\documentclass[reqno]{amsart}

\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2004(2004), No. 61, pp. 1--15.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu  (login: ftp)}
\thanks{\copyright 2004 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}

\title[\hfilneg EJDE-2004/61\hfil Asymptotically almost periodic solutions]
{Asymptotically almost periodic and almost periodic solutions
for a class of evolution equations}

\author[E. Hern\'{a}ndez, M. L. Pelicer, \& J. P. C. dos Santos
\hfil EJDE-2004/61\hfilneg]
{Eduardo Hern\'{a}ndez M., Mauricio L. Pelicer, \& Jos\'{e}  P. C. dos Santos}
 % in alphabetical order

\address{Departamento de Matem\'atica \\
Instituto de Ci\^encias Matem\'aticas de S\~ao Carlos \\
Universidade de S\~ao Paulo \\
Caixa Postal 668 \\
13560-970 S\~ao Carlos, SP. Brazil} 
\email[E. Hern\'{a}ndez]{lalohm@icmc.sc.usp.br} 
\email[M. L. Pelicer]{mpelicer@icmc.sc.usp.br} 
\email[J. P. C. dos Santos]{zepaulo@icmc.sc.usp.br}

\date{}
\thanks{Submitted February 20, 2004. Published April 21, 2004.}
\subjclass[2000]{34K14, 34K30}
\keywords{Almost periodic, asymptotically almost periodic, \hfill\break\indent
semigroup of linear operators}

\begin{abstract}
  In this paper we  study the   existence of asymptotically almost periodic
  and almost periodic solutions for the partial evolution equation
  $$ \frac{d}{dt} (x(t)+g(t,x(t))=Ax(t)+f(t,Bx(t)),
  $$
  where $A$ is the infinitesimal generator of an analytic semigroup on a
  Banach space $X$, $B$ is a closed  linear operator, and $f$, $g$ are
  given functions.
\end{abstract}

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


\section{Introduction}

The existence of almost periodic solutions for abstract evolution
equation defined on  abstract Banach spaces has been studied in
various works, see for instance \cite{bha1,z1,z2,z3,z6}.  By
using the semigroup theory and the contraction mapping principle,
Zaidman studied in \cite{z2} the existence of almost periodic
solutions for the integral equation associated to the  abstract
partial differential equation
\begin{equation}\label{eq1}
x'(t)= Ax(t) + f(t,x(t)),
\end{equation}
 where $A$ is the infinitesimal generator of a
$C_{0}$-semigroup of bounded linear operators on a Banach space.
 Recently,  Bahaj and Sidki   studied in \cite{bha1} the existence of
  almost periodic solution  for  (\ref{eq1}).

  The purpose of this paper is to discuss the  existence
 of  asymptotically almost periodic  and almost periodic solutions for
 partial evolution equations of the form
\begin{gather}
 \frac{d}{dt} (x(t)+g(t,x(t))=Ax(t)+f(t,Bx(t)), \label{eq2}\\
 x(t_{0})=y_0, \label{eq3}
 \end{gather}
where $A$ is the infinitesimal  generator of an analytic semigroup
of linear operators  defined on a Banach space $X$, $B:D(B)\subset
X\to X$ is a special type of closed operator and
$f,g: I \times X \to X$ are give  functions.

 We remark that the  technical framework used in this work allow us,
 for instance, to study  the partial differential equation
$$\frac{ d}{dt}( u(t,\xi) + g(t,u(t,\xi)))=
\frac{\partial^2u(t,\xi)}{\partial^2\xi} + f(t,\frac{
\partial u(t,\xi)}{\partial\xi})).
$$
 In fact, it's well known, see \cite{R2}, that for a class of operators $A$,
there is a bounded linear operator $L:X\to X $ such   that
$\frac{\partial }{\partial \xi}= (-A)^{1/2}\circ L$. Additionally,
we mention that by using the techniques used in this paper,
it's possible to establish the existence of asymptotically almost periodic
solutions for \eqref{eq2}-\eqref{eq3}
without making  additional regularity assumptions on the initial
data. We refer  to Bridges \cite{bri1} and Rankin \cite{R2} for
complementary remarks  about this matter.
\par  The results in this work are  generalizations of the results in
\cite{bha1,z2} and our  ideas and techniques  can be used in the
study of the existence of asymptotically almost periodic and
almost periodic solutions of partial neutral functional
differential  equations and  partial differential equations of
Sobolev type, see Hernandez \cite{He1} for details. In general,
our results are proved by  using the semigroup theory of bounded
linear operators, the theory of fractional power of closed
operators and the contraction mapping principle.

 This  paper has  four  sections.  In section \ref{sec1} we  study the
existence of asymptotically almost periodic and almost periodic solutions
for the  integral equation associated to
\eqref{def1eq} and in  section \ref{ap} we establish conditions
under which these ``mild'' solutions are classical solutions. In
 section \ref{example} an example is considered.

 \section{ Preliminaries}

 In this section we  mention a few results and establish notation
 needed for stating our results. In this paper, $(X, \| \cdot \|)$ is a Banach
space and   $A:D(A)\subset  X\to X$ is the infinitesimal
 generator of a uniformly  exponentially  stable  analytic semigroup of  linear
 operators $(T(t))_{t\geq 0}$   on  $X$ such that $0 \in  \rho(A)$. Throughout
 this work,   $M,\delta$  are positive  constants
 such that  $\| T(t) \|\leq Me^{-\delta t}$ for every $t\geq 0$.
 Under these conditions it  is possible to define the
 fractional power   $(-A)^\alpha$, $0<\alpha\leq 1$, as a closed
 linear operator on its domain  $D((-A)^\alpha)$. Furthermore,
 $D((-A)^\alpha)$ is  dense in $X$ and the expression
 $\| x\|_{\alpha}= \| (-A)^\alpha x \| $
defines a norm in  $D((-A)^\alpha)$. If $X_{\alpha}$ is the space
$D((-A)^\alpha)$ endowed with the norm $\| \cdot \|_{\alpha}$,
then the following properties hold, see \cite{PA}.

\begin{lemma}\label{an}
 Let $0< \gamma \leq \vartheta \leq 1$.
Then $X_{\vartheta}$ is a Banach space and
$X_{\vartheta}\hookrightarrow X_{\gamma}$. Moreover, the function
$t\to (-A)^{\vartheta} T(t)$ is continuous in the uniform
operator topology on $(0,\infty )$ and there exist constants
$C_{\vartheta}, C'_{\vartheta}$ such that
\begin{equation*}
\| (-A)^{\vartheta} T(t)\| \leq \frac{C_{\vartheta}e^{-\delta
t}}{t^{\vartheta}}\quad \hbox{and}\quad\|
(T(t)-I)(-A)^{-\vartheta} \| \leq C'_{\vartheta}t^\vartheta
\end{equation*}
 for every $t>0$.
\end{lemma}

 Let
  $(Z,\| \cdot \| _{Z})$ and $(W,\| \cdot \| _{W})$ be abstract
Banach spaces. In this work, we indicate
by $\mathcal{L}$ $(Z:W)$ the Banach space of  bounded linear
operator of $Z$ into $W$ and we abbreviate to ${\mathcal{L}}(Z)$
whenever $Z=W$.  The notation  $C(I:Z) $ represents the space of
continuous function from $I$ into $Z$ endowed with the uniform
convergence topology. As usual, $C_{b}([0, \infty):Z)$ is the
space of bounded continuous function from $[0, \infty)$ into $Z$
endowed with the uniform convergence topology and $C_0([0,\infty):Z)$ is
the subspace of $C_b([0, \infty):Z)$ formed by the functions
which vanish at infinity.
 Along  this work,   $B_{r}(x:Z)$, $x\in Z$,  will
  denote  the closed  ball with center at $x$ and radius $r>0$ in
 $Z$. For a bounded and  continuous
function $\xi :(a,b)\to Z$ and $t\in (a,b)$, we will employ the
notation $\| \xi \|_{a,t,Z}$ for
\begin{equation}\label{not1}
 \| \xi\| _{a,t,Z}
 =\sup\{ \| \xi(s)\|_{Z}: s \in (a,t]\},
\end{equation}
and we will write simply $\| \xi\|_{t,Z}$  when non confusion
arise.

