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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2013 (2013), No. 103, pp. 1--9.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2013 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2013/103\hfil 
Equi-asymptotically almost periodic functions]
{Equi-asymptotically almost periodic functions and applications
to functional integral equations}

\author[H.-S. Ding, Q.-L. Liu, G. M. N'Gu\'er\'ekata EJDE-2013/103\hfilneg]
{Hui-Sheng Ding, Qing-Long Liu, Gaston M. N'Gu\'er\'ekata}  % in alphabetical order

\address{Hui-Sheng Ding \newline
College of Mathematics and Information Science,
Jiangxi Normal University\\
Nanchang, Jiangxi 330022, China}
\email{dinghs@mail.ustc.edu.cn}

\address{Qing-Long Liu \newline
College of Mathematics and Information Science,
Jiangxi Normal University\\
Nanchang, Jiangxi 330022, China}
\email{758155543@qq.com}

\address{Gaston M. N'Gu\'er\'ekata \newline
Department of Mathematics, Morgan State University,
1700 E. Cold Spring Lane, Baltimore, MD 21251, USA}
\email{nguerekata@aol.com, Gaston.N'Guerekata@morgan.edu}

\thanks{Submitted February 6, 2013. Published April 24, 2013.}
\subjclass[2000]{34K14, 45G10}
\keywords{Asymptotically almost periodic;
equi-asymptotically almost periodic; \hfill\break\indent 
functional integral equation}

\begin{abstract}
 In this article, we introduce the notion of equi-asymptotically almost
 periodicity, and investigate some properties of equi-asymptotically
 almost periodic functions. In addition, by applying the results on
 equi-asymptotically almost periodic functions, we obtain an
 existence theorem for asymptotically almost periodic solutions to a
 class of functional integral equation.
\end{abstract}

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

\section{Introduction and preliminaries}


In 1940s, M. Fr\'{e}chet introduced the notion of asymptotically
almost periodicity, which turns out to be one of the most
interesting and important generalizations of almost periodicity. In
fact, asymptotically almost periodic functions now have been of great
interest for many mathematicians and have been applied to various
branches of pure and applied mathematics, especially to differential
equations and dynamical systems. For example, we refer the reader to
\cite{h1,h2,h3,a2} and references therein for
some recent progress on asymptotically almost periodic functions
and their applications to differential equations.

In a recent work, when studying the existence of
asymptotically almost periodic solutions to a class of Volterra-type
difference equations, Long et al \cite{long} introduced the notion of
equi-asymptotically almost periodic sequences. To the best
of our knowledge, there are only a few publications
about equi-asymptotically almost periodic functions. This motivates
the publication of this work.

For the rest of this paper, if there is no special
statement, we denote by $\mathbb{R}$ the set of real numbers, by
$X$ a Banach space, and by $C(\mathbb{R},X)$ the set of all continuous
functions from $\mathbb{R}$ to $X$. In addition, we denote
$$
C_0(\mathbb{R},X)=\{f\in C(\mathbb{R},X): \lim_{|t|\to \infty}f(t)=0\}.
$$
Next, let us recall some notation and basic results of almost
periodic functions and asymptotically almost periodic functions (for
more details, see \cite{cordu,fink,zhang,Zaidman}).

\begin{definition} \label{def1.1} \rm
A set $E\subset \mathbb{R}$ is said to be relatively dense if there exists
a number $l>0$ such that
$$
(a,a+l)\cap E\neq \emptyset
$$
for every $a\in\mathbb{R}$.
\end{definition}

\begin{definition} \label{def1.2} \rm 
A function $f\in C(\mathbb{R},X)$ is said to be almost periodic if for every 
$\varepsilon>0$ there exists a relatively dense set
$T(f,\varepsilon)\subset \mathbb{R}$ such that
$$
\|f(t+\tau)-f(t)\|<\varepsilon,
$$
for all $t\in\mathbb{R}$ and $\tau \in T(f,\varepsilon)$. We denote the set
of all such functions by $AP(\mathbb{R},X)$.
\end{definition}


\begin{definition}  \label{equi-ap} \rm
A set $F\subset C(\mathbb{R},X)$ is said to be equi-almost periodic if for every
$\varepsilon >0$, there exists a relatively dense set
$T(F,\varepsilon)\subset \mathbb{R}$ such that
$$
\|f(t+\tau)-f(t)\|<\varepsilon,
$$
for all $f\in F$, $t\in\mathbb{R}$, and $\tau\in T(F,\varepsilon)$.
\end{definition}


