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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2011 (2011), No. 72, 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 2011 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2011/72\hfil Almost periodic solutions]
{Almost periodic solutions of neutral functional differential
equations with Stepanov-almost periodic terms}

\author[Md. Maqbul \hfil EJDE-2011/72\hfilneg]
{Md. Maqbul} 

\address{Md. Maqbul \newline
Department  of Mathematics and  Statistics, 
Indian Institute of Technology Kanpur,
Kanpur - 208016, India}
\email{maqbul@iitk.ac.in}

\thanks{Submitted May 4, 2011. Published May 31, 2011.}
\subjclass[2000]{43A60, 34C27, 34K14, 47D06}
\keywords{Stepanov-almost periodic; almost periodic;
$C_0$-semigroup, \hfill\break\indent
neutral differential equation; contraction mapping principle}

\begin{abstract}
 In this paper we study the existence of  almost periodic solutions
 of an autonomous neutral functional differential equation
 with Stepanov-almost periodic terms in a Banach space.
 We use the contraction mapping principle to show the existence
 and the uniqueness of an almost periodic solution of the equation.
\end{abstract}

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

\section{Introduction}

The theory of almost periodic functions was mainly treated and
created by  Bohr during 1924-1926. Bohr's theory was substantially
developed by  Bochner,  Weyl,  Besicovitch,  Farvard,  von
Neumann,  Stepanov, Bogolyubov, and others during the 1920s and
1930s. In 1933,  Bochner defined and studied the almost periodic
functions with values in Banach spaces. Bohr's theory of almost
periodic functions was restricted to the class of uniformly
continuous functions. In 1925,  Stepanov generalized the class of
almost periodic functions in the sense of Bohr without using the
hypothesis of continuity. For more details about almost periodic
functions and Stepanov-almost periodic functions, see
\cite{Amerio,Levitan}. In recent years, the theory of almost
periodic functions has been developed in connection with the
problems of differential equations, dynamical systems, stability
theory and so on.

Functional differential equations arise as models in several
physical phenomena, for example, reaction-diffusion equations,
climate models, population ecology, neural networks etc. More
recently researchers have given special attentions to the study of
equations in which the delay argument occurs in the derivative of
the state variable as well as in the independent variable,
so-called neutral differential equations. Neutral differential
equations have many applications. For example, these equations
arises in many phenomena such as in the study of oscillatory
systems and also in modelling of several physical problems.
Periodicity of solutions of neutral differential equations has
been studied by many authors; see \cite{Chen,Islam}.

Let $(\mathbb{X},\|.\|)$ be a complex Banach space. In this paper,
we study the existence and the uniqueness of an almost periodic
solution to the neutral functional differential equation
\begin{equation} \label{eqn1}
 \frac{d}{dt}[u(t)-F(t,u(t-g(t)))] = Au(t)+G(t,u(t),u(t-g(t)))
\end{equation}
 for $t\in\mathbb{R}$ and $u\in AP(\mathbb{R};\mathbb{X})$,
where $AP(\mathbb{R};\mathbb{X})$ be the set of all almost
periodic functions from $\mathbb{R}$ to $\mathbb{X}$, $A$ is the
infinitesimal generator  of a $C_0$-semigroup $\{T(t)\}_{0\leq
t<\infty}$, and $F:\mathbb{R}\times\mathbb{X}\mapsto\mathbb{X}$,
$G:\mathbb{R}\times \mathbb{X}\times\mathbb{X}\mapsto\mathbb{X}$
are Stepanov-almost periodic functions.

