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\markboth{\hfil Almost periodic solutions of semilinear equations 
\hfil EJDE--2002/98}
{EJDE--2002/98\hfil Mohamed Bahaj \& Omar Sidki \hfil}
\begin{document}
\title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent
{\sc  Electronic Journal of Differential Equations},
Vol. {\bf 2002}(2002), No. 98, pp. 1--11. \newline
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp  ejde.math.swt.edu  (login: ftp)}
 \vspace{\bigskipamount} \\
 %
  Almost periodic solutions of semilinear equations
  with analytic semigroups in Banach spaces
 %
\thanks{ {\em Mathematics Subject Classifications:} 34K05, 34K14, 34G20.
\hfil\break\indent
{\em Key words:} Semilinear equation in Banach spaces,
 almost periodic solution, \hfil\break\indent
 infinitesimal generator of  analytic semigroups,
 fractional powers.
\hfil\break\indent
\copyright 2002 Southwest Texas State University. \hfil\break\indent
Submitted August 15, 2002. Published November 27, 2002.} }
\date{}
%
\author{Mohamed Bahaj \& Omar Sidki}
\maketitle

\begin{abstract}
 We establish the existence and uniqueness of almost
 periodic solutions of a class of semilinear equations having
 analytic semigroups. Our basic tool in this paper is the use
 of fractional powers of operators.
\end{abstract}

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

\section{Introduction}

The existence of almost periodic solutions of abstract
differential equations has been considered in several works;
see for example \cite{a1,a2,b2,h1,y1,z1,z2,z3,z4} and reference
listed therein. There is also an extensive literature for
the same question in semilinear equations. Most of these
works are concerned with equation
\begin{equation}
x'( t) +Ax( t) =f( t,x( t)),  \label{1.1}
\end{equation}
where $f$ is uniformly almost periodic and $-A$ is the infinitesimal
generator of a $\mathcal{C}_{0}-$semigroup \cite{b1,z2,z3,z4}.
Ballotti, Goldstein and Parrott \cite{b2} gave necessary and
sufficient conditions for the existence of almost periodic solutions of the
equation
\[
x'( t) =A( t) x( t),
\]
where $A( t)$ is the generator of a $\mathcal{C}_{0}$ semigroup
on a Banach space. These authors used the mean ergodic theorem.
Zaidman \cite{z1}  proved the existence and uniqueness of an
almost periodic mild solution of the inhomogeneous equation
\begin{equation}
x'( t) +Ax( t) =g( t), \label{1.2}
\end{equation}
where $-A$ is the infinitesimal generator of a $\mathcal{C}_{0}$ semigroup
$S( t)$ satisfying the exponential stability, and $g$ is almost
periodic function from $\mathbb{R}$ into $X$. In this case, the solution is
$x( t) =\int_{-\infty }^{t}S( t-\sigma )g( \sigma ) d\sigma $.
When $A$ generates a $\mathcal{C}_{0}$ semigroup $S(t) $ satisfying the
exponential stability and $f$ is uniformly
Lipschitz continuous with a Lipschitz constant small enough, existence
and uniqueness of an almost periodic mild solution of \eqref{1.1} was
proved in \cite{z2}.

