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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2009(2009), No. 46, pp. 1--8.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2009 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2009/46\hfil Weak almost periodic solutions]
{Weak almost periodic and optimal mild solutions
of fractional evolution equations}

\author[A. Debbouche, M. M. El-Borai \hfil EJDE-2009/46\hfilneg]
{Amar Debbouche, Mahmoud M. El-Borai}  % in alphabetical order

\address{Amar Debbouche \newline
Faculty of Science, Guelma University, Guelma, Algeria}
\email{amar\_debbouche@yahoo.fr}

\address{Mahmoud M. El-Borai \newline
Faculty of Science, Alexandria University, Alexandria, Egypt}
\email{m\_m\_elborai@yahoo.com}

\thanks{Submitted March 10, 2009. Published March 30, 2009.}
\subjclass[2000]{34G10, 26A33, 35A05, 34C27, 35B15}
\keywords{Linear fractional evolution equation;
 Optimal  mild solution; \hfill\break\indent
weak almost periodicity; analytic semigroup}

\begin{abstract}
 In this article, we prove the existence of optimal mild
 solutions for linear fractional evolution equations with
 an analytic semigroup in a Banach space. As in \cite{n2}, we use
 the Gelfand-Shilov principle to prove existence, and
 then the Bochner almost periodicity condition to show 
 that solutions are weakly almost periodic.
 As an application, we study a fractional partial differential
 equation of parabolic type.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{definition}[theorem]{Definition}

\section{Introduction}

 The object of this paper is to study the fractional evolution equation
\begin{equation} \label{e1.1}
\frac{d^\alpha u(t)}{dt^\alpha} + (A-B(t))u(t) = f(t), \quad t>t_{0}
\end{equation}
in a Banach space $X$, where $0<\alpha\leq1$, $u$ is an $X$-valued function
on $\mathbb{R}^{+}=[0,\infty)$, and $f$ is a given abstract function on
$\mathbb{R}^{+}$ with values in $X$. We assume that  -$A$ is a linear
closed operator defined on a dense set $S$ in $X$ into $X$,
$\{ B(t): t\in \mathbb{R}^{+}\}$ is a family of linear bounded operators
defined on $X$ into $X$.
It is assumed that -$A$ generates an analytic semigroup $Q(t)$ such that
$\| Q(t)\|\leq M$ for all $t\in \mathbb{R}^{+}$,
$Q(t)h\in S$, $\| AQ(t)h\|\leq\frac{M}{t}\| h\|$ for every $h\in X$ and
all $t\in (0,\infty)$.

Let $ X $ be a uniformly convex Banach space equipped with a norm
$\| \cdot\|$ and $X^{*}$ its topological dual space. N'Guerekata
\cite{n2}
gave necessary conditions to ensure that the so-called optimal mild
solutions of $ u'(t)=Au(t)+f(t) $ are weakly almost periodic.
Following Gelfand and Shilov \cite{g1}, we  define the fractional integral
of order $\alpha > 0 $ as
$$
I^{\alpha}_{a}f(t) =  \frac{1}{\Gamma(\alpha)} \int_{a}^{t}
(t-s)^{\alpha-1} f(s)ds,
$$
also, the fractional derivative of the function $f$ of order
$ 0<\alpha<1$ as
$$
_{a}D^{\alpha}_{t}f(t)=\frac{1}{\Gamma(1-\alpha)}
\frac{d}{dt} \int_{a}^{t}{f(s)}{(t-s)^{-\alpha}}ds,
$$
where  $ f $ is an abstract continuous function on the interval
$[a,b]$ and ${\Gamma(\alpha)}$ is the Gamma function, see
\cite{m2,p1}.

\begin{definition} \label{def1} \rm
 By a solution of \eqref{e1.1}, we mean a
function $u$ with values in $X$  such that:
\begin{enumerate}
\item  $ u $ is continuous function on  $ \mathbb{R}^{+} $ and
 $ u(t)\in D(A) $,
\item $ \frac{d^{\alpha}u}{dt^{\alpha}} $ exists and continuous on
$ (0,\infty) $, $ 0<\alpha<1 $, and  $ u $ satisfies \eqref{e1.1}
on $ (0,\infty) $.
\end{enumerate}
\end{definition}