 We remark that a  function $f:[a,b] \to Z$ is
 $\sigma$-H\"{o}lder continuous,
 $0<\sigma\leq 1$, if
 there is a  constant $\kappa >0$ such that
\begin{equation*}
\| f(s)-f(t)\| \leq \kappa | t-s |^{\sigma},\quad
 s,t \in [a,b].
\end{equation*}
 We represent by $C^{\sigma}([a,b]; Z)$ the space of
$\sigma$-H\"{o}lder continuous function from $[a,b]$
into $Z$ endowed with the uniform convergence topology. The
notation $C^{\sigma}((a,b]; Z)$ stands for the space  of
continuous function $f:[a,b]\to Z$ such that $f\in
C^{\sigma}([\delta,b]; Z)$ for every $\delta >a$.

 Next we make some remarks concerning   almost periodic
and asymptotically almost periodic functions.

\begin{definition} \label{def2.2} \rm
A continuous function $f: \mathbb{R} \to Z$ is called
almost periodic if for every \,$ \epsilon > 0$ there exists a
relatively dense subset of \,$\mathbb{R}$, denoted by
$\mathcal{H}( \epsilon, f,Z)$, such that
$$ \| f(t+ \xi ) - f(t) \|_{Z} < \epsilon, $$
for every \,$t \in \mathbb{R}$ and every  $ \xi \in \mathcal{H}(
\epsilon, f,Z)$.
\end{definition}

\begin{definition} \label{def2.3} \rm
A continuous function $f: [0, \infty) \to Z $ is the
called asymptotically almost periodic if there exists an almost
periodic function $g( \cdot ):\mathbb{R}\to Z$  and a
function $w( \cdot ) \in C_0([0, \infty):Z)$ such that $f(t) =
g(t) + w(t)$ for every $t \geq 0$.
\end{definition}

  In this paper, $AP(Z)$  and $AAP(Z)$ are the spaces
\begin{gather*}
AP(Z)= \{u\in C_{b}(\mathbb{R}:Z):u
 \hbox{ is almost periodic }\},\\
AAP(Z)=\{u\in C_{b}([0,\infty):Z):u
\hbox{ is asymptotically almost periodic } \},
\end{gather*}
provided with the norm of the uniform convergence. It' s well
known that $AP(Z)$ and $AAP(Z)$ are Banach spaces, see \cite{z5}.

\begin{lemma}[{Characterization of asymptotically almost periodic  function
\cite[Theorem 5]{z5}}] \label{lema2}
Let $F([0,\infty):Z) $ be the  subspace of $ C_{b}([0,\infty):Z)$
 formed by  the functions $f(\cdot)$ which satisfy the following
property: for every $ \epsilon > 0$ there exists
$L( \epsilon,f,Z)> 0$ and a relatively dense subset of  $ [0,\infty) $,
   denoted by $\mathcal{T}(\epsilon, f ,Z) $,  such that
$$ \| f(t+ \xi ) - f(t) \|_Z  < \epsilon,
$$
for every $t \geq L( \epsilon,f,Z)$ and every  $\xi \in
\mathcal{T}( \epsilon,f,Z)$. Then,  $F( [0,\infty):Z) = AAP(Z)$.
\end{lemma}

The next definitions  and properties  are essential for
establishing  our results.

\begin{definition} \label{def2.5}\rm
Let $ \Omega \subset W$ be a  open set and $F: \mathbb{R} \times
\Omega \to Z$ be a continuous function.
\begin{enumerate}
  \item  $F$ is called  pointwise almost periodic ( pointwise a.p.),
  if   $ F(\cdot,x)\in AP(Z)$  for every
$x\in \Omega$.
  \item  $F$ is called uniformly almost periodic (u.a.p.), if for every $\epsilon > 0 $ and every compact $K \subset
\Omega$ there exists a relatively dense subset of $\mathbb{R}$,
denoted by $\mathcal{H}(\epsilon,F, K,Z)$,  such that $$\|F( t +
\xi, y) - F(t,y) \|_{Z} \leq \epsilon,$$ for every $(t, \xi,y) \in
\mathbb{R}\times \mathcal{H}(\epsilon, F,K,Z) \times  K$.
\end{enumerate}
\end{definition}

\begin{definition} \label{def2.6}\rm
Let $ \Omega \subset W$ be a  open set and $F: [0,\infty ) \times
\Omega \to Z$ be a continuous function.
\begin{enumerate}
    \item  $F$ is called  pointwise asymptotically almost periodic
(pointwise a.a.p.), if  $ F(\cdot,x)\in AAP(Z)$ for every $x\in\Omega$.
  \item  $F$ is called uniformly asymptotically almost periodic  (u.a.a.p.),
  if for every $\epsilon > 0 $ and every compact
$K \subset \Omega$ there exists a relatively dense subset of
$[0,\infty)$, denoted  by $\mathcal{T}(\epsilon,F, K,Z) $, and
$L(\epsilon,F, K,Z)> 0$ such that
$$\|F( t + \xi, y) - F(t,y)\|_{Z} \leq \epsilon,
$$
for every $t\geq L(\epsilon,F, K,Z) $ and
every $ (\xi,y) \in \mathcal{T}(F,\epsilon, K,Z)\times K$.
\end{enumerate}
\end{definition}

For details concerning the  next two lemmas, see \cite[Theorem 1.2.7]{Yos1} and
\cite{z2}.

\begin{lemma}\label{lx}
Let  $ \Omega \subset W$ be a open  set and $F: \mathbb{R}
\times \Omega \to Z$ be a continuous function. Then the
following properties hold.
\begin{enumerate}
 \item  If $F$ is pointwise a.p. and
satisfies a local Lipschitz condition at $ x\in \Omega$, uniformly
at $t$, then  $F$ is u.a.p.
 \item If $F$ is u.a.p. and $y\in
AP(W)$ is such that $\overline{\{ y(t): t \in \mathbb{R}\}}^W
\subset\Omega $, then $ F(t, y(t)) \in AP(Z)$.
\end{enumerate}
\end{lemma}

\begin{lemma}\label{28}
Let  $ \Omega \subset W$ be a open  set and $F:[0,\infty)\times
\Omega \to Z$ be a continuous function. Then the following
properties hold.
\begin{enumerate}
 \item If $F$ is pointwise a.a.p. and
satisfies a local Lipschitz condition at $x\in\Omega$, uniformly
at $t$, then $F$ is u.a.a.p.
\item If $F$ is u.a.a.p. and $y \in AAP(W)$ is such that
$\overline{ \{ y(t): t \in [0, \infty)\}}^W
\subset \Omega$, then $F(t,y(t))\in AAP(Z)$.
\end{enumerate}
\end{lemma}

 Throughout this paper, $0<\alpha, \beta\leq 1$ are fixed numbers and
 $(Y,\| \cdot\|_{Y}) $ is  a Banach space such that
$X_{\eta}\hookrightarrow Y \hookrightarrow X$ for every
$\eta\in (0,1)$. To obtain our results we will use  the following technical
conditions.
\begin{enumerate}
\item[(H1)]  The  function $ s\to T(s)y\in C([0,\infty);Y)$ for  every
$y\in Y $ and there are
$\widetilde{M}>0, \widetilde{\delta}>0$ such that $ \|
T(s)\|_{\mathcal{L}(Y)}\leq \widetilde{M}e^{-\tilde{\delta} s}$
for every $s\geq 0$. Moreover, the functions $s\to
(-A)^{1-\beta}T(s)$, $s\to (-A)^{\alpha} T(s)$ defined from  $
(0,\infty)$  into $ \mathcal{L}(X,Y)$ are strongly measurable and
there are  non-decreasing functions $H_{\beta},H_{\alpha}$ and
   numbers $\omega_{i}<0$, $i=1,2,$ such that
    $ e^{\omega_{1}s}H_{\beta}(s)\in  \mathrm{L}^{1}([0,\infty))$,
     $e^{\omega_{2}s}H_{\alpha}(s) \in \mathrm{L}^{1}([0,\infty))$
     and
\begin{gather*}
\| (-A)^{1-\beta} T(s)\|_{\mathcal{L}(X:Y)} \leq
e^{\omega_{1}s}H_{\beta}(s),\quad s>0,\\
 \|(-A)^{\alpha}
T(s)\|_{\mathcal{L}(X:Y)} \leq
e^{\omega_{2}s}H_{\alpha}(s),\quad s>0.
\end{gather*}

\item[(H2)] The function $g(\cdot)$ is
$X_{\beta}$-valued, $(-A)^{\beta}g:\mathbb{R}\times Y\to X$ is
continuous, $ (-A)^{\beta}g(s,0)=0$ for every $s\geq 0 $  and
there is a
 continuous  function $L_{g}:[0,\infty)\to (0,\infty)$ such that
  $L_{g}(0)=0$ and
\begin{equation*}
 \|(-A)^{\beta}g(t_1,y_1)-(-A)^{\beta}g(t_2,y_2)\|
 \leq L_{g}(r) (|t_1 - t_2| +\| y_1-y_2\|_{Y} ),
\end{equation*}
for every $(t_i, y_i)\in\mathbb{R}\times B_{r}(0, Y)$.


