}(n+3)S_{\alpha}(0)$
$\displaystyle\ \ \ \ +h_{\alpha}(n+2)[S_{\alpha}(1)-\alpha
S_{\alpha}(0)]+\Big{[}S_{\alpha}(2)-\alpha S_{\alpha}(1)$ $\displaystyle\ \ \
\ +\frac{\alpha(\alpha-1)}{2}S_{\alpha}(0)\Big{]}h_{\alpha}(n+1)$
$\displaystyle=(h_{\alpha}*\Delta^{\alpha}S_{\alpha})(n)+h_{\alpha}(n+3)+(1-\alpha)h_{\alpha}(n+2)$
$\displaystyle\ \ \ \ +\frac{(\alpha-1)(\alpha-2)}{2}h_{\alpha}(n+1)$
$\displaystyle=A(h_{\alpha}*S_{\alpha})(n)+\gamma(h_{\alpha}*S_{\alpha}^{\lambda})(n)$

when $n\in\mathbb{N}_{0}$. This, incorporating Lemma 3.4 and Lemma 3.6,
further deduces that

(3.16) $\displaystyle\Delta^{\alpha}\big{(}h_{\alpha}*S_{\alpha}*f\big{)}(n)$
$\displaystyle=(\Delta^{\alpha}(h_{\alpha}*S_{\alpha})*f)(n)+(h_{\alpha}*S_{\alpha})(0)f(n+3)$
$\displaystyle\ \ \ \
+\big{[}(h_{\alpha}*S_{\alpha})(1)-\alpha(h_{\alpha}*S_{\alpha})(0)\big{]}f(n+2)$
$\displaystyle\ \ \ \
+\big{[}(h_{\alpha}*S_{\alpha})(2)-\alpha(h_{\alpha}*S_{\alpha})(1)$
$\displaystyle\ \ \ \
+\frac{\alpha(\alpha-1)}{2}(h_{\alpha}*S_{\alpha})(0)\big{]}f(n+1)$
$\displaystyle=(\Delta^{\alpha}(h_{\alpha}*S_{\alpha})*f)(n)+f(n+3)$
$\displaystyle=A(h_{\alpha}*S_{\alpha}*f)(n)+\gamma(h_{\alpha}*S_{\alpha}^{\lambda}*f)(n)$
$\displaystyle\ \ \ \ +f(n+3)$

when $n\in\mathbb{N}_{0}$.

Notice that

$\displaystyle(h_{\alpha}*S_{\alpha}^{\lambda}*f)(n)$
$\displaystyle=\sum_{j=0}^{n}S_{\alpha}^{\lambda}(n-j)(h_{\alpha}*f)(j)$
$\displaystyle=\sum_{j=0}^{n-\lambda}S_{\alpha}(n-\lambda-j)(h_{\alpha}*f)(j)+\sum_{j=n-\lambda+1}^{n}S_{\alpha}(n-\lambda-j)(h_{\alpha}*f)(j)$
$\displaystyle=\sum_{j=0}^{n-\lambda}S_{\alpha}(n-\lambda-j)(h_{\alpha}*f)(j)$
$\displaystyle=(h_{\alpha}*S_{\alpha}*f)(n-\lambda)$

when $n\in\mathbb{N}_{0}$ as $S_{\alpha}(-i)=0$ when $i=1,\cdots,\lambda$.
This combined with (3.16) implies that

(3.17) $\displaystyle\Delta^{\alpha}\big{(}h_{\alpha}*S_{\alpha}*f\big{)}(n)$
$\displaystyle=A(h_{\alpha}*S_{\alpha}*f)(n)+\gamma(h_{\alpha}*S_{\alpha}*f)(n-\lambda)$
$\displaystyle\ \ \ \ +f(n+3)$

when $n\in\mathbb{N}_{0}$.

