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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2010(2010), No. 101, pp. 1--13.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2010 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2010/101\hfil Dependence results]
{Dependence results on almost periodic and almost automorphic
solutions of evolution equations}

\author[J. Blot, P. Cieutat, G. M. N'Gu\'er\'ekata \hfil EJDE-2010/101\hfilneg]
{Jo\"{e}l Blot, Philippe Cieutat, Gaston M. N'Gu\'er\'ekata}  % in alphabetical order

\address{Jo\"{e}l Blot \newline
 Laboratoire SAMM  EA 4543,
Universit\'{e} Paris 1 Panth\'{e}on-Sorbonne, centre P.M.F.,
90 rue de Tolbiac, 75634 Paris cedex 13, France}
\email{blot@univ-paris1.fr}

\address{Philippe Cieutat \newline
Laboratoire de Math\'ematiques de Versailles, UMR-CNRS 8100,
Universit\'{e} Versailles-Saint-Quentin-en-Yvelines,
45 avenue des \'Etats-Unis, 78035 Versailles cedex, France}
\email{Philippe.Cieutat@math.uvsq.fr}

\address{Gaston M. N'Gu\'er\'ekata \newline
 Department of mathematics, Morgan State University,
1700 E. Cold Spring Lane, Baltimore, MD 21252, USA}
\email{gnguerek@jewel.morgan.edu}

\thanks{Submitted May 25, 2010. Published July 21, 2010.}
\subjclass[2000]{47J35, 43A60, 47D06}
\keywords{Semilinear evolution equation; almost periodic function;
\hfill\break\indent almost automorphic function; dependence results}

\begin{abstract}
 We consider the semilinear evolution equations
 $x'(t) = A(t) x(t) + f(x(t), u(t),t)$ and
 $x'(t) = A(t) x(t) + f(x(t), \zeta,t)$ where $A(t)$
 is a unbounded linear operator on a Banach space $X$ and $f$
 is a nonlinear operator. We study the dependence of solutions
 $x$ with respect to the function $u$ in three cases: the continuous
 almost periodic functions, the differentiable almost periodic
 functions, and the almost automorphic functions. We give results
 on the continuous dependence and on the differentiable dependence.
\end{abstract}

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

\section{Introduction}

We consider the  differential equations
\begin{gather}
x'(t) = A(t)x(t) + f(x(t), u(t),t), \label{Eu} \\
x'(t) = A(t)x(t) + f(x(t),\zeta,t), \label{Gzeta}
\end{gather}
where $t \in \mathbb{R} $, $A(t)$ is a unbounded linear operator on a
Banach space and $f$ is a nonlinear operator. The function $u$ can
be seen as a perturbation or as a control term; the term $\zeta$
is a general abstract parameter. Our aim is  to study the
dependence of solutions $x$ of \eqref{Eu} with respect to the
function $u$ and the dependence of the solutions $x$ of
\eqref{Gzeta} with respect to $\zeta$. We consider three classes
of functions: the continuous almost periodic functions, the
differentiable almost periodic functions, and the almost
automorphic functions. In our method we use the following linear
inhomogeneous differential equation. \begin{equation} \label{Lb} x'(t) =
A(t)x(t) + b(t). \end{equation} In the special case where $A$ is independent
of $t$, $A(t) = A$, the  previous equations become the following
equation, respectively,
\begin{gather}
x'(t) = Ax(t) + f(x(t), u(t),t), \label{Ecu} \\
x'(t) = Ax(t) + f(x(t), \zeta,t), \label{Gczeta}\\
x'(t) = Ax(t) + b(t). \label{Lcb}
\end{gather}

Now we describe the contents of this article. In Section 2 we fix
our notations, the various spaces of functions  considered, and
assumptions used later. In Section 3 we establish preliminary
results for the linear case. In Section 4 we treat the continuous
dependence of the solutions of \eqref{Eu} and of \eqref{Ecu} with
respect to $u$ and the continuous dependence of the solutions of
\eqref{Gzeta} and of \eqref{Gczeta} with respect to $\zeta$; we
use a fixed point theorem to realize that. In Section 5 we treat
the differentiable dependence of the solutions of \eqref{Eu} and
of \eqref{Ecu} with respect to $u$ and the differentiable
dependence of the solutions of \eqref{Gzeta} and of \eqref{Gczeta}
with respect to $\zeta$; we use an implicit function theorem to
reach our goal.

\section{Notation}

When $X$ is a Banach space, $AP(X)$ denotes the space of the Bohr
almost periodic functions from $\mathbb{R} $ into $X$, \cite{HNMS},
\cite{LZ,N,Pan,Yo,Z2}. When $n$ is a non
negative integer number, $AP^{(n)}(X)$ denotes the space of the
functions in $AP(X)$ which are of class $C^n$ on $\mathbb{R}$ such that
the derivative of order $k$ belongs to $AP(X)$ for all $k$ between
$0$ and $n$, \cite{BBNP}. $AA(X)$ denotes the space of the Bochner
almost automorphic functions from $\mathbb{R} $ into $X$, \cite{N}.
Endowed with the norm of the uniform convergence, $\| x
\|_{\infty} := \sup_{t \in \mathbb{R} } | x(t)|$, $AP(X)$ and
$AA(X)$ are Banach spaces. Endowed with the norm
\[
\| x \|_{C^n} := \| x \|_{\infty} + \sum_{1 \leq k \leq n}
\| \frac{d^kx}{dt^k} \|_{\infty},
\]
the space $AP^{(n)}(X)$ is a Banach space.

\begin{definition}[{\cite[p. 5-6]{Yo}, \cite[p. 45]{BCNP}}]
\label{def1}\rm
 When $X$ is a Banach space, a continuous mapping
$f : Y \times \mathbb{R} \to X$ is so-called
almost periodic in $t$ uniformly in $y$ when
the following condition holds: for all compact $K \subset Y$ and
for all $\varepsilon > 0$, there exists
${\ell} = {\ell}(K, \varepsilon) > 0$ such that,
for all $r \in \mathbb{R} $, there exists $\tau \in [r, r + {\ell}]$
satisfying $| f(y, t + \tau) - f(y,t) | \leq \varepsilon$
for all $(y,t) \in K \times \mathbb{R} $. We denote by
$APU(Y \times \mathbb{R} , X)$ the space of such mappings.
\end{definition}

\begin{definition}[{\cite[p. 45]{BCNP}}] \label{def2} \rm
 A mapping $f : Y \times \mathbb{R}  \to X$ is so-called
 almost automorphic in $t$ uniformly in $y$ when $f(y,.) \in AA(X)$
for all $y \in Y$ and when, for all compact $K \subset Y$ and
for all $\varepsilon > 0$, there exists
$\delta = \delta( K, \varepsilon) > 0$ satisfying
$| f(y,t) - f(z,t) | \leq \varepsilon$ for all
$t \in \mathbb{R} $ and for all $y,z \in K$ such that
$| y - z | \leq \delta$. We denote by
$AAU(Y \times \mathbb{R} , X)$ the space of such mappings.
\end{definition}

About the continuous almost periodic functions we consider the
following conditions, where $U$ is a Banach space.
\begin{equation} \label{eq1}
f \in APU((X \times U) \times \mathbb{R} , X).
\end{equation}
\begin{equation}\label{eq2}
\begin{gathered}
f \in APU((X \times U) \times \mathbb{R} , X), \\
\forall ( \xi, \zeta, t) \in  X \times U \times \mathbb{R} ,
D_1f(\xi, \zeta,t) \text{ and } D_2f(\xi, \zeta,t)  \text{ exist },\\
D_1f \in APU((X \times U) \times \mathbb{R} , \mathcal{L}(X,X)),\\
D_2f \in APU((X \times U) \times \mathbb{R} , \mathcal{L}(U,X)),
\end{gathered}
\end{equation}
where $D_1f(\xi, \zeta,t)$ is the differential of $f(., \zeta,t)$,
$D_2f (\xi, \zeta,t)$ is the differential of $f(\xi, .,t)$ and
$\mathcal{L}(Y,X)$ denotes the space of the linear bounded
 mappings from $Y$ into $X$.