\item[(H3)] The map $B:D(B)\subset X\to X$ is a closed
linear operator such that  $D((-A)^{\alpha})\subset D(B)$ and
there are continuous functions $\tilde{f}:\mathbb{R}\times
Y\to X$, $L_{\tilde{f}}:[0, \infty) \to [0,\infty)$ such that
$ L_{\tilde{f}}(0)=0$, $\tilde{f}(s,0)=0$ for
every $s\geq 0$,  $\tilde{f}(\mathbb{R}\times X_{\alpha}) \subset
X_{\alpha}$, $(-A)^{\alpha}\tilde{f}(t,x)= f(t,Bx)$ for every
$(t,x) \in \mathbb{R} \times X_{\alpha}$ and
\begin{equation*}
\|\tilde{f}(t_1,y_1) - \tilde{f}(t_2,y_2)\|  \leq
L_{\tilde{f}}(r)( |t_1 - t_2| + \|y_1 - y_2\|_Y),
\end{equation*}
 when  $(t_i, y_i)\in\mathbb{R}\times B_{r}(0, Y)$.
\end{enumerate}

\begin{remark} \label{rmk2.9} \rm
 For examples of semigroups of linear operators and functions
verifying the previous assumption, see Bridges \cite{bri1}, Hagen
\& Turi \cite{Ha1} and  Rankin \cite{R2}.
\end{remark}

 Following Hernandez \cite{He1} and Rankin
 \cite{R2}  we introduce  the next concepts.

\begin{definition}\label{ymild} \rm
A function $u\in C([t_{0},r):Y)$ is a $Y$-mild solution of the
abstract Cauchy problem \eqref{eq2}-\eqref{eq3} if $u(t_{0})=y_0$;
the functions $s\to AT(t-s)g(s,u(s))$,
$s\to(-A)^{\alpha}T(t-s)\tilde{f}(s,u(s))$ belong  to
$\mathrm{L}^{1}([t_{0},t]:Y) $  for every $t_{0}\leq t < r$  and
\begin{equation}\label{def1eq}
\begin{aligned}
u(t)&=T(t-t_{0})(y_0+g(t_{0},y_0))
-g(t,u(t))-\int_{t_{0}}^{t}AT(t-s)g(s,u(s))ds \\
&\quad + \int_{t_{0}}^{t}(-A)^{\alpha}T(t-s)\tilde{f}(s,u(s))ds,\quad t\in
[t_{0},r].
\end{aligned}
\end{equation}
\end{definition}

\begin{definition} \label{def2.11} \rm
A function $u\in C([t_{0},r):X)$ is a  mild solution of
\eqref{eq2}-\eqref{eq3} if $u(t_{0})=y_0$; $u\in
C((t_{0},r):X_{\alpha}) $; the function $ s \to AT(t-s)g
(s,u(s))$ belongs to $ \mathrm{L}^{1}([t_{0},t]:X)$ for every $t
\in [t_{0},r)$ and
\begin{align*}
u(t)  &= T(t-t_{0})(y_0+g(t_{0},y_0))- g(t,u(t)) -
\int^t_{t_{0}}AT(t-s)g(s,u(s))ds  \\
&\quad + \int^t_{t_{0}}T(t-s)f(s,Bu(s))ds, \quad t \in
[t_{0},r).
\end{align*}
\end{definition}

 The next definition has been  introduced in Hernandez  \cite{He1}.

\begin{definition} \label{def2.12} \rm
A function $u\in C([t_{0},r]:X)$ is an S-classical
(Semi-classical) solution of \eqref{eq2}-\eqref{eq3} if
$u(t_{0})=y_0$, $\frac{d}{dt}(u(t)+g(t,u(t)))$ is continuous on
$(t_{0},r)$, $u(t)\in D(A)$  for all $t\in (t_{0},r]$
 and $u(\cdot)$ satisfies \eqref{eq2}-\eqref{eq3} on  $(t_{0},r)$.
\end{definition}

In relation to asymptotically almost periodic and almost periodic solutions
we introduce the following definitions.

\begin{definition} \label{def2.13} \rm
A function  $u \in AP(Y)$ is  an  almost periodic $Y$-mild solution of
\eqref{eq2}-\eqref{eq3} if the functions  $ s \to
AT(t-s)g(s,u(s))$, $s\to (-A)^{\alpha}T(t-s)\tilde{f}(s,u(s))$ belong to
 $ \mathrm{L}^{1}((-\infty,t]:Y)$ for every $t \in \mathbb{R}$ and
$$
u(t) = - g(t,u(t))-\int^t_{- \infty}AT(t-s)g(s,u(s))ds + \int^t_{-
\infty}(-A)^{\alpha}T(t-s)\tilde{f}(s,u(s))ds,
$$
for every $t \in \mathbb{R}$.
\end{definition}

\begin{definition} \rm
A function  $u \in AP(X)$ is an  almost periodic mild solution of
\eqref{eq2}-\eqref{eq3} if $u\in C(\mathbb{R}:X_{\alpha})$, the
function $ s \to AT(t-s)g(s,u(s))$ belongs to $
\mathrm{L}^{1}((- \infty,t]:X)$ for every $t \in \mathbb{R}$  and
$$
u(t) = - g(t,u(t))-\int^t_{- \infty}AT(t-s)g(s,u(s))ds + \int^t_{-
\infty}T(t-s)f(s,Bu(s))ds, \hspace{0.2cm}t \in \mathbb{R}.
$$
\end{definition}

\begin{definition} \rm
A function $u \in AP(X)$ is  a S-classical solution of
\eqref{eq2}-\eqref{eq3} on $\mathbb{R}$,  if $u$ is a S-classical
solution of \eqref{eq2}-\eqref{eq3} on every interval
$[t_{0},t_{0}+\sigma]\subset\mathbb{R}$, with $t_{0}\in
\mathbb{R}$ and $\sigma>0$.
\end{definition}

\begin{definition} \rm
A function  $u \in AAP(Y)$ is  an asymptotically almost periodic  $Y$-mild solution of
\eqref{eq2}-\eqref{eq3} if $u(0)=y_{0}$, the functions   $ s
\to AT(t-s)g(s,u(s))$,  $s\to (-A)^{\alpha
}T(t-s)\tilde{f}(s,u(s))$ belong  to  $ \mathrm{L}^{1}((0,t]:Y)$
for every $t \in [0,\infty)$ and
\begin{align*}
u(t)& = T(t)(y_{0}+g(t_{0},y_{0}))-
g(t,u(t))-\int^t_{0}AT(t-s)g(s,u(s))ds  \\
&\quad + \int^t_{0}(-A)^{\alpha}T(t-s)\tilde{f}(s,u(s))ds,\quad t \in
[0,\infty).
\end{align*}
\end{definition}

\begin{definition}\rm
A function  $u \in AAP(X)$ is a  mild solution of
\eqref{eq2}-\eqref{eq3} if $u(0)=y_{0}$,  $u\in
C((0,\infty):X_{\alpha})$, the function $ s \to
AT(t-s)g(s,u(s))$ belongs to $\mathrm{L}^{1}([0,t]:X)$ for every
$t \in [0,\infty)$  and
\begin{align*}
u(t)& = T(t)(y_{0}+g(t_{0},y_{0}))-
g(t,u(t))-\int^t_{0}AT(t-s)g(s,u(s))ds  \\& +
\int^t_{0}T(t-s)f(s,Bu(s))ds, \quad t \in [0,\infty).
\end{align*}
\end{definition}

\begin{definition} \rm
A function $u \in AAP(X)$ is  a S-classical solution of
\eqref{eq2}-\eqref{eq3}  if  $u$ is a   S-classical solution of
\eqref{eq2}-\eqref{eq3}  on  $[0,r]$ for every $r>0$.
\end{definition}