Let $u:\mathbb{N}\to X$ be defined by $u(0)=u(1)=u(2)=0$ and
$u(n)=(h_{\alpha}*S_{\alpha}*f)(n-3)$ when $n\geq 3$. We claim that

(3.18) $\Delta^{\alpha}u(n)=\Delta^{\alpha}(h_{\alpha}*S_{\alpha}*f)(n-3)$

when $n\geq 3$. Indeed, since $u(0)=u(1)=u(2)=0$, we have

$\displaystyle\Delta^{-(3-\alpha)}u(n)$
$\displaystyle=(k^{3-\alpha}*u)(n)=\sum_{j=0}^{n}k^{3-\alpha}(j)u(n-j)$
$\displaystyle=\sum_{j=0}^{n-3}k^{3-\alpha}(j)(h_{\alpha}*S_{\alpha}*f)(n-3-j)$
$\displaystyle=\Delta^{-(3-\alpha)}(h_{\alpha}*S_{\alpha}*f)(n-3)$

when $n\geq 3$, which clearly implies that (3.18) holds as
$\Delta^{\alpha}=\Delta^{3}\Delta^{-(3-\alpha)}$ by (2.6).

Therefore, (3.17) and (3.18) together deduce that

(3.19) $\Delta^{\alpha}u(n)=Au(n)+\gamma u(n-\lambda)+f(n)$

when $n\geq 3$ as $u(i)=0$ when $i=-\lambda,-\lambda+1,\cdots,2$. This means
that $u$ is a solution of (1.1) when $n\geq 3$.

Next we show the equality (3.19) also holds when $0\leq n\leq 2$. Indeed, It
follows from (3.14) that

(3.20) $\displaystyle u(3)$
$\displaystyle=(h_{\alpha}*S_{\alpha}*f)(0)=(h_{\alpha}*S_{\alpha})(0)f(0)$
$\displaystyle=S_{\alpha}(0)h_{\alpha}(0)f(0)=f(0);$ (3.21) $\displaystyle
u(4)$
$\displaystyle=(h_{\alpha}*S_{\alpha}*f)(1)=(h_{\alpha}*S_{\alpha})(1)f(0)+(h_{\alpha}*S_{\alpha})(0)f(1)$
$\displaystyle=[S_{\alpha}(0)h_{\alpha}(1)+S_{\alpha}(1)h_{\alpha}(0)]f(0)+S_{\alpha}(0)h_{\alpha}(0)f(1)$
$\displaystyle=\alpha f(0)+f(1)$ (3.22) $\displaystyle u(5)$
$\displaystyle=(h_{\alpha}*S_{\alpha}*f)(2)=(h_{\alpha}*S_{\alpha})(2)f(0)$
$\displaystyle\ \ \ \
+(h_{\alpha}*S_{\alpha})(1)f(1)+(h_{\alpha}*S_{\alpha})(0)f(2)$
$\displaystyle=[h_{\alpha}(2)S_{\alpha}(0)+h_{\alpha}(1)S_{\alpha}(1)+h_{\alpha}(0)S_{\alpha}(2)]f(0)$
$\displaystyle\ \ \
+[h_{\alpha}(1)S_{\alpha}(0)+h_{\alpha}(0)S_{\alpha}(1)]f(1)+h_{\alpha}(0)S_{\alpha}(0)f(2)$
$\displaystyle=\frac{\alpha(\alpha+1)}{2}f(0)+\alpha f(1)+f(2).$

