\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2016 (2016), No. 270, pp. 1--12.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2016 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2016/270\hfil Weighted pseudo almost automorphic solutions]
{Weighted pseudo almost automorphic  and $\mathcal{S}$-asymptotically
$\omega$-periodic solutions to fractional
difference-differential equations}

\author[E. Alvarez, C. Lizama \hfil EJDE-2016/270 \hfilneg]
{Edgardo Alvarez, Carlos Lizama}

\address{Edgardo Alvarez \newline
Universidad del Norte,
Departamento de Matem\'aticas y Estad\'istica, Barranquilla, Colombia}
\email{ealvareze@uninorte.edu.co}

\address{Carlos Lizama \newline
Universidad de Santiago de Chile, Facultad de Ciencia,
Departamento de Matem\'atica y Ciencia de la Computaci\'on,
Las Sophoras 173, Estaci\'on Central, Santiago, Chile}
\email{carlos.lizama@usach.cl}

\thanks{Submitted July 5, 2016. Published October 7, 2016.}
\subjclass[2010]{32N05, 65Q10,  47B39}
\keywords{Weyl-like fractional difference; fractional difference equation;
\hfill\break\indent weighted pseudo almost automorphic sequence; 
 $\alpha$-resolvent sequences of operators}

\begin{abstract}
 We study weighted pseudo almost automorphic solutions for the nonlinear
 fractional difference equation
 $$
 \Delta^{\alpha}u(n)=Au(n+1)+f(n, u(n)),\quad n\in \mathbb{Z},
 $$
 for $0<\alpha \leq 1$, where $A$ is the generator of an $\alpha$-resolvent
 sequence $\{S_{\alpha}(n)\}_{n\in\mathbb{N}_0}$ in $\mathcal{B}(X)$.
 We prove the existence and uniqueness of a weighted pseudo almost automorphic
 solution assuming that $f(\cdot, \cdot)$ is weighted almost automorphic
 in the first variable and satisfies a Lipschitz (local and global)
 type condition in the second variable. An analogous result is also proved
 for $\mathcal{S}$-asymptotically $\omega$-periodic solutions.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{example}[theorem]{Example}
\allowdisplaybreaks

\section{Introduction}

In this article, we study  sufficient conditions for the existence and uniqueness 
of discrete weighted pseudo almost automorphic  solutions to the  
semilinear fractional difference - differential equation
\begin{equation}\label{maineq}
\Delta^{\alpha}u(n)=Au(n+1)+f(n, u(n)),\quad \,n\in \mathbb{Z},
\end{equation}
where $0<\alpha \leq 1$, $A$ is a closed  linear operator with domain $D(A)$ 
defined on a Banach space $X$ which generates
an $\alpha$-resolvent sequence $\{S_{\alpha}(n)\}_{n\in\mathbb{N}_0}\subset \mathcal{B}(X)$
and $f:\mathbb{Z}\times X\to X$ is a discrete weighted pseudo almost automorphic function 
in $k\in \mathbb{Z}$ satisfying suitable Lipschitz type conditions with respect 
to $x\in X$. The fractional difference is understood
in the sense defined in \cite{ab-li16}, which is analogous to the Weyl 
fractional derivative in the continuous case. See Definition \ref{def2.4} below.
Difference-differential equations appear in many practical situations, 
for instance in traffic dynamics, theory of probability, theory of
chain processes of chemistry, radioactivity  and in biological models, 
see e.g. \cite{At-Se, Ba, Br-Sa}.
Nonlinear difference equations has been studied by several authors, see 
e.g. \cite{Ab12,Araya,Diagana,GoAMC11,Go14, Li15, Li-Mu16}. 
First studies on extensions of the notion of almost automorphic sequences 
are due to  Fink \cite{Fink}.
The concept of discrete weighted pseudo almost automorphic functions  was
introducing by  Abbas \cite{abbas10} in 2010 as a further generalization 
of almost automorphic sequences.   Agarwal et al.\ \cite{Aga-Cue-Da} obtained 
almost automorphic solutions
to a nonlinear Volterra difference equation.  Ding et al.\ \cite{Ding-Gue-Ni} 
studied the weighted pseudo almost periodic solutions for a class of discrete 
hematopoiesis model. Xia \cite{Xia14} extended the space of weighted pseudo 
almost automorphic sequences with the help of two weights, proved fundamental 
properties of this type of functions and apply his results to Volterra difference 
equations. Then,  Li and  Sun proved applications to some semilinear difference 
equations \cite{Li-Sun11}.  Abbas et al.\ defined the concept of Stepanov
type weighted pseudo almost automorphic sequences and proved an important 
composition theorem \cite{abbas-Chang-Ha14}.  An interesting application 
of almost automorphic sequences to a model of a cellular neural network was  
shown by  Abbas \cite{Ab14}. For a recent application to  discrete delayed 
neutral systems, see \cite{Ad-Ko16}. For other developments, see \cite{Ca-Pi}.

Recently,  Abadias and  Lizama \cite{ab-li16} proved the
existence and uniqueness of almost automorphic solutions for 
 \eqref{maineq}, where $A$ is the generator of a $C_0$-semigroup. 
There the authors introduced the concept of fractional difference in the Weyl-like
sense and the notion of $\alpha$-resolvent sequence.

Our motivation for this article stems from the fact that equations of type 
\eqref{maineq} can arise in many problems of science
and engineering either directly or as part of a discretization process 
\cite{Kac, Ko-Li-Lu}  and that the study of weighted pseudo almost automorphic 
solutions for fractional difference-differential equations does not exist at this 
time. Since the qualitative behavior of the solutions is crucial in order to 
better understand the underlying structure, the existence  of weighted pseudo 
almost automorphic solutions for such equations seems to be highly important. 
 We observe that the research on these properties for fractional difference 
equations is in its early stages. For some limited results we refer the reader 
in particular to \cite{Li15}, \cite{Li-Mu16} and \cite{Li-Ve16}.