\section{Existence results of $Y$-mild solutions}\label{sec1}

 In this section we
establish  the existence of   \,asymptotically almost periodic and  almost
periodic $Y$-mild solutions
for   \eqref{eq2}-\eqref{eq3}. First, we need the next result.

\begin{prop}\label{l4}
Let  $ \mu \in (0,1)$,  $v(\cdot) \in AAP(X_{\mu})$ and assume
that there is   $\omega <0 $  and a non-increasing function
$H_{\mu}(\cdot)$ so that $e^{\omega s}H_{\mu}(s)\in
\mathrm{L}^{1}([0,\infty)) $ and $\|
(-A)^{1-\mu}T(t)\|_{\mathcal{L}(X:Y)}\leq e^{\omega t}H_{\mu}(t)$
for every $t>0$. If $u(\cdot )$ is the function defined by
 \begin{equation}\label{24}
 u(t)=\int_{0}^t AT(t-s)v(s)ds,\quad t\geq 0,
\end{equation}
 then $  u(\cdot)\in AAP( Y)$.
\end{prop}

\begin{proof} From Lemma \ref{lema2},    it's sufficient
to prove that $u \in F( \mathbb{R}^+:Y)$. Let $\epsilon>0$ given and
$\mathcal{T}( \epsilon , v,X_{\mu})$, $L=L( \epsilon,v,X_{\mu})$ be as
in Lemma \ref{lema2}.  If
 $ t \geq L( \epsilon,v,X_{\mu})+1$ and
  $\xi \in \mathcal{T}( \epsilon , v,X_{\mu})$, then
\begin{align*}
&\| u(t + \xi ) - u(t) \|_{Y} \\
& \leq \int_0^\xi \| (-A)^{1 -
\mu}T(t+ \xi-s)(-A)^{\mu}v(s) \|_{Y} ds  \\
&\quad + \int_0^t \|(-A)^{1 - \mu}T(t-s)\left( (-A)^{\mu}v(s+
\xi) - (-A)^{\mu}v(s) \right) \|_{Y} ds  \\
& =  I_1(t, \xi) + I_2(t, \xi).
\end{align*}
Now, we  estimate each term $I_{i}(t, \xi)$  separately. For the
first term we get
\begin{align*}
I_1(t, \xi )
& \leq  \| (-A)^{\mu}v\|_{AAP(X)}\int_0^{\xi}e^{\omega (t + \xi-s)}H_{\mu}
( t + \xi -s) ds \\
& \leq  e^{\omega t}\| (-A)^{\mu}v\|_{AAP(X)} \int_0^{\xi }
e^{\omega (\xi-s)}H_{\mu}( \xi -s)ds,
\end{align*}
 and hence, there exits $d_1 > 0$ independent of $\xi$   such that
 \begin{equation} \label{des2}
 I_1(t,\xi)  \leq c_1 e^{\omega t},
\end{equation}
for every $t \geq L( \epsilon , v,X_{\mu})+ 1$.
On the other hand, for the second term we see that
\begin{align*}
I_2 (t, \xi ) & \leq  \int_0^{L+1}\| (-A)^{1 - \mu}T(t - s)
\left( (-A)^{\mu}v(s+ \xi) - (-A)^{\mu}v(s)
\right) \|_{Y} ds \\
&\quad+ \int_{L+1}^{t}\|(-A)^{1-
\mu}T(t-s)(-A)^{\mu}( v(s + \xi) - v(s) ) \|_{Y} ds \\
& \leq  2\|(-A)^{\mu}v \|_{AAP(X)} e^{\omega (
t-L-1)}\int_{0}^{L+1}
e^{\omega (L+1 -s)} H_{\mu}(L+1 -s )ds  \\
&\quad +\epsilon
 \int_{L+1}^{t}\|(-A)^{1-\mu} T(t-s)\|_{\mathcal{L}(X;Y)}ds \\
& \leq  2 \| (-A)^{\mu}v \|_{AAP(X)}e^{\omega (
t-L-1)}\int_0^{\infty} e^{\omega s}H_{\mu}(s)ds + \epsilon
\int_{0}^{\infty}e^{\omega s}H_{\mu}(s)ds.
\end{align*}
 Thus, there exist positive
constants $d_2, d_3$ independents of $t \geq L( \epsilon ,
v,X_{\mu})+ 1$  and $ \xi \in \mathcal{T}( \epsilon , v,X_{\mu})$
such that
\begin{equation}\label{des1}
 I_2(t, \xi) \leq d_2 e^{\omega t}+
\epsilon d_3.
\end{equation}
 From (\ref{des2})-(\ref{des1})  we have
$$
\| u(t+ \xi ) - u(t) \|_{Y} \leq d_4 e^{\omega t }+ \epsilon d_5,
$$
where $d_{4}, d_{5}$ are positive constants independent of
$t \geq L( \epsilon , v,X_{\mu})+ 1$  and $ \xi \in \mathcal{T}(
\epsilon , v,X_{\mu})$. Thus,  for an appropriate
 $L(\epsilon, u)> L(\frac{\epsilon}{2d_{5}},
 v,X_{\mu})+ 1$, it follows
$$ \| u(t+\xi) - u(t)\|_{Y} \leq \epsilon
$$
for every $t \geq L(\epsilon, u)$ and all
$\xi \in \mathcal{T}(\frac{\epsilon}{2d_{5}} , v,X_{\mu})$,
which shows that $u\in F(\mathbb{R}^+:Y)$ and completes the proof of this result.
\end{proof}

Proceeding as in the previous proof  we can prove the next result.

\begin{corollary} \label{coro3.2}
Let  $ \mu \in (0,1)$ and   $v \in AAP(X_{\mu})$.  If $u(\cdot )$
is the function defined by \ref{24},  then $u(\cdot)\in AAP( X)$.
\end{corollary}

In the next result  we establish the existence
of  asymptotically almost periodic $Y$-mild  solution of \eqref{eq2}-\eqref{eq3}.

\begin{theorem}\label{teo4}
Let $\bf H_1, H_2, \bf H_3$  be verified. Then, there exists
$\epsilon> 0$ such that for every $ y_0 \in B_{\epsilon}(0,Y)$
there exits an  $Y$-mild solution $u( \cdot , y_0)\in
C([0,\infty):Y)$  of \eqref{eq2}-\eqref{eq3}. Moreover, if  the
functions $\tilde{f}, (-A)^{\beta}g:[0,\infty)\times Y\to X $ are
pointwise asymptotically almost periodic, then $u( \cdot, y_0)\in
AAP(Y)$.
\end{theorem}