Thus, using the conditions $u(0)=u(1)=u(2)=0$ and (3.20)-(3.22), we obtain

$\displaystyle\Delta^{\alpha}u(0)$
$\displaystyle=\Delta^{3}\Delta^{-(3-\alpha)}u(0)=(k^{3-\alpha}*u)(3)=u(3)=f(0)$
$\displaystyle\Delta^{\alpha}u(1)$
$\displaystyle=\Delta^{3}\Delta^{-(3-\alpha)}u(1)=(k^{3-\alpha}*u)(4)-3(k^{3-\alpha}*u)(3)$
$\displaystyle=k^{3-\alpha}(0)u(4)+k^{3-\alpha}(1)u(3)-3k^{3-\alpha}(0)u(3)$
$\displaystyle=u(4)-\alpha u(3)=f(1)$ $\displaystyle\Delta^{\alpha}u(2)$
$\displaystyle=\Delta^{3}\Delta^{-(3-\alpha)}u(2)=(k^{3-\alpha}*u)(5)-3(k^{3-\alpha}*u)(4)+3(k^{3-\alpha}*u)(3)$
$\displaystyle=k^{3-\alpha}(0)u(5)+k^{3-\alpha}(1)u(4)+k^{3-\alpha}(2)u(3)-3k^{3-\alpha}(0)u(4)$
$\displaystyle\ \ \ \ -3k^{3-\alpha}(1)u(3)+3k^{3-\alpha}(0)u(3)$
$\displaystyle=u(5)-\alpha u(4)+\frac{\alpha(\alpha-1)}{2}u(3)=f(2).$

Therefore, $\Delta^{\alpha}u(n)=Au(n)+\gamma u(n-\lambda)+f(n)$ when $0\leq
n\leq 2$ as $u(i)=0$ when $i=-\lambda,-\lambda+1,\cdots,2$. We have shown that
$u\in S(\mathbb{N}_{0};X)$ given by (3.14) is a solution of (1.1).

It remains to show that the solution is unique. Clearly, we only need to show
that $0\in S(\mathbb{N}_{0};X)$ is the unique solution of the following
homogeneous equation

(3.23) $\left\\{\begin{array}[]{ll}\Delta^{\alpha}u(n)=Au(n)+\gamma
u(n-\lambda),\ (n\in\mathbb{N}_{0})\\\ u(i)=0~{}\text{
when}~{}i=-\lambda,-\lambda+1,\cdots,2.\end{array}\right.$

Let $u\in S(\mathbb{N}_{0};X)$ be a solution of equation (3.23). We first show
that $u(3)=0$. The identity $\Delta^{\alpha}u(0)=Au(0)+\gamma u(-\lambda)$
implies that $\Delta^{\alpha}u(0)=0$. On the other hand,

$\displaystyle\Delta^{\alpha}u(0)$
$\displaystyle=\Delta^{3}\Delta^{-(3-\alpha)}u(0)$
$\displaystyle=\Delta^{-(3-\alpha)}u(3)-3\Delta^{-(3-\alpha)}u(2)+3\Delta^{-(3-\alpha)}u(1)+\Delta^{\alpha-3}u(0)$
$\displaystyle=(k^{3-\alpha}*u)(3)-3(k^{3-\alpha}*u)(2)+3(k^{3-\alpha}*u)(1)+(k^{3-\alpha}*u)(0)$
$\displaystyle=k^{3-\alpha}(0)u(3)=u(3)$

by the assumption $u(0)=u(1)=u(2)=0$. Consequently, $u(3)=0$

Assume that $u(n)=0$ for all $3\leq n\leq k$ for some $k\geq 3$, we are going
to show that $u(k+1)=0$. Since $k-2<k$, we have $u(k-2)=0$ by assumption. Thus
$\Delta^{\alpha}u(k-2)=Au(k-2)+\gamma u(k-2-\lambda)=0$. On the other hand,

$\displaystyle\Delta^{\alpha}u(k-2)$
$\displaystyle=\Delta^{3}\Delta^{-(3-\alpha)}u(k-2)$
$\displaystyle=\Delta^{-(3-\alpha)}u(k+1)-3\Delta^{-(3-\alpha)}u(k)+3\Delta^{-(3-\alpha)}u(k-1)+\Delta^{-(3-\alpha)}u(k-2)$
$\displaystyle=k^{3-\alpha}*u(k+1)-3k^{3-\alpha}*u(k)+3k^{3-\alpha}*u(k-1)+k^{3-\alpha}*u(k-2)$
$\displaystyle=k^{3-\alpha}(0)u(k+1)=u(k+1).$

by assumption $u(k-2)=u(k-1)=u(k)=0$. Consequently, $u(k+1)=0$ when $3\leq
n\leq k$. Therefore, $u(n)=0$ for all $n\in\mathbb{N}_{0}$. This completes the
proof. ∎