\begin{definition} \label{equi-aap} \rm
A function $f\in C(\mathbb{R},X)$ is said to be asymptotically almost periodic
if for every $\varepsilon $, there exist a constant
$M(\varepsilon)>0$ and a relatively dense set
$T(f,\varepsilon)\subset \mathbb{R}$ such that
$$
\|f(t+\tau)-f(t)\|<\varepsilon,
$$
for all $t\in\mathbb{R}$ with $|t|>M(\varepsilon)$ and $\tau\in
T(F,\varepsilon)$ with $|t+\tau|>M(\varepsilon)$. We denote the set
of all such functions by $AAP(\mathbb{R},X)$.
\end{definition}

The following result can be deduced from \cite[Theorem 2.5]{zhang}.

\begin{lemma}\label{lemma} 
$f\in AAP(\mathbb{R},X)$ if and only if there exists a unique $g\in AP(\mathbb{R},X)$ 
such that $f-g\in C_0(\mathbb{R},X)$.
\end{lemma}

\begin{remark} \label{rmk1.6}\rm
By Lemma \ref{lemma}, we know that every asymptotically almost
periodic function $f$ admits a unique decomposition; i.e., there
exist unique $g\in AP(\mathbb{R},X)$ and $h\in C_0(\mathbb{R},X)$ such that $f=g+h$.
Thus, throughout the rest of this paper, for every $f\in AAP(\mathbb{R},X)$,
we denote by $f_{AP}$ the almost periodic (or principal) component of $f$ and
$f_{C_0}$ be the $C_0(\mathbb{R},X)$ (or corrective) term of $f$.
\end{remark}

Next, we list some basic properties of asymptotically almost
periodic functions. For the proof, we refer the reader to
\cite{Zaidman,zhang}.

\begin{lemma} Let $f,g\in AAP(\mathbb{R},X)$. Then
\begin{itemize}
\item[(a)] $f$ is uniformly continuous on $\mathbb{R}$;
\item[(b)] $\{f_{AP}(t):t\in\mathbb{R}\}\subset \overline{\{f(t):t\in\mathbb{R}\}}$;
\item[(c)] $f+g\in AAP(\mathbb{R},X) $, and $f\cdot g\in AAP(\mathbb{R},\mathbb{R}) $;
\item[(d)] $f(\cdot+a)\in AAP(\mathbb{R},X)$ for any $a\in \mathbb{R}$.
\end{itemize}
\end{lemma}

\section{Equi-asymptotically almost periodic functions}

In this section, we introduce the notion of equi-asymptotically
almost periodicity, and present some basic and interesting
properties for equi-asymptotically almost periodic functions.

\begin{definition} \label{equi-aap2} \rm
A set $F\subset C(\mathbb{R},X)$ is said to be equi-asymptotically almost
periodic if for every $\varepsilon >0$, there exist a constant
$M(\varepsilon)>0$ and a relatively dense set
$T(F,\varepsilon)\subset \mathbb{R}$ such that
$$
\|f(t+\tau)-f(t)\|<\varepsilon,
$$
for all $f\in F$, $t\in\mathbb{R}$ with $|t|>M(\varepsilon)$ and $\tau\in
T(F,\varepsilon)$ with $|t+\tau|>M(\varepsilon)$.
\end{definition}


\begin{theorem}\label{precompact}
Let $F\subset AAP(\mathbb{R},X)$. Then the following
assertions are equivalent:
\begin{itemize}
\item[(i)] $F$ is precompact in $AAP(\mathbb{R},X)$.
\item[(ii)] $F$ satisfies the following three conditions:
\begin{itemize}
\item[(a)]for every $t\in\mathbb{R}$, $\{f(t):f\in F\}$ is precompact in $X$;
\item[(b)]$F$ is equi-uniformly continuous;
\item[(c)]$F$ is equi-asymptotically almost
periodic. \end{itemize}
\item[(iii)] $G$ is precompact in $AP(\mathbb{R},X)$ and $H$ is precompact
in $C_0(\mathbb{R},X)$, where
$$G=\{f_{AP}:f\in F\}\ \mbox{and}\ H=\{f_{C_0}:f\in F\}.$$
\end{itemize}
\end{theorem}