In this paper, we consider the semilinear equation \eqref{1.1} when
$-A$ is the infinitesimal generator of an analytic $\mathcal{C}_{0}$
semigroup $S(t) $ satisfying the exponential stability. We
investigate whether or not the classical solution
inherits uniform almost periodicity from $f$.
We proposed a new method for proving existence whose main component
is the use of fractional powers of operators. More precisely,
we assume that the function $f:\mathbb{R}\times X_{\alpha }\to X$
satisfies the hypothesis:
\begin{enumerate}
\item[(F)] There are numbers $L\geq 0$ and $0\leq \theta \leq 1$ such that
$| f( t_{1},x_{1}) -f(t_{2},x_{2}) |
\leq L( | t_{1}-t_{2}|^{\theta }+| x_{1}-x_{2}| _{\alpha })$ for all
$( t_{1},x_{1})$ $(t_{2},x_{2})$ in $\mathbb{R\times }X_{\alpha }$,
\end{enumerate}
where $X$ is a real or complex Banach space with norm $|\cdot |$,
$A^{\alpha }$ is the fractional power, and $X_{\alpha }$ is the
Banach space $D( A^{\alpha }) $ endowed with the norm
$|x| _{\alpha }=| A^{\alpha }x|$.
We prove first that the map
\[
T\varphi ( t) =\int_{-\infty }^{t}A^{\alpha }S( t-\sigma
) f( \sigma ,A^{-\alpha }\varphi ( \sigma ) )
d\sigma
\]
is a strict contraction. Then we prove the existence of
an almost periodic classical solution over $\mathbb{R}$
of \eqref{1.1}. See Theorem \ref{thm3.1} below.
Our main theorem complements the results in \cite{z2} by considering almost
periodic classical solutions instead of almost periodic mild solutions.

Our work is organized as follows. Section 2 is devoted to a review
of some results on fractional powers of operators and almost periodic functions
with values in a Banach space. In section 3, we state and prove our main result.
The last section is devoted to giving an example of a function satisfying
hypothesis (F).

\section{Preliminary results}

Throughout this work, we use the following notation:
$X$ denotes a real or complex Banach space
endowed with the norm $|\cdot|$ and $\mathcal{L}(X) $
stands for the Banach algebra of bounded linear operators defined on $X$.
For $A$ a linear operator with domain $D(A) $, we denote by $\mathcal{R}
(A)$ the range of $A$.

\subsection*{Fractional powers of operators}

We start by a brief outline of the theory of fractional powers as developed
in \cite{f1,p1}. Let $-A$ is the infinitesimal generator of an analytic semigroup
in a Banach space and $0\in \rho (A)$.  For $\alpha >0$ we define the
fractional power $A^{-\alpha}$ by
\[
A^{-\alpha }=\frac{1}{\Gamma ( \alpha ) }\int_{0}^{\infty
}t^{\alpha -1}S( t) dt
\]
Since $A^{-\alpha }$ is one to one, $A^{\alpha }=( A^{-\alpha }) ^{-1}$.

For $0<\alpha \leq 1$, $A^{\alpha }$ is a closed linear operator whose
domain $D( A^{\alpha }) \supset D( A) $ is dense in $X$. The
closedness of $A^{\alpha }$ implies that $D( A^{\alpha }) $
endowed with the graph norm
\[
| x| _{D( A) }=| x| +| A^{\alpha}x| ,\quad x\in D( A^{\alpha })
\]
is a Banach space. Since $0\in \rho ( A) ,\;A^{\alpha }$ is
invertible, and its graph norm is equivalent to the norm
$| x|_{\alpha }=| A^{\alpha }x| $.
Thus $D( A^{\alpha }) $ equipped with the norm $|\cdot| _{\alpha }$ is a
Banach space which we denote $X_{\alpha }$.

\begin{lemma} \label{lm2.1.1}
Let $-A$ be the infinitesimal generator of an analytic semigroup
$S( t)$. If $0\in \rho ( A)$ then
\begin{enumerate}
\item[(a)] $S( t):X\to D( A^{\alpha}) $ for every $t>0$ and
$\alpha \geq 0$
\item[(b)] For every $x\in D( A^{\alpha }) $,
we have $S( t) A^{\alpha }x=A^{\alpha }S(t) x$
\item[(c)] For every $t>0$ the operator $A^{\alpha }S( t)$
is bounded and $|A^{\alpha }S(t)| _{\mathcal{L}( X) }
\leq M_{\alpha }t^{-\alpha}e^{-\delta t}$
\item[(d)]  For $0<\alpha \leq 1$ and $x\in D( A^{\alpha })$,
we have $| S( t)x-x| \leq C_{\alpha }t^{\alpha }| A^{\alpha }x| $.
\end{enumerate}
\end{lemma}
For more details, see \cite[section 2.6]{p1}.

