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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2011 (2011), No. 117, pp. 1--12.\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-2011/117\hfil S-asymptotically periodic solutions]
{S-asymptotically periodic solutions for partial
differential equations with finite delay}

\author[W. Dimbour,  G. Mophou, G. M. N'Gu\'er\'ekata\hfil EJDE-2011/117\hfilneg]
{William Dimbour,  Gis\`ele Mophou, Gaston M. N'Gu\'er\'ekata}  % in alphabetical order

\address{William Dimbour\newline
Laboratoire C.E.R.E.G.M.I.A.,
Universit\'e des Antilles et de la Guyane, 
Campus Fouillole 97159 Pointe-\`a-Pitre  Guadeloupe (FWI)} 
\email{William.Dimbour@univ-ag.fr}

\address{Gis\`ele Mophou \newline
Laboratoire C.E.R.E.G.M.I.A.,
Universit\'e des Antilles et de la Guyane,  
Campus Fouillole 97159 Pointe-\`a-Pitre  Guadeloupe (FWI)}
\email{gisele.Mophou@univ-ag.fr}

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

\thanks{Submitted August 11, 2011. Published September 14, 2011.}
\subjclass[2000]{34K05, 34A12, 34A40}
\keywords{S-asymptotically periodic function; mild solution;
\hfill\break\indent
exponentially stable semigroup;  fractional power operator}

\begin{abstract}
 In this article, we give some sufficient conditions for the
 existence and uniqueness of S-asymptotically periodic
 (mild) solutions for some partial functional differential
 equations. To illustrate our main result, we study a  diffusion
 equation with delay.
\end{abstract}

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

\section{Introduction}

The main purpose of this work is to study the existence and uniqueness
of S-asymptotically periodic solutions in the $\alpha$-norm
for the partial differential equation
\begin{equation}\label{eq22}
 \begin{gathered}
 \frac{d}{dt} u(t)=-Au(t)+L(u_t)+f(t,u(t))\quad\text{for }t \ge 0,\\
 u_0=\varphi
\end{gathered}
\end{equation}
where  $-A$ is the  infinitesimal generator of an analytic semigroup
 $T(t)$, $t\geq 0$  on a Banach space $\mathbb{X}$.

For $0<\alpha\le 1$, let $ A^{\alpha}$ be the
fractional power of $A$ with domain $D(A^{\alpha})$, which endowed
with the norm $|x|_{\alpha}=\|A^{\alpha}x\|$ forms a Banach space
$\mathbb{X}_{\alpha}$.
Let $\mathcal{C}_{\alpha}=C([-r,0],\mathbb{X}_\alpha)$
be the Banach space of all continuous functions from $[-r,0]$ to
$\mathbb{X}_{\alpha}$ endowed with the norm
$$
| \phi|_{\mathcal{C}_{\alpha}}
=\sup_{-r\le\theta\le0}|\phi(\theta)|_{\alpha}.
$$
Let $L$ be a bounded linear operator from $\mathcal{C}_{\alpha}$ to
$\mathbb{X}_{\alpha}$, and
$f:\mathbb{R}\times \mathbb{X}_\alpha\to \mathbb{X}_\alpha$
a continuous function.
As usual the history function $x_t \in\mathcal{C}_{\alpha}$
is defined by
$$
x_t(\theta)=x(t+\theta)\quad\text{for }\theta\in]-r,0].
$$

The theory of partial functional differential
equations and its applications are an active are of research;
see for instance \cite{Hale,traweb,Wu} and the references therein.
Several articles study the existence and uniqueness of almost periodic,
almost automorphic, and weighted pseudo almost periodic solutions
of various differential equations. In \cite{Elazzouzi}, the author
deals with the existence of $C^{(n)}$-almost periodic
and $C^{(n)}$-automorphic solution of  the  equation
\begin{equation}\label{eq20}
 \begin{gathered}
  \frac{d}{dt} u(t)=-Au(t)+L(u_t)+f(t)\quad\text{for }t \ge 0,\\
 u_0=\varphi
\end{gathered}
\end{equation}
To achieve his goal, the author uses the the variation of constants
formula and the reduction method developed
 by Adimy et al. \cite{adimy0}.
Ezzinbi  and Boukli-Hacene \cite{Ezzinbi0} studied
the existence and uniqueness of weighted pseudo-almost automorphic
solution for \eqref{eq20}, using the variation of constants
formula developed  by Ezzinbi and N'Gu\'er\'ekata \cite{Ezzinbi}.

The literature relative to S-asymptotically periodic
functions remains limited due to the novelty of the concept.
Qualitative properties of such functions are discussed for instance
in \cite{Blot,Henriquez1,Lizama}. In \cite{Blot}, the authors
present a new composition theorem for such functions.
Various properties of S-asymptotically periodic functions
are also investigated in a general study of classes of bounded
continuous functions taking values in a Banach space $\mathcal{X}$.
In \cite{Cuevas}, a new concept of weighted S-asymptotically
periodic functions is introduced generalizing in a natural
way the one studied here. There  are some papers
dealing with the existence of S-asymptotically periodic
solutions of differential equations and fractional
differential equations in finite as well as infinite dimensional
spaces; see  \cite{Blot,Henriquez1,Henriquez2,Lizama,Nicola}.
In this paper, motivated by all these works, we first
reconsider \eqref{eq20} and prove that if $f$ is  an
S-asymptotically periodic function in the $\alpha$-norm
then its has a unique solution on $[-r,+\infty[$ .
Moreover, the restriction of the solution on $\mathbb{R}^+$ is
S-asymptotically periodic solutions in the $\alpha$-norm.
This allow us to study the existence and uniqueness of an
S-asymptotically  periodic solution in the $\alpha$-norm,
for \eqref{eq22}.

This work is organized as follows.  In Section \ref{prem},
we recall some fundamental properties of S-asymptotically
periodic functions  and  fractional powers of a closed
operator.  Section \ref{main} is devoted to the
main result. We illustrate our main result in
Section \ref{appl}  by examining  the existence and uniqueness
of S-asymptotically periodic (mild) solutions for
some diffusion equations with delay.