\begin{proof}
(i) $\Rightarrow$ (ii): Let $F$ be precompact in
$AAP(\mathbb{R},X)$. Then, obviously, for every $t\in\mathbb{R}$, $\{f(t):f\in F\}$
is precompact in $X$. In addition, for every $\varepsilon>0$, there
exist $f_1,f_2,\dots,f_k\in F$ such that for every $f\in F$,
$$\min_{1\leq i\leq k}\|f-f_i\|<\varepsilon,$$
where $k$ is a positive integer dependent on $\varepsilon$.
Combining this with the fact that $\{f_i\}^k_{i=1}$ is
equi-uniformly continuous and equi-asymptotically almost periodic,
we know that (b) and (c) hold.

(ii) $\Rightarrow$ (iii): Let $\{g_n\}_{n=1}^{\infty}\subset G$.
For every $n$, there exist $f_n\in F$ and $h_n\in H$ such that
$f_n=g_n+h_n$. By (a) and (b), applying Arzela-Ascoli theorem and
choosing diagonal sequence, we can get a subsequence of
$\{f_n\}_{n=1}^{\infty}$, which we still denote by
$\{f_n\}_{n=1}^{\infty}$ for convenience, such that
$\{f_n(t)\}_{n=1}^{\infty}$ is uniformly convergent on every compact
subsets of $\mathbb{R}$.

Since $\{f_n\}_{n=1}^{\infty}$ is equi-asymptotically almost
periodic, for every $\varepsilon>0$, there exists
$l(\varepsilon),M(\varepsilon)>0$ such that for every $t\in\mathbb{R}$ with
$|t|>M(\varepsilon)$, there is a $$\tau_t\in
[M(\varepsilon)+1-t,M(\varepsilon)+1-t+l(\varepsilon)]$$ satisfying
\begin{equation}\label{3}
\|f_n(t+\tau_t)-f_n(t)\|<\frac{\varepsilon}{3}
\end{equation}
for all $n\in \mathbb{N}$. Noting that $\{f_n(t)\}_{n=1}^{\infty}$ is
uniformly convergent on
$$
[-M(\varepsilon)-l(\varepsilon)-1,M(\varepsilon)+l(\varepsilon)+1],
$$
for the above $\varepsilon>0$, there exists $N\in \mathbb{N}$ such that for
all $m\geq n\geq N$ and 
$t\in [-M(\varepsilon)-l(\varepsilon)-1,M(\varepsilon)+l(\varepsilon)+1]$,
\begin{equation}\label{4}
\|f_m(t)-f_n(t)\|<\frac{\varepsilon}{3}
\end{equation}
Combining \eqref{3} and \eqref{4}, for all $m\geq n\geq N$ and
$t\in\mathbb{R}$ with $|t|>M(\varepsilon)$, we have
\begin{align*}
&\|f_m(t)-f_n(t)\|\\
&\leq \|f_m(t)-f_m(t+\tau_t)\|+\|f_m(t+\tau_t)-f_n(t+\tau_t)\|
 +\|f_n(t+\tau_t)-f_n(t)\|\\
&\leq& \varepsilon,
\end{align*}
this and \eqref{4} yield  $\{f_n(t)\}_{n=1}^{\infty}$ is
uniformly convergent on $\mathbb{R}$. In view of
$$
\{g_m(t)-g_n(t):t\in\mathbb{R}\}\subset \overline{\{f_m(t)-f_n(t):t\in\mathbb{R}\}}
$$
for all $m,n\in\mathbb{N}$, we conclude that $\{g_n(t)\}_{n=1}^{\infty}$ is
also uniformly convergent on $\mathbb{R}$; i.e., $\{g_n\}_{n=1}^{\infty}$ is
convergent in $AP(\mathbb{R},X)$. So $G$ is precompact in $AP(\mathbb{R},X)$. In
addition, it follows from the above proof that $F$ is precompact,
and thus $H$ is also precompact.