About the differentiable almost periodic functions we consider
the following conditions.
\begin{equation} \label{eq3}
\begin{gathered}
f \in APU((X \times U) \times \mathbb{R} , X) \cap C^n((X \times U)
\times \mathbb{R} , X),\\
\forall k=0,\dots ,n,\; D^kf \in APU((X \times U) \times \mathbb{R} ,
\mathcal{L}_k(((X \times U) \times \mathbb{R} )^k, X),
\end{gathered}
\end{equation}
where $D^kf$ denotes the differential of order $k$ of $f$ and
where $\mathcal{L}_k(Y^k,X)$ denotes the space of the $k$-linear
continuous mappings from $Y^k$ into $X$.
\begin{equation} \label{eq4}
\begin{gathered}
f \in APU((X \times U) \times \mathbb{R} , X) \cap C^{n+1}((X \times U)
 \times \mathbb{R} , X),\\
\forall k=0,\dots ,n+1,\;  D^kf \in APU((X \times U)
\times \mathbb{R}, \mathcal{L}_k(((X \times U) \times \mathbb{R})^k, X))).
\end{gathered}
\end{equation}

About almost automorphic functions we consider the following conditions.
\begin{equation} \label{eq5}
f \in AAU((X \times U) \times \mathbb{R} , X).
\end{equation}
\begin{equation}\label{eq6}
\begin{gathered}
f \in AAU((X \times U) \times \mathbb{R} , X),\\
 \forall ( \xi, \zeta, t) \in  X \times U \times \mathbb{R},\;
 D_1f(\xi, \zeta,t)  \text{ and }  D_2f(\xi, \zeta,t) \text{ exist,}\\
 D_1f \in AAU((X \times U) \times \mathbb{R} , \mathcal{L}(X,X)),\\
D_2f \in AAU((X \times U) \times \mathbb{R} , \mathcal{L}(U,X)).
\end{gathered}
\end{equation}

For a linear operator $A$ on $X$, not necessarily bounded, we denote
by $\mathcal{D}(A)$ its domain, by $\varrho (A)$ its resolvent
set and by $R( \lambda; A)$ its resolvent operators
(cf. \cite[p. 8]{Paz}).

\begin{definition}\label{def23} \rm
A family $(F(t,s))_{t \geq s}$ of bounded linear operators on $X$
is called an {\rm evolution family} when $F(t,t) = I$
(the identity operator on $X$) for all $t \in \mathbb{R}$,
$F(t,s)F(s,r)= F(t,r)$ for all $t \geq s \geq r$ and
$(t,s) \mapsto F(t,s)x$ is continuous for all $x \in X$.
\end{definition}


\begin{definition}\label{def24} \rm
We say that the evolution family  $(F(t,s))_{t \geq s}$ in
$\mathcal{L}(X,X)$ is {\rm exponentially stable} when there
exist $c > 0$ and $\omega > 0$ such that
$\| F(t,s) \| \leq c\cdot e^{- \omega(t-s)}$ for all $t \geq s$.
\end{definition}

For all $t \in \mathbb{R}$, let $A(t) : \mathcal{D}(A(t)) \subset X \to X$
be a unbounded linear operator.


\begin{definition}[{\cite{AT}, \cite[p. 269]{BBNP}}]\label{def25}
 We say that $(A(t))_t$ satisfies the  Acquistapace-Terrini conditions
when there exist $\lambda_0 \geq 0$, $\theta \in (\frac{\pi}{2},\pi)$,
$L \geq 0$, $K \geq 0$, $\alpha \in (0,1]$, $\beta \in (0, 1]$,
such that $\alpha + \beta > 1$, satisfying
$\Sigma_{\theta} \cup \{ 0 \} \subset \varrho (A(t)
- \lambda_0 I)$
(where $ \Sigma_{\theta} := \{ \lambda \in {\mathbb{C}} \setminus \{ 0 \} :
 | \arg \lambda | \leq \theta \}$) for all $t \in \mathbb{R}$,
$\| R(\lambda ; A(t) - \lambda_0 I) \|
\leq \frac{K}{1+ | \lambda |}$ for all
$t \in \mathbb{R}$, and $\| (A(t) - \lambda_0 I)R(\lambda ; A(t)
- \lambda_0 I) [R(\lambda_0 ; A(t)) -R(\lambda_0 ; A(s)) ]
\| \leq L | t-s |^{\alpha}| \lambda |^{- \beta}$
for all $t,s \in \mathbb{R} $, for all $\lambda \in \Sigma_{\theta}$.
\end{definition}


\begin{remark}\label{rem26} \rm
Under these Acquistapace-Terrini conditions, the family $(A(t))_t$
generates a unique evolution family $(F(t,s))_{t \geq s}$ in
$\mathcal{L}(X,X)$ such that, for all $s \in \mathbb{R}$ and for all
$x_0 \in \overline{\mathcal{D}(A(s))}$, the function
$t \mapsto F(t,s)x_0$ is continuous at $t = s$ and it is the
unique solution in $C([s, \infty),X) \cap C^1((s, \infty),X)$
of the following Cauchy problem: $x'(t) = A(t)x(t)$ for $t > s$
and $x(s) = x_0$ (cf. \cite{AT}).
\end{remark}

We consider the following condition.
\begin{equation} \label{eq7}
\parbox{10cm}{
$(A(t))_t$ satisfies  the  Acquistapace-Terrini   conditions
$R(\lambda_0,A(.)) \in AP(\mathcal{L}(X,X))$ for $\lambda_0$
given  in  Definition \ref{def25}
and  the evolution  family  $(F(t,s)_{t \geq s}$ is exponentially
stable.}
\end{equation}
We also consider the  following condition which are the assumptions
\cite[(A1)-(A2)]{DLN}.