It is suitable to rewrite equation \eqref{e1.1} in the form
\begin{equation}
u(t)=u(t_{0})+\frac{1}{\Gamma(\alpha)}\int_{t_{0}}^{t}(t-s)^{\alpha -1}
[(B(s)-A)u(s)+ f(s)]ds.\label{e1.2}
\end{equation}
According to \cite{e1,e2,e3,e4,e5}, a solution of  equation \eqref{e1.2} can be
formally represented by
\begin{equation}
\begin{aligned}
  u(t) &=  \int_{0}^{\infty}\zeta_{\alpha}(\theta)Q((t-t_{0})^{\alpha}
   \theta)u(t_{0})d\theta\\
&\quad + \alpha\int_{t_{0}}^{t}\int_{0}^{\infty}\theta(t-s)^{\alpha-1}
\zeta_{\alpha}(\theta)Q((t-s)^{\alpha}\theta)F(s)d\theta ds,
   \end{aligned} \label{e1.3}
\end{equation}
where $ F(t)=B(t)u(t)+f(t) $ and  $ \zeta_{\alpha} $ is a probability
density function defined on $ (0,\infty) $  such that its Laplace
transform is given by
$$
\int_{0}^{\infty}e^{-\theta x}\zeta_{\alpha}(\theta)d\theta
=\sum_{j=0}^{\infty}\frac{(-x)^{j}}{\Gamma(1+\alpha j)}, \quad
0<\alpha\leq1, x>0,
$$
A continuous solution of the integral equation \eqref{e1.3} is called
a mild solution of \eqref{e1.1}.

The theory of almost periodic functions with values in a Banach space
was  developed by  Bohr,  Bochner, von Neumann, and others
\cite{a1,b2}.
See also \cite{b1,d1,m1,n2,n3,y1}.

\begin{definition} \label{def1.2} \rm
 A function  $ f:\mathbb{R} \to X  $ is called (Bochner) almost periodic if
\begin{itemize}
\item[(i)]  $f$ is strongly continuous, and
\item[(ii)] for each  $ \epsilon >0 $  there exists $ l(\epsilon)>0 $,
such that every interval  $ I $ of length  $ l(\epsilon) $  contains
a number  $ \tau $ such that
 $\sup_{t\in \mathbb{R}}\| f(t+\tau)-f(t)\|<\epsilon $.
\end{itemize}
\end{definition}

\section{Optimal mild solutions}

As in N'Guerekata \cite{n2}, let  $ \Omega_{f} $ denote the set of mild
solutions  $ u(t) $ of \eqref{e1.1} which are bounded  over $\mathbb{R}$;
that is
\begin{equation}
\mu(u)=\sup_{t\in\mathbb{R}}\| u(t)\|<\infty, \label{e2.1}
\end{equation}
 where $\mathbb{R}=(-\infty,\infty)$. We assume here that
$ \Omega_{f}\neq\emptyset $, and  recall that
a bounded mild solution $\tilde{u}(t) $  of \eqref{e1.1} is
 called optimal mild solution of \eqref{e1.1} if
\begin{equation}
\mu(\tilde{u})\equiv\mu^{*}=\inf_{u\in\Omega_{f}}\mu(u). \label{e2.2}
\end{equation}

\begin{theorem} \label{thm2.1}
 Assume that  $ \Omega_{f}\neq\emptyset $  and $ f:\mathbb{R}\to X $
is a nontrivial strongly continuous function, then \eqref{e1.1} has a
unique optimal mild solution.
\end{theorem}

Compare with \cite[Theorem 1.1, p.138]{z3} and  \cite[Theorem 1. p. 673]{n2}.
Our proof is based on the following lemma.