##  4\. A Characterization of the $\ell^{p}$-Maximal Regularity

Let $A\in B(X)$ and $f\in S(\mathbb{N}_{0};X)$, where $X$ is a Banach space.
In this section, we study the $\ell^{p}$-maximal regularity for the fractional
difference equation with delay $\lambda\in\mathbb{N}$:

(4.1) $\left\\{\begin{array}[]{ll}&\Delta^{\alpha}u(n)=Au(n)+\gamma
u(n-\lambda)+f(n),\ (n\in\mathbb{N}_{0})\\\
&u(i)=0~{}\text{when}~{}i=-\lambda,-\lambda+1,\cdots,2,\end{array}\right.$

where $1<p<\infty$, $2<\alpha<3$ and $\gamma\in\mathbb{R}$.

Let $f\in S(\mathbb{N}_{0};X)$ be given. By Theorem 3.8, the unique solution
of (4.1) can be represented by $u(0)=u(1)=u(2)=0$ and

(4.2) $u(n)=(h_{\alpha}*S_{\alpha}*f\big{)}(n-3)$

when $n\geq 3$. This means that $\Delta^{\alpha}u(0)=f(0)$,
$\Delta^{\alpha}u(1)=f(1)$, $\Delta^{\alpha}u(2)=f(2)$ and

(4.3) $\displaystyle\Delta^{\alpha}u(n)$
$\displaystyle=A(h_{\alpha}*S_{\alpha}*f\big{)}(n-3)+\gamma(h_{\alpha}*S_{\alpha}*f\big{)}(n-3-\lambda)+f(n)$
$\displaystyle=A(h_{\alpha}*S_{\alpha}*f\big{)}(n-3)+\gamma(h_{\alpha}*S_{\alpha}^{\lambda}*f\big{)}(n-3)+f(n)$

when $n\geq 3$ as $u(i)=0$ when $i=-\lambda,-\lambda+1,\cdots,2$. In analogy
to cases $\alpha=1$ and $\alpha=2$ (see for instance [5]), we introduce the
following definition concerning maximal regularity.

######  Definition 4.1.

Let $1<p<\infty$, $2<\alpha<3$, $\lambda\in\mathbb{N}$ and let $A\in B(X)$ be
given. We say that (4.1) has the $\ell^{p}$-maximal regularity if

$(\mathcal{E}_{\alpha}f)(n):=A(h_{\alpha}*S_{\alpha}*f)(n)=A\sum_{j=0}^{n}(h_{\alpha}*S_{\alpha})(n-j)f(j)$
$(\mathcal{F}_{\alpha}f)(n):=(h_{\alpha}*S_{\alpha}^{\lambda}*f)(n)=\sum_{j=0}^{n}(h_{\alpha}*S_{\alpha}^{\lambda})(n-j)f(j)$

when $n\in\mathbb{N}_{0}$, define two bounded linear operators
$\mathcal{E}_{\alpha},\mathcal{F}_{\alpha}$ on $\ell^{p}(\mathbb{N}_{0};X)$.