\begin{definition}\label{def27} \rm
We say that $(A(t))_t$ satisfies the Ding-Long-N'Gu\'{e}r\'{e}kata
conditions when $(A(t))_t$ generates an evolution family
$(F(t,s))_{t \geq s}$ and
 there exists $P \in C(\mathbb{R} , \mathcal{L}(X,X))$ such that
$P(t)$ is a projection for all $t \in \mathbb{R}$,  there exist
$c \geq 0$, $\omega > 0$ such that $F(t,s)P(s) = P(t)F(t,s)$
for all $t \geq s$, and denoting $Q := I-P$, the restriction
$F_Q(t,s) : Q(s)X \to Q(t)X$ is invertible for all $t \geq s$,
and $\| F(t,s)P(s) \| \leq c .e^{- \omega. (t-s)}$,
$\| F_Q(t,s)Q(t) \| \leq c. e^{- \omega. (t-s)}$ for all
$t \geq s$. Setting $\Gamma(t,s) := F(t,s)P(s)$ when $t \geq s$
and $\Gamma(t,s) := - F_Q(t,s)Q(s)$ when $t < s$, for all real
sequence $(s'_m)_m$, there exists a subsequence $(s_m)_m$ of
$(s'_m)_m$ such that $\Lambda(t,s)x := \lim_{m \to \infty}
 \Gamma(t + s_m, s + s_m)x $ is well defined for all $x \in X$
and for all $t,s \in \mathbb{R} $, and moreover
$\lim_{m \to \infty} \Lambda (t- s_m, s-s_m)x = \Gamma(t,s)x$
for all $x \in X$ and for all $t,s \in \mathbb{R} $.
\end{definition}

Note that
\begin{equation} \label{eq8}
(A(t))_t \text{ satisfies  the Ding-Long-N'Gu\'er\'ekata  conditions}.
\end{equation}
In the special case where $A(t) = A$ is constant with respect to $t$,
we consider the following notion, see  \cite[p. 56]{N}.

\begin{definition}\label{def5} \rm
We say that the linear unbounded operator
$A : \mathcal{D}(A) \subset X \to X$ generates a $C_0$-semigroup
$(T(t))_{t \geq 0}$ in $\mathcal{L}(X,X)$ which is
exponentially stable when there exist $c > 0$ and $\omega > 0$
such that $\| T(t) \| \leq c\cdot e^{- \omega t}$ for all
$t \geq 0$.
\end{definition}

Note that
\begin{equation} \label{eq9}
A: \mathcal{D}(A) \subset X \to X
\text{ generates  an  exponentially  stable $C_0$-semigroup}.
\end{equation}

\begin{definition}[{\cite[pp. 106, 146, 184]{Paz}}]\label{def6} \rm
 When $x : \mathbb{R}  \to X$ is a continuous function, $x$
is so-called a  mild solution of \eqref{Eu}
(respectively of \eqref{Lb} respectively of \eqref{Gzeta}, respectively
of \eqref{Ecu}, respectively of \eqref{Lcb}, respectively of \eqref{Gczeta})
when the following condition holds for all
$t \geq s$:\\
$x(t) = F(t,s)x(s) + \int_s^t F(t,r)f(x(r),u(r),r) dr$
(respectively  $x(t) = F(t,s)x(s) + \int_s^t F(t,r)b(r) dr$,
respectively
$x(t) = F(t,s) x(s) + \int_s^t F(t,r)f(x(r), \zeta,r)dr$,
respectively $x(t) = T(t-s)x(s) + \int_s^t T(t-r)f(x(r),u(r),r) dr$,
respectively
$x(t) = T(t-s)x(s) + \int_s^t T(t-r)b(r) dr$,
respectively  $x(t) = T(t-s) x(s) + \int_s^t\! T(t-r) f(x(r), \zeta, r)dr$).
\end{definition}

\begin{definition}[{\cite[pp. 105, 146, 184]{Paz}}]\label{def7} \rm
A function $x : \mathbb{R} \to X$ is so-called a classical solution
of \eqref{Ecu} (respectively of \eqref{Lcb}, respectively of
 \eqref{Gczeta})) if $x$ is continuously differentiable on
$\mathbb{R}$, $x(t) \in \mathcal{D}(A)$ for all
$t \in \mathbb{R}$, and \eqref{Ecu} (respectively of \eqref{Lcb},
respectively of \eqref{Gczeta})) is satisfied on $\mathbb{R}$.
\end{definition}

\section{The linear case}

About the linear equations, we consider the following conditions.
\begin{gather}
\text{For all $b \in AP(X)$,  \eqref{Lb} has a unique  mild
 solution in $AP(X)$},\label{eq10}
\\
\text{For all $b \in AP^{(n)}$,  \eqref{Lb} has a unique mild
 solution in $AP^{(n)}(X)$}, \label{eq11}
\\
\text{For  all $b \in AA(X)$,  \eqref{Lb} has a unique mild
 solution in $AA(X)$}. \label{eq12}
\end{gather}
In \cite[Theorem 3.6]{BBNP} it is shown that \eqref{eq10} and
\eqref{eq11} are fulfilled when \eqref{eq7} is satisfied.
In \cite[Theorem 2.2]{DLN} it is shown that \eqref{eq12} is fulfilled
when \eqref{eq8} is satisfied.


\begin{theorem}\label{th1}
 Under \eqref{eq7} (respectively under \eqref{eq8}) we define the
operators $T_{ap} : AP(X) \to AP(X)$ and
$T_{apn} : AP^{(n)}(X) \to AP^{(n)}(X)$
(respectively $T_{aa} : AA(X) \to AA(X)$) in the following way:
$T_{ap}(b)$ (respectively $T_{apn}(b)$, respectively
$T_{aa}(b)$) is the unique mild solution of \eqref{Lb} in $AP(X)$
(respectively $AP^{(n)}(X)$, respectively $AA(X)$) for all
$b \in AP(X)$ (respectively $AP^{(n)}(X)$, respectively $AA(X)$).
Then $T_{ap}$, $T_{apn}$ and $T_{aa}$ are linear bounded operators.
\end{theorem}

\begin{proof}
The conditions \eqref{eq10}-\eqref{eq12} ensure that the operators
$T_{ap}$, $T_{apn}$ and $T_{aa}$ are well defined and their
linearity is clear. We prove that the graph of $T_{ap}$,
$\mathcal{G}(T_{ap})$, is closed in $AP(X) \times AP(X)$.
Let $(b_m, x_m)_m$ be a sequence in $\mathcal{G}(T_{ap})$ which
(uniformly) converges to $(b,x) \in AP(X) \times AP(X)$.
And so, for all $m \in {\mathbb{N}}$, and for $t\geq s$, the following
equality holds.
$$
x_m(t) = F(t,s)x_m(s) + \int_s^t F(t,r)b_m(r) dr.
$$
Since the uniform convergence implies the pointwise converge,
$\lim_{m \to \infty}x_m(t) = x(t)$ and
$\lim_{m \to \infty}x_m(s) = x(s)$.
Since $F(t,s)$ is a bounded linear operator,
we have $\lim_{m \to \infty}F(t,s)x_m(s) = F(t,s)x(s)$.
Note that, for all $r \in [s,t]$, we have
\begin{align*}
| F(t,r) b_m(r) - F(t,r) b(r) |
&\leq \| F(t,r) \|. | b_m(r) - b(r) |\\
&\leq c e^{- \omega (t-s)} \| b_m - b \|_{\infty}
\leq c \| b_m - b \|_{\infty},
\end{align*}
and consequently we obtain that the sequence $(F(t,.)b_m)_m$ converges
 uniformly  to $F(t,.)b$ on $[s,t]$, and then, \cite{Die}, we have
$$
\lim_{m \to \infty}\int_s^t F(t,r)b_m(r) dr = \int_s^t F(t,r)b(r) dr.
$$
Then, when $m \to \infty$, we obtain the  equality
$$
x(t) = F(t,s)x(s) + \int_s^t F(t,r)b(r) dr
$$
for all $t \geq s$. This proves that $(b,x) \in \mathcal{G}(T_{ap})$.
Since $T_{ap}$ is closed and since
$\mathcal{D} (T_{ap}) = AP(X)$, by using the Closed Graph
Theorem (Theorem II.1.9 in \cite[p. 45]{Go} ), we deduce that
$T_{ap}$ is bounded.
The reasoning is similar for $T_{apn}$ and $T_{aa}$.
\end{proof}