  The existence of almost periodic solutions of abstract
differential equations has been considered by many authors; see
\cite{Amerio,ASRao,Zaidman,Zaidman1,Zaidman2}. Zaidman
\cite{Zaidman} proved the existence and uniqueness of almost
periodic solution to the nonhomogeneous differential equation
\begin{equation}
\frac{d}{dt}u(t)=Au(t)+f(t),\label{1}
\end{equation}
where $A$ is a linear unbounded operator in $\mathbb{X}$  which is
the infinitesimal generator of a $C_0$-semigroup with
exponential decay as $t\rightarrow\infty$, and
$f:\mathbb{R}\mapsto\mathbb{X}$ is an almost periodic function.
Zaidman \cite{Zaidman1} considered the same equation \eqref{1} in a
Hilbert space $\mathbb{H}$ and proved the existence and the
uniqueness of almost periodic solution provided that $A$ is a
bounded linear operator in $\mathbb{H}$ such that $\|e^{-tA}\|\leq
e^{-\omega t}$, for all $t>0$ and for some $\omega>0$, and
$f:\mathbb{R}\mapsto\mathbb{H}$ is a continuous function which is
in $S^2_{ap}(\mathbb{R};\mathbb{H})$, Rao \cite{ASRao} also
considered the same equation \eqref{1} in a Banach space
$\mathbb{X}$ and proved the existence and the uniqueness of almost
periodic solution provided that $A$ is the infinitesimal generator
of a continuous semigroup $\{T(t): 0\leq t<\infty\}$, with $T(t)$
satisfying $\|T(t)\|\leq Me^{-\beta t}$ for some $M>0$, for some
$\beta>0$ and all $t\geq0$, and $f:\mathbb{R}\mapsto\mathbb{X}$ is
an $S^1$-almost periodic continuously differentiable function,
with $f'$ being $S^1$-bounded on $\mathbb{R}$.

In this paper, we extend the previous-mentioned results to the
equation \eqref{eqn1}.  We use the contraction mapping principle
to prove the existence and uniqueness of an almost periodic
solution of the equation \eqref{eqn1}.

  \section{Preliminaries}

In this section we give some basic definitions, notation, and
results.  In the rest of this paper, $(\mathbb{X},\|\cdot\|)$
stands for a complex Banach  space.

\begin{definition} \label{def1}\rm
A one parameter family $\{T(t)\}_{0\leq t<\infty}$ of bounded
linear operators from $\mathbb{X}$ into $\mathbb{X}$ is called a
$C_0$-semigroup of bounded linear operators on $\mathbb{X}$ if
\begin{itemize}
\item[(i)] $T(0)=I$, where $I$
is the identity operator on $\mathbb{X}$.
\item[(ii)] $T(t+s)=T(t)T(s)$ for
every $t,s\geq0$.
\item[(iii)] $\lim_{t\downarrow
0}{T(t)x}=x$ for every $x\in\mathbb{X}$.
\end{itemize}
\end{definition}

The linear operator $A$ defined by
\begin{gather*}
D(A)=\big\{x\in
\mathbb{X}:\lim_{t\downarrow 0}\frac{T(t)x-x}{t}\text{ exists }\big\}\\
Ax=\lim_{t\downarrow 0}\frac{T(t)x-x}{t}
=\frac{d^+T(t)x}{dt}|_{t=0}\quad\text{for } x\in D(A)
\end{gather*}
is the infinitesimal generator of the semigroup
$\{T(t)\}_{0\leq t<\infty}$, where $D(A)$ is the domain of $A$.

\begin{theorem} \label{thm1}
Let $\{T(t)\}_{0\leq t<\infty}$ be a $C_0$-semigroup of bounded
linear operators on $\mathbb{X}$. Then
\begin{itemize}
\item[(i)] there exists $\omega\in\mathbb{R}$ and $M\geq1$ such that
$$\|T(t)\|\leq Me^{t\omega},\quad \forall t\geq0;$$
\item[(ii)] the mapping $(t,x)\mapsto T(t)x$ is jointly continuous
from $[0,\infty)\times\mathbb{X}$ to $\mathbb{X}$.
\end{itemize}
\end{theorem}

For a detailed proof of the above theorem,
see \cite[theorem 2.3.1]{Vrabie},  and
\cite[corollary 2.3.1]{Vrabie}.

\begin{definition} \label{def2.3}\rm
A continuous function $f : \mathbb{R} \mapsto \mathbb{X}$ is said to be  almost periodic  if for every $\epsilon>0$ there exists a positive number $l$  such that every interval of length $l$ contains a number $\tau$ such that
$$
\|f(t+\tau)-f(t)\|<\epsilon \quad \forall t\in \mathbb{R}.
$$
\end{definition}