\begin{lemma}[{\cite[Corollary 8.2.1]{l1}}] \label{lem2.2}
 If $ K $  is a non-empty convex and closed subset of a uniformly
convex Banach space $ X $  and  $ v\notin K $, then there exists a
unique  $ k_{0}\in K $ such that $ |v-k_{0}|=\inf_{k\in K}|v-k| $.
\end{lemma}

\begin{proof}[Proof of Theorem \ref{thm2.1}]
 It suffices to prove that  $ \Omega_{f} $ is a convex and closed set
because the trivial solution  $ 0\notin \Omega_{f} $, then we use
lemma \ref{lem2.2} to deduce the uniqueness of the optimal mild  solution,
see \cite{n2}.
For the convexity of  $ \Omega_{f} $, we consider two distinct
bounded mild solutions  $ u_{1}(t) $  and  $ u_{2}(t) $, and a real
number $ 0\leq\lambda\leq1 $ and let
$ u(t)=\lambda u_{1}(t)+(1-\lambda)u_{2}(t), t\in \mathbb{R}$.
For every $ t_{0}\in \mathbb{R} $,  $ u(t) $ is continuous and (
see \cite{n2})
has the integral representation
\begin{equation}
u(t)=T(t-t_{0})u(t_{0})+\int_{t_{0}}^{t}S(t-s)F(s)ds, \quad
t\geq t_{0}, \label{e2.3}
\end{equation}
where
$$
T(t)=\int_{0}^{\infty}\zeta_{\alpha}(\theta)Q(t^{\alpha}\theta)d\theta  , \quad
S(t)=\alpha\int_{0}^{\infty}\theta t^{\alpha-1}\zeta_{\alpha}(\theta)
 Q(t^{\alpha}\theta)d\theta.
$$
We have  $ u(t_{0})=\lambda u_{1}(t_{0})+(1-\lambda)u_{2}(t_{0}) $,
then  $ u(t) $ is a mild solution of \eqref{e1.1}.
We note that $ u(t) $ is bounded over  $\mathbb{R}$ since
$ \mu(u)=\sup_{t\in\mathbb{R}}\| u(t)\|\leq\lambda
\mu(u_{1})+(1-\lambda)\mu(u_{2})<\infty $, we conclude that
$ u(t)\in \Omega_{f} $.
Now we show that $ \Omega_{f} $ is closed. Let
$ u_{n}\in \Omega_{f} $  a sequence such that
$ \lim_{n\to\infty}u_{n}(t)=u(t), t\in \mathbb{R} $.
For all  $ t_{0}\in \mathbb{R} $ and  $ t\geq t_{0} $ we have
\begin{equation}
u_{n}(t)=T(t-t_{0})u_{n}(t_{0})
+\int_{t_{0}}^{t}S(t-s)[B(s)u_{n}(s)+f(s)]ds, \label{e2.4}
\end{equation}
It is clearly that  $ T(t-t_{0}) $ and  $ S(t-s) $ are continuous operators,
then for every fixed  $ t $ and  $ t_{0} $ with  $ t\geq t_{0} $, we have
 \begin{align*}
\lim_{n\to\infty}T(t-t_{0})u_{n}(t_{0})
&= \lim_{n\to\infty}\int_{0}^{\infty}\zeta_{\alpha}(\theta)Q((t-t_{0})^{\alpha}\theta)u_{n}(t_{0})d\theta\\
&= \int_{0}^{\infty}\zeta_{\alpha}(\theta)Q((t-t_{0})^{\alpha}\theta)d\theta\lim_{n\to\infty}u_{n}(t_{0})\\
&=  T(t-t_{0})\lim_{n\to\infty}u_{n}(t_{0})\\
&=  T(t-t_{0})u(t_{0}).
\end{align*}
Similarly we have
 \begin{align*}
\lim_{n\to\infty}\int_{t_{0}}^{t}S(t-s)[B(s)u_{n}(s)+f(s)]ds
&= \int_{t_{0}}^{t}S(t-s)[\lim_{n\to\infty}B(s)u_{n}(s)+f(s)]ds\\
&= \int_{t_{0}}^{t}S(t-s)F(s)ds.
 \end{align*}
Then we deduce that
$$
u(t)=T(t-t_{0})u(t_{0})+\int_{t_{0}}^{t}S(t-s)F(s)ds,
$$
for all $ t_{0}\in \mathbb{R}, t\geq t_{0} $, which means that
$ u(t) $  is a mild solution of \eqref{e1.1}. Finally we show that
$ u(t) $ is bounded over $ \mathbb{R}$.  We can write \eqref{e2.3} as
\begin{align*}
u(t)&= T(t-t_{0})u(t_{0})+\int_{t_{0}}^{t}S(t-s)F(s)ds-u_{n}(t)+u_{n}(t)\\
&= T(t-t_{0})[u(t_{0})-u_{n}(t_{0})]+\int_{t_{0}}^{t}S(t-s)(B(u-u_{n}))(s)ds+u_{n}(t),
\end{align*}
for   $ n=1,2,\dots, $ and every $ t_{0}\in \mathbb{R} $  such that
$ t\geq t_{0} $.
  Since $ \int_{0}^{\infty}\zeta_{\alpha}(\theta)d\theta=1$, it follows
that $\| T(t)\|\leq M$, again, since
$ \int_{0}^{\infty}\theta\zeta_{\alpha}(\theta)d\theta=1$
(see \cite[p. 54]{e5}), it follows that $ \| S(t)\|\leq\alpha M t^{\alpha-1} $.
Let $\|  B\|\leq C$.
These estimates lead to
$$
\| u(t)\|\leq M\| u(t_{0})-u_{n}(t_{0})\|
+\alpha MC\int_{t_{0}}^{t}( t-s)^{\alpha-1}\| u(s)-u_{n}(s)\| ds
+\| u_{n}(t)\|.
 $$
Choose  $ n $ large enough, for every  $ \epsilon_{1}, \epsilon_{2}>0 $
we get
$$
\mu(u)\leq\epsilon_{1}+\epsilon_{2}+\mu(u_{n})<\infty .
$$
Thus $ u\in\Omega_{f} $. This completes the proof.
\end{proof}

