\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} \begin{thebibliography}{00} \bibitem{AMR} R. Abraham, J. E. Marsden, and T. Ratiu; \emph{Manifolds, tensor analysis, and applications}, Addison-Wesley Publishing Company, Inc., Reading, Massachussets, 1983. \bibitem{AT} P. Acquistapace and B. Terrini; \emph{A unified approach to abstract linear nonaotonomous parabolic equations}, Rend. Sem. Mat. Univ. Padova \textbf{78}, 1987, 47-107. \bibitem{BBNP} J.-B. Baillon, J. 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