\subsection*{Almost periodic functions in Banach spaces}

The theory of almost periodic functions with values in a Banach space was
developed by H. Bohr, S. Bochner, J. von Neumann, and others; cf., e.g.,
\cite{a1,b3}. From their results, we will mention several results which
will be used in this work.

Let $C_{b}( \mathbb{R},X) $ denote the usual Banach space of
bounded continuous functions from $\mathbb{R}$ into $X$ under the 
supremum norm $|\cdot| _{\infty }$. 
Given a function $f:\mathbb{R}\to X$ and
$\omega \in \mathbb{R}$, we define the $\omega$-translate of $f$ as
$f_{\omega }( t) =f( t+\omega)$, $t\in \mathbb{R}$. We will denote
by $H( f) =\{ f_{\omega }:\omega \in \mathbb{R}\} $  the set
of all translates of $f$.

\paragraph{Definition.} %2.2.1
(Bochner's characterization of almost periodicity)
A function $f\in C_{b}( \mathbb{R},X) $ is said to
be almost periodic if and only if $H( f) $ is
relatively compact in $C_{b}( \mathbb{R},X)$.

Of course, almost periodic functions can as well be characterized in terms
of relatively dense sets in $\mathbb{R}$ of $\tau $-almost periods.

\paragraph{Definition.} % 2.2.2
A function $f:R\to X$ is called almost periodic if
\begin{enumerate}
\item[(i)] $f$ is continuous, and
\item[(ii)] for each $\varepsilon >0$ there exists
$l(\varepsilon ) >0$, such that every interval $I$  of
length $l(\varepsilon )$ contains a number $\tau $
such that
$|f( t+\tau )-f( t) | <\varepsilon$ for all $t\in R$.
\end{enumerate}
Let $Y$ denote a Banach space and $\Omega $ an open subset of $Y$.

\paragraph{Definition.} %2.2.3
A continuous function $f:\mathbb{R}\times \Omega \to X$
is called uniformly almost periodic if for every $\varepsilon >0$
and every compact set $K\subset \Omega$ there exists a relatively dense
set $P_{\varepsilon }$ in $\mathbb{R}$ such that
$| f( t+\tau ,x) -f(t,x) | \leq \varepsilon $
for all $t\in \mathbb{R}$, $\tau \in P_{\varepsilon }$ and all
$y\in K$.

The following is essential for our results and is proven in
\cite[Theorem I.2.7]{y1}.

\begin{lemma} \label{lm2.2.1}
Let $f:R\times \Omega \to X$ be uniformly almost periodic and
$y:R\to \Omega $ be an almost periodic function such that
$\overline{\mathcal{R}(y)}\subset \Omega$,  then the function
$t\to f( t,y( t) ) $ also is almost periodic.
\end{lemma}
We are now in position to state and prove the main result of this paper.

\section{Main Result}

\paragraph{Definition.}%  3.1
A function $x:[ 0,T[\to X$ is a (classical) solution of \eqref{1.1}
on $[ 0,T[$ if $x$ is continuous on $[ 0,T[ $, continuously differentiable
on $]0,T[$, $x( t) \in D( A)$ for $0<t<T$ and \eqref{1.1} is satisfied.

\paragraph{Definition.} %3.2
A continuous solution $x$ of the integral equation
\begin{equation}
x( t) =S( t-t_{0}) x( t_{0})
+\int_{t_{0}}^{t}S( t-\sigma ) f( \sigma ,x( \sigma
) ) d\sigma  \label{ast}
\end{equation}
will be called a mild solution of \eqref{1.1}.


\paragraph{Remark.}
When $A$ generates a semigroup with negative exponent, it is
easy to see that if $x(.) $ is a bounded mild solution of
\eqref{1.1} on  $\mathbb{R}$. Then we can take the limit as
$t_{0}\to -\infty $ on the right-hand of \eqref{ast} to
obtain
\begin{equation}
x(t) =\int_{-\infty }^{t}S( t-\sigma ) f( \sigma,x( \sigma ) ) d\sigma\,.
\label{astast}
\end{equation}
Conversely, if $x(.) $ is a bounded continuous function and
\eqref{astast} is verified, then $x( .) $ is a mild
solution of \eqref{1.1}.