It is easy to see that when (4.1) has the $\ell^{p}$-maximal regularity, then
for all $f\in\ell^{p}(\mathbb{N}_{0};X)$, the unique solution $u$ of (4.1)
given by (4.2) satisfies $\Delta^{\alpha}u\in\ell^{p}(\mathbb{N}_{0};X)$ by
(4.3). We will need the following hypothesis:

$(\mathcal{C}_{\alpha}):~{}~{}~{}~{}\sup_{n\in\mathbb{N}_{0}}\big{\|}S_{\alpha}(n)\big{\|}<\infty~{}~{}{\rm
and}~{}~{}z^{3-\alpha}(z-1)^{\alpha}-\gamma z^{-\lambda}\in\rho(A)~{}~{}{\rm
for~{}~{}all}~{}~{}|z|=1,\ z\not=\pm 1.$

Now we are ready to state the main result of this section.

######  Theorem 4.2.

Let $X$ be a UMD space, $2<\alpha<3$, $\gamma\in\mathbb{R}$,
$\lambda\in\mathbb{N}$ and let $1<p<\infty$. Assume that $A\in B(X)$ and the
assumption $(\mathcal{C}_{\alpha})$ holds. Then the following statements are
equivalent:

  1. $(i)$

$(4.1)$ has the $\ell^{p}$-maximal regularity;

  2. $(ii)$

the sets

(4.4)
$\big{\\{}e^{(3-\alpha)it}(e^{it}-1)^{\alpha}\big{[}e^{(3-\alpha)it}(e^{it}-1)^{\alpha}-\gamma
e^{-\lambda it}-A\big{]}^{-1}:\ t\in\mathbb{T}\big{\\}}$

and

(4.5) $\big{\\{}e^{-\lambda it}[e^{(3-\alpha)it}(e^{it}-1)^{\alpha}-\gamma
e^{-\lambda it}-A\big{]}^{-1}:\ t\in\mathbb{T}\big{\\}}$

are $R$-bounded.

###### Proof.

We first prove that the implication (ii) $\Rightarrow$ (i) is true. Let

$\displaystyle g_{\alpha}(t)$ $\displaystyle=e^{3it}(1-e^{-it})^{\alpha},\
G_{1}(t)=g_{\alpha}(t)\big{[}g_{\alpha}(t)-\gamma e^{-\lambda
it}-A\big{]}^{-1},$ $\displaystyle\ G_{2}(t)=e^{-\lambda
it}\big{[}g_{\alpha}(t)-\gamma e^{-\lambda it}-A\big{]}^{-1}$

when $t\in\mathbb{T}$. Then

$\displaystyle g_{\alpha}^{\prime}(t)=3ig_{\alpha}(t)+\frac{\alpha
ig_{\alpha}(t)}{e^{it}-1}=(3i+\frac{\alpha i}{e^{it}-1})g_{\alpha}(t)$
$\displaystyle G_{1}^{\prime}(t)=(3i+\frac{\alpha
i}{e^{it}-1})\Big{(}G_{1}(t)-G_{1}^{2}(t)\Big{)}-\gamma\lambda
iG_{1}(t)G_{2}(t)$ $\displaystyle G_{2}^{\prime}(t)=-\gamma
iG_{2}(t)-(3i+\frac{\alpha i}{e^{it}-1})G_{1}(t)G_{2}(t)-\gamma\lambda
iG_{2}^{2}(t)$

when $t\in\mathbb{T}$. Thus the sets

$\big{\\{}(e^{it}-1)(e^{it}+1)G_{1}^{\prime}(t):\ t\in\mathbb{T}\big{\\}}$

and

$\big{\\{}(e^{it}-1)(e^{it}+1)G_{2}^{\prime}(t):\ t\in\mathbb{T}\big{\\}}$

are $R$-bounded by assumption and Remark 2.3. By Theorem 2.4, there exist
$T_{\alpha},\ L_{\alpha}\in B(\ell^{p}(\mathbb{Z};X))$ such that

(4.6) $\widehat{T_{\alpha}f}(e^{it})=G_{1}(t)\hat{f}(e^{it})$

and

(4.7) $\widehat{L_{\alpha}f}(e^{it})=G_{2}(t)\hat{f}(e^{it})$

when $t\in\mathbb{T}$ for all $f\in\ell^{p}(\mathbb{Z};X)$ with finite
support. From (4.6), (4.7) and the trivial identity