In this article we prove the following new results:
 Let $0<\alpha<1$ and $A$ be the generator of an exponentially stable  
$C_0$-semigroup $T(t)$ with growth bound $\omega_0(A)$. Suppose that
 $f:\mathbb{Z}\times X \to X$ can be decomposed as $f=g+\varphi$ where 
$f$ is discrete almost automorphic and $\varphi$ is weight mean ergodic. 
Assume also that $f$  is globally Lipschitz with  constant satisfying the 
estimate $L< \frac{1}{\omega_0(A)}$. Then, there exists a unique solution 
in a mild sense, that can also be decomposed as $u=v+ \nu $ where $v$ and 
$\nu$ have the same regularity as $g$ and $\varphi$, respectively. 
See Corollary \ref{cor3.3} for a precise description of this result. 
Using the same methods, we obtain an analogous result on the regularity of 
solutions for those data $f$ with the property that there exists an integer 
number $\omega$ such that $f(n+\omega)-f(n)$ goes to zero in weighted mean. 
Nonlocal versions of this results  are also provided. See Theorem \ref{th5.4} 
and Corollary \ref{cor3.5}. In order to prove this kind of results, we  
introduce a new convolution theorem, which provides regularity for the linearized 
version of \eqref{maineq}. See Theorem \ref{conv1}.

This article is organized as follows: Section 2 is devoted to preliminaries,
 where we prove a convolution theorem which is new in the context of 
$\alpha$-resolvent families associated with the operator that appears in 
 \eqref{maineq}.  In Section 3 we present our  main results 
on the  existence and uniqueness  of weighted pseudo almost automorphic 
and $\mathcal{S}$-asymptotically $\omega$-periodic solutions. 
Finally, in Section 4 we give a concrete example to illustrate the main findings.

\section{Preliminaries}

In this section, we give the basic definitions and essential results that 
we will be used later.
We first introduce the following spaces of sequences.

\begin{itemize}
\item[(i)] $s(\mathbb{Z},X)$ is the vector space of all vector valued sequences.
\item[(ii)] $BS(\mathbb{Z},X):=\{f:\mathbb{Z}\to X:\|f\|_{\infty}:=\sup_{n\in\mathbb{Z}}\|f(n)\|<\infty\}$.
\item[(iii)] $l^p_{\rho}(\mathbb{Z},X):=\big\{f:\mathbb{Z}\to X:\|f\|_{l^p_{\rho}}
:=\sum_{n=-\infty}^{\infty}\| f(n)\|^p \rho(n) <\infty\big\}$
where $\rho: \mathbb{Z} \to (0, \infty)$ is a positive sequence.
\item[(iv)] $C_0(\mathbb{Z},X):=\{f\in BS(\mathbb{Z},X):\lim_{n\to\infty}\|f(n)\|=0\}$.
\item[(v)] $C_{\omega}(\mathbb{Z},X)=\{f\in BS(\mathbb{Z},X):\text{$f$ is $\omega$-periodic}\}$.
\item[(vi)] $\mathfrak{U}\mathfrak{C}(\mathbb{Z}\times X,X)$ is the set of all functions
$f:\mathbb{Z}\times X\to X$  satisfying that for all $\epsilon>0$ there exists
$\delta>0$ such that
  \begin{equation*}
    \|f(k,x)-f(k,y)\|\leq\epsilon
  \end{equation*}
for all $k\in \mathbb{Z}$ and for all $x,y\in X$ with $\|x-y\|<\delta$.
\end{itemize}

Let $U$ be the set of all
sequences $\rho:\mathbb{Z}\to(0,\infty)$ which are locally summable over $\mathbb{Z}$.
For  a given $\rho \in U$ and $K\in \mathbb{Z}_{+}$ we denote
$$
m_d(K,\rho) =\sum_{k=-K}^{K}\rho(k).
    $$
  Define
\begin{gather*}
    U_{\infty}=\{\rho\in U:\lim_{K\to\infty}m_d(K,\rho)  =\infty\}, \\
U_{b}=\{\rho\in U_{\infty}:0<\inf_{k\in\mathbb{Z}}\rho(k)\leq\sup_{k\in\mathbb{Z}}
\rho(k)<\infty\}.
\end{gather*}
We have that $U_b\subset U_{\infty}\subset U$.

Let $\rho_1,\rho_2\in U_{\infty}$. $\rho_1$ is said to be equivalent to $\rho_2$ if
$\rho_1/\rho_2\in U_{b}$. In this case we write $\rho_1\sim\rho_2$. 
It can be proved that
$U_{\infty}=\cup_{\rho\in U_{\infty}}\{\varrho\in U_{\infty}:\rho\sim\varrho\}$.

Let $\rho\in U_{\infty}$ and $m\in\mathbb{Z}$. We define $\rho_m(n)=\rho(n+m)$ 
for $n\in Z$ and
\begin{equation*}
    U_T=\{\rho\in U_{\infty}:\rho\sim\rho_m \text{  for each $m\in \mathbb{Z}$}\}.
\end{equation*}

A sequence $f:\mathbb{Z}\to X$ is called almost automorphic if for every integer sequence 
$\{ k'_n \}$, there exists a subsequence $\{ k_n \}$ such that 
$$
\bar{f}(k):=\lim_{n\to \infty}f(k+k_n)
$$ 
is well defined for each $k\in\mathbb{Z}$ and 
$\lim_{n\to\infty}\bar{f}(k-k_n)=f(k)$, see \cite[Definition 2.1]{Araya} 
and references therein. We denote by $AA_{d}(\mathbb{Z},X)$ the set of almost automorphic 
sequences. It is well known that the set $AA_{d}(\mathbb{Z},X)$ endowed with 
the norm $\| f \|_{\infty}:=\sup_{k\in\mathbb{Z}}\| f(k)\|$ is a Banach space, see 
\cite[Theorem 2.4]{Araya}. A typical example is
$f(k) = \sin\big(\frac{1}{2+\cos(k) + \cos (\sqrt{2}k)}\big)$, $k \in \mathbb{Z}$.
A sequence $f:\mathbb{Z}\times X\to X$ is said to be almost automorphic if $f(k,x)$ is
almost automorphic in $k\in\mathbb{Z}$ for any $x\in X$. We denote this space 
by $AA_d(\mathbb{Z}\times X,X)$.