The main result of this paper is the following theorem.

\begin{theorem} \label{thm3.1}
Let $-A$ be the infinitesimal generator of an analytic
semigroup $\{ S( t)\} _{t\geq 0}$ satisfying
$| S( t) | _{\mathcal{L}( X) }\leq M\exp ( \beta t)$, for all $t>0$
($\beta <0$).
If $f:\mathbb{R}\times X\to X$ is uniformly almost periodic
and $f$ satisfies the assumption (F)
Then for $L$ sufficiently small enough, there exists one
and only one almost periodic solution over $\mathbb{R}$
of the semilinear equation \eqref{1.1}.
\end{theorem}

\paragraph{Remark.}
Assumption (F) is commonly used for this type of equations, as seen in
 \cite{m1,p1}).
 
In the proof of our main result, we will need the following technical lemma.

\begin{lemma} \label{lm3.1}
If $g:\mathbb{R}\to X$ is almost periodic and locally H\"{o}lder continuous,
then there exists one and only one almost periodic (classical) solution over
$\mathbb{R}$ of the equation \eqref{1.2}. The solution is
$x( t)=\int_{-\infty }^{t}S( t-\sigma ) g( \sigma ) d\sigma$.
\end{lemma}

\paragraph{Proof.}
In \cite{z1}, it is proved the existence of the almost periodic mild solution
of \eqref{1.2}.
It is known, see \cite{p1}, that in the case of
H\"{o}lder continuity of $g$ and if $A$ generates an analytic semigroup,
then the mild solution is a classical solution of the differential equation
\eqref{1.2}. \hfill$\square$


We define the set
$$
AP( X) =\{ \varphi :\mathbb{R}\to X,\;\varphi
\mbox{ is almost periodic }\}
$$
with the usual supremum norm over $\mathbb{R}$ which we denote by
$|\cdot| _{\infty }$. On the set $AP( X) $, we define a mapping
\begin{equation}
T\varphi (t) =\int_{-\infty }^{t}A^{\alpha }S( t-\sigma
) f( \sigma ,A^{-\alpha }\varphi ( \sigma ) )
d\sigma  \label{3.1}
\end{equation}
First, we show that $T$ is well defined.
Let $\varphi \in AP( X) $, using a standard properties of the
almost-periodicity, we have
\[
N=\sup_{t\in \mathbb{R}}| f( t,A^{-\alpha }\varphi ( t) ) | <\infty .
\]
by Lemma \ref{lm2.1.1}.c, we have
\[
| T\varphi ( t) | \leq M_\alpha N\int_{-\infty
}^t( t-\sigma ) ^{-\alpha }\exp ( -\delta ( t-\sigma) ) d\sigma .
\]
With the change variable $s=t-\sigma $, we obtain
\[
| T\varphi ( t) | \leq M_{\alpha }N\int_{0}^{+\infty
}s^{-\alpha }\exp ( -\delta s) ds
\]
which shows that $T\varphi $ exists.

\begin{lemma} \label{lm3.2}
The operator $T$ is well defined, and maps $AP(X)$ into itself.
\end{lemma}