$A\big{[}g_{\alpha}(t)-\gamma e^{-\lambda
it}-A\big{]}^{-1}=(g_{\alpha}(t)-\gamma e^{-\lambda
it})\big{[}g_{\alpha}(t)-\gamma e^{-\lambda it}-A\big{]}^{-1}-I$

we deduce that

(4.8) $\displaystyle A\big{[}g_{\alpha}(t)-\gamma e^{-\lambda
it}-A\big{]}^{-1}\hat{f}(e^{it})$ $\displaystyle=(g_{\alpha}(t)-\gamma
e^{-\lambda it})\big{[}g_{\alpha}(t)$ $\displaystyle\ \ \ \ -\gamma
e^{-\lambda it}-A\big{]}^{-1}\hat{f}(e^{it})-\hat{f}(e^{it})$

defines a bounded linear operator $K_{\alpha}\in B(\ell^{p}(\mathbb{Z};X))$
given by $K_{\alpha}f(n)=T_{\alpha}f(n)-\gamma L_{\alpha}f(n)-f(n)$ for all
$f\in\ell^{p}(\mathbb{Z};X)$ with finite support.

It follows from Remark 3.3 and (3.13) that

(4.9)
$\displaystyle\widehat{h_{\alpha}*S_{\alpha}}(e^{it})=e^{3it}\big{[}g_{\alpha}(t)-\gamma
e^{-\lambda it}-A\big{]}^{-1},$
$\displaystyle\widehat{S_{\alpha}^{\lambda}*h_{\alpha}}(e^{it})=e^{-\lambda
it}\big{[}g_{\alpha}(t)-\gamma e^{-\lambda it}-A\big{]}^{-1}$

when $t\in\mathbb{T}$. These together with (4.8) imply that

(4.10)
$\widehat{(\mathcal{E}_{\alpha}f)}(e^{it})=e^{3it}A\big{[}g_{\alpha}(t)-\gamma
e^{-\lambda it}-A\big{]}^{-1}\hat{f}(e^{it})$

and

(4.11) $\widehat{(\mathcal{F}_{\alpha}f)}(e^{it})=e^{-\lambda
it}\big{[}g_{\alpha}(t)-\gamma e^{-\lambda
it}-A\big{]}^{-1}\hat{f}(e^{it})=\widehat{L_{\alpha}f}(e^{it})$

when $t\in\mathbb{T}$ for all $f\in\ell^{p}(\mathbb{N}_{0};X)$ with finite
support. Hence $\mathcal{E}_{\alpha}$ and $\mathcal{F}_{\alpha}$ are bounded
linear operators on $\ell^{p}(\mathbb{N}_{0};X)$ by (4.7) and (4.8),
respectively. Here we have used the fact that the set of all
$f\in\ell^{p}(\mathbb{N}_{0};X)$ with finite support is dense in
$\ell^{p}(\mathbb{N}_{0};X)$. We have shown that (4.1) has $\ell^{p}$-maximal
regularity. Hence the implication (ii) $\Rightarrow$ (i) is valid.

Now assume that (i) holds true. Then

$(\mathcal{E}_{\alpha}f)(n)=\left\\{\begin{aligned}
&A(h_{\alpha}*S_{\alpha}*f)(n),\ (n\geq 3);\\\ &0,\ \ \ \ \
\text{otherwise}.\end{aligned}\right.$

and

$(\mathcal{F}_{\alpha}f)(n)=\left\\{\begin{aligned}
&(h_{\alpha}*S_{\alpha}^{\lambda}*f)(n),\ (n\geq 3);\\\ &0,\ \ \ \ \ \
\text{otherwise}.\end{aligned}\right.$

define two bounded linear operators
$\mathcal{E}_{\alpha},\mathcal{F}_{\alpha}\in B(\ell^{p}(\mathbb{N}_{0};X))$.
It follows from (4.10)-(4.11) that