Now we treat the autonomous case. We need some lemmas.
We consider the following conditions.
\begin{gather}
\text{For all $b \in AP(X)$, \eqref{Lcb} has a unique mild solution
 in $AP(X)$}, \label{eq13} \\
\text{For  all $b \in AP^{(n)}$, \eqref{Lcb} has a unique  mild solution
 in $AP^{(n)}(X)$}, \label{eq14} \\
\text{For all $b \in AA(X)$, \eqref{Lcb} has  a unique mild  solution
  in $AA(X)$}. \label{eq15}
\end{gather}

\begin{lemma}[{\cite[p. 332 ]{Z1}}]\label{lem1}
 Under assumption \eqref{eq9}, if $b : \mathbb{R} \to X$ is bounded
and continuous, if $x_1 : \mathbb{R} \to X$ and
$x_2 : \mathbb{R} \to X$ are  bounded continuous mild solutions
of \eqref{Lcb} then we
have $x_1 = x_2$.
\end{lemma}

\begin{lemma}\label{lem2}
Under assumption \eqref{eq9}, for all $b \in AP^{(n)}(X)$ there
exists a unique mild solution of \eqref{Lcb} in $AP^{(n)}(X)$.
And moreover, for $n \geq 1$, the mild solution is a classical solution.
\end{lemma}

\begin{proof}
The uniqueness is a consequence of Lemma \ref{lem1}.
To prove the existence, we consider the function
$x : \mathbb{R} \to X$ defined by
$x(t) := \int_{- \infty}^t T(t-r)b(r) dr$ for all
$ t \in \mathbb{R}$. First, we note that
$| T(t-r)b(r) - T(t-s)b(s) | \leq | T(t-r)(b(r)
 - b(s)) | + | (T(t-r) -T(t-s))b(s) |
\leq c\cdot e^{- \omega.(t-r)} | b(r) - b(s) |
+ | (T(t-r) -T(t-s))b(s) |$; when
$r \to s$ the first term converges to zero by using the continuity
of $b$, and the second term converges to zero by
using \cite[Corollary 2.3]{Paz}. And so the function
$r \mapsto   T(t-r) b(r)$ is continuous on $(-\infty, t]$,
and consequently it is Lebesgue measurable.

We note that $| T(t-r) b(r) | \leq c.e^{- \omega.(t-r)}
\| b \|_{\infty}$ for all $r \in (-\infty, t]$.
It is well-known that the function
$r \mapsto ce^{- \omega.(t-r)}$ is Lebesgue integrable on $(-\infty, t]$
and consequently $r \mapsto T(t-r)b(r)$ is Lebesgue integrable
on $(-\infty, t]$, see \cite[Proposition 2.4.8]{Mar}.
And so the function $x$ is well defined on $\mathbb{R}$.
By using the change of variable formula,
\cite[Proposition 8.4.10]{Mar}, since $r \mapsto t-r$ is a
$C^1$-diffeomorphism from $(- \infty, t)$ on $(0, \infty)$,
we obtain $x(t) = \int_0^{\infty} T(s) b(t-s) ds$.

Reasoning as at the beginning of this proof we verify that the
function $s \mapsto T(s)b(t-s)$ is continuous on $\mathbb{R}_+$, and
therefore it is Lebesgue measurable on $\mathbb{R}_+$. Since it is
well-known that the function
$s \mapsto ce^{- \omega s} \| b \|_{\infty}$ is Lebesgue integrable
on $\mathbb{R}_+$, and since the inequality
$| T(s) b(t-s) | \leq c\cdot e^{- \omega s} \| b
\|_{\infty}$ holds when $s \in \mathbb{R}_+$, we can use the first
part of \cite[Proposition 2.4.10]{Mar} that permits us to say
that $x$ is continuous on $\mathbb{R}$.

 From the last formula of $x$ it is easy to obtain the inequalities
$| x(t + \tau) - x(t) | \leq \int_0^{\infty} \| T(s)
\|.| b(t+ \tau -s) - b(t-s) | ds
\leq \int_0^{\infty} c\cdot e^{- \omega s} | b(t+ \tau -s)
- b(t-s) | ds $ from which we easily verify that
$x \in AP(X)$ by using the definition of the Bohr almost
periodicity or the Bochner criterion, \cite[p. 4.]{LZ}.
When $b \in AP^{(n)}$ with $n \geq 1$, since $T(s)$ is bounded and
consequently it is differentiable, and so the function
$t \mapsto T(s)b(t-s)$ is of class $C^1$ on $\mathbb{R}$,
and its derivative satisfies the inequality
$| T(s)b'(t-s) |  \leq c e^{- \omega s}
\| b' \|_{\infty}$ where the function
$s \mapsto c e^{- \omega s}\| b' \|_{\infty}$
is Lebesgue integrable on $\mathbb{R}$, that permits us to
use the second part of \cite[Proposition 2.4.10]{Mar}, and then
to say that the function $x$ is differentiable on $\mathbb{R}$,
and that its derivative is $x'(t) = \int_0^{\infty}T(s)b'(t-s)ds$
for all $t \in \mathbb{R}$.  From the inequality
$| x'(t + \tau) - x'(t) |
\leq \int_0^{\infty} c e^{- \omega s} | b'(t+ \tau -s)
- b'(t-s) | ds $ it is easy to see that $x' \in AP(X)$ when
$b' \in AP(X)$. Iterating this reasoning we obtain that
$x^{(k)} \in AP(X)$ when $b^{(k)}  \in AP(X)$ for all
$k=1,\dots ,n$. And so we obtain $x \in AP^{(n)}(X)$ when
$b \in AP^{(n)}(X)$.

To verify that $x$ is a mild solution of \eqref{Lcb},
the reasoning is similar to this one given in \cite{Z1}.
To prove that the mild solution is a classical solution
when $n \geq 1$, it remains to prove that
$x(t) \in \mathcal{D}(A)$ and $x$ satisfies \eqref{Lcb}
when $t \in \mathbb{R}$. Recall that the mild solution $x$
of \eqref{Lcb} is given by $x(t) = \int_{-\infty}^t T(t-r)b(r) dr$.
It is easy to verify the following equality, for $h > 0$:
\begin{equation} \label{eq116}
\frac{T(h)x(t) - x(t)}{h} = \frac{x(t+h) - x(t)}{h} - \frac {1}{h}\int_t^{t+h} T(t+h-r)b(r) dr.
\end{equation}
 From the continuity of $b$ it is clear that the second term
of the right-hand of \eqref{eq116} has the limit $b(t)$ when
$h \to 0$. Since $x$ is differentiable on $\mathbb{R}$,
it follows from \eqref{eq116} that $x(t) \in \mathcal{D}(A)$
and $Ax(t) = x'(t) - b(t)$ for all $t \in \mathbb{R}$;
consequently $x$ is a classical solution of \eqref{Lcb}.
\end{proof}