\paragraph{Proof.}
For $\varphi \in AP( X)$, it follows from Lemma \ref{lm2.2.1} that
$t\to f( t,A^{-\alpha }\varphi ( t) )$ is
almost periodic. Hence, for each $\varepsilon >0$ there exists a set
$P_{\varepsilon }$ relatively dense in $\mathbb{R}$ such that
$$| f( t+\tau ,A^{-\alpha }\varphi ( t+\tau )
) -f( t,A^{-\alpha }\varphi ( t) ) | \leq \varepsilon
$$
for all $t\in \mathbb{R}$ and $\tau \in P_{\varepsilon }$.
Therefore, the map $T$ defined by \eqref{3.1} satisfies
\begin{align*}
|T&\varphi ( t+\tau ) -T\varphi ( t) | \\
=&\Big| \int_{-\infty }^{t+\tau }A^{\alpha }S( t+\tau -\sigma )
 f( \sigma ,A^{-\alpha }\varphi ( \sigma ) ) d\sigma \\
&-\int_{-\infty }^{t}A^{\alpha }S( t-\sigma ) f( \sigma
,A^{-\alpha }\varphi ( \sigma ) ) d\sigma \Big|\\
=&\Big| \int_{-\infty }^{t}A^{\alpha }S( t-\sigma ) f(
\sigma +\tau ,A^{-\alpha }\varphi ( \sigma +\tau ) ) d\sigma\\
&-\int_{-\infty }^{t}A^{\alpha }S( t-\sigma ) f(
\sigma ,A^{-\alpha }\varphi ( \sigma ) ) d\sigma \Big|\\
\leq& \int_{-\infty }^{t}| A^{\alpha }S( t-\sigma ) |
_{\mathcal{L}( X) }| f( \sigma +\tau ,A^{-\alpha
}\varphi ( \sigma +\tau ) ) -f( \sigma ,A^{-\alpha
}\varphi ( \sigma ) ) | d\sigma \\
\leq& \varepsilon M_{\alpha }\int_{-\infty }^{t}( t-\sigma )
^{-\alpha }\exp ( -\delta ( t-\sigma ) ) d\sigma
\end{align*}
Which shows that the function $T\varphi $ also is almost periodic and that
$T:AP( X) \to AP( X)$. \hfill$\square$