(iii) $\Rightarrow$ (i): The proof is straightforward; we omit it.
\end{proof}

\begin{remark} \label{rmk2.3}\rm
Theorem \ref{precompact} can be seen as an extension of
the corresponding compactness criteria for the subsets of $AP(\mathbb{R},X)$
(see, e.g., \cite{cordu,Zaidman}).
\end{remark}

\begin{definition} \label{def2.4} \rm
$F\subset C_0(\mathbb{R},X)$ is called equi-$C_0$ if
$$
\lim_{|t|\to\infty}\sup_{f\in F}\|f(t)\|=0.
$$
\end{definition}

\begin{theorem}\label{th1}
The following two assertions are equivalent:
\begin{itemize}
\item[(I)] $F$ is equi-asymptotically almost periodic;
\item[(II)] $G$ is equi-almost periodic and $H$ is equi-$C_0$, where
$$
G=\{f_{AP}:f\in F\}\ \mbox{and}\ H=\{f_{C_0}:f\in F\}.
$$
\end{itemize}
\end{theorem}

\begin{proof}
The proof from (II) to (I) is straightforward. We will only give the
 proof from (I) to (II)
by using the idea in the proof of \cite[Theorem 2.5]{zhang}.

Since $F$ is equi-asymptotically almost periodic, for every
$k\in\mathbb{N}$, there exist a constant $M_k>0$ and a relatively dense set
$T(F,k)\subset \mathbb{R}$ such that
\begin{equation}\label{1}
\|f(t+\tau)-f(t)\|<\frac{1}{k},
\end{equation}
for all $f\in F$, $t\in\mathbb{R}$ with $|t|>M_k$ and $\tau\in T(F,k)$ with
$|t+\tau|>M_k$. Moreover, for every $f\in F\subset AAP(\mathbb{R},X)$,
noting that $f$ is uniformly continuous, for the above $k\in\mathbb{N}$,
there exists $\delta^f_k>0$ such that
\begin{equation}\label{2}
\|f(t_1)-f(t_2)\|<\frac{1}{k}
\end{equation}
for all $t_1,t_2\in \mathbb{R}$ with $|t_1-t_2|<\delta^f_k$.

Now, for every $t\in\mathbb{R}$ and $k\in\mathbb{N}$, we choose $\tau_k^t\in T(F,k)$
with $t+\tau_k^t>M_k$. Also, we denote
$$
g^f_k(t)=f(t+\tau_k^t),\quad t\in\mathbb{R},\ k\in\mathbb{N},\ f\in F.
$$ 
Next, we divide the remaining proof into eight steps.
\smallskip

\noindent\textbf{Step 1.} For every $f\in F$, there holds
\begin{equation}\label{step1}
\|g^f_k(t_1)-g^f_k(t_2)\|<\frac{5}{k}
\end{equation}
for all $k\in\mathbb{N}$, and $t_1,t_2\in \mathbb{R}$ with $|t_1-t_2|<\delta^f_k$.
In fact, by \eqref{1} and \eqref{2}, we have
\begin{align*}
&\|g^f_k(t_1)-g^f_k(t_2)\|\\
&= \|f(t_1+\tau_k^{t_1})-f(t_2+\tau_k^{t_2})\|\\
&\leq
\|f(t_1+\tau_k^{t_1})-f(t_1+\tau_k^{t_1}+\tau)\|+\|f(t_1+\tau_k^{t_1}+\tau)-f(t_2+\tau_k^{t_1}+\tau)\|\\
&\quad+\|f(t_2+\tau_k^{t_1}+\tau)-f(t_2+\tau)\|
 +\|f(t_2+\tau)-f(t_2+\tau+\tau_k^{t_2})\|\\
&\quad +\|f(t_2+\tau+\tau_k^{t_2})-f(t_2+\tau_k^{t_2})\|
< \frac{5}{k},
\end{align*}
where $\tau\in T(F,k)$ satisfies
$$
\min\{t_1+\tau_k^{t_1}+\tau,t_2+\tau_k^{t_1}+\tau,t_2+\tau,t_2
+\tau+\tau_k^{t_2}\}>M_k.
$$


\noindent\textbf{Step 2.} For every $k\in\mathbb{N}$, there holds
\begin{equation}\label{step2}
\|g^f_k(t+\tau)-g^f_k(t)\|<\frac{5}{k}
\end{equation}
for all $f\in F$, $\tau\in T(F,k)$, and $t\in\mathbb{R}$.
In fact, by using \eqref{1}, we have
\begin{align*}
&\|g^f_k(t+\tau)-g^f_k(t)\|\\
&=\|f(t+\tau+\tau_k^{t+\tau})-f(t+\tau_k^{t})\|\\
&\leq \|f(t+\tau+\tau_k^{t+\tau})-f(t+\tau+\tau_k^{t+\tau}+\tau')\|\\
&\quad +\|f(t+\tau+\tau_k^{t+\tau}+\tau')-f(t+\tau_k^{t+\tau}+\tau')\|\\
&\quad +\|f(t+\tau_k^{t+\tau}+\tau')-f(t+\tau')\|
 +\|f(t+\tau')-f(t+\tau'+\tau_k^t)\|\\
&\quad +\|f(t+\tau'+\tau_k^t)-f(t+\tau_k^{t})\|
< \frac{5}{k},
\end{align*}
where $\tau'\in T(F,k)$ satisfies
$$
\min\{t+\tau+\tau_k^{t+\tau}+\tau',t+\tau_k^{t+\tau}
+\tau',t+\tau',t+\tau'+\tau_k^t\}>M_k.
$$