\section{PRELIMINARIES \label{prem}}

Let $(\mathbb{X},\|\cdot\|)$ be a Banach space.
Denote by $C(\mathbb{R}^+, \mathbb{X})$, the space of all continuous
functions from $\mathbb{R}^+$ to
$ \mathbb{X}$, and  by $BC(\mathbb{R}^+ , \mathbb{X})$ the space of
all bounded continuous functions  $\mathbb{R}^+ \to \mathbb{X}$.
The space $BC(\mathbb{R}^+ , \mathbb{X})$ endowed with the
supremum norm $\| f \|_{\infty}:=\sup_{t \ge 0} | | f(t) | |$
is a Banach space.

\subsection*{S-asymptotically periodic functions}

\begin{definition} \rm
For a function $f$ in $BC(\mathbb{R}^+, \mathbb{X})$, we say that
$f$ belongs to $C_0(\mathbb{R}^+, \mathbb{X})$ if
$ \lim_{ t\to \infty}\| f(t)\|=0$.

Let $\omega$ be a fixed positive number and
$f\in BC(\mathbb{R}^+, \mathbb{X})$. We say that
$f$ is $\omega$-periodic, denoted by $f \in P_{\omega}(\mathbb{X})$, if
$f$ has period $\omega$.
Note that  $P_{\omega}(\mathbb{X})$  is a Banach  subspace of
$BC(\mathbb{R}^+ , \mathbb{X})$ under the supremum norm.

\end{definition}


\begin{definition}[\cite{Blot,Lizama}] \rm
Let $f \in BC(\mathbb{R}^+ , \mathbb{X})$ and $\omega>0$.
We say that $f$ is asymptotically $\omega$-periodic if $f=g+h$
where $g \in P_{\omega}(\mathbb{X})$ and
$h\in C_0(\mathbb{R}^+ ,\mathbb{X})$.

We denote by $AP_{\omega}(\mathbb{X})$ the set of all asymptotically
$\omega$-periodic functions from $\mathbb{R}^+$ to $\mathbb{X}$.
Note that $AP_{\omega}(\mathbb{X})$ is a Banach space under the
supremum norm.
\end{definition}

From the above definitions, it follows that
$AP_{\omega}(\mathbb{X})=P_{\omega}(\mathbb{X})
\bigoplus C_0(\mathbb{R}^+ ,\mathbb{X})$; cf. \cite{Lizama}.

\begin{definition}[\cite{Henriquez1}] \rm
A function $f \in BC(\mathbb{R}^+ , \mathbb{X})$ is called
S-asymptotically $\omega$-periodic if there exists $\omega$
such that $\lim_{ t \to \infty}(f(t+\omega)-f(t))=0$.
In this case we say that $\omega$ is an asymptotic period of
$f$ and that $f$ is S-asymptotically $\omega$-periodic.
\end{definition}

We will denote by $SAP_{\omega}(\mathbb{X})$, the set of
all S-asymptotically $\omega$-periodic functions from
$\mathbb{R}^+ to \mathbb{X}$. Then we have
$$
AP_{\omega}(\mathbb{X}) \subset SAP_{\omega}(\mathbb{X}).
$$
Note that the inclusion above is strict.
Consider the function $f:\mathbb{R}^+ \to c_0$ where
$c_0=\{ x=(x_n)_{n \in \mathbb{N}}: \lim_{n \to \infty}x_n=0 \}$
equipped with the norm $\| x \| = \sup_{n \in \mathbb{N}} | x(n) |$,
and $f(t)= (\frac{2nt^2}{t^2 + n^2})_{n \in \mathbb{N}}$.
Then $f \in SAP_{\omega}(c_0)$ but $ f \notin AP_{\omega}(c_0)$;
see \cite[Example 3.1]{Henriquez1}.

The following result is due to Henriquez-Pierri-T\`aboas;
\cite[Proposition 3.5]{Henriquez1}.

\begin{theorem} \label{thm1}
The space $SAP_{\omega}(\mathbb{X})$ endowed with the
norm $\| \cdot \|_{\infty}$  is a Banach space.
\end{theorem}

\begin{theorem}[{\cite[Theorem 3.7]{Blot}}] \label{thm2}
Let $\phi:\mathbb{X} \to \mathbb{Y}$ be a function which is uniformly
continuous on  bounded subsets of $\mathbb{X}$ and such that
$\phi$ maps bounded subsets of $\mathbb{X}$ into bounded subsets
of $\mathbb{Y}$. Then for all $f\in SAP_{\omega}(\mathbb{X})$,
the composition
$\phi \circ f:=[t\to\phi(f(t))]\in SAP_{\omega}(\mathbb{X})$.
\end{theorem}

\begin{corollary}[{\cite[Corollary 3.10]{Blot}}] \label{coro}
Let $\mathbb{X}$ and $\mathbb{Y}$ be two Banach spaces, and denote
by  $ \mathbb{B}(\mathbb{X},\mathbb{Y})$, the space of all bounded
linear operators from $\mathbb{X}$ into $\mathbb{Y}$.
Let  $A \in \mathbb{B}(\mathbb{X},\mathbb{Y})$.
Then when $f \in SAP_{\omega}(\mathbb{X})$, we have
$Af:=[t \to Af(t)] \in SAP_{\omega}(\mathbb{Y})$.
\end{corollary}

Next we consider asymptotically $\omega$-periodic
functions with parameters.