Let $AP(\mathbb{R};\mathbb{X})$ be the set of all almost periodic
functions from $\mathbb{R}$ to $\mathbb{X}$.
Then $(AP(\mathbb{R};\mathbb{X}),\|.\|_{\infty})$ is a Banach space
with supremum norm given by
$$
\|u\|_{\infty} = \sup_{t\in \mathbb{R}}\|u(t)\|.
$$

\begin{theorem} \label{thm2}
If $f\in AP(\mathbb{R};\mathbb{X})$, then $f$ is uniformly
continuous.
\end{theorem}

\begin{theorem}[Bochner's Criterion] \label{thm2.5}
 A continuous function $f:\mathbb{R}\mapsto\mathbb{X}$ is an
almost periodic function if and only if for every sequence of real
numbers $(s'_n)$, there exists a subsequence $(s_n)$ such that
$(f(t+s_n))$ converges uniformly for $t\in\mathbb{R}$.
\end{theorem}

\begin{lemma} \label{lem2.6}  %\label{lemma}
If $u\in AP(\mathbb{R};\mathbb{X})$ and $g\in
AP(\mathbb{R};\mathbb{R})$, then $u(.-g(.))\in
AP(\mathbb{R};\mathbb{X})$.
\end{lemma}

For a detailed proof of the above lemma see \cite[Lemma 2.4]{Chen}.
Let $\mathbb{Y},\;\mathbb{W}$ be Banach spaces. We define the set
$AP(\mathbb{R}\times \mathbb{X};\mathbb{Y})$ which consists of all
continuous functions
$f:\mathbb{R}\times\mathbb{X}\mapsto\mathbb{Y}$ such that $f(.,
x)\in AP(\mathbb{R};\mathbb{Y})$ uniformly for each $x\in E$,
where $E$ is any compact subset of $\mathbb{X}$.

\begin{proposition}[{\cite[Proposition 1]{Amir}}] \label{prop2.7}
If $f\in AP(\mathbb{R}\times \mathbb{X};\mathbb{Y})$ and $h\in
AP(\mathbb{R};\mathbb{X})$, then the function $f(.,h(.))\in
AP(\mathbb{R};\mathbb{Y})$.
\end{proposition}

Throughout the rest of the paper we fix $p, 1 \leq p <\infty$.
Denote by $L^{p}_{\rm loc}(\mathbb{R};\mathbb{X})$ the space of all
functions from $\mathbb{R}\;\mbox{into}\;\mathbb{X}$ which are
locally $p$-integrable in Bochner-Lebesgue sense. We say that a
function, $f\in L^{p}_{\rm loc}(\mathbb{R};\mathbb{X})$ is
$p$-Stepanov bounded ($S^{p}$-bounded) if
$$
\|f\|_{S^{p}} = \sup_{t\in\mathbb{R}}
\Big(\int_{t}^{t+1}\|f(s)\|^{p}ds\Big)^{1/p}<\infty.
$$
We indicate by $L^{p}_{s}(\mathbb{R};\mathbb{X})$ the set of
$S^{p}$-bounded functions.

\begin{definition} \label{stepanovap}\rm
A function $f\in L^{p}_{s}(\mathbb{R};\mathbb{X})$ is said to be
almost periodic in the sense of Stepanov ($S^{p}$-almost periodic)
if for every $\epsilon>0$ there exists a positive number $l$  such
that every interval of length $l$ contains a number $\tau$ such that
$$
\sup_{t\in\mathbb{R}}\Big(\int_{t}^{t+1}\|f(s+\tau)-f(s)\|^{p}ds
\Big)^{1/p}<\epsilon.
$$
\end{definition}

Let $S^{p}_{ap}(\mathbb{R};\mathbb{X})$ be the set of all
$S^{p}$-almost periodic functions.

It is clear that $f(t)$ almost periodic implies
$f(t)$ is $S^{p}$-almost periodic; that is,
 $AP(\mathbb{R};\mathbb{X})\subset S^{p}_{ap}(\mathbb{R};\mathbb{X})$.
Moreover, if $1\leq m<p$, then $f(t)$ is $S^{p}$-almost periodic
implies $f(t)$ is $S^{m}$-almost periodic.

\begin{lemma}[\cite{Amerio}, Bochner] \label{lemma1}
If $f\in S^{p}_{ap}(\mathbb{R};\mathbb{X})$ and uniformly
continuous, then $f$ is almost periodic.
\end{lemma}

\begin{lemma}[Bochner]  \label{lemma2}
$f\in S^{p}_{ap}(\mathbb{R};\mathbb{X})$ if and only if
$f^{b}\in AP(\mathbb{R};L^{p}([0,1];\mathbb{X}))$, where
$f^{b}(t)=\{f(t+s):s\in[0,1]\}$, $t\in\mathbb{R}$.
\end{lemma}

For a detailed proof of the above lemma, see
\cite[pp. 78, 79]{Amerio}.