\section{Weak almost periodic solutions}

To formulate a property of almost periodic functions, which is equivalent
to Definition \ref{def1.2}, we discuss the concept of normality of almost periodic
functions. Namely, let $f(t)$ be almost periodic in
$ t\in \mathbb{R} $, then for every sequence  of real numbers
$ (s'_{n}) $  there exists a subsequence  $ (s_{n}) $ such that
$ f(t+s_{n}) $ is uniformly convergent in $t\in \mathbb{R}$.
see Hamaya  \cite[p. 188]{h1}. It is well known
\cite{n1,n2,z2,z3} that:
$ f:\mathbb{R}\to X $ is weakly almost periodic if for every sequence
of real numbers  $ (s'_{n}) $   there  exists a subsequence
$ (s_{n}) $   such that every $ (f(t+s_{n})) $  is convergent in
the weak  sense, uniformly in $t\in \mathbb{R}$. In other words,
for every $ u^{*}\in X^{*} $, the sequence
 $ (\langle u^{*},f(t+s_{n})\rangle) $  is uniformly convergent in
$t\in \mathbb{R}$, where $ \langle\cdot,\cdot\rangle$ denotes duality
$\langle X^{*},X\rangle $. For each $ Q(t), t\in \mathbb{R}^{+} $,
 $ Q^{*}(t) $ denotes the adjoint operator of $ Q(t) $.