\begin{definition}[\cite{Henriquez1}]  \label{def5}
A continuous function $f:[0,\infty[\times \mathbb{X} \to \mathbb{X}$
is said to be uniformly S-asymptotically $\omega$-periodic on
bounded sets if for every bounded set $K \subset \mathbb{X}$,
the set $\{f(t,x):t\ge 0,x \in K \}$ is bounded and
$  \lim_{ t \to \infty}(f(t,x)-f(t+\omega, x))=0$ uniformly in
$x \in K$.
\end{definition}

\begin{definition}[\cite{Henriquez1}] \label{def6} \rm
A continuous function $f:[0,\infty[\times \mathbb{X} \to \mathbb{X}$
 is said to be asymptotically uniformly continuous on bounded sets
if for every $\epsilon>0$ and every bounded set $K \subset \mathbb{X}$,
 there exist $L_{\epsilon, K}>0$ and $\delta_{\epsilon, K}>0$ such
that $| | f(t,x)-f(t,y)\|<\epsilon$ for all $t\ge L_{\epsilon, K}$
and all $x,y \in K$ with $\| x-y \| < \delta_{\epsilon, K}$.
\end{definition}

\begin{theorem}[\cite{Henriquez1}] \label{thm3}
Let $f:[0,\infty[\times \mathbb{X} \to \mathbb{X}$ be a function
which uniformly S-asymptotically $\omega$-periodic on bounded sets
and asymptotically uniformly continuous on bounded sets.
Let $u:[0,\infty[$ be S-asymptotically $\omega$-periodic function.
Then the Nemytskii operator $\phi(\cdot):=f(\cdot,u(\cdot))$
is S-asymptotically $\omega$-periodic function.
\end{theorem}

\subsection*{Fractional powers of the operator $A$}

Let $\varrho (A)$ denote the resolvent set of $A$.
We assume without loss of generality that
\begin{equation}\label{rho0}
0\in \varrho(A).
\end{equation}
This allows us, on the one hand, to say that  there exist constants
$M>1$ and $\delta>0$ such that
\begin{equation}\label{bornT}
\| T(t) x\|\le Me^{-\delta t}\| x\| ,\quad \forall t \ge 0,\;
x \in\mathbb{X}\,,
\end{equation}
and  on the other hand, to define the fractional power $A^\alpha$
for  $0 <\alpha <1$,  as a closed linear operator on its domain
$ D(A^\alpha)$ with inverse $A^{-\alpha} $ given  by
$$
A^{-\alpha}=\frac{1}{\Gamma(\alpha)}\int_0^t t^{\alpha-1}T(t)dt
$$
where $\Gamma$ denotes the Gamma function
$$
\Gamma(\alpha)=\int_0^t t^{\alpha-1}e^{-\alpha t}dt.
$$
We have the following basic properties for $A^{\alpha}$.

\begin{theorem}[{\cite[pp. 69-75]{Pazy}}] \label{thm4}
For $0<\alpha< 1$, the following properties hold.
\begin{itemize}
\item[(i)] $\mathbb{X}_\alpha=D(A^{\alpha})$ is a Banach space
with the norm $|x|_\alpha=\| A^\alpha x\| $
for $ x\in D(A^{\alpha})$;

\item[(ii)] $A^{-\alpha}$ is the closed linear operator
 with $Im(A^{-\alpha})=D(A^{\alpha})$  and we have
 $A^{\alpha}=(A^{-\alpha})^{-1}$;

\item[(iii)] $A^{-\alpha}\in \mathbb{B}(\mathbb{X},\mathbb{X})$;

\item[(iv)] $T(t):\mathbb{X}\to \mathbb{X}_{\alpha}$ for every $t>0$;

\item[(v)] $A^{\alpha}T(t)x=T(t)A^{\alpha}x$ for each
$x \in D(A^{\alpha})$ and $t\ge0$;

\item[(vi)]  $0<\alpha\le\beta$ implies
$D(A^{\beta})\hookrightarrow D(A^{\alpha})$;

\item[(vii)]  There exists $M_\alpha>1$ such that
$$
\| A^{\alpha}T(t)x| | \le M_{\alpha}\frac{e^{-\delta t}}{t^{\alpha}}
\| x\|\quad\text{for }x \in \mathbb{X},\; t>0.
$$
where  $\delta >0$ is given by \eqref{bornT}
\end{itemize}
\end{theorem}

\begin{remark} \label{rmk1}\rm
Observe as in \cite{hsiang,Mophou} that from Theorem \ref{thm4}
(iv) and (v), the restriction $T_\alpha(t)$ of $T(t)$ to
$\mathbb{X}_\alpha$ is  exactly the part of $T(t)$ in
$\mathbb{X}_\alpha$.

Let $x \in\mathbb{X}_{\alpha}$.
$$
| T(t)x|_{\alpha}=\| A^{\alpha}T(t)x\|
=\| T(t)A^{\alpha} x\|\le | T(t)|\,| | A^{\alpha} x\|
=| T(t)|\, | x|_{\alpha},
$$
and as $t$ decreases to $0$,
$$
| T(t)x - x|_\alpha=\| A^\alpha T(t)x - A^\alpha x\|
= \| T(t)A^\alpha x - A^\alpha x\| \to  0,
$$
for all $x\in \mathbb{X}_\alpha;$  it follows that
$(T(t))_{t\geq 0}$ is a family of strongly continuous semigroup
on $\mathbb{X}_\alpha$ and $| T_\alpha (t)| \leq | T(t)|$
for all $t\geq 0$.
\end{remark}


\begin{proposition}[\cite{Elazzouzi,Travis}]
$((T(t)_{t\geq 0})$ is a strongly continuous semigroup on
$\mathcal{C}_\alpha$; that is,
\begin{itemize}
\item[(i)] for all  $t\geq 0$ $T(t)$ is a bounded linear operator
on $\mathcal{C}_\alpha$;
\item[(ii)] $T(0)=I$;
\item[(iii)] $T(t+s)=T(t)T(s)$ for all $t,\, s \geq 0$;
\item[(iv)] for all $\varphi \in \mathcal{C}_\alpha$,
$T(t) \varphi$  is a continuous function of $t\geq 0$
with values in $\mathcal{C}_\alpha$.
\end{itemize}
\end{proposition}

\section{Applications to partial  differential
equations with finite delay} \label{main}

\begin{definition}\label{def7} \rm
Let $ \varphi\in \mathcal{C}_{\alpha}$.
A function $u:[-r, +\infty[\to\mathbb{X}_{\alpha}$ is said to
be a mild solution of \eqref{eq20} if the following conditions
hold:
\begin{itemize}
\item[(i)]  $u:[-r, +\infty[\to\mathbb{X}_{\alpha}$ is continuous;