\begin{proof}
Let $J:[0,\infty)\to \mathbb{\mathbb{R}}$ be the function defined by
$$
 J(r) =  L_g(r) \Big(\|
(-A)^{-\beta}\|_{\mathcal{L}(X:Y)} +\int_0^\infty
e^{\omega_{1}s}H_{\beta}(s)ds \Big)+
L_{\tilde{f}}(r)\int_0^\infty e^{\omega_{2}s}H_{\alpha}(s)ds
$$
and let   $r> 0$, $\gamma \in (0,1)$ be such that
\begin{equation}\label{26}
\widetilde{M}\left( 1 + r \|  (-A)^{-\beta}\|_{\mathcal{L}(X:Y)}
L_g(r)\right) \gamma r + J(r)r < r.
\end{equation}
Note that the assertion holds for $ \epsilon= \gamma r$. To
prove this statement  we fix $ y_0 \in B_{\epsilon}(0,Y)$ and
define the operator $ \Gamma : B_r(0,C_b([0,\infty):Y))
\to C( [0,\infty):Y)$ by
\begin{align*}
\Gamma x(t)& = T(t) ( y_0 + g(0,y_0) ) - g(t,x(t))
  +\int_0^t
(-A)^{1- \beta}T(t-s) ( -A)^{\beta}g(s, x(s)) ds \\
&\quad +\int_0^t (-A)^{\alpha}T(t-s) \tilde{f}(s,x(s))ds.
\end{align*}
 From  the assumptions  on the functions   $ s \to
(-A)^{\alpha }T(s)$ and $ s \to (-A)^{1-\beta }T(s)$, the
estimates
\begin{align*}
\|(-A)^{ 1 - \beta}T(s)(-A)^{\beta}g(s,x(s))\|_{Y}
 &\leq  \|(-A)^{1-\beta}T(s)\|_{\mathcal{L}(X;Y)} L_{g}(r) r\\
 &\leq   e^{\omega_{1}s}H_{\beta}(s)L_{g}(r) r,
\end{align*}
\begin{equation*}
   \|(-A)^{\alpha}T(s)\tilde{f}(s,x(s))\|_{Y}
\leq \|(-A)^{\alpha}T(s)\|_{\mathcal{L}(X;Y)}L_{\tilde{f}} (r)r
\leq e^{\omega_{2}s}H_{\alpha }(s)  L_{\tilde{f}}(r) r,
\end{equation*}
and the Bochner Theorem, we  infer  that $\Gamma x(t)$ is well
defined and that $\Gamma x \in C([0,\infty);Y)$. Moreover, for
$t\geq 0$ we get
\begin{align*}
&\| \Gamma x(t) \|_{Y} \\
& \leq  \widetilde{M}(\|
y_0 \|_{Y} + L_g( r) \| (-A)^{-\beta}\|_{\mathcal{L}(X:Y)}\| y_0
\|_{Y}) + L_g ( r) \|
(-A)^{-\beta}\|_{\mathcal{L}(X:Y)}\| x(t) \|_{Y}   \\
&\quad + \int_0^t\|(-A)^{1-\beta} T(s)\|_{\mathcal{L}(X;Y)}L_g(r) r ds
 + \int_0^t \|(-A)^{\alpha} T(s)\|_{\mathcal{L}(X;Y)}L_{\tilde{f}} (
r) r ds  \\
& \leq  \widetilde{M}(\gamma r + L_g( r) \|
(-A)^{-\beta}\|_{\mathcal{L}(X:Y)}\gamma r)+ J(r)r,
\end{align*}
which from (\ref{26}) implies that  $\Gamma ( B_{r}( 0, C_b( [0,
\infty): Y)))\subset  B_{r}( 0, C_b( [0, \infty): Y)) $.

 Next,  we prove that $ \Gamma$ is a
contraction on $ B_{r}( 0, C_b( [0, \infty):Y)) $. For functions
$u, v \in B_{r}( 0, C_b( [0, \infty): Y)) $ we get
\begin{align*}
\| \Gamma u (t) - \Gamma v(t) \|_{Y}
& \leq  L_g(r)\|
(-A)^{-\beta}\|_{\mathcal{L}(X:Y)} \| u(t) - v(t) \|_{Y} \\
&\quad +L_g(r)
\int_0^t\|(-A)^{1-\beta} T(t-s)\|_{\mathcal{L}(X;Y)}\| u (s) - v(s) \|_Y ds  \\
&\quad +  L_{\tilde{f}}(r) \int_0^t \|(-A)^{\alpha}
T(t-s)\|_{\mathcal{L}(X;Y)}\| u (s) - v(s) \|_Y ds \\
&  \leq   L_g(r) \Big(\| (-A)^{-\beta}\|_{\mathcal{L}(X:Y)}
+\int_0^\infty e^{\omega_{1}s}H_{\beta}(s)ds \Big)\| u - v
\|_{0,t,Y}\\
&\quad + \Big(L_{\tilde{f}}(r) \int_0^\infty
e^{\omega_{2}s}H_{\alpha}(s)ds\Big)\,\| u - v\|_{0,t,Y}\\
&\leq J(r)\| u - v \|_{0,t,Y},
\end{align*}
which proves that $\Gamma$ is a contraction on
$ B_{r}( 0, C_b([0, \infty): Y)) $ and that $ \Gamma$ has a unique fixed point
$u(\cdot,y_{0})\in  B_{r}( 0, C_b( [0, \infty): Y)) $. Clearly,
$u(\cdot,y_{0})$ is a  $Y$-mild solution  of
\eqref{eq2}-\eqref{eq3}.

Since   $(-A)^{\beta}g$ and $\tilde{f}$ are pointwise
asymptotically almost periodic, it  follows from Lemma \ref{28}
and  Proposition  \ref{l4} that each solution $u(\cdot,y_{0})$, $
y_0 \in B_{\epsilon}(0,Y)$,   is  an asymptotically almost periodic 
$Y$-mild solution of \eqref{eq2}-\eqref{eq3}.  The proof
is now complete. \end{proof}


 In the next result,  we discuss the existence of
 almost periodic $Y$-mild solutions.

\begin{theorem}\label{teo2}
If the  assumptions ${\bf H_{1},H_{2}}, {\bf H_{3}}$ are satisfied
and the functions  $(-A)^{\beta}g, \tilde{f}$ are pointwise almost
periodic,  then there exits an almost periodic $Y$-mild solution of
\eqref{eq2}-\eqref{eq3}.
\end{theorem}

\begin{proof} Let $\Gamma :AP(Y)\to AP(Y) $ be  the map defined by
$$
\Gamma u(t) =- g(t,u(t)) - \int^t_{- \infty}AT(t-s)g(s,u(s))ds
   + \int^t_{-
   \infty}(-A)^{\alpha}T(t-s)\tilde{f}(s,u(s))ds.
$$
 The same arguments used in the proof of Theorem
\ref{teo4} proves that $\Gamma u(t)$ is well defined and that
$\Gamma u \in C_b(\mathbb{R};Y)$. In order to prove that $\Gamma$
is $AP(Y) $-valued, we fix  $u\in AP(Y)$ and  $\epsilon>0$. We
know from Zaidman \cite[pp. 30 ]{z5} and Lemma \ref{lx}, that
$z(t)=(\tilde{f}(t,u(t)),(-A)^\beta g(t,u(t)))\in AP(X\times X)$.
If $\xi \in \mathcal{H}(\epsilon , z(\cdot),X\times X)$ we get
\begin{align*}
&\|\Gamma u(t+ \xi) -\Gamma u(t)  \|_Y \\
& \leq \| (-A)^{-\beta}\|_{\mathcal{L}(X;Y)} \|
(-A)^{\beta}g(t+\xi,u(t+\xi))-(-A)^{\beta}g(t,u(t))\|\\
&\quad + \int^t_{- \infty}\|(-A)^{1-\beta}T(t-s)
\left( (-A)^{\beta}g(s+\xi,u(s+\xi))-(-A)^{\beta}g(s,u(s))
  \right) \|_{Y} ds \\
& \quad+  \int^t_{- \infty}
\|(-A)^{\alpha}T(t-s)\left(\tilde{f}(s+\xi,u(s+\xi))-\tilde{f}(s,u(s))\right)
 \|_{Y}ds \\
&\leq  \epsilon[\| (-A)^{-\beta}\|_{\mathcal{L}(X;Y)} +
 \int_{0}^{\infty}( \|(-A)^{ 1 -\beta}T(s)\|_{\mathcal{L}(X;Y)}+
     \|(-A)^{\alpha}T(s)\|_{\mathcal{L}(X;Y)})ds] \\
& \leq \epsilon[\| (-A)^{-\beta}\|_{\mathcal{L}(X;Y)} +
 \int_0^\infty\left(e^{\omega_{1}s}H_{\alpha}(s)
 +e^{\omega_{2}s}H_{\beta}(s)\right)ds],
\end{align*}
 which shows that  $\Gamma u \in AP(Y)$. Thus, $\Gamma$ is well
defined and with values in $ AP(Y)$.