 We define the set $S^{p}_{ap}(\mathbb{R}\times
\mathbb{X};\mathbb{Y})$ which consists of all functions
$f:\mathbb{R}\times\mathbb{X}\mapsto\mathbb{Y}$ such that $f(.,
x)\in S^{p}_{ap}(\mathbb{R};\mathbb{Y})$ uniformly for each $x\in
E$, where $E$ is any compact subset of $\mathbb{X}$.

We define the set $S^{p}_{ap}(\mathbb{R}\times
\mathbb{X}\times\mathbb{Y};\mathbb{W})$ which consists of all
functions
$f:\mathbb{R}\times\mathbb{X}\times\mathbb{Y}\mapsto\mathbb{W}$
such that $f(., x,y)\in S^{p}_{ap}(\mathbb{R};\mathbb{W})$
uniformly for each $(x,y)\in E$, where $E$ is any compact subset
of $\mathbb{X}\times\mathbb{Y}$.

\begin{example}\label{exmp1} \rm
$(\mathbb{R}^2,\|\cdot\|)$ is a Banach space, where
$$
\|x\|=|x_{1}|+|x_2|,\quad x=(x_{1},x_2)\in\mathbb{R}^2.
$$
The functions
$F:\mathbb{R}\times\mathbb{R}^2\mapsto\mathbb{R}^2$,
$G:\mathbb{R}\times\mathbb{R}^2\times\mathbb{R}^2\mapsto\mathbb{R}^2$
are defined by
\begin{gather*}
F(t,x)=(f(t),\sin x_{1}-\sin x_2),\\
G(t,x,y)=K_2(f(t),e^{-x_{1}}-e^{-x_2}+\cos y_{1}-\cos y_2),
\end{gather*}
where $K_2>0$, $x=(x_{1},x_2),\;y=(y_{1},y_2)$, and
$$
f(t)=\begin{cases} n & t=n\pi,\; n\in\mathbb{Z}\\
\sin t & \mbox{otherwise.}\end{cases}
$$
Notice that $f\in S^{p}_{ap}(\mathbb{R};\mathbb{R})$ but
$f\notin AP(\mathbb{R};\mathbb{R})$, as it is unbounded and
discontinuous.
Hence it is easy to see that
$F\in S^{p}_{ap}(\mathbb{R}\times \mathbb{R}^2;\mathbb{R}^2)$,
$G\in S^{p}_{ap}(\mathbb{R}\times \mathbb{R}^2\times\mathbb{R}^2;
 \mathbb{R}^2)$ but
$F\notin AP(\mathbb{R}\times \mathbb{R}^2;\mathbb{R}^2)$,
$G\notin AP(\mathbb{R}\times \mathbb{R}^2\times
\mathbb{R}^2;\mathbb{R}^2)$.
\end{example}

Throughout the rest of the paper we assume that $A$ is the
infinitesimal generator of $C_0$-semigroup $\{T(t)\}_{0\leq
t<\infty}$. In the view of theorem \ref{thm1}(i), we also assume
that there exists constants $\omega>0$ and $M\geq1$ such that
\begin{equation}
\|T(t)\|\leq Me^{-\omega t}\;\;\mbox{for}\;\;0\leq t<\infty.
\label{2.1}
\end{equation}

\begin{definition} \label{def2.12}\rm
By an almost periodic mild solution $u :\mathbb{R} \mapsto
\mathbb{X}$ of the differential equation \eqref{eqn1} we mean that
$u\in AP(\mathbb{R};\mathbb{X})$, and $u(t)$ satisfies
\begin{equation}
u(t)=F(t,u(t-g(t)))+\int_{-\infty}^{t}T(t-s)G(s,u(s),u(s-g(s)))ds,
\quad t\in\mathbb{R}.       \label{sol1}
\end{equation}
\end{definition}

Throughout the rest of the paper we consider the
following assumptions.
\begin{itemize}
\item[(H1)] $g\in AP(\mathbb{R};\mathbb{R})$,
$F\in S^{p}_{ap}(\mathbb{R}\times\mathbb{X};\mathbb{X})$ and
$G\in S^{p}_{ap}(\mathbb{R}\times\mathbb{X}\times\mathbb{X};\mathbb{X})$.