\item[(ii)]  $u(t)= T(t)\varphi(0)+\int_{0}^t T(t-s)[L(u_s)+f(s)]ds$
for $t\ge 0$;
\item[(iii)] $u_{0}=\varphi$.
\end{itemize}
\end{definition}

For the rest of this article, we define
$$
\Omega=\{u:[-r,\,+\infty[\to \mathbb{X}_\alpha\text{ such that }
u|_{[-r,0]}\in \mathcal{C}_\alpha  \text{ and }
u|_{\mathbb{R}^+}\in SAP_{\omega}(\mathbb{X}_{\alpha}) \}.
$$
Note that if $u\in \Omega$ then $u $ is bounded on
$[-r,\,+\infty[$. We set
\begin{equation}\label{bornu}
\| u\|_\Omega = {\sup_{ s\in [-r,+\infty[}}| u(s)|_\alpha.
\end{equation}
It is clear that
$\| u\|_\infty \leq \| u\|_\Omega $.


\begin{lemma}\label{lem1}
Under assumption \eqref{rho0}, the function $l$ defined by
$$
l(t)=T(t)\varphi(0)
$$
belongs to  $SAP_{\omega}(\mathbb{X}_{\alpha})$.
\end{lemma}

\begin{proof}
Since  $\varphi(0)\in \mathbb{X}_\alpha$,  we have  on the one hand
that   $(T(t))_{t\geq 0}$  is a family of strongly continuous
semigroup on $\mathbb{X}_\alpha$ (see Remark  \ref{rmk1}),
and on the other hand that
$| l(t)|_\alpha\leq M|\varphi(0)|_\alpha$ because \eqref{bornT} holds.
Consequently $l\in BC(\mathbb{R}^+,\mathbb{X}_\alpha)$.

Now using  \eqref{bornT} and Remark \ref{rmk1}, we obtain
for $t\geq 0$,
\begin{align*}
| l(t+\omega)-l(t)|_{\alpha}
&=  |T(t+\omega)\varphi(0)-T(t)\varphi(0)|_{\alpha}\\
&\le |T(t+\omega)\varphi(0)|_{\alpha}+ | T(t)\varphi(0)|_{\alpha}\\
&\le | T(t+\omega)| | \varphi(0)|_{\alpha}+| T(t)|
 | \varphi(0)|_{\alpha}\\
&\le Me^{-\delta (t+\omega)}| \varphi(0)|_{\alpha}+Me^{-\delta t}|
 \varphi(0)|_{\alpha}.
\end{align*}
As $\delta >0$, we deduce that
$$
\lim_{t \to \infty}| l(t+\omega)-l(t)|_{\alpha} = 0.
$$
Thus  $l \in SAP_{\omega}(\mathbb{X}_{\alpha})$.
\end{proof}

\begin{lemma}\label{lem2}
If $u \in \Omega$, then
\begin{gather}
| u_t|_{\mathcal{C}_\alpha} \leq  \| u\|_\Omega,\label{est1}\\
| L(u_t)|_\alpha \leq | L|_{\mathbb{B}(\mathcal{C}_\alpha,
 \mathbb{X}_\alpha)}\| u\|_\Omega\label{est2}\\
 {\lim_{ t\to +\infty}}| u_{t+\omega}-u_t|_{\mathcal{C}_\alpha}
= 0 \label{est3}.
\end{gather}
\end{lemma}

\begin{proof}
For any $\theta\in [-r,0]$ and $t\geq 0$, we have
$$
| u_{t}(\theta)|_{\alpha}=| u(t+\theta)|_{\alpha}.
$$
Since $u_t$ is continuous  on  $[-r,0]$  which is  compact,
we know that there exists $ \theta^* \in [-r,0]$ such that
\[
| u_{t}|_{\mathcal{C}_\alpha}
=  \sup_{ -r\le\theta\le0}| u(t+\theta)|_{\alpha}
= | u(t+\theta^*)|_{\alpha}.
\]
 Since $u \in \Omega$,  we deduce that  \eqref{est1} holds.
As $L\in \mathbb{B}(\mathcal{C}_\alpha,\mathbb{X}_\alpha)$,  we
can write
$$
| L( u_{t})|_{\mathbb{X}_\alpha} \leq
 | L|_{\mathbb{B}(\mathcal{C}_\alpha,\mathbb{X}_\alpha)}
| u_t|_{\mathcal{C}_\alpha}.
$$
Therefore, using  \eqref{est1}, we obtain \eqref{est2}.

To complete the proof of the lemma, it suffices to prove \eqref{est3}.
As  $u_t$ is continuous on  $[-r,0]$  which is  compact,
there exists $ \theta^* \in [-r,0]$ such that
\begin{align*}
| u_{t+\omega}-u_{t}|_{\mathcal{C}_\alpha}
&= {\sup_{ -r\le\theta\le0}}| u(t+\theta+\omega)-u(t+\theta)|_{\alpha}\\
&= | u(t+\theta^*+\omega)-u(t+\theta^*)|_{\alpha}.
\end{align*}
Set $s=t+\theta$. Then, as $t$ tends to $+\infty$ we have $s$ tends
to $+\infty$. Consequently
 $$
 {\lim_{ t \to \infty}}
| u(t+\theta^*+\omega)-u(t+\theta^*)|_{\alpha}
= {\lim_{ s \to \infty}}| u(s+\omega)-u(s)|_{\alpha}
= 0
$$
since  $u \in \Omega$.   Hence,
$\lim_{t \to \infty}| u_{t+\omega}-u_{t}|_{\mathcal{C}_\alpha}=0$.
\end{proof}

\begin{lemma}\label{lem3}
Assume that \eqref{rho0}  holds.  Let
$f \in SAP_{\omega}(\mathbb{X}_{\alpha})$ and  $\phi \in \Omega $.
Then the function $\Phi: t \mapsto L(\phi_t)+f(t)$ belongs to
$SAP_{\omega}(\mathbb{X}_{\alpha})$.
\end{lemma}