 Note that there exists  $r_{0}>0$  small enough such that
$\Gamma$  is a contraction from $B_{r_{0}}(0, AP(Y))$ into
$B_{r_{0}}(0, AP(Y))$.  Let $r>0$ and  $u\in B_{r}(0,AP(Y))$. If
$t\in \mathbb{R}$ we see that
\begin{align*}
\| \Gamma u(t)\|_{Y}
& \leq \| (-A)^{-\beta}\|_{\mathcal{L}(X;Y)}\|  g(t,u(t)) \| \\
&\quad + L_{g}(r)\int^t_{-
\infty}\|(-A)^{1 -\beta}T(t-s)\|_{\mathcal{L}(X;Y)} \|u(s) \|_Y ds \\
&\quad + L_{\tilde{f}}(r) \int^t_{-
\infty}\|(-A)^{\alpha}T(t-s)\|_{\mathcal{L}(X;Y)}\|u(s) \|_Y ds \\
& \leq   \| (-A)^{-\beta}\|_{\mathcal{L}(X;Y)} L_{g}(r)r +
L_{g}(r)r\int^{\infty}_{0}e^{\omega_{1}s}H_{\beta}(s)ds\\
&\quad + L_{\tilde{f}}(r)r \int^{ \infty}_{0}e^{\omega_{2}s}H_{\alpha}(s)ds,
\end{align*}
and so that
$ \| \Gamma u \|_{AP(Y)}  \leq r J(r)$,
where
$$ J(r)= L_{g}(r)\Big(\|
(-A)^{-\beta}\|_{\mathcal{L}(X;Y)} + \int^{
\infty}_{0}e^{\omega_{1}s}H_{\beta}(s)ds\Big) +
L_{\tilde{f}}(r)\int^{\infty}_{0}e^{\omega_{2}s}H_{\alpha}(s)\,ds.
$$
Since $J(\cdot)$ is continuous and  $J(0)=0$, we can fix $r_0> 0 $
such that $J(r_0) < 1$. Obviously,
$ \Gamma(B_{r_{0}}(0,AP(Y))) \subseteq B_{r_{0}}(0,AP(Y))$. Moreover, for
$u , v \in B_{r_{0}}(0,AP(Y))$  we get
\begin{align*}
&\|\Gamma u(t) - \Gamma v(t) \|_{Y} \\
& \leq  \| (-A)^{-\beta}\|_{\mathcal{L}(X;Y)}\|
(-A)^{\beta}g(t, u(t))- (-A)^{\beta}g(t,v(t)) \|\\
& \quad + \int^{t}_{-
\infty}\|(-A)^{1 -\beta}T(t-s)\|_{\mathcal{L}(X;Y)}
  \|(-A)^{\beta}g(s,u(s))-(-A)^{\beta}g(s,v(s))\|ds  \\
& \quad +  \int^{t}_{-
\infty}\|(-A)^{\alpha}T(t-s)\|_{\mathcal{L}(X;Y)}
\|\tilde{f}(s,u(s))-\tilde{f}(s,v(s))\| ds  \\
& \leq  L_{g}(r_{0})\| (-A)^{-\beta}\|_{\mathcal{L}(X;Y)} \|u -v\|_{AP(Y)}  \\
&\quad  + \|u - v
\|_{AP(Y)}\Big(L_{g}(r_{0})\int^{\infty}_{0}
e^{\omega_{1}s}H_{\beta}(s) ds + L_{\tilde{f}}(r_{0})
\int^{\infty}_{0}e^{\omega_{2}s}H_{\alpha}(s)ds\Big), \\
&\leq J(r_{0})\| u - v \|_{AP(Y)},
\end{align*}
which proves that  $\Gamma $ is a contraction on
$B_{r_{0}}(0,AP(Y))$  and that
 there exists an almost periodic $Y$-mild  solution
  of    \eqref{eq2}-\eqref{eq3}. The proof  is finished.
 \end{proof}

\section{Existence and regularity of mild solutions}\label{ap}

In this section we  establish conditions under which an $Y$-mild
solution of \eqref{eq2}-\eqref{eq3} is a mild solution. Then, we
apply theses results to prove the  existence of asymptotically almost
periodic and almost periodic solutions for \eqref{eq2}-\eqref{eq3}.

In the next
results, $u(\cdot)\in C([0,b]:Y)$ is a $Y$-mild solution of
\eqref{eq2}- \eqref{eq3} on $ [0,b]$ and  the next condition is
always assumed. \smallskip

\noindent\textbf{Assumption (Afg).}
There are constants $0 < \sigma_1, \sigma_2 < 1$ such that
\begin{gather*}
\|(-A)^{\beta}g(t,x) - (-A)^{\beta}g(s,y)\|  \leq  L_{g}(r)
\left( |t-s|^{\sigma_1} + \|x-y\|_Y\right), \\
\|\tilde{f}(t,x) - \tilde{f}(s,y)\| \leq  L_{\tilde{f}}(r)
\left( |t-s|^{\sigma_2} + \|x-y\|_Y \right),
\end{gather*}
 for each  $(t,s)\in \mathbb{R}^2 $ and every $ x,y\in B_{r}(0,Y)$.
Moreover, $0<\alpha <\beta \leq 1 $ and  $\| (-A)^{-\beta}
\|_{\mathcal{L}(X:Y)}L(\| u\| _{0,b,Y} ) <1$.


\begin{remark}\label{30} \rm
 Observe that the solutions given by the Theorems
 \ref{teo4} and \ref{teo2} are such that
 $\| (-A)^{-\beta} \|_{\mathcal{L}(X:Y)}L(\| u\| _{\sigma, \sigma +\mu,Y} )
 <1$ for all $\sigma \in \mathbb{R}$ and all  $ \mu >0$.
  \end{remark}

\begin{prop}\label{prop1}
Let condition (Afg) be satisfied  and assume that
there are positive constants $d, d_{1},d_{2}$;
$0<\xi_1,\xi_{2}<1$ such that
$\|(-A)^{ 1 - \beta +\mu}T(s) \|_{\mathcal{L}(X;Y)} \leq  \frac{d_1 }{s^{\xi_1}}$
and $\|(-A)^{\alpha + \mu}T(s) \|_{\mathcal{L}(X;Y)} \leq \frac{ d_2}{s^{\xi_{2}}}$
for every $s\in (0,b]$ and  every $\mu \in [0,d]$.  Then
$u \in C^\sigma((0,b];Y)$ for
  $\sigma= \min\{d,1-\alpha, \sigma_{1}, 1- \xi_1,   1-\xi_2\}$.
 \end{prop}