\item[(H2)] The functions $F,G$  satisfy the property that
there exists $K_{1},K_2>0$ such that
$$
\|F(t,u(t))-F(s,v(s))\|\leq K_{1}\|u(t)-v(s)\|
$$
for all $t,s\in\mathbb{R}$ and for each $u,v\in
AP(\mathbb{R};\mathbb{X})$, and
$$
\|G(t,x_{1},\phi(t))-G(t,x_2,\varphi(t))\|
\leq K_2(\|x_{1}-x_2\|+\|\phi-\varphi\|_{\infty})
$$
for all $t\in\mathbb{R}$ and for $(x_{1},\phi),(x_2,\varphi)\in
\mathbb{X}\times AP(\mathbb{R};\mathbb{X})$.
\end{itemize}

\begin{example}\label{exmp2} \rm
Consider the function $G$ defined in example \ref{exmp1}.
For $x,y\in\mathbb{R}^2$ and $u,v\in
AP(\mathbb{R};\mathbb{R}^2)$, we observe that
\begin{align*}
&\|G(t,x,u(t))-G(t,y,v(t))\|\\
&\leq K_2|e^{-x_{1}}-e^{-x_2}+\cos
u_{1}(t)-\cos u_2(t)-e^{-y_{1}}+e^{-y_2}-\cos v_{1}(t)-\cos v_2(t)|\\
&\leq K_2(|x_{1}-y_{1}|+|x_2-y_2|+|u_{1}(t)-v_{1}(t)|+|
u_2(t)-v_2(t)|)\\
&= K_2(\|x-y\|+\|u(t)-v(t)\|)\\
&\leq K_2(\|x-y\|+\|u-v\|_{\infty})\quad \forall t\in\mathbb{R}.
\end{align*}
Thus $G$ satisfies the assumption (H2).

Define $F:\mathbb{R}\times\mathbb{R}^2\mapsto\mathbb{R}^2$ by
$F(t,x)=K_{1}(0,\sin x_{1}-\sin x_2)$, where $K_{1}>0$.
Clearly $F\in S^{p}_{ap}(\mathbb{R}\times\mathbb{R}^2;\mathbb{R}^2)$.

For $u,v\in AP(\mathbb{R};\mathbb{R}^2)$ and $t,s\in\mathbb{R}$,
we observe that
\begin{align*}
\|F(t,u(t))-F(s,v(s))\|&\leq K_{1}|\sin u_{1}(t)-\sin
u_2(t)-\sin v_{1}(s)+\sin v_2(s)|\\
&\leq K_{1}(|\sin u_{1}(t)-\sin v_{1}(s)|+|\sin u_2(t)-\sin
v_2(s)|)\\
&\leq K_{1}(|u_{1}(t)-v_{1}(s)|+|u_2(t)-v_2(s)|)\\
&= K_{1}\|u(t)-v(s)\|.
\end{align*}
Thus $F$ satisfies the assumption (H2).
\end{example}

  \section{Main results}

 In this section we prove the existence and uniqueness of almost
periodic mild solution for \eqref{eqn1}.
We define two mappings $\Lambda$ and $L$ by
\begin{gather}
(\Lambda u)(t)= F(t,u(t-g(t)))+\int_{-\infty}^{t}T(t-s)G(s,u(s),
u(s-g(s)))ds,   \label{3.1}\\
(Lf)(t)= \int_{-\infty}^{t}T(t-s)f(s)ds,\quad t\in\mathbb{R}.
\label{3.2}
\end{gather}
Throughout the rest of the paper we indicate the conjugate
index of $p$ by $q$; that is, $\frac{1}{p}+\frac{1}{q}=1$.
We show the following.