\begin{remark}\label{rem1} \rm
The proof of Lemma \ref{lem2} is an extension at the cases
$n \geq 1$ of the proof of a theorem in \cite{Z1} done when $n = 0$.
This proof in \cite{Z1} is itself an extension of the proof of
the Neugebauer-Bohr theorem, for the finite-dimensional systems,
for instance given in \cite{Ros} p. 206-207. In \cite{Z1}
or in \cite{Ros}, the authors use the Riemann improper integral;
in the previous proof we only use the Lebesgue integral.
\end{remark}

\begin{remark}\label{rem35} \rm
When $A(t)=A$ is constant with respect to $t$, the
condition \eqref{eq7} is reduced to the following condition:
\begin{equation} \label{eq37}
\begin{gathered}
\exists \lambda_0 \in \mathbb{R}_+, \exists \theta
\in ( \frac{\pi}{2}, \pi), \exists K \in \mathbb{R}_+, \\
\Sigma_{\theta}  \cup \{ 0 \} \subset \rho (A - \lambda_0 I),\\
\forall \lambda \in \Sigma_{\theta}, \| R(\lambda + \lambda_0; A) \| \leq \frac{K}{1 + | \lambda | }.
\end{gathered}
\end{equation}
If an infinitesimal generator of a $C_0$-semigroup $(T(t))_{t \geq 0}$
 satisfies this last condition, then $(T(t))_{t \geq 0}$ is
differentiable (and even it can be extended to an
analytic semigroup), see \cite[Theorem 5.2]{Paz};
 therefore condition \eqref{eq37} is not a consequence of
 \eqref{eq9} and it is not necessary to obtain the conclusion
of Lemma \ref{lem2}.
\end{remark}

\begin{lemma}[{\cite[Theorem 2.17]{N}, \cite[Theorem 3.1]{Nsf}}]
\label{lem34}
 Under assumption \eqref{eq9}, for all $b \in AA(X)$ there exists
a unique mild solution of \eqref{Lcb}.
\end{lemma}

\begin{theorem}\label{th2}
Under assumption \eqref{eq9} we can define the operators
$T^c_{ap}$, $T^c_{apn}$, and $T^c_{aa}$ as follows:
for all $b \in AP(X)$ (respectively $AP^{(n)}(X)$,
 respectively $AA(X)$) $T^c_{ap}(b)$ (respectively $T^c_{apn}(b)$,
respectively $T^c_{aa}(b)$) is the unique mild solution
of \eqref{Lcb} in $AP(X)$ (respectively $AP^{(n)}(X)$,
respectively $AA(X)$). Then $T^c_{ap} : AP(X) \to AP(X)$,
$T^c_{apn} : AP^{(n)}(X) \to AP^{(n)}(X)$ and
$T^c_{aa} : AA(X) \to AA(X)$ are linear and bounded.
\end{theorem}

\begin{proof}
Theorem in \cite{Z1} ensures that \eqref{eq9} implies \eqref{eq13}.
Lemma \ref{lem2} ensures that \eqref{eq9} implies \eqref{eq14}.
Lemma \ref{lem34} ensures that \eqref{eq9} implies \eqref{eq15}.
 And so the three operators $T^c_{ap}$, $T^c_{apn}$, and $T^c_{aa}$
are well defined. The rest of the proof is similar to this one
of Theorem \ref{th1}.
\end{proof}


\subsection*{Notation}
$\| T_{ap} \|_\mathcal{L}$
(respectively $\| T_{apn} \|_\mathcal{L}$, respectively
$\| T_{aa} \|_\mathcal{L}$, respectively
$\| T^c_{ap} \|_\mathcal{L}$, respectively
$\| T^c_{apn} \|_\mathcal{L}$, respectively
$\| T^c_{aa} \|_\mathcal{L}$) denotes the norm of the
linear bounded operator $T_{ap}$ (respectively $T_{apn}$,
respectively $T_{aa}$, respectively $T^c_{ap}$,
respectively $T^c_{apn}$, respectively $T^c_{aa}$).

\section{The continuous dependence}

\subsection{Solutions of equations \eqref{Eu} and \eqref{Gzeta}}
First we formulate the following conditions:
\begin{equation} \label{eq16}
\begin{gathered}
\exists c_{ap} \in (0, \| T_{ap} \|_{\mathcal{L}^{-1})},
\forall t \in \mathbb{R}, \forall \xi, \xi_1 \in X, \forall \zeta \in U,\\
| f(\xi, \zeta,t) - f(\xi_1, \zeta,t) | \leq  c_{ap} | \xi - \xi_1 |.
\end{gathered}
\end{equation}
%
\begin{equation} \label{eq17}
\begin{gathered}
\exists c_{apn} \in (0, \| T_{apn} \|_{\mathcal{L}^{-1})},
\forall t \in \mathbb{R}, \forall \xi, \xi_1 \in X, \forall \zeta \in U,\\
| f(\xi, \zeta,t) - f(\xi_1, \zeta,t) | \leq  c_{apn} | \xi - \xi_1 |.
\end{gathered}
\end{equation}
%
\begin{equation} \label{eq18}
\begin{gathered}
\exists c_{aa} \in (0, \| T_{aa} \|_{\mathcal{L}^{-1})},
\forall t \in \mathbb{R}, \forall \xi, \xi_1 \in X, \forall \zeta \in U,\\
| f(\xi, \zeta,t) - f(\xi_1, \zeta,t) | \leq  c_{aa} | \xi - \xi_1 |.
\end{gathered}
\end{equation}
We recall the following parametrized fixed point theorem.

\begin{theorem}[{\cite[Th\'eor\`eme 46-bis ]{Sc}}] \label{th41}
Let $(Z,d)$ be a complete metric space and let $W$ be a topological
space. Let $\Phi : Z \times W \to Z$ be a mapping such that the
partial mappings $w \mapsto \Phi(z,w)$ are continuous for all
$z \in Z$, and such that there exists $c \in [0,1)$ satisfying
$d(\Phi(z,w), \Phi(z_1,w)) \leq c. d(z,z_1)$ for all $z,z_1 \in Z$
 and for all $w \in W$. Then, for all $w \in W$, there exists a
unique $\underline{z}(w) \in Z$ such that
$\Phi(\underline{z}(w), w) = \underline{z}(w)$, and moreover
the mapping $w \mapsto \underline{z}(w)$ is continuous from $W$
into $Z$.
\end{theorem}

Now we state the result on the continuous dependence for
\eqref{Eu}.