\noindent\textbf{Step 3.} For every $n\in\mathbb{N}$, there holds
\begin{equation}\label{step3}
\|g^f_m(t)-g^f_n(t)\|<\frac{4}{n}
\end{equation}
for all $f\in F$, $t\in\mathbb{R}$, and $m,n\in\mathbb{N}$ with $m\geq n$.
In fact, without loss of generality, we can assume that 
$M_{k+1}\geq M_k$ for all $k\in\mathbb{N}$. Then, by using \eqref{1}, we have
\begin{align*}
&\|g^f_m(t)-g^f_n(t)\|\\
&=\|f(t+\tau_m^{t})-f(t+\tau_n^{t})\|\\
&\leq \|f(t+\tau_m^{t})-f(t+\tau_m^{t}+\tau)\|
 +\|f(t+\tau_m^{t}+\tau)-f(t+\tau_m^{t}+\tau+\tau_n^{t})\|\\
&\quad +\|f(t+\tau_m^{t}+\tau+\tau_n^{t})-f(t+\tau+\tau_n^{t})\|
 +\|f(t+\tau+\tau_n^{t})-f(t+\tau_n^{t})\|\\
&< \frac{1}{n}+\frac{1}{n}+\frac{1}{m}+\frac{1}{n}\leq \frac{4}{n},
\end{align*}
where $\tau\in T(F,n)$ satisfies
$$
\min\{t+\tau_m^{t}+\tau,t+\tau_m^{t}+\tau+\tau_n^{t},t+\tau+\tau_n^{t}\}>M_m.
$$

\noindent\textbf{Step 4.}
 Let
$$
g^f(t)=\lim_{n\to \infty}g^f_n(t),\quad t\in\mathbb{R},\ f\in F.
$$
By Step 3, we know that for every $f\in F$, $g^f$ is well-defined.
Moreover, it follows from Step 3 that for every $n\in\mathbb{N}$, there
holds
\begin{equation}\label{step4}
\|g^f(t)-g^f_n(t)\|\leq \frac{4}{n}
\end{equation}
for all $f\in F$, $t\in\mathbb{R}$, and $n\in\mathbb{N}$.
\smallskip

\noindent\textbf{Step 5.}
 For every $f\in F$, $g^f$ is uniformly continuous on $\mathbb{R}$.
In fact, by \eqref{step1} and \eqref{step4}, we have
\begin{align*}
\|g^f(t_1)-g^f(t_2)\|
&\leq \|g^f(t_1)-g^f_n(t_1)\|+
\|g^f_n(t_1)-g^f_n(t_2)\|+\|g^f_n(t_2)-g^f(t_2)\|\\
&\leq \frac{4}{n}+ \frac{5}{n} +\frac{4}{n}=\frac{13}{n},
\end{align*}
for all $n\in \mathbb{N}$ and $t_1,t_2\in \mathbb{R}$ with $|t_1-t_2|<\delta^f_n$.
\smallskip

\noindent\textbf{Step 6.} $\{g^f\}_{f\in F}$ is equi-almost periodic.
By \eqref{step2} and \eqref{step4}, for every $n\in \mathbb{N}$, we obtain
\begin{align*}
&\|g^f(t+\tau)-g^f(t)\|\\
&\leq \|g^f(t+\tau)-g_n^f(t+\tau)\|+
\|g_n^f(t+\tau)-g_n^f(t)\|+\|g_n^f(t)-g^f(t)\|\\
&\leq \frac{4}{n}+ \frac{5}{n} +\frac{4}{n}=\frac{13}{n},
\end{align*}
for all $f\in F$, $\tau\in T(F,n)$, and $t\in\mathbb{R}$. Then, it follows
that $\{g^f\}_{f\in F}$ is equi-almost periodic.
\smallskip