\begin{proposition} \label{prop2}
If $f\in S^{p}_{ap}(\mathbb{R}\times\mathbb{X};\mathbb{Y})$ and
$g\in AP(\mathbb{R};\mathbb{X})$, then
$f(.,g(.))\in S^{p}_{ap}(\mathbb{R};\mathbb{Y})$.
\end{proposition}

\begin{proof}
From  Lemma \ref{lemma2}, it follows that $f^{b}\in
AP(\mathbb{R}\times\mathbb{X};L^{p}([0,1];\mathbb{Y}))$, where
$f^{b}(t,x)=\{f(t+s,x):s\in[0,1]\},\;t\in\mathbb{R},x\in\mathbb{X}$.
From proposition \ref{prop2.7}, it follows that
$f^{b}(.,g(.))\in AP(\mathbb{R};L^{p}([0,1];\mathbb{Y}))$. Again
from  Lemma \ref{lemma2}, we get $f(.,g(.))\in
S^{p}_{ap}(\mathbb{R};\mathbb{Y})$.
\end{proof}

\begin{proposition} \label{prop3}
If $f\in S^{p}_{ap}(\mathbb{R}\times\mathbb{X}
\times\mathbb{Y};\mathbb{W})$,
$g\in AP(\mathbb{R};\mathbb{X})$ and
$h\in AP(\mathbb{R};\mathbb{Y})$, then
$f(.,g(.),h(.))\in S^{p}_{ap}(\mathbb{R};\mathbb{W})$.
\end{proposition}

\begin{proof}
From the Bochner's Criterion, it follows that $(g(.),h(.))\in
AP(\mathbb{R};\mathbb{X}\times\mathbb{Y})$. Hence from the
proposition \ref{prop2}, we get $f(.,g(.),h(.))\in
S^{p}_{ap}(\mathbb{R};\mathbb{W})$.
\end{proof}

\begin{lemma}  \label{lemma31}
If $f(t)$ is an $S^{p}$-almost periodic function, then the function
$(Lf)(t)$ is an almost periodic function.
\end{lemma}

\begin{proof}
We consider
$$
(Lf)_{k}(t)= \int_{t-k}^{t-k+1}T(t-s)f(s)ds,\quad
k\in\mathbb{N},\;t\in\mathbb{R}.
$$
Then
\begin{equation}
\begin{split}
\|(Lf)_{k}(t)\|
&\leq  \int_{t-k}^{t-k+1}\|T(t-s)\|\|f(s)\|ds \\
&\leq  M\int_{t-k}^{t-k+1}e^{-\omega(t-s)}\|f(s)\|ds.
\end{split}\label{3.3}
\end{equation}
\textbf{Case 1:} $1<p<\infty$.
Then $1<q<\infty$. Using the H\"{o}lder's inequality, we have
\begin{align*}
& M\int_{t-k}^{t-k+1}e^{-\omega(t-s)}\|f(s)\|ds\\
&\leq M\Big(\int_{t-k}^{t-k+1}e^{-\omega q(t-s)}ds\Big)^{1/q}
\Big(\int_{t-k}^{t-k+1}\|f(s)\|^{p}ds\Big)^{1/p}\\
&\leq \frac{M}{\sqrt[q]{q\omega}}\left(e^{-q\omega(k-1)}-e^{-q\omega k}\right)^{1/q}\|f\|_{S^{p}}\\
&= M\frac{e^{-\omega
k}}{\sqrt[q]{q\omega}}(e^{q\omega}-1)^{1/q}\|f\|_{S^{p}}.
\end{align*}
Since the series $\sum_{k=1}^{\infty}e^{-\omega k}$ is convergent,
therefore from the Weierstrass test the sequence of functions
$\sum_{k=1}^{n}(Lf)_{k}(t)$ is uniformly convergent on $\mathbb{R}$.
Hence we have
\begin{equation*}
(Lf)(t)=\sum_{k=1}^{\infty}(Lf)_{k}(t).
\end{equation*}
From  theorem \ref{thm1}(ii), $(Lf)(.)$ is continuous.
Let $\epsilon>0$. Then there exists a positive number $l$ such that
every interval of length $l$ contains a number $\tau$ such that
$$
\sup_{t\in\mathbb{R}}\Big(\int_{t}^{t+1}\|f(s+\tau)-f(s)\|^{p}ds
\Big)^{1/p}<\epsilon_{1},
$$
where
$$
0<\epsilon_{1}<\frac{\sqrt[q]{q\omega}(e^{\omega}-1)\epsilon}
{M(e^{q\omega}-1)^{1/q}}.
$$
Now we consider $\|(Lf)_{k}(s+\tau)-(Lf)_{k}(s)\|$
\begin{align*}
&= \big\|\int_{s+\tau-k}^{s+\tau-k+1}T(s+\tau-z)f(z)dz
-\int_{s-k}^{s-k+1}T(s-z)f(z)dz\big\|\\
&\leq \int_{s-k}^{s-k+1}\|T(s-z)\|\|f(\tau+z)-f(z)\|dz\\
&\leq M\int_{s-k}^{s-k+1}e^{-\omega(s-z)}\|f(\tau+z)-f(z)\|dz\\
&\leq M\Big(\int_{s-k}^{s-k+1}e^{-\omega q(s-z)}dz\Big)^{1/q}
\Big(\int_{s-k}^{s-k+1}\|f(z+\tau)-f(z)\|^{p}dz\Big)^{1/p}\\
&< \epsilon_{1}M\frac{e^{-\omega
k}}{\sqrt[q]{q\omega}}\left(e^{q\omega}-1\right)^{1/q}.
\end{align*}
Therefore,
\begin{align*}
\sum_{k=1}^{\infty}\|(Lf)_{k}(s+\tau)-(Lf)_{k}(s)\|
&\leq \frac{\epsilon_{1}M}{\sqrt[q]{q\omega}}
\sum_{k=1}^{\infty}e^{-\omega k}\left(e^{q\omega}-1\right)^{1/q}\\
&= \frac{\epsilon_{1}M(e^{q\omega}-1)^{1/q}}{\sqrt[q]{q\omega}
(e^{\omega}-1)}
<\epsilon.
\end{align*}
Hence $(Lf)(t)$ is an almost periodic function.