\paragraph{Proof of Theorem \ref{thm3.1}}
Consider the mapping from the Banach space $AP( X) $ into itself
defined by
\[
T\varphi =\psi ( t) =\int_{-\infty }^{t}A^{\alpha }S( t-\sigma )
f( \sigma ,A^{-\alpha }\varphi ( \sigma ) ) d\sigma .
\]
We will show that $T$ has a fixed point.
Let $\varphi _{1},\varphi _{2}\in AP( X) $. Then
$$
| T\varphi _{1}( t) -T\varphi _{2}( t) |
\leq \int_{-\infty }^{t}| A^{\alpha }S( t-\sigma )| _{\mathcal{L}( X) }
| f( \sigma ,A^{-\alpha }\varphi_{1}( \sigma ) )
-f( \sigma ,A^{-\alpha }\varphi_{2}( \sigma ) ) | d\sigma .
$$
From assumption (F), we have
\begin{align*}
| T\varphi _{1}( t) -T\varphi _{2}( t) |
\leq &L| \varphi _{1}-\varphi _{2}| _{\infty }\int_{-\infty
}^{t}| A^{\alpha }S( t-\sigma ) | _{\mathcal{L}(
X) }d\sigma \\
\leq &L| \varphi _{1}-\varphi _{2}| _{\infty }\int_{-\infty
}^{t}( t-\sigma ) ^{-\alpha }\exp ( -\delta ( t-\sigma
) ) d\sigma
\end{align*}
and by the change of variable $s=t-\sigma $, we have
\begin{align*}
| T\varphi _{1}-T\varphi _{2}| _{\infty }
\leq &LM_{\alpha
}| \varphi _{1}-\varphi _{2}| _{\infty }\int_{0}^{+\infty
}s^{-\alpha }e^{-\delta s}ds \\
=&LM_{\alpha }\delta ^{\alpha }\Gamma ( 1-\alpha ) |
\varphi _{1}-\varphi _{2}| _{\infty },
\end{align*}
where $\Gamma ( .) $ is the classical gamma function.
We use the well known identity
\[
\Gamma ( \alpha ) \Gamma ( 1-\alpha )
=\frac{\pi }{\sin \pi \alpha }\quad \text{for }0<\alpha <1.
\]
Then, we can deduce that $T$  is a strict contraction, provided $L$ is
sufficiently small, $L<\frac{\sin \pi \alpha }{\alpha }\frac{\Gamma
( \alpha ) }{M_{\alpha }\delta ^{\alpha }}$.
By the contraction mapping theorem there exists $\varphi \in AP(X)$
such that
\begin{equation}
\varphi =\int_{-\infty }^{t}A^{\alpha }S( t-\sigma ) f(
\sigma ,A^{-\alpha }\varphi ( \sigma ) ) d\sigma .
\label{3.2}
\end{equation}
Since $A^{\alpha }$ is closed,
\begin{equation}
\varphi =A^{\alpha }\int_{-\infty }^{t}S( t-\sigma ) f(
\sigma ,A^{-\alpha }\varphi ( \sigma ) ) d\sigma .
\label{3.3}
\end{equation}
Applying the operator $A^{-\alpha }$ on both sides of \eqref{3.3},
\begin{equation}
A^{-\alpha }\varphi =\int_{-\infty }^{t}S( t-\sigma ) f(
\sigma ,A^{-\alpha }\varphi ( \sigma ) ) d\sigma .
\label{3.4}
\end{equation}
Next, we show that $t\to f( t,A^{-\alpha }\varphi
( t) ) $ is H\"{o}lder continuous on $\mathbb{R}$.
To this end we show first that the solution $\varphi $ of \eqref{3.4} 
is H\"{o}lder continuous on $\mathbb{R}.$
By Lemma \ref{lm2.1.1}.d We note that for every $\beta $ satisfying $0<\beta
<1-\alpha $ and for every $h>0$, we have
\begin{equation}
| ( S( h) -I) A^{\alpha }S( t-\sigma )
| \leq C_{\beta }h^{\beta }| A^{\alpha +\beta }S( t-\sigma
) |  \label{3.5}
\end{equation}
and
\begin{equation}
\begin{aligned}
| \varphi ( t+h) -\varphi ( t) |
\leq & \Big| \int_{-\infty }^{t}( S( h) -I) A^{\alpha
}S( t-\sigma ) f( \sigma ,A^{-\alpha }\varphi ( \sigma) ) d\sigma \Big|  \\
&+\Big| \int_{t}^{t+h}A^{\alpha }S( t+h-\sigma ) f( \sigma
 ,A^{-\alpha }\varphi ( \sigma ) ) d\sigma \Big|
\end{aligned} \label{3.6}
\end{equation}
Let $K=A^{-\alpha }\varphi ( \mathbb{R}) $ and
$N=\sup_{( t,x) \in \mathbb{R}\times K} | f( t,x) |$.
Clearly $K$ is compact.
Using Lemma \ref{lm2.1.1}.c and \eqref{3.5} we can estimate each of the terms
of \eqref{3.6} separately:
\begin{multline*}
\Big| \int_{-\infty }^{t}( S( h) -I) A^{\alpha
}S( t-\sigma ) f( \sigma ,A^{-\alpha }\varphi ( \sigma
) ) d\sigma \Big| \\
\leq M_{\alpha +\beta }NC_{\beta }h^{\beta }\int_{-\infty }^{t}(
t-\sigma ) ^{-( \alpha +\beta ) }\exp ( -\delta (t-\sigma ) ) \,d\sigma.
\end{multline*}
By Lemma \ref{lm2.1.1}.c,
\begin{align*}
\Big| \int_{t}^{t+h}A^{\alpha }S( t+h-\sigma ) f( \sigma
,A^{-\alpha }\varphi ( \sigma ) ) d\sigma \Big|
&\leq M_{\alpha }N\int_{t}^{t+h}( t+h-\sigma ) ^{-\alpha }d\sigma \\
&\leq M_{\alpha }N\frac{h^{1-\alpha }}{1-\alpha }.
\end{align*}
Combining \eqref{3.8} with these estimates, it follows that there
is a constant $C$ such that
\[
| \varphi ( t+h) -\varphi ( t) | \leq
Ch^{\beta }\;
\]
and therefore $\varphi $ is H\"{o}lder continuous on $\mathbb{R}$.