\noindent\textbf{Step 7.} $\{h^f\}_{f\in F}$ is equi-$C_0$, 
where $h^f(t)=f(t)-g^f(t)$ for all $f\in F$ and $t\in\mathbb{R}$.
In fact, firstly, by Step 5, $h^f\in C(\mathbb{R},X)$ for every $f\in F$;
secondly, for every $n\in\mathbb{N}$, by \eqref{step4} and the definition of
$\tau_n^t$, we have
\begin{align*}
\|h^f(t)\|
&= \|f(t)-g^f(t)\|\\
&\leq \|f(t)-g_n^f(t)\|+\|g^f_n(t)-g^f(t)\|\\
&\leq \|f(t)-f(t+\tau_n^t)\|+\frac{4}{n}\\
&\leq \frac{1}{n}+\frac{4}{n}=\frac{5}{n},
\end{align*}
for all $f\in F$, $t\in\mathbb{R}$ with $|t|>M_n$. Thus, $\{h^f\}_{f\in F}$
is equi-$C_0$.
\smallskip

\noindent\textbf{Step 8.}
 It follows from the above steps that $G=\{g^f\}_{f\in F}$
and $H=\{h^f\}_{f\in F}$. This completes the proof.
\end{proof}


\section{Application to functional integral equations}

In this section, we apply some results in Section 2 to discuss the
existence of asymptotically almost periodic solutions for the
following functional integral equation:
\begin{equation}\label{fun-int-eq}
x(t)=\sum_{i=1}^n f_i(t,x(t))\cdot\int_{\mathbb{R}}k_i(t,s)g_i(s,x(s))ds,\quad
t\in\mathbb{R},
\end{equation}
where $n$ is a fixed positive integer, and $f_i:\mathbb{R}\times \mathbb{R}\to\mathbb{R}$,
$k_i:\mathbb{R}\times\mathbb{R}\to\mathbb{R}$, $g_i:\mathbb{R}\times \mathbb{R}\to\mathbb{R}$ ($i=1,2,\dots,n$)
satisfy some conditions stated below.

The study on the existence of solutions to various kinds
of functional integral equations has been one of the most
attractive topics in the theory of integral equations. We refer the
reader to \cite{a1,b1,d1,ding2,long2} and references
therein for some of recent developments on this topic. Especially,
in \cite{ding2,long2}, the authors obtained some results concerning
periodic solutions and almost periodic solutions for some variants
of equation \eqref{fun-int-eq}. In addition, we
would like to note that by a recent work \cite{z2},
in the sense of category, the ``amount'' of almost periodic functions
(not periodic) is far more than the ``amount" of continuous periodic
functions. Thus, studying the existence of almost periodic solutions
for differential equations is necessary.

Next, we will extend the results in \cite{ding2,long2} to
asymptotically almost periodic case for equation \eqref{fun-int-eq}. 
Before establishing our existence theorem, we
first recall a fixed point theorem in a Banach algebra, which is due
to \cite[Theorem 2.1]{long2} (see also \cite[Theorem 2.1]{d2}).


\begin{theorem}\label{fp}
Let $n$ be a positive integer, and $C$ be a nonempty, closed, convex 
and bounded subset of a Banach algebra $X$. Assume that the operators
 $A_i:X\to X$ and $B_i:C\to X$, $i=1,2,\dots,n$, satisfy
\begin{itemize}
\item[(a)] for each $i\in \{1,2,\dots,n\}$, there exists $L_i>0$
such that 
\[
\|A_ix-A_iy\|\leq L_i\|x-y\|\quad\text{for all }x,y\in X;
\]
\item[(b)] for each $i\in \{1,2,\dots,n\}$, $B_i$ is continuous
and $B_i(C)$ is precompact;
\item[(c)] for each $y\in C$, $x=\sum_{i=1}^n A_i x\cdot B_i
y$ implies that $x\in C$;
\end{itemize}
Then, the operator equation $x=\sum_{i=1}^n A_i x\cdot B_i x$
has a solution provided that $\sum_{i=1}^n (L_i\cdot M_i)<1$,
where $M_i=\sup_{x\in C}\|B_i x\|$, $i=1,2,\dots,n$.
\end{theorem}

Now, we present our existence theorem.