\textbf{Case 2:} $p=1$.
Then $q=\infty$ and using the H\"{o}lder's inequality, we have
\begin{align*}
M\int_{t-k}^{t-k+1}e^{-\omega(t-s)}\|f(s)\|ds
&\leq M\Big(\sup_{t-k\leq s\leq t-k+1}
e^{-\omega(t-s)}\Big)\Big(\int_{t-k}^{t-k+1}\|f(s)\|ds\Big)\\
&\leq Me^{-\omega(k-1)}\|f\|_{S^{1}}.
\end{align*}
Since the series $\sum_{k=1}^{\infty}e^{-\omega(k-1)}$ is
convergent, therefore from the Weierstrass test and
from \eqref{3.3}, the sequence of functions
$\sum_{k=1}^{n}(Lf)_{k}(t)$ is uniformly convergent on
$\mathbb{R}$. Hence we have
\begin{equation*}
(Lf)(t)=\sum_{k=1}^{\infty}(Lf)_{k}(t).
\end{equation*}
Notice that $(Lf)(.)$ is continuous.
Let $\epsilon>0$. Then there exists a positive number $l$ such that
every interval of length $l$ contains a number $\tau$ such that
$$
\sup_{t\in\mathbb{R}}\Big(\int_{t}^{t+1}\|f(s+\tau)-f(s)\|ds\Big)
<\epsilon_2,
$$
where
$$
0<\epsilon_2<\frac{(e^{\omega}-1)\epsilon}{Me^{\omega}}.
$$
Now we consider
\begin{align*}
&\|(Lf)_{k}(s+\tau)-(Lf)_{k}(s)\|\\
&= \big\|\int_{s+\tau-k}^{s+\tau-k+1}T(s+\tau-z)f(z)dz
-\int_{s-k}^{s-k+1}T(s-z)f(z)dz\big\|\\
&\leq \int_{s-k}^{s-k+1}\|T(s-z)\|\|f(\tau+z)-f(z)\|dz\\
&\leq M\int_{s-k}^{s-k+1}e^{-\omega(s-z)}\|f(\tau+z)-f(z)\|dz\\
&\leq M\Big(\sup_{s-k\leq z\leq s-k+1}e^{-\omega(s-z)}\Big)
\Big(\int_{s-k}^{s-k+1}\|f(z+\tau)-f(z)\|dz\Big)\\
&< \epsilon_2Me^{-\omega(k-1)}.
\end{align*}
Therefore,
\begin{align*}
\sum_{k=1}^{\infty}\|(Lf)_{k}(s+\tau)-(Lf)_{k}(s)\|
\leq \epsilon_2M\sum_{k=1}^{\infty}e^{-\omega(k-1)}
= \epsilon_2M\frac{e^{\omega}}{e^{\omega}-1}
< \epsilon.
\end{align*}
Hence $(Lf)(t)$ is an almost periodic function.
\end{proof}