\begin{theorem} \label{thm3.1}
 Let  $ f:\mathbb{R}\to X $ be almost periodic and a nontrivial strongly
 continuous function, also assume that  $ f\in L^{1}(R) $ and
$ Q^{*}(t)\in L(X^{*}) $ for every $ t\in \mathbb{R}^{+} $, then the
optimal mild solution of   \eqref{e1.1} is weakly almost periodic.
\end{theorem}

\begin{proof} As in  N'Guerekata \cite{n2}, let   $u(t) $ be the unique optimal
 mild solution of \eqref{e1.1}, by Theorem \ref{thm2.1}
$$
u(t)=T(t-t_{0})u(t_{0})+\int_{t_{0}}^{t}S(t-s)F(s)ds,
$$
for all $ t_{0}\in \mathbb{R}$, $t\geq t_{0} $. Let  $ (s'_{n}) $ be
an arbitrary sequence of real numbers. Since  $ f $ is  almost periodic,
we can extract a subsequence  $ (s_{n})\subset(s'_{n}) $  such that
$ \lim_{n\to \infty}f(t+s_{n})=g(t) $ uniformly in $t\in \mathbb{R}$.
We note that  $ g(t) $ is also strongly continuous. For fixed
$ t_{0}\in \mathbb{R} $, we can  obtain a subsequence of  $ (s_{n}) $,
which again we will denote $  (s_{n}) $, such that
$$
\mathop{\rm weak\text{-}lim}_{n\to \infty}u(t_{0}+s_{n})=v_{0}\in X.
$$
Since  $ X $ is a reflexive Banach space, then the function
$$
y(t)=T(t-t_{0})v_{0}+\int_{t_{0}}^{t}S(t-s)(Bu+g)(s)ds,
$$
is strongly continuous. It is a mild solution of
$$
\frac{d^{\alpha}u(t)}{dt^{\alpha}}+(A-B(t))u(t)=g(t),\quad t\in \mathbb{R}.
$$
\end{proof}

We need the following lemmas.

\begin{lemma} \label{lem3.2}
 For each  $ t\in \mathbb{R} $, we have
$$
\mathop{\rm weak\text{-}lim}_{n\to \infty}u(t+s_{n})=y(t).
$$
\end{lemma}

\begin{proof}
 We can write
$$
u(t+s_{n})=T(t-t_{0})u(t_{0}+s_{n})+\int_{t_{0}}^{t}S(t-s)[(Bu)(s)
+f(s+s_{n})]ds,
$$
$ n=1,2,\dots $ (see for instance  \cite[p. 721]{z1}).
Let $ u^{*}\in X^{*} $, then we have
$$
\langle u^{*},T(t-t_{0})u(t_{0}+s_{n})\rangle
-\langle u^{*},T(t-t_{0})v_{0}\rangle = \langle T^{*}(t-t_{0})u^{*},u(t_{0}+s_{n})-v_{0}
\rangle,
$$
for every $ n=1,2,\dots, $ we deduce that the sequence
$ (T(t-t_{0})u(t_{0}+s_{n})) $  converges to  $ T(t-t_{0})v_{0} $
in the weak sense. Also we have
\begin{align*}
&\int_{t_{0}}^{t}S(t-s)[(Bu)(s)+f(s+s_{n})]ds - \int_{t_{0}}^{t}S(t-s)[(Bu)(s)+g(s)]ds\\
&\leq \|\int_{t_{0}}^{t}S(t-s)[f(s+s_{n})-g(s)]ds\|\\
&\leq  \alpha M\int_{t_{0}}^{t}(t-s)^{\alpha-1}\| f(s+s_{n})-g(s)\| ds.
 \end{align*}
This leads to
$$
\lim_{n\to \infty}\int_{t_{0}}^{t}S(t-s)[(Bu)(s)+f(s+s_{n})]ds
=\int_{t_{0}}^{t}S(t-s)[(Bu)(s)+g(s)]ds,
$$
in the strong sense, then consequently in the weak sense in  $ X $.
\end{proof}