\begin{theorem}\label{th}
Let $p\geq 1$ and $\frac{1}{p}+\frac{1}{q}=1$. 
Assume that the following assumptions hold:
\begin{itemize}
\item[(H1)] For each $i\in \{1,2,\dots,n\}$, $f_i(\cdot,x)\in AAP(\mathbb{R})$ for any fixed $x\in\mathbb{R}$ and there exists a
constant $L_i>0$ such that
$$
|f_i(t,x)-f_i(t,y)|\leq L_i|x-y|,\quad \forall t\in \mathbb{R},\ \forall x,y \in\mathbb{R}.
$$
\item[(H2)] For each $i\in \{1,2,\dots,n\}$, $g_i(\cdot,x)$
 is measurable for all $x\in\mathbb{R}$,
$g_i(t,\cdot)$ is continuous for almost all $t\in \mathbb{R}$, and for each
$r>0$, there exists a function $\mu^r_{i}\in L^{p}(\mathbb{R})$ such that
$|g_i(t,x)|\leq\mu_{i}^r(t)$ for all $|x|\leq r$ and almost all
$t\in \mathbb{R}$.
\item[(H3)] For each $i\in \{1,2,\dots,n\}$, 
$\widetilde{k}_{i}\in AAP(\mathbb{R},L^q(\mathbb{R}))$, where 
$[\widetilde{k}_{i}(t)](s)=k(t,s)$, $\forall
t,s\in\mathbb{R}$.
\item[(H4)] There exists a constant $r_0>0$ such that
\begin{equation}\label{11}
\sum_{i=1}^nL_iK_i \|\mu_i^{r_0}\|_p<1,
\end{equation}
 where
$K_i=\sup_{t\in \mathbb{R}}\|\widetilde{k}_{i}(t)\|_q$; and
\begin{equation}\label{12}
\sum_{i=1}^n\big[\sup_{t\in\mathbb{R},|x|\leq r}|f_i(t,x)| 
\cdot K_i \cdot\|\mu_i^{r_0}\|_p\big]<r,\quad \forall r>r_0.
\end{equation}
\end{itemize}
Then  \eqref{fun-int-eq} has an asymptotical
 almost periodic solution.
\end{theorem}

\begin{proof}
For each $i\in \{1,2,\dots,n\}$, we denote
$$
(A_ix)(t)=f_i(t,x(t)),\quad x\in AAP(\mathbb{R}),\ t\in\mathbb{R},
$$
and
$$
(B_ix)(t)=\int_{\mathbb{R}}k_i(t,s)g_i(s,x(s))ds,\quad x\in AAP(\mathbb{R}),\ t\in\mathbb{R}.
$$

Now, we show that $A_i,B_i$ map $AAP(\mathbb{R})$ into $AAP(\mathbb{R})$. Fix
 $x\in AAP(\mathbb{R})$. By using (H1), we can get $A_ix\in AAP(\mathbb{R})$. In addition,
since (H2) holds, for all $t_1,t_2\in \mathbb{R}$, we have
\begin{equation}\label{5}
|(B_ix)(t_1)-(B_ix)(t_2)|\leq
\|\widetilde{k}_{i}(t_1)-\widetilde{k}_{i}(t_2)\|_{q}\cdot
\|\mu_i^{\|x\|_{AAP(\mathbb{R})}}\|_p.
\end{equation}
Combining \eqref{5} and the fact that 
$\widetilde{k}_{i}\in AAP(\mathbb{R},L^q(\mathbb{R}))$, we conclude that $B_ix\in AAP(\mathbb{R})$.

Let $C=\{x\in AAP(\mathbb{R}):\|x\|\leq r_0 \}$, where $r_0$ is the constant
in (H4). Obviously, $C$ is a nonempty, closed, convex and bounded
subset in $AAP(\mathbb{R})$. In addition, it follows from (H1) that for each
$i\in \{1,2,\dots,n\}$, $\|A_ix-A_iy\|\leq L_i\|x-y\|$ for all
$x,y\in X$, i.e., the assumption (a) of Theorem \ref{fp} holds.