\begin{theorem}\label{th42}
 Under assumptions \eqref{eq1}, \eqref{eq7} and \eqref{eq16}
(respectively \eqref{eq3}, \eqref{eq7} and \eqref{eq17},
respectively \eqref{eq5}, \eqref{eq8} and \eqref{eq18}),
for all $u \in AP(U)$ (respectively $AP^{(n)}(U)$, respectively $AA(U)$)
there exists a unique $\underline{x}(u) \in AP(X)$
(respectively $AP^{(n)}(X)$, respectively $AA(X)$) which is a
mild solution of \eqref{Eu}. Moreover the mapping
$u \mapsto \underline{x}(u)$ is continuous from $AP(U)$
(respectively $AP^{(n)}(U)$, respectively $AA(U)$)
into $AP(X)$ (respectively $AP^{(n)}(X)$, respectively $AA(X)$).
\end{theorem}

\begin{proof}
First we treat the case of the continuous almost periodic functions.
When $u \in AP(U)$, note that $x \in AP(X)$ is a mild solution
of \eqref{Eu} if and only if we have
$x = T_{ap} \circ \mathcal{N}_f(x,u)$, where
$\mathcal{N}_f : AP(X) \times AP(U) \to AP(X)$ is the superposition
operator (or the Nemytskii operator) built on $f$; i.e.,
 $\mathcal{N}_f (x,u) := [ t \mapsto f(x(t), u(t),t)]$.
By using \cite[Lemma 3.4]{BCNP} we know that $\mathcal{N}_f $
is well defined.  From \eqref{eq16} it is easy to verify that
we have $\| \mathcal{N}_f(x,u) - \mathcal{N}_f(x_1,u)
\|_{\infty} \leq c_{ap} \| x - x_1 \|_{\infty}$ for all
$x,x_1 \in AP(X)$ and for all $u \in AP(U)$.

We set $\Phi_{ap} := T_{ap} \circ \mathcal{N}_f :AP(X)\times
AP(U) \to AP(X)$. For all $x,x_1 \in AP(X)$ and for all
$u \in AP(U)$, we have
$$
\|  \Phi_{ap}(x,u) - \Phi_{ap}(x_1,u) \|_{\infty}
\leq \| T_{ap} \|_\mathcal{L} c_{ap}\| x-x_1 \|_{\infty}
= d_{ap}\| x-x_1 \|_{\infty},
$$
where $d_{ap} := \| T_{ap} \|_\mathcal{L} c_{ap} \in [0,1)$.
Moreover, by using Theorem 3.5 in \cite{BCNP} we know that
$\mathcal{N}_f$ is continuous and consequently the partial
mapping $u \mapsto  \Phi_{ap}(x,u)$ is continuous
(as a composition of continuous mappings) on $AP(U)$ for all
$x \in AP(X)$.
And so we can use Theorem \ref{th41} and we obtain the announced
result for the continuous almost periodic case.
For the mild solution in $AP^{(n)}(X)$ (respectively $AA(X)$),
the reasoning is similar by using Theorem 7.2
(respectively \cite[Lemma 9.4 and Theorem 9.6 ]{BCNP})
instead of \cite[Lemma 3.4 and Theorem 3.5]{BCNP}.
\end{proof}

Now we establish the theorem on the continuous dependence for
\eqref{Gzeta}.

\begin{theorem}\label{th43}
 Under assumptions \eqref{eq1}, \eqref{eq7} and \eqref{eq16}
(respectively \eqref{eq3}, \eqref{eq7} and \eqref{eq17},
respectively \eqref{eq5}, \eqref{eq8} and \eqref{eq18}),
for all $\zeta \in U$ there exists a unique
$\underline{x}(\zeta) \in AP(X)$ (respectively $AP^{(n)}(X)$,
respectively $AA(X)$) which is a mild solution of \eqref{Gzeta}.
Moreover the mapping $\zeta \mapsto \underline{x}(\zeta)$
is continuous from $U$ into $AP(X)$ (respectively $AP^{(n)}(X)$,
respectively $AA(X)$).
\end{theorem}

\begin{proof}
Let $\phi$ be the operator from $AP(U)$ into $AP(X)$ defined as
follows: $\phi(u)$ is the unique mild solution of \eqref{Eu} in
$AP(X)$ provided by Theorem \ref{th42}. By using Theorem
\ref{th42} we obtain that $\phi$ is well defined and continuous.
We consider $U$ as the Banach subspace of the constant functions
in $AP(U)$. And so we define the operator $\psi : U \to
AP(X)$ as the restriction of $\phi$ at $U$. Then $\psi(\zeta)$ is
the unique mild solution of \eqref{Gzeta} in $AP(X)$ and $\psi$ is
continuous. The reasoning is similar for the other cases.
\end{proof}

\subsection{Solutions of equations \eqref{Ecu} and
\eqref{Gczeta}}
When $A(t) =A$ does not depend on $t$, we consider the following
conditions.
\begin{equation} \label{eq19}
\begin{gathered}
\exists c^1_{ap} \in (0, \| T^c_{ap} \|_{\mathcal{L}^{-1})}, \forall t \in \mathbb{R}, \forall \xi, \xi_1 \in X, \forall \zeta \in U,\\
| f(\xi, \zeta,t) - f(\xi_1, \zeta,t) | \leq  c^1_{ap} | \xi - \xi_1 |.
\end{gathered}
\end{equation}
%
\begin{equation} \label{eq20}
\begin{gathered}
\exists c^1_{apn} \in (0, \| T^c_{apn} \|_{\mathcal{L}^{-1})}, \forall t \in \mathbb{R}, \forall \xi, \xi_1 \in X, \forall \zeta \in U,\\
| f(\xi, \zeta,t) - f(\xi_1, \zeta,t) | \leq  c^1_{apn} | \xi - \xi_1 |.
\end{gathered}
\end{equation}
%
\begin{equation} \label{eq21}
\begin{gathered}
\exists c^1_{aa} \in (0, \| T^c_{aa} \|_{\mathcal{L}^{-1})}, \forall t \in \mathbb{R}, \forall \xi, \xi_1 \in X, \forall \zeta \in U,\\
| f(\xi, \zeta,t) - f(\xi_1, \zeta,t) | \leq  c^1_{aa} | \xi - \xi_1 |.
\end{gathered}
\end{equation}

\begin{theorem}\label{th44}
We assume \eqref{eq9} fulfilled. Under \eqref{eq1} and
\eqref{eq19} (respectively \eqref{eq3} and \eqref{eq20},
respectively \eqref{eq5} and \eqref{eq21}), for all $u \in AP(U)$
(respectively $AP^{(n)}(U)$, respectively $AA(U)$) there exists a
unique $\underline{x}(u) \in AP(X)$ (respectively $AP^{(n)}(X)$,
respectively $AA(X)$) which is a mild solution of \eqref{Ecu}.
Moreover the mapping $u \mapsto \underline{x}(u)$ is continuous
from $AP(U)$ (respectively $AP^{(n)}(U)$, respectively $AA(U)$)
into $AP(X)$ (respectively $AP^{(n)}(X)$, respectively $AA(X)$).
Moreover, for $n \geq 1$, the mild solution $\underline{x}(u) \in
AP^{(n)}(X)$ is a classical solution.
\end{theorem}

\begin{proof}
For the mild solution the proof is similar to this one of
Theorem \ref{th42}. Remark that $\underline{x}(u)$ is a mild
solution of \eqref{Lcb} with
$b(t) := f(\underline{x}(u)(t), u(t),t)$. If $f$ satisfies
\eqref{eq3} and if $u \in AP^{(n)}(X)$, then we have
$\underline{x}(u) \in AP^{(n)}(X)$ and by using
\cite[Theorem 7.2]{BCNP}, we obtain that $b \in AP^{(n)}(X)$.
In this case, by help of Lemma \ref{lem2}, we deduce that
$\underline{x}(u)$ is a classical solution.
\end{proof}

\begin{theorem}\label{th45}
We assume \eqref{eq9} fulfilled. Under \eqref{eq1} and \eqref{eq19}
(respectively \eqref{eq3} and \eqref{eq20},
 respectively \eqref{eq5} and \eqref{eq21}), for all $\zeta \in U$
there exists a unique $\underline{x}(\zeta) \in AP(X)$
(respectively $AP^{(n)}(X)$, respectively $AA(X)$) which
is a mild solution of \eqref{Gczeta}. Moreover the mapping
$\zeta \mapsto \underline{x}(\zeta)$ is continuous from $U$
into $AP(X)$ (respectively $AP^{(n)}(X)$, respectively $AA(X)$).
Moreover, for $n \geq 1$, the mild solution
$\underline{x}(u) \in AP^{(n)}(X)$ is a classical solution.
\end{theorem}

The proof of Theorem \ref{th45} is similar to the proof of
Theorem \ref{th43} and it is omitted.