\begin{proof} We follow the ideas in Rankin \cite{R2}.
Let $t\in (0,b) $ and  $0 < h < 1$  such that $t+h\in(0,b] $. Then
\begin{align*}
&\|u(t+h) - u(t) \|_{Y}\\
& \leq \| (-A)^{\alpha}T(\frac{t}{2})\|_{\mathcal{L}(X;Y)}
 \|(T(h)-I) T(\frac{t}{2})(-A)^{-\alpha}(y_{0}+g(0,y_{0}))\| \\
&\quad + \|(-A)^{ - \beta}\|_{\mathcal{L}(X;Y)} \|
(-A)^{\beta}g(t+h,u(t+h)) - (-A)^{\beta}g(t,u(t))\| \\
&\quad + \int^t_{ 0}\|(-A)^{1 - \beta + \mu }T(t-s)\left( T(h)-I
\right) (-A)^{\beta-\mu}g(s,u(s))\|_Y ds \\
&\quad +  \int^{t+h}_t\|(-A)^{1 -
\beta}T(t+h-s)\|_{\mathcal{L}(X;Y)}\|(-A)^{\beta}g(s,u(s))\| ds \\
&\quad + \int^t_{ 0}\|(-A)^{\alpha + \mu }T(t-s)\left( T(h)-I
\right)  (-A)^{-\mu}\tilde{f}(s,u(s))\|_Y ds \\
&\quad + \int^{t+h}_t\|(-A)^{\alpha}T(t+h-s)\|_{\mathcal{L}(X;Y)}
\| \tilde{f}(s,u(s))\| ds  \\
& \leq  \frac{2^{\xi_{2}}d_{2}}{t^{\xi_{2}}}MC'_{\alpha}h^{1-\alpha} \|
y_{0}+g(0,y_{0})\| \\
&\quad + \|(-A)^{-
\beta}\|_{\mathcal{L}(X;Y)}L_{g}(\| u\| _{b,Y})
\left( h^{\sigma_{1}} + \|u(t+h) - u(t) \|_Y \right) \\
&\quad + \int^t_{ 0} \frac{d_{1}}{(t-s)^{\xi_1}}\|( T(h)-I)
(-A)^{\beta-\mu}g(s,u(s))\| ds
+L_{g}(\| u\| _{b,Y})\| u\| _{b,Y}\frac{d_1h^{1-\xi_1}}{1-\xi_1}  \\
&\quad +   \int^t_{ 0} \frac{d_2}{(t-s)^{\xi_2}}\|( T(h)-I)
(-A)^{- \mu}\tilde{f}(s,u(s))\| ds
+L_{\tilde{f}}(\| u\| _{b,Y})\| u\| _{b,Y}\frac{d_2 h^{1-\xi_2}}{1-\xi_2}\\
& \leq \frac{2^{\xi_{2}}d_{2}}{t^{\xi_{2}}}MC'_{\alpha}h^{1-\alpha}\|
y_{0}+g(0,y_{0})\|\\
&\quad + \|(-A)^{ - \beta}\|_{\mathcal{L}(X;Y)}L_{g}(\| u\|_{b,Y})
\left( h^{\sigma_{1}} + \|u(t+h) - u(t) \|_Y \right) \\
& \quad +  \tilde{d}_1 h^{\mu} \int^t_{0}
\frac{ds}{(t-s)^{\xi_1}}  +\tilde{d}_2h^{1-\xi_1} +  \tilde{d}_3
h^{\mu}\int^{t}_{0}\frac{ds }{(t-s)^{\xi_2}} +\tilde{d}_4
h^{1-\xi_2},
\end{align*}
and then
\begin{align*}
 \| u(t+h) - u(t) \|_Y & \leq  L_{g}(\| u\| _{b,Y})\|(-A)^{-
 \beta}\|_{\mathcal{L}(X;Y)} \| u(t+h) - u(t) \|_Y +
\tilde{d}_{5}h^{1-\alpha}\\
 & \quad+  \tilde{d}_6h^{\sigma_{1}} + \tilde{d}_7h^{\mu}+
 \tilde{d}_2 h^{1- \xi_1}+ \tilde{d}_4 h^{1- \xi_2},
\end{align*}
 where the constants  $\tilde{d}_i,\, i =1,2,\dots 7,$  are
independent of $t,h $ and  $\mu\in [0,d]$. Since $\|
(-A)^{-\beta} \|_{\mathcal{L}(X;Y)}L(\| u\| _{b,Y} ) < 1$ and
$t,h,\mu$  are arbitrary, the last inequality proves that
$u(\cdot)\in C^{\sigma}((0,b];Y)$ for
$\sigma=\min\{d,1-\alpha,\sigma_{1}, 1- \xi_1, 1- \xi_2\}$. The proof   is
complete
\end{proof}

\begin{prop}\label{prop2}
Under the assumptions  of Proposition  \ref{prop1},
$u(\cdot) \in C((0,b];X_{\gamma})$ for $\gamma =\min\{1-\alpha,
\beta\}$.
\end{prop}

\begin{proof} First we introduce the decomposition
$u= \sum ^{3}_{i=1}u_i$ where
\begin{gather*}
u_1(t)  =  T(t)( u(0)+g(0,u(0)))- g(t,u(t)),  \\
u_2(t)  =  \int^t_{ 0}(-A)^{ 1 - \beta}T(t-s)(-A)^{\beta}g(s,u(s))ds , \\
u_{3}(t) = \int^t_{0}(-A)^{\alpha}T(t-s)\tilde{f}(s,u(s))ds.
\end{gather*}
It is obvious that $u_{1}\in C((0,b]; X_{\beta})$. On the
other hand, from Proposition \ref{prop1} we know that $u(\cdot)\in
C^{\sigma}((0,b];Y)$ for $\sigma= \min\{d,1-\alpha, \sigma_{1}, 1-
\xi_1, 1- \xi_2\}$ which from the estimate
\begin{align*}
&\| (-A)^{\gamma +1 }T(t-s)\left(g(s,u(s)) -
g(t,u(t)\right)\| \\
 &\hspace{0.5cm} \leq \| (-A)^{\gamma +1 -
\beta}T(t-s)\|_{\mathcal{L}(X)}
 \|(-A)^{\beta}g(s,u(s)) -(-A)^{\beta}g(t,u(t)\| \\
 & \hspace{0.5cm}\leq \frac{C_{\gamma+1-\beta}}{(t-s)^{ \gamma +  1 -
\beta}}L_{g}(\| u\| _{b,Y})\left( |t-s|^{\sigma_1} + \| u(s) -
u(t) \|_Y\right)   \\
& \hspace{0.5cm}\leq  \frac{\tilde{d}_1 }{(t-s)^{ \gamma +  1 -
\beta- \sigma_1}} + \frac{\tilde{d}_2}{(t-s)^{ \gamma + 1 - \beta
-\sigma }},
\end{align*}
 implies that  the function
$$ v(s)= (-A)^{ \gamma +1}T(t-s)
 \left(g(s,u(s)) -
g(t,u(t))\right),$$ is integrable on $[ 0 , t)$, $t\in [a,b]$,
when $\gamma < \min\{\beta+\sigma_{1},\beta+\sigma \}$. In
particular, for $\gamma =\beta$  we find that
\begin{align*}
&\int^t_{ 0} v(s)ds  + (-A)^{ \beta}g(t,u(t)) \\
& =  \int_0^t v(s)ds + (-A)\int^t_{ 0}(-A)^{\beta}T(t-s)g(t,u(t))ds
 +T(t)(-A)^{ \beta}g(t,u(t))\\
& =  \int^t_{ 0}(-A)^{ \beta +1}T(t-s)\left(g(s,u(s)) -g(t,u(t))\right)ds  \\
&\quad +\int^t_{ 0}(-A)^{  \beta +1}T(t-s)g(t,u(t))ds +T(t)(-A)^{\beta}g(t,u(t)),
\end{align*}
which shows   that  $ u_2(\cdot) \in C([0,b]; X_{\beta})$ since
$(-A)^{\beta}$ is a closed operator.

 Proceeding as in the previous case, we can prove that
 $ u_3(\cdot) \in C([0,b]; X_{1 -\alpha})$.
From theses remarks we  conclude  that $u(\cdot)\in C((0,b];
X_{\gamma})$  for $ \gamma = \min\{ 1-\alpha,\beta \}$. The proof
is complete
\end{proof}

\begin{theorem}\label{27}
Under the hypotheses of Proposition \ref{prop2}, if $ \alpha\leq  1 - \alpha$,
then $u(\cdot)$ is a mild solution of \eqref{eq2}-\eqref{eq3}.
\end{theorem}

The assertion of this theorem is a consequence
 of Assumption (H3) and  Lemma \ref{an}.
 Next we establish conditions  under which $u(\cdot)$ is a
 S-classical  solution.