Finally, it remains to proved that $t\to f( t,A^{-\alpha
}\varphi ( t) ) $ is H\"{o}lder continuous on $\mathbb{R}$.
From assumption $( F) $ we have
\[
| f( t,A^{-\alpha }\varphi ( t) ) -f(
s,A^{-\alpha }\varphi ( s) ) | \leq L\big( |
t-s| ^{\theta }+| \varphi ( t) -\varphi (s) | \big);
\]
therefore, $t\to f( t,A^{-\alpha }\varphi ( t)
) $ is H\"{o}lder continuous on $\mathbb{R}$.
Let $\varphi $ be the solution of \eqref{3.2} and consider the
equation
\begin{equation}
\frac{dx( t) }{dt}+Ax( t) =f( t,A^{-\alpha
}\varphi ( t) ).  \label{3.7}
\end{equation}
From Lemma \ref{lm3.1} this equation has a unique solution
given by
\begin{equation}
\psi( t) =\int_{-\infty }^{t}S( t-\sigma ) f(
\sigma ,A^{-\alpha }\varphi ( \sigma ) ) d\sigma . \label{3.8}
\end{equation}
Moreover, we have $\psi ( t) \in D( A) $ for all $t\in
\mathbb{R\;}$and a fortiori $\psi ( t) \in D( A^{\alpha
}) .$\newline
Applying the operator $A^{\alpha }$ on both sides of \eqref{3.8}, we have
\begin{equation}
A^{\alpha }\psi ( t) =\int_{-\infty }^{t}A^{\alpha }S(
t-\sigma ) f( \sigma ,A^{-\alpha }\varphi ( \sigma )) d\sigma
=\varphi ( t) \label{3.9}
\end{equation}
From \eqref{3.7} and \eqref{3.9} we readily see that $\psi
( t) =A^{-\alpha }\varphi ( t) $ is solution of \eqref{1.1}.
The uniqueness of $\psi $ follows easily from the uniqueness of the solution
of \eqref{3.2} and \eqref{3.7}. Therefore, the proof of Theorem \ref{thm3.1} is
complete.\hfill$\square$

\section{Example}

Let $X=L^{2}(( 0,1) ;\mathbb{R}) $ and
\begin{equation}
Au=-u''\quad\text{with}\quad  u\in D( A) =\{ u\in
H_{0}^{1}(( 0,1) ;\mathbb{R}) ;u''\in X\}.  \label{4.1}
\end{equation}
Then $A$ is self-adjoint, with compact resolvent and is the infinitesimal
generator of an analytic semigroup $S( t) $. We take
$\alpha =1/2$, that is
$X_{1/2}=( D( A^{1/2}),|\cdot| _{1/2})$.
Define the function $f:\mathbb{R}\times X_{1/2}\to X$, by
$ f( t,u) =h( t) g(u')$
for each $t\in \mathbb{R}$ and  $u\in X_{1/2}$,
where $h:\mathbb{R\to R}$ is almost periodic in $\mathbb{R}$ and there
exist $k_{1}>0$ and $\theta \in ] 0,1[ $ such that
\begin{equation}
| h( t) -h( s) | \leq k_{1}|t-s| ^{\theta },\quad
\text{for all }t,s\in \mathbb{R}. \label{4.2}
\end{equation}
and $g:X\to X$ is Lipschitz continuous on $X$. Concrete example of
the function $g$ are
\[
g( u) =\sin ( u), \quad g( u) =ku, \quad g( u) =\arctan ( u)
\]
We give first some known results for the operators $A$ and
$A^{1/2}$ defined by \eqref{4.1}.