\section{The differentiable dependence}

\subsection{Solutions of equations \eqref{Eu} and \eqref{Gzeta}}
In this subsection, first we provide conditions to ensure the
 differentiability of the dependence of the solution $x$ with
respect to $u$ for \eqref{Eu}.

\begin{theorem}\label{th51}
Under assumption \eqref{eq2} and \eqref{eq7}
(respectively \eqref{eq4} and \eqref{eq7}, respectively \eqref{eq6}
and \eqref{eq8}) we assume that there exist $u_0 \in AP(U)$
(respectively $AP^{(n)}(U)$,respectively $AA(U)$) and
$x_0 \in AP(X)$ (respectively $AP^{(n)}(X)$, respectively
$AA(X)$) which is a mild solution of $(E,u_0)$.
We also assume that the following inequality hold:
\begin{gather*}
\sup_{t \in \mathbb{R}} \| D_1f(x_0(t), u_0(t),t) \|
 < \| T_{ap} \|_{\mathcal{L}^{-1}}\\
\text{(respectively }
\sup_{t \in \mathbb{R}} \| D_1f(x_0(t), u_0(t),t) \|
 < \| T_{apn} \|_{\mathcal{L}^{-1}},\\
\text{respectively }
\sup_{t \in \mathbb{R}} \| D_1f(x_0(t), u_0(t),t) \|
 < \| T_{aa} \|_{\mathcal{L}^{-1})}.
\end{gather*}
Then there exist an open neighborhood $\mathcal{U}$ of $u_0$ in
$AP(U)$ (respectively $AP^{(n)}(U)$, respectively $AA(U)$), an
open neighborhood $\mathcal{X}$ of $x_0$ in $AP(X)$ (respectively
$AP^{(n)}(X)$, respectively $AA(X)$), and a $C^1$-mapping $u
\mapsto \underline{x}(u)$ from  $\mathcal{U}$ into $\mathcal{X}$
such that, for all $u \in \mathcal{U}$, $\underline{x}(u)$ is a
mild solution of \eqref{Eu}. Moreover $\underline{x}(u)$ is the
unique mild solution of \eqref{Eu} in $\mathcal{X}$; notably we
have $\underline{x}(u_0) = x_0$.
\end{theorem}

\begin{proof}
We do the proof for the almost periodic case. The proofs of the
other cases are similar.
In the proof of Theorem \ref{th2}, we have seen that, when
$u \in AP(U)$, $x \in AP(X)$ is a mild solution of \eqref{Eu}
if and only if we have $x = T_{ap} \circ \mathcal{N}_f(x,u)$.
We denote by $\pi_1 : AP(X) \times AP(U) \to AP(X)$ the first projection,
 $\pi_1(x,u) := x$. Clearly $\pi_1$ is a linear bounded operator.

We introduce the nonlinear operator
$\Psi_{ap} : AP(X) \times AP(U) \to AP(X)$ by setting
$\Psi_{ap}(x,u) := \pi_{1}(x,u) - T_{ap} \circ \mathcal{N}_f(x,u)$.
And so, $x \in AP(X)$ is a mild solution of \eqref{Eu} if and only
if we have $\Psi_{ap}(x,u) = 0$.
By using \eqref{eq2} and \cite[Theorem 5.1]{BCNP}, we know that
$\mathcal{N}_f$ is of class $C^1$ from $AP(X) \times AP(U)$ into
$AP(X)$. Since $T_{ap}$ and $\pi_1$ are linear bounded, they are
of class $C^1$. Consequently $\Psi_{ap}$ is of class $C^1$
as a composition of operators of class $C^1$.
Since $x_0$ is a mild solution of $(E,u_0)$ we have
$\Psi_{ap}(x_0, u_0) = 0$.
By using the chain rule, the partial differential of $\Psi_{ap}$
with respect to the first variable at $(x_0,u_0)$ is
$D_x \Psi_{ap}(x_0,u_0) = I - T_{ap} \circ D_x \mathcal{N}_f(x_0,u_0)$
where $I$ is the identity operator of $\mathcal{L}(AP(X), AP(X))$.
After Theorem 5.1 in \cite{BCNP} we know that, for all
$h \in AP(X)$,
$D_x \mathcal{N}_f(x_0,u_0).h = [t \mapsto D_1f(x_0(t), u_0(t),t).h(t)]$,
and then by using the assumption on $D_1f$ we obtain
$$
\| D_x \mathcal{N}_f(x_0,u_0) \|_\mathcal{L}
\leq \sup_{ t \in \mathbb{R}} \| D_1f(x_0(t), u_0(t),t) \|
< \| T_{ap} \|_{\mathcal{L}^{-1}}.
$$
Consequently we have $\| T_{ap} \circ D_x \mathcal{N}_f (x_0,u_0)
\| < 1$. Then by using a classical argument on the Neumann series
(Proof of \cite[Lemma 2.5.4]{AMR},  or  \cite[Th\'eor\`eme 1.7.2]{Ca})
we know that $I-  T_{ap} \circ D_x \mathcal{N}_f (x_0,u_0)$
is invertible. 
And so we can use the implicit function theorem
(\cite[Th\'eor\`eme 4.7.1]{Ca}, or \cite[Th\'eor\`eme 2.5.7]{AMR}) and we can
assert that there exist a neighborhood $\mathcal{U}$ of $u_0$
in $AP(U)$, a neighborhood $\mathcal{X}$ of $x_0$ in $AP(X)$
and a $C^1$-mapping $u \mapsto \underline{x}(u)$, from $\mathcal{U}$
into $\mathcal{X}$, such that $\underline{x}(u_0) = x_0$,
and such that
$\{(x,u) \in \mathcal{X} \times \mathcal{U} : \Psi_{ap}(x,u) = 0 \}
= \{ ( \underline{x}(u),u) : u \in \mathcal{U} \}$.
The conclusion of the theorem is just a translation of these properties.
\end{proof}

The following theorem treats the differentiable dependence for the
equations \eqref{Gzeta}.