Let $\rho_1\in U_{\infty}$.  Define the ergodic space (see \cite{abbas10}) by
\begin{equation*}
PAA_0S(\mathbb{Z},X,\rho_1)=
\big\{f\in BS(\mathbb{Z},X):\lim_{K\to\infty}\frac{1}{m_d(K,\rho_1)}
\sum_{k=-K}^{K}\|f(k)\|\rho_1(k)=0\big\}.
\end{equation*}
Particularly, for $\rho_1,\rho_2\in U_{\infty}$ (see \cite{Xia14}),
\begin{equation*}
PAA_0S(\mathbb{Z},X,\rho_1,\rho_2)
=\big\{f\in BS(\mathbb{Z},X):\lim_{K\to\infty}\frac{1}{m_d(K,\rho_1)}
\sum_{k=-K}^{K}\|f(k)\|\rho_2(k)=0\big\}.
\end{equation*}

\begin{remark} \label{rmk2.1} \rm
Note that if $\rho_1\sim\rho_2$ then 
$$
PAA_0S(\mathbb{Z},X,\rho_1,\rho_2)=PAA_0S(\mathbb{Z},X,\rho_1)=PAA_0S(\mathbb{Z},X,\rho_2).
$$
\end{remark}

Let $\rho_1,\rho_2\in U_{\infty}$. A sequence $f:\mathbb{Z}\to X$ is called 
discrete weighted pseudo almost automorphic if it can be expressed as 
$f=g+\varphi$, where $g\in AA_d(\mathbb{Z},X)$ and
$\varphi\in PAA_0S(\mathbb{Z},X,\rho_1,\rho_2)$, see \cite[Definition 8]{Xia14} 
and references therein. The set of such functions is denoted by 
$WPAA_d(\mathbb{Z},X)$. It is well known that the set $WPAA_d(\mathbb{Z},X)$ is a Banach space
 with the norm $\|f\|_{\infty}=\sup_{k\in\mathbb{Z}}\|f(k)\|$ 
(see \cite[Lemma 10]{Xia14}). A classical example is the
function $f(k)=\text{signum}(\cos 2\pi k\theta)+e^{-|k|}$ with  
$\rho_1(k)=\rho_2(k)=1+k^2$ for $k\in\mathbb{Z}$ (see \cite{abbas10}).


\begin{remark} \label{rmk2.2} \rm
If $\rho_1\sim\rho_2$, then $WPAA_d(\mathbb{Z},X)$ coincide with the discrete weighted
pseudo almost automorphic functions $WPAAS(\mathbb{Z})$ defined in \cite{abbas10}.
\end{remark}

Similarly, we define (see \cite{Xia14})
\begin{align*}
&PAA_0S(\mathbb{Z}\times X,X,\rho_1,\rho_2) \\
&= \Big\{f\in BS(\mathbb{Z}\times X,X):\lim_{K\to\infty}\frac{1}{m_d(K,\rho_1)}
\sum_{k=-K}^{K}\|f(k,x)\|\rho_2(k)=0 \\
&\quad \text{uniform in $x\in X$}\big\}.
\end{align*}
A function $f:\mathbb{Z}\times X\to X$ is said to be discrete weighted pseudo 
almost automorphic in $k\in \mathbb{Z}$ for each $x\in X$, if it can be 
decomposed as $f=g+\varphi$, where $g\in AA_d(\mathbb{Z}\times X)$
and $\varphi\in PAA_0S(\mathbb{Z}\times X,X,\rho_1,\rho_2)$. Denote by 
$WPAA_d(\mathbb{Z}\times X,X)$ the set of such functions.

Throughout the rest of this paper, we denote by $V_{\infty}$ the set of all
functions $\rho_1,\rho_2\in U_{\infty}$ satisfying that there exists an 
unbounded set $\Omega\subset \mathbb{Z}$ such for all $m\in \mathbb{Z}$,
\begin{gather*}
\lim_{|k|\to \infty,k\in\Omega}\sup\frac{\rho_2(k+m)}{\rho_1(k)}<\infty, \\
\lim_{K\to\infty}\frac{\sum_{k\in([-K,K]\setminus\Omega)+m}\rho_2(k)}{m_d(K,\rho_1)}=0.
\end{gather*}
Xia \cite{Xia14} proved the following composition theorem.

\begin{theorem}[{\cite[Th. 16]{Xia14}}] \label{comp1}
Assume that $\rho_1,\rho_2\in V_{\infty}$, and that  
$f\in WPAA_d(\mathbb{Z}\times X, X)\cap \mathfrak{U}\mathfrak{C}(\mathbb{Z}\times X, X)$ 
and $h\in WPAA_d(\mathbb{Z},X)$. Then $f(\cdot,h(\cdot))\in WPAA_d(\mathbb{Z},X)$.
\end{theorem}

We recall that a function $f:\mathbb{Z}\times X \to X$ is said to be locally Lipschitz
with respect to the second variable if for each positive number $r$, for all 
$k \in\mathbb{Z}$ and for all $x,y \in X$ with $\|x\| \leq r$ and $\|y\| \leq r$,
 we have   $\| f(k,x)-f(k,y)\| \leq L(r) \| x - y \|$, where 
$L:\mathbb{R}_+ \to \mathbb{R}_+$ is a nondecreasing function.