\begin{lemma} \label{lem3.3}
$\mu(y)=\mu(u)=\mu^{*}$.
\end{lemma}

\begin{proof}
 Since  $ u(t) $ is an optimal mild solution of \eqref{e1.1}, we have
$ \mu^{*}=\mu(u)=\sup_{t\in\mathbb{R}}\| u(t)\|$.
Let $ u^{*}\in X^{*} $, then by lemma \ref{lem3.2} we obtain
$$
\lim_{n\to\infty}\langle u^{*},u(t+s_{n})\rangle
=\langle u^{*},y(t)\rangle,
$$
for every  $ t\in \mathbb{R} $. For each  $ n=1,2,\dots$, we have
$$
\|\langle u^{*},u(t+s_{n})\rangle\|\leq\| u^{*}\| \| u(t+s_{n})\|
\leq\| u^{*}\|\mu^{*} .
$$
Therefore, $ \|\langle u^{*},y(t)\rangle\|\leq\| u^{*}\|\mu^{*} $
for every $ t\in \mathbb{R} $, and consequently
$ \| y(t)\|\leq\mu^{*} $ for every $ t\in \mathbb{R} $, so that
$ \mu(y)<\mu^{*} $.
We suppose that $ \mu(y)<\mu^{*} $.
Note that $ \lim_{n\to\infty}g(t-s_{n})=f(t) $  uniformly in
$t\in \mathbb{R}$ because  $ f(t) $ is almost periodic. Since $X$  is
a reflexive Banach  space, we can extract from the sequence $ (s_{n}) $,
a subsequence which we still denote   $ (s_{n}) $  such that
$ (y(t_{0}-s_{n})) $ is weakly convergent to  $ z\in X $. We have
$$
\lim_{n\to \infty}y(t-s_{n})=T(t-t_{0})z+\int_{t_{0}}^{t}S(t-s)F(s)ds
$$
in the weak sense for every $ t\in \mathbb{R} $. Now we consider
the function
$$
Z(t)=T(t-t_{0})z+\int_{t_{0}}^{t}S(t-s)F(s)ds.
$$
It is a bounded mild solution of equation \eqref{e1.1}.
Similarly as above, we have  $ \mu(Z)\leq\mu(y) $; therefore,
$ \mu(Z)<\mu^{*} $, which is absurd by definition of $ \mu^{*} $.
\end{proof}


\begin{lemma} \label{lem3.4}
 $\mu(y)=\inf_{v\in \Omega_{g}}\mu(v)$;
i.e.,  $ y(t) $ is an optimal mild solution of the equation
\begin{equation}
\frac{d^{\alpha}u(t)}{dt^{\alpha}}+(A-B(t))u(t)=g(t), \quad
t\in \mathbb{R}.\label{e3.1}
\end{equation}
\end{lemma}