Next, we show that the assumption (b) of Theorem \ref{fp} holds. We
first show that every $B_i$ is continuous. Let $x_k\to x$ in
$AAP(\mathbb{R})$. Since
$$
|(B_ix_k)(t)-(B_ix)(t)|\leq K_i\|g_i(\cdot,x_k(\cdot))-g_i(\cdot,x(\cdot))\|_p,
$$
and $g_i(t,\cdot)$ is continuous for almost all $t\in \mathbb{R}$, by using
Lebesgue's dominated convergence theorem, we conclude that
$B_ix_k\to B_ix$ in $AAP(\mathbb{R})$. So $B_i$ is continuous. In addition,
by a direct calculation, we have
$$
|(B_ix)(t)|\leq K_i\|\mu_i^{r_0}\|, \quad t\in\mathbb{R},\ x\in C,
$$
which means that every $B_i(C)$ is uniformly bounded. Also, it
follows from \eqref{5} that for all $t_1,t_2\in \mathbb{R}$,
$$ 
|(B_ix)(t_1)-(B_ix)(t_2)|
\leq \|\widetilde{k}_{i}(t_1)-\widetilde{k}_{i}(t_2)\|_{q}\cdot
\left\|\mu_i^{r_0}\right\|_p,
$$ 
which yields that every $B_i(C)$ is
equi-uniformly continuous and equi-asymptotically almost periodic.
Then, by using Theorem \ref{precompact}, we know that every $B_i(C)$
is precompact in $AAP(\mathbb{R})$.

Now, let us verify the assumption (c) of Theorem \ref{fp}. 
Let $y\in C$ and $x=\sum_{i=1}^n A_i x\cdot B_i y$. We claim that $x\in
C$. Otherwise, $\|x\|> r_0$. Then, by using \eqref{12}, we obtain
$$
\|x\|=\|\sum_{i=1}^n A_i x\cdot B_i y\|
\leq \sum_{i=1}^n\big[\sup_{t\in\mathbb{R}}|f_i(t,x(t))| \cdot 
K_i \cdot\|\mu_i^{r_0}\|_p\big]<\|x\|,
$$
which is a contradiction.

It follows from \eqref{11} that
$$
\sum_{i=1}^n (L_i\cdot
M_i)=\sum_{i=1}^n (L_i\cdot \sup_{x\in C}\|B_ix\|)\leq
\sum_{i=1}^nL_iK_i \|\mu_i^{r_0}\|_p<1.
$$ 
So all the assumptions of Theorem \ref{fp} hold. Thus, the operator equation
$x=\sum_{i=1}^n A_i x\cdot B_i x$ has a solution in
$AAP(\mathbb{R})$; i.e., Equation \eqref{fun-int-eq} has an
asymptotical almost periodic solution.
\end{proof}

Next, we give an example, which does not aim to generality but
illustrate our existence theorem.

\begin{example} \label{ex3.3} \rm
Let $n=1$, $p=1$, $q=\infty$,
\begin{gather*}
f_1(t,x)=\frac{\arctan x\cdot (\sin t+\sin (\pi t)+e^{-t^2})}{10},\quad
g_1(t,x)=\frac{x\sin(  xe^{t^2})}{4(1+t^2)},\\
k_1(t,s)=(\sin t+\sin (\sqrt{2}t)+\frac{1}{1+t^2})e^{-s^{2}}.
\end{gather*}
Since for each $x\in \mathbb{R}$, $f_1(\cdot,x)\in AAP(\mathbb{R})$ and
$$
|f_1(t,x)-f_1(t,y)|\leq \frac{3}{10}|x-y|,\quad t\in\mathbb{R}, \ x,y\in\mathbb{R},
$$
we know that (H1) holds with $L_1=3/10$. It is easy to
verify that (H2) holds with 
$$
\mu_1(r)=\frac{r}{4(1+t^2)},\ r>0.
$$
Obviously, (H3) holds and 
$K_1=\sup_{t\in \mathbb{R}}\|\widetilde{k}_{1}(t)\|_{\infty}\leq 3$. 
Moreover, by a direct calculation, one can show that (H4) holds 
with $r_0=1$. So, by
Theorem \ref{th}, the following functional integral equation
$$
x(t)= f_1(t,x(t))\cdot\int_{\mathbb{R}}k_1(t,s)g_1(s,x(s))ds,\quad
t\in\mathbb{R},
$$
 has an asymptotical almost periodic solution.
\end{example}

\subsection*{Acknowledgements}
This work was partially supported by grants 11101192 from
the NSF of China, 211090 from the Key Project of Chinese Ministry of Education,
20114BAB211002 from the NSF of Jiangxi Province, and GJJ12173 from
the Jiangxi Provincial Education Department.

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\end{document}