$\displaystyle\widehat{(\mathcal{E}_{\alpha}f)}(e^{it})=e^{3it}T\big{[}g_{\alpha}(t)-\gamma
e^{-\lambda it}-A\big{]}^{-1}\hat{f}(e^{it}),$
$\displaystyle\widehat{(\mathcal{F}_{\alpha}f)}(e^{it})=e^{-\lambda
it}\big{[}g_{\alpha}(t)-\gamma e^{-\lambda it}-A\big{]}^{-1}\hat{f}(e^{it})$

when $t\in\mathbb{T}$ for all $f\in\ell^{p}(\mathbb{N}_{0};X)$ with finite
support. Since the convolution operators $\mathcal{E}_{\alpha}$ and
$\mathcal{F}_{\alpha}$ are translation invariant on $\ell^{p}(\mathbb{Z};X)$,
$\mathcal{E}_{\alpha}$ and $\mathcal{F}_{\alpha}$ can extend to bounded linear
operators on $\ell^{p}(\mathbb{Z};X)$. It follows from (2.9) that the
functions $t\to e^{3it}A\big{[}g_{\alpha}(t)-\gamma e^{-\lambda
it}-A\big{]}^{-1}$ and $t\to e^{-\lambda it}\big{[}g_{\alpha}(t)-\gamma
e^{-\lambda it}-A\big{]}^{-1}$ are $\ell^{p}$-Fourier multiplier.

Thus the sets

$\big{\\{}e^{3it}A\big{[}g_{\alpha}(t)-\gamma e^{-\lambda it}-A\big{]}^{-1}:\
t\in\mathbb{A}\big{\\}}~{}{\rm and}~{}\big{\\{}e^{-\lambda
it}\big{[}g_{\alpha}(t)-\gamma e^{-\lambda it}-A\big{]}^{-1}:\
t\in\mathbb{T}\big{\\}}$

are $R$-bounded by Theorem 2.5.

This combined with Remark 2.3 and the trivial equality

$g_{\alpha}(t)\big{[}g_{\alpha}(t)-\gamma e^{-\lambda
it}-A\big{]}^{-1}=T\big{[}g_{\alpha}(t)-\gamma e^{-\lambda
it}-A\big{]}^{-1}+\gamma e^{-\lambda it}\big{[}g_{\alpha}(t)-\gamma
e^{-\lambda it}-A\big{]}^{-1}+I$

implies that the set

$\big{\\{}g_{\alpha}(t)\big{[}g_{\alpha}(t)-\gamma e^{-\lambda
it}-A\big{]}^{-1}:\ t\in\mathbb{T}\big{\\}}$

is $R$-bounded. This completes the proof. ∎

Since the second condition in Theorem 4.2 does not depend on the parameter
$1<p<\infty$, we have the following immediate consequence.

######  Corollary 4.3.

Let $X$ be a UMD space, $2<\alpha<3$, $\gamma\in\mathbb{R}$ and
$\lambda\in\mathbb{N}$. Assume that $A\in B(X)$ and the assumption
$(\mathcal{C}_{\alpha})$ holds. If $(4.1)$ has the $\ell^{p}$-maximal
regularity for some $1<p<\infty$, then it has the $\ell^{p}$-maximal
regularity for all $1<p<\infty$.

If the underlying Banach space $X$ is a Hilbert space, then the
$R$-boundedness coincides with the norm boundedness [3]. This together with
Theorem 4.2 give the following result.