\begin{proof}
By lemma \ref{lem3.3},  $ y(t) $ is bounded over $\mathbb{R}$. Also  $ y(t) $
is a mild solution of \eqref{e3.1} which implies $ y(t)\in \Omega_{g} $.
It remains to prove that $ y(t) $ is optimal. Suppose it is not.
Since  $ \Omega_{g}\neq\emptyset $, by Theorem \ref{thm2.1}, there exists a unique
optimal solution  $ v(t) $ of \eqref{e3.1}.  We have  $ \mu(v)<\mu(y) $ and
$$
v(t)=T(t-t_{0})v(t_{0})+\int_{t_{0}}^{t}S(t-s)(Bu+g)(s)ds,
$$
for all $ t_{0}\in \mathbb{R}, t\geq t_{0} $.  We can find a subsequence
$ (s_{n_{k}})\subset(s_{n}) $  such that
$$
\mathop{\rm weak\text{-}lim}_{k\to\infty}v(t-s_{n_{k}})
=T(t-t_{0})z+\int_{t_{0}}^{t}S(t-s)F(s)ds\equiv V(t).
$$
Noting that $ V(t)\in \Omega_{f} $ and  $ \mu(V)\leq\mu(v)<\mu(y) $,
which is absurd. Therefore,  $ y(t) $ is an optimal mild solution
of \eqref{e3.1}, and in fact the only one by Theorem \ref{thm2.1}.
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm3.1}]
To prove that  $ u(t) $ is weakly almost periodic, it suffices to show that
$$
\mathop{\rm weak\text{-}lim}_{n\to\infty}u(t+s_{n})=y(t)
$$
uniformly in  $ t\in \mathbb{R} $.  Suppose that this does not hold;
then there exists  $ u^{*}\in X^{*} $   such that
$$
\lim_{n\to\infty}\langle u^{*},u(t+s_{n})\rangle = \langle u^{*},y(t)\rangle
$$
is not uniform in $ t\in \mathbb{R} $. Consequently, we can find a
number $ \gamma>0 $, and a sequence  $ (t_{k}) $ with two subsequences
$ (s'_{k}) $ and  $ (s''_{k}) $ of  $ (s_{n}) $  such that
\begin{equation}
|\langle u^{*},u(t+s'_{k})-u(t+s''_{k})\rangle|>\gamma \label{e3.2}
\end{equation}
for all $ k=1,2,\dots $.
 Again, let us extract two subsequences of   $ (s'_{k}) $ and
$ (s''_{k}) $ respectively, with the same  notation, such that
\[
\lim_{k\to\infty}f(t+t_{k}+s'_{k})=g_{1}(t), \quad{text}\quad
\lim_{k\to\infty}f(t+t_{k}+s''_{k})=g_{2}(t)
\]
both uniformly in $t\in \mathbb{R}$, because  $ f $  is almost periodic.
As we did previously, we  may obtain
$$
\mathop{\rm weak\text{-}lim}_{k\to\infty}f(t+t_{k}+s'_{k})=T(t-t_{0})z_{1}
+\int_{t_{0}}^{t}S(t-s)[(Bu)(s)+g_{1}(s)]ds\equiv y_{1}(t),
$$
and
$$
\mathop{\rm weak\text{-}lim}_{k\to\infty}f(t+t_{k}+s''_{k})
=T(t-t_{0})z_{2}+\int_{t_{0}}^{t}S(t-s)[(Bu)(s)+g_{2}(s)]ds\equiv y_{2}(t)
$$
for each  $ t\in \mathbb{R} $, where  $ y_{1}(t) $  and  $ y_{2}(t) $
are optimal mild solutions in  $ \Omega_{g_{1}} $ and  $ \Omega_{g_{2}} $,
respectively.
Since $ \lim_{k\to\infty}f(t+t_{k}+s_{k}) $ exists uniformly
in $t\in \mathbb{R}$, and $ (s'_{k}), (s''_{k}) $ are  two subsequences of
$ (s_{k}) $, we will get
$$
\sup_{s\in \mathbb{R}}\| f(s+s'_{k})-f(s+s''_{k})\|<\epsilon
$$
if $ k\geq k_{0}(\epsilon) $ and consequently
$$
\sup_{s\in \mathbb{R}}\| f(t+t_{k}+s'_{k})-f(t+t_{k}+s''_{k})\|<\epsilon
$$
for $ k\geq k_{0}(\epsilon) $, which shows that  $ g_{1}(s)=g_{2}(s) $
for all  $ s\in \mathbb{R} $. By the  uniqueness of the optimal mild
solution we get $ y_{1}(t)=y_{2}(t)$, $t\in \mathbb{R} $. But
$ y_{1}(0)=\mathop{\rm weak\text{-}lim}_{k\to\infty}u(t_{k}+s'_{k})$ and
$ y_{2}(0)=\mathop{\rm weak\text{-}lim}_{k\to\infty}u(t_{k}+s''_{k})$.
 Clearly $y_{1}(0)=y_{2}(0)$ contradicts the inequality \eqref{e3.2} above.
This completes the proof.
\end{proof}