The previous theorem admits a new version with local conditions on the function $f$.

\begin{corollary}\label{cor2.7}
Let $\rho_1,\rho_2\in V_{\infty}$. Let $f:\mathbb{Z} \times X  \to X$ 
be a discrete weighted pseudo almost automorphic  function in the 
first variable and locally Lipschitz in the second variable. 
Then the conclusion of the previous theorem holds.
\end{corollary}

A function $f\in BS(\mathbb{Z},X)$ is called discrete asymptotically $\omega$-periodic 
if there exist $g\in C_{\omega}(\mathbb{Z},X)$, $\varphi\in C_{0}(\mathbb{Z},X)$ such that 
$f=g+\varphi$.  The collection of such functions is denoted by $AP_{\omega}(\mathbb{Z},X)$.
A function $f\in BS(\mathbb{Z},X)$ is called discrete $\mathcal{S}$-asymptotic 
$\omega$-periodic if there exists $\omega\in\mathbb{Z}^+\setminus\{0\}$ such that 
$\lim_{n\to\infty}(f(n+\omega)-f(n))=0$, see \cite[Definition 5]{Xia14-2} 
and references therein. The collection of such functions is denoted by 
$\mathcal{S}AP_{\omega}(\mathbb{Z},X)$.

Let $\rho\in U_{\infty}$. A function $f\in BS(\mathbb{Z},X)$ is called discrete 
pseudo-$\mathcal{S}$-asymptotic
$\omega$-periodic if there exists $\omega\in\mathbb{Z}^+\setminus\{0\}$ such that
$$
\lim_{n\to\infty}\frac{1}{2n}\sum_{k=-n}^{n}\|f(k+\omega)-f(k)\|=0;
$$
see \cite[Definition 6]{Xia14-2}.
The collection of such functions is denoted by $P\mathcal{S}AP_{\omega}(\mathbb{Z},X)$.

Let $\rho\in U_{\infty}$. A function $f\in BS(\mathbb{Z},X)$ is called discrete weighted 
pseudo-$\mathcal{S}$-asymptotic
$\omega$-periodic if there exists $\omega\in\mathbb{Z}^+\setminus\{0\}$ such that
$$
\lim_{n\to\infty}\frac{1}{m_d(n,\rho)}\sum_{k=-n}^{n}\|f(k+\omega)-f(k)\|\rho(k)=0,
$$
see \cite[Definition 7]{Xia14-2}.
The set of such functions is denoted by $WP\mathcal{S}AP_{\omega}(\mathbb{Z},X,\rho)$.
It is clear that 
$AP_{\omega}(\mathbb{Z},X)\subset P\mathcal{S}AP_{\omega}(\mathbb{Z},X)
\subset WP\mathcal{S}AP_{\omega}(\mathbb{Z},X,\rho)$.
It is well known that the set $WP\mathcal{S}AP_{\omega}(\mathbb{Z},X,\rho)$ 
is a Banach space with the norm
$\|f\|_{\infty}:=\sup_{k\in \mathbb{Z}}\|f(k)\|$ (see \cite[Lemma 8]{Xia14-2}).

\begin{remark}[\cite{Xia14-2}] \label{rmk} 
If $\rho_1,\rho_2\in U_{\infty}$ and $\rho_1\sim \rho_2$ then
\begin{gather*}
WP\mathcal{S}AP_{\omega}(\mathbb{Z},X,\rho_1)=WP\mathcal{S}AP_{\omega}(\mathbb{Z},X,\rho_2), \\
WP\mathcal{S}AP_{\omega}(\mathbb{Z},X,\rho_1/\rho_2)=P\mathcal{S}AP_{\omega}(\mathbb{Z},X).
\end{gather*}
\end{remark}

\begin{theorem}[{\cite[Th. 12]{Xia14-2}}] \label{comp2}
Assume that $\rho\in U_{\infty}$ and that  
$f\in WP\mathcal{S}AP_{\omega}(\mathbb{Z}\times X, X,\rho)\cap
\mathfrak{U}\mathfrak{C}(\mathbb{Z}\times X, X)$ and 
$h\in WP\mathcal{S}AP_{\omega}(\mathbb{Z},X,\rho)$. 
Then 
\[
f(\cdot,h(\cdot))\in WP\mathcal{S}AP_{\omega}(\mathbb{Z},X,\rho).
\]
\end{theorem}

Now, we present an alternative version of the preceding theorem 
with local conditions on $f$.

\begin{corollary}\label{cor2.8}
Let $\rho\in U_{\infty}$. Let $f:\mathbb{Z} \times X  \to X$ be a discrete 
$\mathcal{S}$-asymptotic $\omega$-periodic  function in the first 
variable and locally Lipschitz in the second variable. 
Then, the conclusion of the previous theorem is true.
\end{corollary}