\begin{proof}
It is clear  that $\Phi\in C(\mathbb{R}^+, \mathbb{X}_\alpha)$.
Using Lemma \ref{lem2}, we obtain
\[
| \Phi(t)|_\alpha \leq | L(\phi_t)|_\alpha+| f(t)|_\alpha
\leq | L|_{\mathbb{B}(\mathcal{C}_\alpha,\mathbb{X}_\alpha)}
\| \phi\|_\Omega +\| f\|_{\infty}.
\]
This implies  that  $\Phi\in BC(\mathbb{R}^+, \mathbb{X}_\alpha)$.
Hence
\begin{equation}\label{est4}
\| \Phi\|_\infty\leq  | L|_{\mathbb{B}(\mathcal{C}_\alpha,
\mathbb{X}_\alpha)}  | | \phi\|_\Omega+\| f\|_{\infty}.
\end{equation}
On the other hand, for all $t\geq 0$,
\begin{align*}
| \Phi_{t+\omega}-\Phi_{t}|_{\alpha}
&\leq | L( \phi_{t+\omega}-\phi_{t})|_{\alpha}+| f(t+\omega)
 -f(t)|_{\alpha}\\
&\leq | L|_{\mathbb{B}(\mathcal{C}_\alpha,\mathbb{X}_\alpha)}|
  \phi_{t+\omega}-\phi_{t}|_{\mathcal{C}_\alpha}+| f(t+\omega)
 -f(t)|_{\alpha},
\end{align*}
Since $\phi\in \Omega$, using  Lemma \ref{lem2}-\eqref{est3}
and the fact that $f\in SAP_{\omega}(\mathbb{X}_{\alpha})$,
we deduce that
 \begin{equation}\label{lim1}
 {\lim_{ t \to \infty}}| \Phi_{t+\omega}-\Phi_{t}|_{\alpha}=0.
\end{equation}
This completes the proof.
\end{proof}

\begin{proposition}\label{prop2}
Assume that \eqref{rho0}  holds.  Let
$f \in SAP_{\omega}(\mathbb{X}_{\alpha})$. For each $\phi \in \Omega $,
define the nonlinear operator $\wedge_0$ by
$$
(\wedge_0 \phi)(t)=\begin{cases}
\varphi(t) & \text{if }  t\in [-r,0],\\
 T(t)\varphi(0)+  \int_{0}^t T(t-s)[L(\phi_s)+f(s)]ds &\text{if }
 t\geq 0.
\end{cases}
$$
Then $\wedge_0$ maps $\Omega$ into itself.
\end{proposition}

\begin{proof}
It is clear that $(\wedge_0 \phi)$ is defined on $[-r,+\infty[$
and  because $\varphi \in \mathcal{C}_\alpha$, we have
$(\wedge_0 \phi)|_{[-r,0]}\in \mathcal{C}_\alpha$.
Thus it suffices to show that the function
$$
v: t \to  \int_{0}^t T(t-s)[L(\phi_s)+f(s)]ds
\in SAP_{\omega}(\mathbb{X}_{\alpha})$$
to complete the proof, since by Lemma \ref{lem1},
$T(t)\varphi(0)\in  SAP_{\omega}(\mathbb{X}_{\alpha})$.