######  Corollary 4.4.

Let $H$ be a Hilbert space, $\gamma\in\mathbb{R}$, $\lambda\in\mathbb{N}$ and
let $1<p<\infty$. Assume that $A\in B(X)$ and the assumption
$(\mathcal{C}_{\alpha})$ holds. Then $(4.1)$ has the $\ell^{p}$-maximal
regularity if and only if there exists $C>0$ such that

$\sup_{t\in\mathbb{T}}\big{\|}e^{(3-\alpha)it}(e^{it}-1)^{\alpha}\big{[}e^{(3-\alpha)it}(e^{it}-1)^{\alpha}-\gamma
e^{-\lambda it}-A\big{]}^{-1}\big{\|}<\infty$
$\sup_{t\in\mathbb{T}}\big{\|}e^{-\lambda
it}[e^{(3-\alpha)it}(e^{it}-1)^{\alpha}-\gamma e^{-\lambda
it}-A\big{]}^{-1}\big{\|}<\infty$

In the following example, served as an application of Theorem 4.1, we provide
a criterion that ensures the $\ell^{p}$-maximal regularity of equation $(4.1)$
under a suitable assumption on the operator $A$. We observe that this example
was studied by many authors [15, 16] in the case $0<\alpha<2$. The argument
used in [15, 16] works as well in the case $2<\alpha<3$.

######  Example 4.5.

Let $1<p<\infty$, $2<\alpha<3$, $\gamma\in\mathbb{R}$, $\lambda\in\mathbb{N}$.
Let $H$ be a Hilbert space and $A\in B(H)$ satisfying the following condition

$(\mathfrak{C})\ \ \ \
\|A\|<\omega_{f}:=\min_{t\in\mathbb{T}}|f_{\alpha,\gamma,\lambda}(t)|<1$

where $f_{\alpha,\gamma,\lambda}(t)=e^{(3-\alpha)it}(e^{it}-1)^{\alpha}-\gamma
e^{-\lambda it}$. It follows form the condition $\mathfrak{C}$ and Theorem
7.3-4 in [10] that $f_{\alpha,\gamma,\lambda}\in\rho(A)$,

$(f_{\alpha,\gamma,\lambda}-A)^{-1}=\sum_{j=0}^{\infty}\frac{A^{j}}{\Big{(}f_{\alpha,\gamma,\lambda}(t)\Big{)}^{j+1}}~{}~{}~{}{\rm
and}~{}~{}~{}\|(f_{\alpha,\gamma,\lambda}-A)^{-1}\|\leq\frac{1}{\omega_{f}-\|A\|}$

when $t\in\mathbb{T}$.

Again using condition $\mathfrak{C}$, there exists a circle $\Gamma$ centered
at the origin of the complex plane with radius $r<1$ such that

$S_{\alpha}(n)=\left\\{\begin{aligned} &\frac{1}{2\pi
i}\int_{\|z\|=r}z^{n}\big{[}z^{2}+(1-\alpha)z+\frac{(\alpha-1)(\alpha-2)}{2}\big{]}\big{[}z^{3-\alpha}(z-1)^{\alpha}\\\
&\ \ \ \ \ -\gamma z^{-\lambda}-A\big{]}^{-1}dz,\ (n\geq 3)\\\
&I,~{}\text{when}~{}n=0,1,2,\\\
&0,~{}\text{when}~{}n=-\lambda,-\lambda+1,\cdots,-1.\end{aligned}\right.$

It follows that

$\|S_{\alpha}(n)\|<\frac{4}{\omega_{f}-\|A\|}$

when $n\in\mathbb{N}_{0}$. It is easy to verify that the sets

$\big{\\{}e^{(3-\alpha)it}(e^{it}-1)^{\alpha}\big{[}e^{(3-\alpha)it}(e^{it}-1)^{\alpha}-\gamma
e^{-\lambda it}-A\big{]}^{-1}:\ t\in\mathbb{T}\big{\\}}$

and

$\big{\\{}e^{-\lambda it}[e^{(3-\alpha)it}(e^{it}-1)^{\alpha}-\gamma
e^{-\lambda it}-A\big{]}^{-1}:\ t\in\mathbb{T}\big{\\}}$

are bounded. Hence (4.1) has the $\ell^{p}$-maximal regularity by Corollary
4.4.

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