\section{Application}
     Consider the partial differential equation of fractional order
\begin{equation}
\frac{\partial^{\alpha}u(x,t)}{\partial t^{\alpha}}
+\sum_{\vert q\vert\leq2m}a_{q}(x)D^{q}_{x}u(x,t)
=\int_{\mathbb{R}^{n}}K(x,\eta,t)u(\eta,t)d\eta+f(x,t), \label{e4.1}
\end{equation}
where $t\in \mathbb{R}^{+}$, $x\in \mathbb{R}^{n}$,
$D^{q}_{x}=D^{q_{1}}_{x_{1}}\dots D^{q_{n}}_{x_{n}}$,
$D_{x_{i}}=\frac{\partial}{\partial x_{i}}$,
$q=(q_{1},\dots,q_{n})$ is an $n$-dimensional multi-index,
$\vert q\vert=q_{1}+\dots+q_{n}$.
Let $L_{2}(\mathbb{R}^{n})$ be the set of all square integrable
functions on $\mathbb{R}^{n}$. We denote by $C^{m}(\mathbb{R}^{n})$
the set of all continuous real-valued functions defined on
$\mathbb{R}^{n}$ which have continuous partial derivatives of order
less than or equal to $m$. By $C^{m}_{0}(\mathbb{R}^{n})$ we denote the
set of all functions $f\in C^{m}(\mathbb{R}^{n})$ with compact supports.
Let $H^{m}_{0}(\mathbb{R}^{n})$ be the completion of
$C^{m}_{0}(\mathbb{R}^{n})$ with respect to the norm
$$
\| f\|^{2}_{m}=\sum_{\vert q\vert\leq m}
\int_{\mathbb{R}^{n}}\vert D^{q}_{x}f(x)\vert^{2}dx.
$$
It is supposed that

\noindent(i) The operator $A=-\sum_{\vert
q\vert=2m}a_{q}(x)D^{q}_{x}$  is uniformly parabolic on
$\mathbb{R}^{n}$. In other words, all the coefficients $a_{q},
\vert q\vert=2m$, are continuous and bounded on $\mathbb{R}^{n}$
and
$$(-1)^{m}\sum_{\vert q\vert=2m}a_{q}(x)\xi^{q}\geq
c\vert\xi\vert^{2m}, \quad c>0,
$$
for all $x\in \mathbb{R}^{n}$ and all $\xi\ne0, \xi\in \mathbb{R}^{n}$,
where $\xi^{q}=\xi^{q_{1}}_{1}\dots\xi^{q_{n}}_{n}$ and
 $\vert\xi\vert^{2}=\xi^{2}_{1}+\dots+\xi^{2}_{n}$.

\noindent(ii) All the coefficients $a_{q}, \vert q\vert=2m$, satisfy
 a uniform H\"older condition on $\mathbb{R}^{n}$,
$\int_{\mathbb{R}^{n}}K^{2}(x,\eta,t)d\eta<\infty$. It's proved,
see \cite[p. 438]{e1}, that the operator $A$ defined by (i) with domain of
definition $S=H^{2m}(\mathbb{R}^{n})$ generates an analytic
semigroup $Q(t)$ defined on $L_{2}(\mathbb{R}^{n})$, and that
$H^{2m}(\mathbb{R}^{n})$ is dense in $X=L_{2}(\mathbb{R}^{n})$.
Which achieves the proof of the existence of (bounded) mild
solutions of the equation \eqref{e4.1}.

\noindent(iii) $f$ is a nontrivial strongly continuous function
defined on $\mathbb{R}^{n}\times \mathbb{R}^{+}$ satisfying: For
every $\epsilon>0$ there exists $\beta>0$ such that every interval
$[a,a+\beta]$  contains at least a point $\tau$ such that
$$
\int_{\mathbb{R}^{n}}\vert f(x,t+\tau)-f(x,t)\vert^{2}dx<\epsilon,
$$
for all $t\in \mathbb{R}^{+}$ and all $x\in \mathbb{R}^{n}$.
Applying Theorems \ref{thm2.1}, \ref{thm3.1}, stated above, we deduce that \eqref{e4.1}
has a  unique optimal mild solution which is weakly almost periodic.

\subsection*{Acknowledgements}
The authors are grateful to the anonymous referee for his or her carefully
reading of the original manuscript and for the valuable suggestions.

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