For $t\ge0$, let $\Phi(t)=L(\phi_t)+f(t)$. Then
\begin{align*}
v(t+\omega)-v(t)
&=  \int_0^{t+\omega}T(t+\omega-s)\Phi(s)\,ds-\int_0^tT(t-s)\Phi(s)\,ds\\
&=  \int_0^{\omega}T(t+\omega-s)\Phi(s)\,ds
 +\int_{\omega}^{t+\omega}T(t+\omega-s)\Phi(s)\,ds\\
&\quad-  \int_0^tT(t-s)\Phi(s)\,ds.
\end{align*}
Then
$$
| v(t+\omega)-v(t)|_{\alpha} \le | I_1(t)|_{\alpha}
+ | I_2(t)|_{\alpha},
$$
where
\begin{gather*}
I_1(t)=  \int_0^{\omega}T(t+\omega-s)\Phi(s)\,ds,\\
I_2(t)= \int_{\omega}^{t+\omega}T(t+\omega-s)\Phi(s)\,ds
-\int_0^tT(t-s)\Phi(s)\,ds,\\
|I_1(t)|_{\alpha}=  \big| \int_0^{\omega}T(t+\omega-s)\Phi(s)\,ds
 \big|_{\alpha}
\leq  \int_0^{\omega}| T(t+\omega-s)\Phi(s)|_{\alpha}ds
\end{gather*}
Since
\begin{align*}
   \int_0^{\omega}| T(t+\omega-s)\Phi(s)|_{\alpha}ds
&=  \int_0^{\omega}| |  A^\alpha T(t+\omega-s)\Phi(s)| | ds\\
&=   \int_0^{\omega}| |   T(t+\omega-s) A^\alpha\Phi(s)| | ds\\
&\leq  \int_0^{\omega}Me^{-\delta (t+\omega-s)}\| A^{\alpha} \Phi(s)\|
\,ds,
\end{align*}
using \eqref{est4} we deduce that
\begin{align*}
| I_1(t)|_{\alpha}
&\leq Me^{-\delta (t+\omega)}  \int_0^{\omega}e^{\delta s}
 | \Phi(s)| _\alpha ds\\
&\leq Me^{-\delta (t+\omega)}\| \Phi\|_\infty
 \int_0^{\omega}e^{\delta s} ds\\
&\leq {\frac{1}{\delta}}Me^{-\delta (t+\omega)}\| \Phi\|_\infty
 (e^{\delta w} -1)\\
&\leq {\frac{1}{\delta}}M\| \Phi\|_\infty e^{-\delta t}
\end{align*}
Consequently,
$  \lim_{ t \to \infty}| I_1(t)|_{\alpha}=0$
In view of \eqref{lim1},  we can find $T_{\epsilon}$ sufficiently
large such that
$$
| \Phi(t+\omega)-\Phi(t)|_{\alpha} < \frac{\delta}{M}\epsilon,
\quad\text{for } t>T_{\epsilon}.
$$
After a change of variable, we obtain
$$
I_2(t)= \int_0^t T(t-s) (\Phi(s + \omega)-\Phi(s))\,ds .
$$
Thus we obtain
$$
| I_2(t) |_{\alpha} \le \big| \int_0^{T_{\epsilon}}T(t-s)
(\Phi(s + \omega)-\Phi(s))\,ds\big|_{\alpha}
+\big| \int_{T_{\epsilon}}^{t}T(t-s) (\Phi(s + \omega)-\Phi(s))\,ds
\big|_{\alpha}.
$$
Observing that
\begin{align*}
\big| \int_0^{T_{\epsilon}}T(t-s) (\Phi(s + \omega)
 -\Phi(s))\,ds\big|_{\alpha}
&\le  \int_0^{T_{\epsilon}}\big|T(t-s)\big(\Phi(s + \omega)
 -\Phi(s) \big)\big|_{\alpha}ds\\
&\leq  \int_0^{T_{\epsilon}} M e^{-\delta (t-s)} | \Phi(s
 + \omega)-\Phi(s) |_\alpha ds\\
&\leq 2  \int_0^{T_{\epsilon}} M e^{-\delta (t-s)}
 \| \Phi \|_{\infty} ds\\
&\leq 2M\|\Phi| |_{\infty} e^{-\delta t}
 \int_0^{T_{\epsilon}}e^{\delta s}ds\\
&\leq 2M  \| \Phi| |_{\infty} e^{-\delta t}
\big(\frac{e^{\delta T_{\epsilon} }}{\delta}-\frac{1}{\delta}\big),
\end{align*}
we deduce that
$$
\lim_{t \to \infty}\big| \int_0^{T_{\epsilon}}T(t-s) (\Phi(s
 + \omega)-\Phi(s))\,ds\big|_{\alpha} =0
$$
since   $ \lim_{t \to \infty} [ 2M   \| \Phi| |_{\infty} e^{-\delta t}
(\frac{e^{\delta T_{\epsilon} }}{\delta}-\frac{1}{\delta})] =0$.
Also we have
\begin{align*}
\big| \int_{T_{\epsilon}} ^{t}T(t-s) (\Phi(s + \omega)-\Phi(s))\,ds
\big|_{\alpha}
& \le  \int_{T_{\epsilon}} ^{t}
\big| T(t-s) (\Phi(s + \omega)-\Phi(s))\big|_{\alpha}ds\\
&\le  \int_{T_{\epsilon}} ^{t}| T(t-s)| |(\Phi(s
 + \omega)-\Phi(s))|_{\alpha} ds\\
&\le  \int_{T_{\epsilon}}^t
Me^{-\delta(t-s)} \frac{\delta}{M}\epsilon
\leq \epsilon.
\end{align*}
Therefore
$$
{\lim_{t \to \infty}} \int_{T_{\epsilon}} ^{t}T(t-s)
(\Phi(s + \omega)-\Phi(s))\,ds =0.
$$
Finally, we obtain
${\lim_{t \to \infty}}  I_2(t) =0$ and we have
 $t \to    \int_{0}^t T(t-s)[L(\phi_s)+f(s)]ds \in
 SAP_{\omega}(\mathbb{X}_{\alpha})$.
In summary, we have proved that
\begin{itemize}
\item  $(\wedge_0 \phi)$ is defined $[-r,+\infty[$,
\item $(\wedge_0 \phi)|_[-r,0]\in \mathcal{C}_\alpha$,
\item $(\wedge_0 \phi)|_{\mathbb{R}^+}\in SAP_{\omega}
(\mathbb{X}_{\alpha})$;
\end{itemize}
that is, $(\wedge_0 \phi)\in \Omega$.
\end{proof}

\begin{theorem} \label{thm5}
Suppose that  \eqref{rho0} holds and
$f \in SAP_{\omega}(\mathbb{X}_{\alpha})$. Let $v$ be the restriction
of the  mild solution of \eqref{eq20}
on $\mathbb{R}^+$. Then $v \in SAP_{\omega}(\mathbb{X}_{\alpha})$.
\end{theorem}

\begin{proof}
According to the definition of mild solution of \eqref{eq20}
given by Definition \ref{def7}, we have for any $t\geq 0$,
$$
v(t)=T(t)\varphi(0)+\int_{0}^t T(t-s)[L(u_s)+f(s)]ds.
$$
Hence it suffices to apply Proposition \eqref{prop2}, with $u=\phi$,
to  obtain that  $v$ belongs to $SAP_{\omega}(\mathbb{X}_{\alpha})$.
\end{proof}

We make the following assumption.
\begin{itemize}
\item[(H1)] The function $g:{R}^+ \times \mathbb{X}_{\alpha}
\to \mathbb{X}_{\alpha}$, $t \to g(t,u)$  is continuous and
there exists a constant $K_f \ge 0$ such that
$$
| g(t,u)-g(t,v)|_\alpha \le K_g | u-v |_\alpha \quad
\text{for all }t \in \mathbb{R}^+ \; ( u,v )\in \mathbb{X}^2.
$$

\item[(H2)]   $M\big(| L|_{\mathbb{B}(\mathcal{C}_\alpha,
\mathbb{X}_\alpha)}+K_g\big)/\delta<1$.
\end{itemize}

\begin{definition}\label{def8}
Let $ \varphi\in\mathcal{C}_{\alpha}$. A function
$u:[-r, +\infty[\to\mathbb{X}_{\alpha}$ is said to be a mild
solution of  \eqref{eq22}  if the following conditions hold:
\begin{itemize}
\item[(i)] $u:[-r, +\infty[\to\mathbb{X}_{\alpha}$ is continuous;
\item[(ii)]  $u(t)= T(t)\varphi(0)+ \int_{0}^t T(t-s)[L(u_s)
+g(s,u(s))]ds \quad\text{for }t\ge 0$;
\item[(iii)] $u_{0}=\varphi$.
\end{itemize}
\end{definition}