\begin{lemma} \label{lemma32}
 The operator $\Lambda$ maps $AP(\mathbb{R};\mathbb{X})$ into
itself.
\end{lemma}

\begin{proof}
Let $u\in AP(\mathbb{R};\mathbb{X})$. From  Lemma
\ref{lem2.6}, we get $u(.-g(.))\in AP(\mathbb{R};\mathbb{X})$.
Hence from theorem \ref{thm2}, $u(t-g(t))$ is uniformly
continuous on $\mathbb{R}$.
For given $\epsilon>0$ there exists $\delta>0$ such that
$\|u(t_{1}-g(t_{1}))-u(t_2-g(t_2))\|<\epsilon/K_{1}$ whenever
$|t_{1}-t_2|<\delta$.
From the assumption (H2), we obtain
\begin{align*}
&\|F(t_{1},u(t_{1}-g(t_{1})))-F(t_2,u(t_2-g(t_2)))\|
&\leq  K_{1}\|u(t_{1}-g(t_{1}))-u(t_2-g(t_2))\|\\
&< \epsilon\quad \text{whenever }|t_{1}-t_2|<\delta.
\end{align*}
Hence  $F(t,u(t-g(t)))$ is uniformly continuous.
And also from the proposition \ref{prop2}, we have
 $F(.,u(.-g(.)))\in S^{p}_{ap}(\mathbb{R};\mathbb{X})$.
Thus from Lemma \ref{lemma1}, we obtain
 $F(.,u(.-g(.)))\in AP(\mathbb{R};\mathbb{X})$.
From  proposition \ref{prop3}, we obtain
$G(.,u(.),u(.-g(.)))\in S^{p}_{ap}(\mathbb{R};\mathbb{X})$.
Hence from  Lemma \ref{lemma31}, we obtain
$(LG)(.,u(.),u(.-g(.)))\in AP(\mathbb{R};\mathbb{X})$.
Thus  $(\Lambda u)(.)\in AP(\mathbb{R};\mathbb{X})$.
\end{proof}

\begin{theorem} \label{thm35}
Suppose $(K_{1}+\frac{2M}{\omega}K_2)<1$. Then
\eqref{eqn1} has unique  almost periodic mild solution.
\end{theorem}

\begin{proof}
Let $u,v\in AP(\mathbb{R};\mathbb{X})$. We observed that
\begin{align*}
&\|(\Lambda u)(t)-(\Lambda v)(t)\|\\
&\leq \|F(t,u(t-g(t)))-F(t-v(t-g(t)))\|\\
&\quad +\int_{-\infty}^{t}\|T(t-s)\|\|G(s,u(s),u(s-g(s)))-G(s,v(s),v(s-g(s)))\|ds\\
&\leq K_{1}\|u(t-g(t))-v(t-g(t))\|\\
&\quad + MK_2\int_{-\infty}^{t}e^{-\omega(t-s)}(\|u(s)-v(s)\|
 +\|u-v\|_{\infty})ds\\
&\leq K_{1}\|u-v\|_{\infty}+2MK_2
\Big(\int_{-\infty}^{t}e^{-\omega(t-s)}ds\Big)\|u-v\|_{\infty}\\
&\leq \Big(K_{1}+\frac{2M}{\omega}K_2\Big)\|u-v\|_{\infty}.
\end{align*}
Thus
\[
\|\Lambda u-\Lambda
v)\|_{\infty}\leq \big(K_{1}+\frac{2M}{\omega}K_2\big)
\|u-v\|_{\infty}.
\]
Thus $\Lambda$ is a contraction map on
$AP(\mathbb{R};\mathbb{X})$. Therefore,
 $\Lambda$ has unique fixed
point in $AP(\mathbb{R};\mathbb{X})$, that is, there exist unique
$\psi\in AP(\mathbb{R};\mathbb{X})$ such that $\Lambda\psi=\psi$.
Therefore the equation \eqref{eqn1} has unique almost periodic
mild solution.
\end{proof}


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\end{document}