The following definition of discrete derivative in Weyl sense is due to 
 Abadias and  Lizama \cite{ab-li16}.
We define the forward Euler operator $\Delta:s(\mathbb{Z},X)\to s(\mathbb{Z},X)$ by 
$$
\Delta f(n)=f(n+1)-f(n),\quad n\in\mathbb{Z}.
$$ 
Recursively we define 
$$
\Delta^{k+1}=\Delta^k\Delta=\Delta\Delta^k,\quad k\in\mathbb{N},
$$ 
and $\Delta^0=I$ is the identity operator. 
It is easy to see that 
$$
\Delta^k f(n)=\sum_{j=0}^k(-1)^{k-j}\binom{k}{j}f(n+j).
$$ 
In particular $\Delta^1=\Delta$. In addition, for $\alpha>0$, 
we consider the scalar sequence $\{k^{\alpha}(n)\}_{n\in\mathbb{N}_0}$
defined by 
\begin{equation*}
k^{\alpha}(n):=\frac{\Gamma(n+\alpha)}{\Gamma(\alpha)\Gamma(n+1)}.
\end{equation*}
We note that the kernel $k^{\alpha}$ satisfies the semigroup property 
in $\mathbb{N}_0$, that is,
$$
(k^{\alpha}*k^{\beta})(n)=\sum_{j=0}^{n}k^{\alpha}(n-j)k^{\beta}(j)
=k^{\alpha+\beta}(n)
$$ 
with $n\in\mathbb{N}_0$ and $\alpha,\beta>0$.

\begin{definition} \label{def} \rm
 Let $\alpha>0$ be given  and $\rho(n)=|n|^{\alpha-1}, n \in \mathbb{Z}$. 
The $\alpha$-th fractional sum of a sequence $f\in l^1_{\rho}(\mathbb{Z},X)$ 
is defined by 
$$
\Delta^{-\alpha}f(n):=\sum_{j=-\infty}^n k^{\alpha}(n-j)f(j),\quad n\in\mathbb{Z}.
$$
\end{definition}

See also \cite{Li15} for related work on a slight variant of this definition.

\begin{remark} \label{rem2.3} \rm
The previous definition can be numerically compared with the continuous 
fractional integral in the sense of Weyl, see \cite[Section 3.3]{Or-Co-Tr15}.
 Moreover, we observe that as a consequence of the semigroup property 
of the kernel $k^{\alpha}$ we have that 
$\Delta^{-\alpha}\Delta^{-\beta}=\Delta^{-(\alpha+\beta)}
=\Delta^{-\beta}\Delta^{-\alpha}$.
\end{remark}

\begin{definition} \label{def2.4}  \rm
Let $\alpha>0$ be given and $\rho(n)=|n|^{\alpha-1}$ for $n \in \mathbb{Z}$.
 The $\alpha$-th fractional difference of a sequence $f\in l^1_{\rho}(\mathbb{Z},X)$ 
is defined by 
$$
\Delta^{\alpha}f(n):=\Delta^{m}\Delta^{-(m-\alpha)}f(n),\quad n\in\mathbb{Z},
$$ 
with $m=[\alpha]+1$.
\end{definition}

A sequence $\{S(n)\}_{n\in\mathbb{N}_0}\subset\mathcal{B}(X)$ is called
summable if $\| S\|_1:=  \sum_{n=0}^{\infty}\| S(n)\|<\infty$.
The following definition is introduced in \cite{Li15}.

\begin{definition} \label{DefResol} \rm
Let $\alpha>0$ and $A$ be a closed linear operator with domain $D(A)$ 
defined on a Banach space $X$. An operator-valued sequence 
$\{S_{\alpha}(n)\}_{n\in\mathbb{N}_0}\subset \mathcal{B}(X)$ is called a
discrete $\alpha$-resolvent family generated by $A$ if it satisfies the 
following conditions
\begin{itemize}
\item[(i)] $S_{\alpha}(n)Ax=AS_{\alpha}(n)x$ for $n\in\mathbb{N}_0$ and $x\in D(A);$
\item[(ii)] $S_{\alpha}(n)x=k^{\alpha}(n)x+A(k^{\alpha}*S_{\alpha})(n)x$, for all $n\in\mathbb{N}_0$ and $x\in X$.
\end{itemize}
\end{definition}

We recall the following practical criteria for summability of 
$\alpha$-resolvent families.

\begin{theorem}[{\cite[Th. 3.5]{ab-li16}}] \label{Theorem3.6}
Let $0<\alpha <1$ and $A$ be the generator of an exponentially stable 
$C_0$-semigroup $\{T(t)\}_{t\geq 0}$ defined on a Banach space $X$. 
Then $A$ generates a discrete $\alpha$-resolvent family 
$\{ S_{\alpha}(n) \}_{n\in \mathbb{N}_0}$  defined by
\begin{equation}\label{3.3}
S_{\alpha}(n)x  := \int_0^{\infty}\int_0^{\infty}  e^{-t} 
\frac{t^n}{n!} f_{s,\alpha}(t) T(s)x\,ds\,dt, \quad n \in \mathbb{N}_0, \;
 x \in X.
\end{equation}
Moreover, $\{ S_{\alpha}(n) \}_{n\in \mathbb{N}_0}$ is summable.
 (Here $f_{s,\alpha}(t)$ is the function called
stable L\'{e}vy process, see \cite{ab-li16}).
\end{theorem}

Let $\mathcal{M}(\mathbb{Z},X):=\{WPAA_{d}(\mathbb{Z},X), WPSAP_{\omega}(\mathbb{Z},X,\rho)\}$ and
$\mathcal{M}(\mathbb{Z}\times X,X):=\{WPAA_{d}(\mathbb{Z}\times X,X),
 WPSAP_{\omega}(\mathbb{Z}\times X,X,\rho)\}$. 
The following is our main result on regularity under convolution 
of the above mentioned spaces.