\begin{proposition}\label{prop1}
Suppose that  \eqref{rho0} holds. Assume also that the function
$g$ is uniformly S-asymptotically $\omega$-periodic on bounded
sets and {\rm(H1)} hold.  For each $\phi \in \Omega$, define
the nonlinear operator $\wedge_1$ by
$$
(\wedge_1 \phi)(t)= \begin{cases}
\varphi(t) & \text{if }  t\in [-r,0],\\
 T(t)\varphi(0)+  \int_{0}^t T(t-s)[L(\phi_s)+g(s,\phi(s))]ds
&\text{if } t\geq 0.
\end{cases}
$$
Then $\wedge_1$ maps $\Omega$ into itself.
\end{proposition}

\begin{proof}
We have $\phi|_{\mathbb{R}^+} \in  SAP_{\omega}(\mathbb{X}_{\alpha})$
since $\phi\in \Omega$. Since $g$ satisfying  (H1),
it follows from  Theorem  \ref{thm3} that the function
$h: t\mapsto g(t,\phi(t))$ belongs to
$ SAP_{\omega}(\mathbb{X}_{\alpha})$. Hence, it  suffices
to proceed exactly as for the proof of the Proposition \ref{prop2}
replacing $f(\cdot)$ by $h(\cdot)$ to obtain that $\wedge_1$ maps
$\Omega$ into itself.
\end{proof}


\begin{theorem} \label{thm6}
Suppose that  \eqref{rho0}  and {\rm(H2)}  hold.
Also assume that the function $g$ is uniformly S-asymptotically
$\omega$-periodic on bounded sets and {\rm (H1)} hold. Then for
all $\varphi \in \mathcal{C}_{\alpha}$, Equation \eqref{eq22}
has a unique mild solution in $\Omega$.
\end{theorem}

\begin{proof}
Consider the operator $Q: \Omega \to \Omega$ defined by:
$$
(Q u)(t)= \begin{cases}
\varphi(t) & \text{if }  t\in [-r,0],\\
 T(t)\varphi(0)+  \int_{0}^t T(t-s)[L(u_s)+g(s,u(s))]ds
&\text{if } t\geq 0.
\end{cases}
$$
 Observe that in view of Proposition \ref{prop1}, $Q$ is well defined.
Consider $u,v\in \Omega$.  For all $t\in [-r,\,+\infty[$, we have
\begin{align*}
&| (Qu)(t)-(Qv)(t)|_\alpha\\
&=\big|    \int_{0}^t T(t-s)[(L(u_s)-L(v_s))+(g(s,u(s))-g(s,v(s)))]ds
 \big|_\alpha\\
&\leq    \int_{0}^t \big| T(t-s)[(L(u_s)-L(v_s))+(g(s,u(s))-g(s,v(s)))]
\big|_\alpha ds.
\end{align*}
Therefore, using \eqref{bornT} and \eqref{est1}, we obtain
\begin{align*}
&| (Qu)(t)-(Qv)(t)|_\alpha\\
&\leq   \int_{0}^t Me^{-\delta(t-s)}
[| L(u_s)-L(v_s)|_\alpha +| g(s,u(s))-g(s,v(s))|_\alpha] ds \\
&\leq Me^{-\delta t}| L|_{\mathbb{B}(\mathcal{C}_\alpha,
 \mathbb{X}_\alpha)}  \int_{0}^t e^{\delta s}| u_s
 - v_s|_{\mathcal{C}_\alpha} ds\\
&\quad + Me^{-\delta t}K_g   \int_{0}^t e^{\delta s}| u(s)
-v(s)|_\alpha ds \\
&\leq Me^{-\delta t}| L|_{\mathbb{B}(\mathcal{C}_\alpha,
 \mathbb{X}_\alpha)}\| u-v\|_{\Omega}  \int_{0}^t e^{\delta s} ds\\
&\quad +  Me^{-\delta t}K_g\| u-v\|_\infty
 \int_{0}^t e^{\delta s} ds.
\end{align*}
Since $\| u-v\|_\infty\leq  \| u-v\|_{\Omega}$,  we deduce
that for all $t\geq -r$,
\begin{align*}
| (Qu)(t)-(Qv)(t)|_\alpha
&\leq Me^{-\delta t}| L|_{\mathbb{B}(\mathcal{C}_\alpha,\mathbb{X}_\alpha)}\| u-v\|_{\Omega}  \int_{0}^t e^{\delta s} ds\\
&\quad +  Me^{-\delta t}K_g\| u-v\|_\infty
  \int_{0}^t e^{\delta s} ds\\
&\leq {\frac{Me^{-\delta t}}{\delta}}
\big(| L|_{\mathbb{B}(\mathcal{C}_\alpha,\mathbb{X}_\alpha)}+K_g\big)
 \| u-v\|_{\Omega}(e^{\delta t}-1)\\
&\leq {\frac{M}{\delta}}\big(| L|_{\mathbb{B}
(\mathcal{C}_\alpha,\mathbb{X}_\alpha)}+K_g\big)\| u-v\|_{\Omega}.
\end{align*}
 Hence
$$
\| (Qu)(t)-(Qv)(t)\|_\Omega\leq  {\frac{M}{\delta}}
\big(| L|_{\mathbb{B}(\mathcal{C}_\alpha,\mathbb{X}_\alpha)}
+K_g\big)\| u-v\|_{\Omega}.
$$
 Hence assumption (H2) allows us to conclude  in view of the
contraction mapping  principle that Q has a unique point
fixed in $u\in \Omega$.  The proof is now complete.
\end{proof}