Let $u\in D( A) $ and $\lambda \in \mathbb{R}$, such that
$Au=-u''=\lambda u$; that is,
\begin{equation}
u''+\lambda u=0  \label{4.3}
\end{equation}
We have $\langle Au,u\rangle =\langle \lambda u,u\rangle$;
that is,
\[
\left\langle -u'',u\right\rangle =| u'|
_{L^{2}}^{2}=\lambda | u| _{L^{2}}^{2}
\]
so $\lambda \in \mathbb{R}_{+}^{\ast }$.
The solutions of \eqref{4.3} have the form
\[
u( x) =C\cos ( \sqrt{\lambda }x) +D\sin ( \sqrt{\lambda }x)
\]
we have $u( 0) =u( 1) $, so, $C=0$ and
$\sqrt{\lambda }=n\pi $, $n\in \mathbb{N}^{\ast }$.
Put $\lambda _{n}=n^{2}\pi^{2}$.
The solutions of equation \eqref{4.3} are
\[
u_n( x) =D\sin ( \sqrt{\lambda _n}x) ,\;\;n\in \mathbb{N}^{*}.
\]
We have $\langle u_n,u_m\rangle =0$, for $n\neq m$ and
$\langle u_n,u_n\rangle =1$. So $D=\sqrt{2}$ and
\[
u_{n}( x) =\sqrt{2}\sin ( \sqrt{\lambda _{n}}x).
\]
For $u\in D( A) $, there exists a sequence of reals $( \alpha_{n}) $
such that
\begin{gather*}
u( x) =\sum_{n\in \mathbb{N}^{\ast }}
\alpha_{n}u_{n}( x) ,\\
\sum_{n\in \mathbb{N}^{\ast }}(\alpha _{n}) ^{2}<+\infty ,\quad
\sum_{n\in \mathbb{N}^{\ast }}( \lambda_{n}) ^{2}( \alpha _{n}) ^{2} <+\infty
\end{gather*}
We have
\[
A^{1/2}u( x) =\sum_{n\in \mathbb{N}^{*}}\sqrt{\lambda _n}\alpha _nu_n( x)
\]
with $u\in D( A^{1/2}) $; that is,
$\sum_{n\in \mathbb{N}^{\ast }}(\alpha _{n}) ^{2}<+\infty $ and
$\sum_{n\in \mathbb{N}^{\ast }}\lambda _{n}( \alpha _{n}) ^{2}<+\infty $.

We show now that $f$ satisfies the hypothesis (F). In fact,
for $t_{1},t_{2}\in \mathbb{R}$ and $u_{1},u_{2}\in X_{1/2}$,
we have
\begin{align*}
f( t_1,u_1) -f( t_2,u_2)
=&h( t_1) g(u_1') -h( t_2) g( u_2') \\
=&[ h( t_1) -h( t_2) ] g( u_1') +h( t_2) [ g( u_1') -g(u_2') ]
\end{align*}
So,
\begin{equation}
\begin{aligned}
| f( t_{1},u_{1}) -f( t_{2},u_{2}) |
_{L^{2}} \leq &| h( t_{1}) -h( t_{2}) | | g( u_{1}') | _{L^{2}}
+| h( t_{2}) | | g( u_{1}')-g( u_{2}') | _{L^{2}}  \\
\leq &| g| _{\infty }| h( t_{1}) -h(t_{2}) | +| g| _{{\rm Lip}}| h( t_{2}) | |
u_{1}'-u_{2}'| _{L^{2}}.
\end{aligned} \label{4.4}
\end{equation}
Since $h$ is almost periodic, there exists $k_{2}>0$, such that
\begin{equation}
| h( t_{2}) | \leq k_{2}  \label{4.5}
\end{equation}
Therefore, from \eqref{4.2}, \eqref{4.4}, \eqref{4.5}, and the fact that $g( u') $ is Lipschitz on
$X_{1/2}$ (see for instance \cite[p.~75]{h2}), we have
\begin{align*}
| f( t_{1},u_{1}) -f( t_{2},u_{2}) | _{X}
\leq &k_{1}| g| _{\infty }| t_{1}-t_{2}| ^{\theta
}+k_{2}| g| _{Lip}| u_{1}-u_{2}| _{1/2} \\
\leq &L( | t_{1}-t_{2}| ^{\theta }+|u_{1}-u_{2}| _{1/2}) .
\end{align*}
Therefore, $f$ satisfies the hypothesis (F), with
$L=\max (k_{1}| g| _{\infty },k_{2}| g| _{{\rm Lip}})$.

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\noindent\textsc{Mohamed Bahaj}\\
Faculty of Sciences and Technology,\\
Settat, Morocco \\
email: bahajm@caramail.com \smallskip

\noindent\textsc{Omar Sidki} \\
Faculty of Sciences and Technology\\
Fez, Morocco \\
email:osidki@hotmail.com

\end{document}