\begin{theorem}\label{conv1}
Let $0<\alpha<1$, $\rho_1,\rho_2\in V_{\infty}$ and $\rho\in U_T$. 
Assume that $A$ generates a summable discrete $\alpha$-resolvent 
family $\{S_{\alpha}(n)\}_{n\in\mathbb{N}_0}\subset \mathcal{B}(X)$.
If $f$ belongs to one of the spaces $\Omega \in \mathcal{M}(\mathbb{Z},X)$ then
\begin{equation*}
u(n)=\sum_{j=-\infty}^{n-1}S_{\alpha}(n-1-j)f(j)\quad (n\in\mathbb{Z})
\end{equation*}
belongs to the same space $\Omega$.
\end{theorem}

\begin{proof}
First note that $u$ is well defined since 
$\| u(n) \| \leq \| S_{\alpha} \|_1 \| f\|_{\infty}$, for all $n \in \mathbb{Z}$.
First, we consider $f\in WPAA_d(\mathbb{Z},X)$. Let $u=f_1+f_2$, where 
$f_1\in AA_d(\mathbb{Z},X)$ and
$f_2\in PAA_0S(\mathbb{Z},X,\rho_1,\rho_2)$.  Then
$$
u(n)=\sum_{j=-\infty}^{n-1}S_{\alpha}(n-1-j)f_1(j)+
\sum_{j=-\infty}^{n-1}S_{\alpha}(n-1-j)f_2(j)=:u_1(n)+u_2(n).
$$
It follows from \cite[Theorem 4.5]{ab-li16} that $u_1\in AA_d(\mathbb{Z},X)$. 
 It remains to prove that $u_2\in PAA_0S(\mathbb{Z},X,\rho_1,\rho_2)$.  Indeed,
\begin{align*}
&\frac{1}{m_d(K,\rho_1)}\sum_{k=-K}^{K}\|u_2(k)\|\rho_2(k) \\
&= \frac{1}{m_d(K,\rho_1)}\sum_{k=-K}^{K}
\big\|\sum_{j=-\infty}^{k-1}S_{\alpha}(k-1-j)f_2(j)\big\|\rho_2(k)\\
&\leq\sum_{m=0}^{\infty}\|S_{\alpha}(m)\|
\Big(\frac{1}{m_d(K,\rho_1)}\sum_{k=-K}^{K}\|f_2(k-1-m)\|\rho_2(k)\Big).
\end{align*}
Since $PAA_0S(\mathbb{Z},X,\rho_1,\rho_2)$ is invariant under translation by 
\cite[Lemma 10]{Xia14} we obtain that
$f_2(\cdot-m)\in PAA_0S(\mathbb{Z},X,\rho_1,\rho_2)$. 
By Lebesgue dominated convergence theorem, we have 
\[
\lim_{K\to\infty}\frac{1}{m_d(K,\rho_1)}\sum_{-K}^{K}\|u_2(k)\|\rho_2(k)=0.
\]
Hence $u\in WPAA_d(\mathbb{Z},X)$. It proves the claim for such space. 
Now, let $f\in WPSAP_{\omega}(\mathbb{Z},X,\rho)$. Then,
\begin{align*}
& \frac{1}{m_d(K,\rho)}\sum_{k=-K}^{K}\|u(k+\omega)-u(k)\|\rho(k) \\ &=
\frac{1}{m_d(K,\rho)}\sum_{k=-K}^{K}\bigg\|\sum_{j
=-\infty}^{k+\omega-1}S_{\alpha}(k+\omega-1-j)f(j)\\
&-\sum_{j=-\infty}^{k-1}S_{\alpha}(k-1-j)f(j)\bigg\|\rho(k)\\
&\leq\frac{1}{m_d(K,\rho)}\sum_{k=-K}^{K}
\sum_{j=-\infty}^{k-1}\|S_{\alpha}(k-1-j)\|\|f(j+\omega)-f(j)\|\rho(k)\\
&\leq\sum_{m=0}^{\infty}\|S_{\alpha}(m)\|
\Big(\frac{1}{m_d(K,\rho)}\sum_{k=-K}^{K}\|f(k-1-m+\omega)-f(k-1-m)\|\rho(k)\Big).
\end{align*}
Since $WPSAP_{\omega}(\mathbb{Z},X,\rho)$ is invariant under translation by 
\cite[Lemma 10]{Xia14-2} we obtain that
$f(\cdot-1-m)\in WPSAP_{\omega}(\mathbb{Z},X,\rho)$. 
By Lebesgue dominated convergence theorem, we have 
\[
\lim_{K\to\infty}\frac{1}{m_d(K,\rho)}\sum_{-K}^{K}\|u(k+\omega)-u(k)\|\rho(k)=0.
\]
Hence $u\in WPSAP_{\omega}(\mathbb{Z},X,\rho)$. The proof is complete.
\end{proof}

\section{Solutions for nonlinear fractional difference equations}

We consider the  fractional difference equation
\begin{equation}\label{FractEquSemlinear}
\Delta^{\alpha}u(n)=Au(n+1)+f(n, u(n)),\quad n\in \mathbb{Z},
\end{equation}
for $0<\alpha<1$, where $A$ is the generator of a discrete $\alpha$-resolvent
 family $\{S_{\alpha}(n)\}_{n\in\mathbb{N}_0}$ in $\mathcal{B}(X)$, and
$\Delta ^{\alpha}$ is the fractional difference of the sequence $u$ 
of order $\alpha$.