\begin{theorem}\label{th52}
Under assumptions \eqref{eq2} and \eqref{eq7}
(respectively \eqref{eq4} and \eqref{eq7},
respectively \eqref{eq6} and \eqref{eq8}) we assume that
there exist $\zeta_0 \in U$ , and $x_0 \in AP(X)$
(respectively $AP^{(n)}(X)$, respectively $AA(X)$) which
is a mild solution of \eqref{Gzeta} with $\zeta=\zeta_0$.
We also assume that   the following inequality holds:
\begin{gather*}
\sup_{t \in \mathbb{R}} \| D_1f(x_0(t), \zeta_0,t) \|
 < \| T_{ap} \|_{\mathcal{L}^{-1}}\\
\text{(respectively }
\sup_{t \in \mathbb{R}} \| D_1f(x_0(t), \zeta_0),t) \|
 < \| T_{apn} \|_{\mathcal{L}^{-1}}, \\
\text{respectively }
\sup_{t \in \mathbb{R}} \| D_1f(x_0(t), \zeta_0,t) \|
 < \| T_{aa} \|_{\mathcal{L}^{-1})}.
\end{gather*}
Then there exist an open neighborhood $Z$ of $\zeta_0$ in $AP(U)$,
an open neighborhood $\mathcal{X}$ of $x_0$  in $AP(X)$
(respectively $AP^{(n)}(X)$, respectively $AA(X)$), and a
$C^1$-mapping $\zeta \mapsto \underline{x}(\zeta)$ from
$Z$ into $\mathcal{X}$ such that, for all $\zeta \in Z$,
$\underline{x}(\zeta)$ is a mild solution of \eqref{Gzeta}.
Moreover $\underline{x}(\zeta)$ is the unique mild solution
of \eqref{Gzeta} in $\mathcal{X}$; notably we have
$\underline{x}(\zeta_0) = x_0$.
\end{theorem}

\begin{proof}
Let $\Phi$ be the operator from $\mathcal{U}$ into $\mathcal{X}$
defined as follows: $\Phi (u)$ is the unique mild solution of
\eqref{Eu} in $\mathcal{X} \cap AP(X)$ provided by Theorem
\ref{th51}. By using Theorem \ref{th51} we obtain $\Phi$ is of
class $C^1$. We consider $U$ as the Banach subspace of the
constant functions in $AP(U)$. And so we define the operator $\Psi
: \mathcal{U} \cap U \to \mathcal{X} \cap U$ as the
restriction of $\Phi$ to $U$. Then $\Psi(\zeta)$ is the unique
mild solution of \eqref{Gzeta} in $AP(X)$ and $\Psi$ is of class
$C^1$. The reasoning is similar for the other cases.
\end{proof}

\subsection{Solutions of equations \eqref{Ecu} and \eqref{Gczeta}}
 Now we establish a result of differentiability in the
special case where $A(t) =A$ is constant with respect to $t$; i.e.,
 for the equations \eqref{Ecu}.

\begin{theorem}\label{th53}
We assume \eqref{eq9} fulfilled. Under assumption \eqref{eq2}
(respectively \eqref{eq4}, respectively \eqref{eq6}) we assume that
there exist $u_0 \in AP(U)$ (respectively $AP^{(n)}(U)$,
respectively $AA(U)$), and $x_0 \in
AP(X)$ (respectively $AP^{(n)}(X)$, respectively $AA(X)$)
which is a mild solution of $(E_c,u_0)$. We also assume that
the following inequality holds
\begin{gather*}
\sup_{t \in \mathbb{R}} \| D_1f(x_0(t), u_0(t),t) \|
 < \| T^c_{ap} \|_{\mathcal{L}^{-1}}\\
\text{(respectively }
\sup_{t \in \mathbb{R}} \| D_1f(x_0(t), u_0(t),t) \|
 < \| T^c_{apn} \|_{\mathcal{L}^{-1}},\\
\text{respectively }
\sup_{t \in \mathbb{R}} \| D_1f(x_0(t), u_0(t),t) \|
< \| T^c_{aa} \|_{\mathcal{L}^{-1})}.
\end{gather*}
Then there exist an open neighborhood $\mathcal{U}$ of $u_0$ in
$AP(U)$ (respectively $AP^{(n)}(U)$, respectively $AA(U)$), an
open neighborhood $\mathcal{X}$ in $AP(X)$ (respectively
$AP^{(n)}(X)$, respectively $AA(X)$), and a $C^1$-mapping $u
\mapsto \underline{x}(u)$ from  $\mathcal{U}$ into $\mathcal{X}$
such that, for all $u \in \mathcal{U}$, $\underline{x}(u)$ is a
mild solution of \eqref{Ecu}. Moreover $\underline{x}(u)$ is the
unique mild solution of \eqref{Ecu} in $\mathcal{X}$; notably we
have $\underline{x}(u_0) = x_0$.
\end{theorem}

The proof of Theorem \ref{th53} is similar to the proof of
Theorem \ref{th51} and it is omitted.
One of the main tools used in the proofs of Theorem \ref{th51}
and Theorem \ref{th53} is the implicit function theorem.
The use of the implicit function theorem in a functional
analytic framework was done in \cite{Bl} for periodic solutions
of ordinary differential equations, and in \cite{BCM} for
almost periodic solutions of ordinary differential equations.

The following theorem is a differentiability result for
 \eqref{Gczeta}.

\begin{theorem}\label{th54}
We assume \eqref{eq9} fulfilled. Under \eqref{eq2}
(respectively \eqref{eq4}, respectively \eqref{eq6}) we assume
that there exist $\zeta_0 \in U$ , and $x_0 \in AP(X)$
(respectively $AP^{(n)}(X)$, respectively $AA(X)$) which is
a mild solution of \eqref{Gczeta} with $\zeta=\zeta_0$.
 We also assume that the following inequality holds:
\begin{gather*}
\sup_{t \in \mathbb{R}} \| D_1f(x_0(t), \zeta_0,t) \|
 < \| T^c_{ap} \|_{\mathcal{L}^{-1}}\\
\text{(respectively }
\sup_{t \in \mathbb{R}} \| D_1f(x_0(t), \zeta_0),t) \|
 < \| T^c_{apn} \|_{\mathcal{L}^{-1}},\\
\text{respectively }
\sup_{t \in \mathbb{R}} \| D_1f(x_0(t), \zeta_0,t) \|
 < \| T^c_{aa} \|_{\mathcal{L}^{-1})}.
\end{gather*}
Then there exist an open neighborhood $Z$ of $\zeta_0$ in $AP(U)$,
an open neighborhood $\mathcal{X}$ of $x_0$  in $AP(X)$
(respectively $AP^{(n)}(X)$, respectively $AA(X)$), and
a $C^1$-mapping $\zeta \mapsto \underline{x}(\zeta)$ from
$Z$ into $\mathcal{X}$ such that, for all $\zeta \in Z$,
$\underline{x}(\zeta)$ is a mild solution of \eqref{Gczeta}.
Moreover $\underline{x}(\zeta)$ is the unique mild solution
of \eqref{Gczeta} in $\mathcal{X}$; notably we have
$\underline{x}(\zeta_0) = x_0$.
\end{theorem}

The proof of Theorem \ref{th54} is similar to the proof
of Theorem \ref{th52} and it is omitted.

\begin{remark}\label{rem55} \rm
For reasons similar to these ones used about Theorem \ref{th44},
the mild solution $\underline{x}(u)$ of \eqref{Ecu} (respectively
\eqref{Gczeta}) in $AP^{(n)}(X)$, for $n \geq 1$, provided by
Theorem \ref{th53} (respectively Theorem \ref{th54}) is a
classical solution.
\end{remark}

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\end{document}