\section{Application \label{appl}}

Consider the functional partial differential  equation
\begin{equation}\label{syst2}
\begin{gathered}
 \frac{\partial }{\partial t}u(t,x)
=\frac{\partial ^2}{\partial x^2}u(t,x)
+\int_{-r}^0 q(\theta)y(t+\theta,x)d\theta+g(t ,u(t,x))\quad
 t\in \mathbb{R}^+, x\in [0,\pi]\\
u(t,0)=u(t,\pi)=0\quad t\in\mathbb{R}^+\\
u(\theta,x)=\phi(\theta,x),\quad\text{for $\theta\in[-r,0]$ and
 $x\in[0,\pi]$}
\end{gathered}
\end{equation}
where $q:[-r,0]\to\mathbb{R}$ is continuous.
To study this system in the abstractt form \eqref{eq22},
 we choose $\mathbb{X}=L^2([0,\,\pi])$ and the operator
$A:D(A)\subset \mathbb{X}\to \mathbb{X}$  is given by
$Au=-u''$ with domain
$$
D(A)=\{u\in \mathbb{X}: u'\in \mathbb{X}, \,u''\in \mathbb{X},\,
u(0)=u(\pi)=0\}.
$$
Then $-A$ generates an analytic semigroup $T(\cdot)$ such that
$\|T(t)\|\leq e^{-t}$, $t\geq 0$ (\cite{Lunardi}).
 Moreover, the eigenvalues of $A$ are $n^2\pi^2$ and the
corresponding normalized eigenvectors are
$e_n(x)=\sqrt{2}\sin(n\pi x),\, n=1,2,\cdots$. Hence, we have
\begin{itemize}

\item[(a)] $Au= \sum_{n=1}^\infty n^2\pi^2\langle u,e_n\rangle e_n$
  if $ u\in D(A)$;

\item[(b)] $A^{-1/2}u= \sum_{n=1}^\infty \frac{1}{n}\langle u,
 e_n\rangle  e_n$ if $ u\in \mathbb{X}$;

\item[(c)] The operator $A^{1/2}$ is given by
$$
A^{1/2}u= \sum_{n=1}^\infty  n\langle u,e_n\rangle e_n
$$
for each $u\in D(A^{1/2})=\{u\in \mathbb{X}:
 \sum_{n=1}^\infty \frac{1}{n}\langle u,e_n\rangle e_n\in
\mathbb{X}\} $.
\end{itemize}

Let $\mathbb{X}_{1/2}= \big(D(A^{1/2}),|\cdot|_{1/2}\big)$ where
$| x |_{1/2}=\| A^{1/2} x\|_2$ for each $x\in D(A^{1/2})$.
Let  $\mathcal{C}_\alpha$ be the Banach space
$C([-r\,,0],\mathbb{X}_{1/2})$ equipped with  norm $|\cdot|_\infty$.
We define $g:\mathbb{R}^+\times \mathbb{X}_{1/2}\to \mathbb{X}_{1/2}$
and $\varphi :[-r,0]\times[0,\pi]\to \mathbb{X}_{1/2} $
 by $g(t,u(t))(x)=g(t,u(t,x))$ and $\phi(\theta)(x)=\phi(\theta,x)$
respectively.
We define the operator $L$ by
$$
L(\phi)(x)=  \int_{-r}^0 q(\theta)\phi(\theta)(x) d\theta
\quad\text{for } x\in [0,\,\pi],\; \phi \in \mathcal{C}_{1/2}.
$$
we have $ A^{1/2} \phi(\theta)(x)\in  L^2([-r,\,0])$ since
$\phi \in \mathcal{C}_{1/2}$ . It follows that
\begin{align*}
\big| A^{1/2} L(\phi)(x)\big|^2
&\leq  \int_{-r}^0q(\theta)^2 d\theta
   \int_{-r}^0\big| A^{1/2} \phi(\theta)(x)\big|^2d\theta\\
&\leq r \big( {\sup_{ -r\leq \theta\leq 0}} q(\theta)\big)^2
 \int_{-r}^0| A^{1/2} \phi(\theta)(x)|^2 d\theta
\end{align*}
since $q$ is continuous on $[-r,\,0]$ which is a compact set
of $\mathbb{R}$. Therefore we deduce that
\begin{align*}
  \int_0^\pi | A^{1/2} L(\phi)(x)|^2 \, dx
&\leq r \big( {\sup_{ -r\leq \theta\leq 0}} q(\theta)\big)^2
   \int_0^\pi\int_{-r}^0| A^{1/2} \phi(\theta)(x)|^2 d\theta\, dx\\
&=  r \big( {\sup_{ -r\leq \theta\leq 0}} q(\theta)\big)^2
  \int_{-r}^0   \int_0^\pi | A^{1/2} \phi(\theta)(x)|^2 dx \, d\theta.
\end{align*}
Hence, we obtain
$$
| L(\phi)|_{1/2} \leq r^2 \big( {\sup_{ -r\leq \theta\leq 0}}
q(\theta)\big)^2   |  \phi|_{\mathcal{C}_{1/2}}^2 .
$$
This means that $L$ is a bounded linear operator from
$\mathcal{C}_{1/2} $ to $\mathbb{X}_{1/2}$. Therefore,
 \eqref{syst2} takes the abstract form  \eqref{eq22}.

Assume   $\int_{-r}^0 | q(\theta)| d\theta<1$ and   that  the
function $g:{R}^+ \times \mathbb{X}_{\alpha} \to \mathbb{X}_{\alpha}$,
$t \to g(t,u)$  is continuous and  there exists a constant
$K_f \ge 0$ such that
$$
| g(t,u)-g(t,v)|_\alpha \le K_g | u-v |_\alpha  \quad\text{for all }
t \in \mathbb{R}^+,\; ( u,v )\in \mathbb{X}^2.
$$
Note that such a function exists. Take for instance
Let $f(t,x)= e^{-t} x$ then
$| f(t,x)-f(t,y)|_{1/2} \leq | x-y|_{1/2}$.

\begin{theorem} \label{thm7}
Assume  that  $g$  is uniformly S-asymptotically $\omega$-periodic
on bounded sets and asymptotically uniformly continuous on bounded sets. Then
System \eqref{syst2} has a unique solution  defined on
$[-r,\infty[$ such that its restriction on $\mathbb{R}^+$ belongs
to $ SAP_{\omega}(\mathbb{X}_{\alpha})$
provided $(r^2 \left( {\sup_{ -r\leq \theta\leq 0}} q(\theta)\right)^2
+K_g)<1$.
\end{theorem}

\begin{proof}
It suffices to apply Theorem \ref{thm6}, observing that
(H2) is satisfied since $r^2 \big( {\sup_{ -r\leq \theta\leq 0}}
q(\theta)\big)^2 +K_g<1$ and $M=\delta=1$.
\end{proof}


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