Since our objective is to study the solubility of \eqref{FractEquSemlinear} 
in the spaces $\mathcal{M}(\mathbb{Z},X)$, where the forcing term $f$ is only bounded, 
we need to use the definition of mild solution introduced by  
Abadias and  Lizama in \cite{ab-li16}.



\begin{definition} \label{def3.1} \rm 
Let $0<\alpha<1$, $A$ be the generator of a discrete $\alpha$-resolvent 
family $\{S_{\alpha}(n)\}_{n\in\mathbb{N}_0}\subset \mathcal{B}(X)$, and
$f: \mathbb{Z} \times X \to X$. We say that a sequence $u:\mathbb{Z}\to X$ is a mild 
solution of \eqref{FractEquSemlinear} if $m \to S_{\alpha}(m)f(n-m)$ is summable 
on $\mathbb{N}_0$, for each $n\in\mathbb{Z}$  and $u$ satisfies 
$$
u(n+1)=\sum_{j=-\infty}^{n}S_{\alpha}(n-j)f(j,u(j)),\quad n\in\mathbb{Z}.
$$
\end{definition}

Our first result in this section provides a simple criterion for the existence
 and uniqueness of discrete weighted pseudo almost automorphic
and discrete $\mathcal{S}$-asymptotic $\omega$-periodic mild solutions. 
The proof is based on the Banach fixed point theorem.

\begin{theorem}\label{SolSemCase} 
Let $0<\alpha<1$, $\rho_1,\rho_2\in V_{\infty}$ and $\rho\in U_T$. 
Assume that $A$ generates a summable discrete $\alpha$-resolvent 
family $\{S_{\alpha}(n)\}_{n\in\mathbb{N}_0}\subset \mathcal{B}(X)$.
If $f \in \Omega \in \mathcal{M}(\mathbb{Z}\times X,X)$ and it is globally 
Lipschitz in the following sense: 
$$
\| f(n,x)-f(n,y) \|\leq L\| x-y\|,\quad \text{for all $n\in\mathbb{Z}$  and  all }x,y\in X,
$$  
where $L < \frac{1}{\| S_{\alpha} \|_1}$, then \eqref{FractEquSemlinear} has
a unique mild solution $u$ which belongs to the corresponding  subset 
$ \Omega \in  \mathcal{M}(\mathbb{Z},X)$.
\end{theorem}

\begin{proof}
Let $f\in WPAA_d(\mathbb{Z}\times X,X)$ and consider the operator 
$T: WPAA_{d}(\mathbb{Z},X)\to WPAA_{d}(\mathbb{Z},X)$ defined by 
\begin{equation}\label{T}
(Tu)(n):=\sum_{j=-\infty}^{n-1}S_{\alpha}(n-1-j)f(j,u(j)),\quad n\in \mathbb{Z}.
\end{equation}
Since $u\in WPAA_d(\mathbb{Z},X)$ it follows from Theorem \ref{comp1} that 
$f(\cdot,u(\cdot))$ belongs to $WPAA_d(\mathbb{Z},X)$.
Now, from Theorem \ref{conv1} we have that $Tu\in WPAA_d(\mathbb{Z},X)$.
Hence $T$ is well-defined.
In addition, for $u,v\in WPAA_{d}(\mathbb{Z},X)$ and $n\in\mathbb{Z}$ the 
following inequality holds,
\begin{align*}
  \|(Tu)(n)-(Tv)(n)\| 
&\leq \sum_{j=-\infty}^{n-1}\| S_{\alpha}(n-1-j)(f(j,u(j))-f(j,v(j))) \|\\
&\leq \sum_{j=-\infty}^{n-1}\| S_{\alpha}(n-1-j)\| \| u(j)-v(j) \| \\
&\leq  L \| S_{\alpha}\|_1 \| u-v \|_{\infty}.
\end{align*}

By hypothesis we conclude that $T$ is a contraction, and using Banach fixed 
point theorem we get that there exists a unique discrete weighted pseudo 
almost automorphic mild solution of \eqref{FractEquSemlinear} .

The proof for the space of  $\mathcal{S}$-asymptotic $\omega$-periodic 
sequences is analogous.
\end{proof}

Using \cite[Remark 3.6]{ab-li16} we  have a precise  estimate for 
$ \|S_{\alpha}\|_1$ that can be used to prove the following Corollary.
The proof use Theorem \ref{Theorem3.6} and the proof of Theorem \ref{SolSemCase}.

\begin{corollary}\label{cor3.3}
Let $0<\alpha<1$, $\rho_1,\rho_2\in V_{\infty}$, $\rho\in U_T$ and $A$
 be the generator of a $C_0$-semigroup $T(t)$ such that
 $\|T(t)\| \leq Me^{-\omega t}$, for some $M>0$ and $\omega >0$.  
If $f\in \mathcal{M}(\mathbb{Z}\times X,X)$  is globally Lipschitz with constant 
$L < 1/\omega$
 then \eqref{FractEquSemlinear} has
a unique mild solution $u$  which belongs to the same space as $f$.
\end{corollary}

Next, we show that the conclusion of the previous theorem holds with a 
local Lipschitz condition on $f$.

\begin{theorem}\label{th5.4} 
Let $0<\alpha<1$, $\rho_1,\rho_2\in V_{\infty}$ and $\rho\in U_T$. 
Assume that $A$ generates a summable discrete $\alpha$-resolvent 
family $\{S_{\alpha}(n)\}_{n\in\mathbb{N}_0}\subset \mathcal{B}(X)$.
Let $f\in \mathcal{M}(\mathbb{Z}\times X,X)$  that satisfies a local Lipschitz condition. 
If there exist $r_0>0$ such that
$$
\|S_{\alpha}\|_1 \Big (L(r_0) + \frac{\sup_{k}\| f(k,0)\|}{r_0} \Big)<1,
$$
then \eqref{FractEquSemlinear} has
a unique mild solution $u$  which belongs to the same space as 
$f$ with $\| u \|_{\infty} := \sup_{k}\|u(k)\| \leq r_0$.
\end{theorem}

\begin{proof}
First, we consider $f\in WPAA_{d}(\mathbb{Z}\times X,X)$.
 Note that  $T: WPAA_{d}(\mathbb{Z},X)\to WPAA_{d}(\mathbb{Z},X)$ given by \eqref{T} 
is well defined by Corollary \ref{cor2.7} and Theorem \ref{conv1}.

Let $B_{r_0} (0):= \{ u \in WPAA_d(\mathbb{Z},X) : \| u \|_{\infty} < r_0 \}$ 
be the ball of radius $r_0$ on $WPAA_d(\mathbb{Z},X)$. We show that 
$T \big( B_{r_0} (0) \big) \subset B_{r_0} (0)$. Indeed, let $u$ be in 
$B_{r_0} (0)$. Since $f$ is locally Lipschitz, we obtain
$$
\|f(k,u(k))\| \leq \|f(k,u(k))-f(k,0)\| + \|f(k,0)\| 
\leq L(r_0) \|u(k) \| + \| f(k,0) \|,
$$
for $k \in \mathbb{Z}$.
Moreover, we have the estimate
\begin{align*}
&\| T(u)(n) \| \\
&\leq \sum_{j=-\infty}^{n-1}\|S_{\alpha}(n-1-j)\| \| f(j,u(j)) -f(j,0) \| 
+ \sum_{j=-\infty}^{n-1}\|S_{\alpha}(n-1-j)\| \|f(j,0)\| \\ 
& \leq L(r_0) \sum_{j=-\infty}^{n-1}\|S_{\alpha}(n-1-j)\| \| u(j) \| 
 + \|S_{\alpha}\|_1 \sup_{k} \|f(k,0)\| \\ 
&\leq \|S_{\alpha}\|_1 \Big (L(r_0) + \frac{\sup_{k}\| f(k,0)\|}{r_0} \Big)r_0 
\leq  r_0,
\end{align*}
proving the claim. On the other hand, for $u,v \in B_{r_0}(0)$ we have that
\begin{align*}
&\| Tu(n) -Tv(n) \| \\
&\leq \sum_{j=-\infty}^{n-1}\|S_{\alpha}(n-1-j)\| \| f(j,u(j)) -f(j,v(j)) \|\\ 
& \leq L(r_0) \sum_{j=-\infty}^{n-1}\|S_{\alpha}(n-1-j)\| \| u(j) -v(j) \| \\
&\leq \|S_{\alpha}\|_1 L(r_0) \|u - v \|_{\infty}.
\end{align*}
Observing that $ \|S_{\alpha}\|_1 L(r_0) <1$, it follows that $T$ is a 
contraction in $B_{r_0}(0)$. Then there is a unique $u \in B_{r_0}(0)$ 
such that $Tu=u$.

The proof for $f\in WPSAP_{\omega}(\mathbb{Z}\times X,X,\rho)$ is similar, we 
just have to take 
$B_{r_0} (0):= \{ u \in WPSAP_{\omega}(\mathbb{Z},X,\rho) : \| u \|_{\infty} < r_0 \}$ 
be the ball of radius $r_0$ on $WPSAP_{\omega}(\mathbb{Z},X,\rho)$.
\end{proof}

The following corollary is an immediate consequence of the previous results.

\begin{corollary}\label{cor3.5}
Let $0<\alpha<1$, $\rho_1,\rho_2\in U_{\infty}$, $\rho\in U_T$ and $A$ 
be the generator of a $C_0$-semigroup $T(t)$ such that 
$\|T(t)\| \leq Me^{-\omega t}$, for some $M>0$ and $\omega >0$.  
If $f\in \mathcal{M}(\mathbb{Z}\times X,X)$  is locally Lipschitz and satisfy
$$
\frac{1}{\omega} \Big (L(r_0) + \frac{\sup_{k}\| f(k,0)\|}{r_0} \Big)<1,
$$
for some $r_0>0$, then \eqref{FractEquSemlinear} has
a unique mild solution $u$  which belongs to the same space as $f$.
\end{corollary}

We finish this article with a simple example to illustrate how our abstract
 results apply.

\begin{example}\label{examp3.6} \rm
We consider the fractional difference equation
\begin{equation}\label{eq6.2}
\Delta^{\alpha} u(k)= Au(k+1) +  \frac{\nu g(k) u(k)}{1+ \sup_k| u(k)|}, 
\quad k \in \mathbb{Z},
\end{equation}
where $0<\alpha <1$ act as a tuning parameter for the  difference 
Equation \eqref{eq6.2}, the operator $A$ is the generator of an exponentially 
stable $C_0$-semigroup on a Banach space $X$, $\nu$ is a parameter and  
$g(k)=\operatorname{signum}(\cos 2\pi k\theta)+e^{-|k|}$.  
We know by \cite{abbas10} that $g\in WPAA_d(\mathbb{Z},X)$ where $\rho=1+k^2$. 
Now, it can be shown that the function
$$
f(k,x):= \frac{\nu g(k) x}{1+\|x\|_{\infty}}, \quad k \in \mathbb{Z},\,x \in X, 
$$
is a discrete weighted pseudo almost automorphic function on $\mathbb{Z}\times X$. 
 We have the estimate
\[
\| f(k,x) - f(k, y)\|_{\infty}\leq \nu \|g\|_{\infty}  (1+ \|y\|)\|x-y\|_{\infty}.
\]

Therefore, we can choose
$L(r)=  \nu \|g\|_{\infty}(1+r)$, $r>0$,
to deduce that $f(k,x)$ is locally Lipschitz. Since $f(k,0)=0$,
we obtain that for sufficiently small $\nu$ the condition
$\|S_{\alpha}\|_1 L(r) <1$
is satisfied. We conclude, by Theorem \ref{th5.4}, that the fractional 
model \eqref{eq6.2} admits a unique discrete weighted pseudo almost 
automorphic solution.
\end{example}

\subsection*{Acknowledgments}
Edgardo Alvarez was partially supported by Direcci\'on
de Investigaciones Universidad del Norte, Project number 2016-011.
Carlos Lizama was  partially supported by CONICYT, under Fondecyt Grant
number 1140258 and CONICYT - PIA - Anillo ACT1416.

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