\begin{prop}\label{prop3}
Let assumption in Theorem   \ref{27} be satisfied and assume that
 $ L_{g}(\| u\| _{b,Y})\|(-A)^{\alpha-
\beta}\|_{\mathcal{L}(X:Y)}\|(-A)^{-\alpha}\|_{\mathcal{L}(X)}
<1$. Then  $u\in C^{\sigma} ((0 ,b]:X_{\alpha} )$ for
$\sigma=\min \{ \beta-\alpha,\sigma_{1}\}$.
\end{prop}

\begin{proof}
Using the fact that  $u\in C((0,b]:X_{\alpha})$ and Lemma
\ref{an},  for   $0<\delta<t<b$ and $h>0$
 such that $t+h<b$ we find that
\begin{align*}
&\|u(t+h) - u(t) \|_{\alpha}\\
& \leq  \| (-A)^{\alpha} (T(h)-I)T(t)(y_{0}+g(0,y_{0}))\|\\
&\quad +\|(-A)^{ \alpha- \beta}\|_{\mathcal{L}(X)} \|
(-A)^{\beta}g(t+h,u(t+h)) -(-A)^{\beta}g(t,u(t))\| \\
&\quad + \int^t_{ 0}\|(-A)^{1 - \beta + \alpha }\left(
T(h)-I \right) T(t-s)(-A)^{\beta}g(s,u(s))\| ds \\
&\quad +  \int^{t+h}_t\|(-A)^{1 - \beta+ \alpha}T(t+h-s)\|
\|(-A)^{\beta}g(s,u(s))\| ds\\
&\quad + \int^t_{ 0}\|(-A)^{ \alpha }\left(
T(h)-I \right)T(t-s)f(s,u(s))\| ds \\
&\quad +  \int^{t+h}_t\|(-A)^{\alpha} T(t+h-s)\| \|f(s,u(s))\| ds\\
&\leq \frac{C'_{1-\alpha}h^{1-\alpha}}{\delta^{\alpha}}\| y_{0}+g(0,y_{0}) \| \\
&\quad +   \|(-A)^{\alpha- \beta}\|_{\mathcal{L}(X)}L_{g}(\| u\| _{b,Y})
\left[ h^{\sigma_{1}} + \|u(t+h) - u(t) \| _{Y} \right] \\
&\quad + \int^t_{ 0}C'_{\beta-\alpha}h^{\beta-\alpha}\| (-A)^{1
- \beta + \alpha }T(t-s)(-A)^{\beta}g(s,u(s))\| ds\\
&\quad +\int^{t+h}_{t} \frac{C_{1- \beta+\alpha }}{(t+h-s)^{1-
\beta+\alpha}}\|(-A)^{\beta}g(s,u(s))\| ds \\
&\quad +\int^t_{0}C'_{1-\alpha}h^{1-\alpha}\|
(-A)^{\alpha}T(t-s)f(s,u(s))\| ds \\
&\quad +\int^{t+h}_{t}\frac{C_{\alpha}}{(t+h-s)^{\alpha}}\| f(s,u(s))\|\,ds
\end{align*}
and hence
\begin{align*}
&\| u(t+h) - u(t) \|_{\alpha}\\
& \leq \|(-A)^{\alpha- \beta}\|_{\mathcal{L}(X)} L_{g}(\| u\| _{b,Y})
 \|(-A)^{-\alpha}\|_{\mathcal{L}(X,Y)}
 \|u(t+h) - u(t) \|_{\alpha}\\
&\quad  +  \tilde{d}_{1}h^{\sigma_{1}} +\tilde{d}_2h^{\beta-\alpha}+
  \tilde{d}_3h^{1-\alpha},
\end{align*}
where the constants $\tilde{d}_{i}$ are independent of $t\geq
\delta$ and $h$. This inequality completes the proof of this
Proposition since $\beta>\alpha$.
\end{proof}

  The next result  is consequence of Proposition
 \ref{prop3},  \cite[Theorem 4.3.2]{PA} and \cite[Lemma 2]{He1}.

\begin{theorem}\label{teo3}
 Assume that the hypotheses of   Proposition \ref{prop3} are satisfied.
 If  $g\in C(\mathbb{R}\times X:X_{1})$ and
  $\beta+\min\{\beta -\alpha, \sigma_{1}\}>1$, then $u(\cdot)$ is a
  S-classical solution of \eqref{eq2}-\eqref{eq3}.
\end{theorem}

\begin{remark}\label{31} \rm
It is clear that the previous   results of regularity of $Y$-mild
solutions are valid for every $Y$-mild solution $ u\in C(
[\sigma,\sigma+\mu]; Y),\sigma\in \mathbb{R}, \mu>0$.
 \end{remark}

  As consequence of the Theorems   \ref{27}, \ref{teo3}   and
Remarks \ref{30} and  \ref{31}, we obtain the following existence
result of asymptotically almost periodic and almost periodic solutions
of \eqref{eq2}-\eqref{eq3}.
The proof  of the next result will be omitted.


\begin{theorem}\label{29}
Let assumptions  (H1)--(H3) and condition
(Afg1) be satisfied; also assume that $\alpha\leq
1-\alpha$ and $\beta+\min\{\beta -\alpha, \sigma_{1}\}>1$.  Then
the following properties are satisfied.
\begin{enumerate}
    \item If the functions
     $f,(-A)^{\beta}g:[0,\infty)\times Y\to X $
    are pointwise asymptotically almost periodic,
   then there exists $\epsilon> 0$ such that for every $ y_0 \in
B_{\epsilon}(0,Y)$ there exits an  asymptotically almost periodic S-classical solution,
$u(\cdot,y_{0})$, of the system \eqref{eq2}-\eqref{eq3} such that
  $u(0,y_{0})=y_{0}$.
  \item  If the functions $f,(-A)^{\beta}g:[0,\infty)\times Y\to X $ are
   pointwise  almost periodic, then there exits an  almost periodic S-classical solution of
  the equation  \eqref{eq2}-\eqref{eq3}.
    \end{enumerate}
\end{theorem}


\section{Example}\label{example}

In this section we illustrate some of our results.
 Consider the first order   evolution equation
\begin{gather}
\frac{d}{d t}\Big[u(t,\xi) +\int_{0}^{\pi}  a(t) b(\eta,\xi)
u(t,\eta)d \eta\Big]
=\frac{\partial^{2} u(t,\xi)} {\partial\xi^{2}} + F(t,u(t,\xi)),\quad
\xi\in I=[0,\pi] \label{H2} \\
 u(t, 0)  =  u(t, \pi)  =  0, \quad t \in  \mathbb{R},\label{H3}
 \end{gather}
 where $a( \cdot )\in C_{b}( \mathbb{R},\mathbb{R})$, $a(0)=0$ and
\begin{enumerate}
  \item[(a)] The functions  $   b( \eta, \xi)$,
   $ {\frac{\partial^{i} b( \eta, \xi)} {\partial\xi^{i} } },i=1,2$,
are  measurable, $b(\eta, \pi)= b(\eta, 0) = 0$ for every $\eta
\in \mathbb{R}$ and
\begin{equation}
L_{g}= | a(\cdot)|_{C_{b}(\mathbb{R}:\mathbb{R})}
\max\big\{(\int_{0}^{\pi}\int_{0}^{\pi}
 (\frac{\partial^{i}
 b(\eta,\xi) }{\partial\xi^{i}})^{2}d\eta d\xi)^{1/2}:i=0,1,2\big\}<1;
\end{equation}
\item[(b)] $F:\mathbb{R}^{2}\to \mathbb{R}$ is continuous and
 there is  $\mu\in C_{b}( \mathbb{R},\mathbb{R}^{+})$ such that
 $\mu (0)=0$ and
   $$ | F(t,x)-F(t, y)|\leq  \mu(t)| x-y |,$$ for
every $t\in \mathbb{R}$ and every $(x, y) \in  \mathbb{R}^{2} $.
  \end{enumerate}
Let $X=L^{2}([0,\pi])$ and $A:D(A)\subset X\to X$  be the
operator $Ax=x''$ where
$$ D(A) := \{x(\cdot) \in L^{2}([0, \pi]) : x''(\cdot) \in
L^{2}([0, \pi]), \;x(0) = x(\pi) = 0 \}.
$$
It is well known that
$A$ is the infinitesimal generator $C_{0}$-semigroup
$(T(t))_{t\geq 0}$ on $X$. Moreover, the next Theorem is  valid.

\begin{theorem}\label{calorprop}
Under the above conditions, the following properties  hold
\begin{enumerate}
\item $A$ has discrete spectrum, the eigenvalues are $- n^{2},\;n
\in \mathbb{N},$ with corresponding eigenvectors $z_{n} (\xi) :=
\left(\frac{2}{\pi}\right)^{1/2} \sin (n \xi)$ and  the set
$\{z_{n} : n \in \mathbb{N} \}$ is an orthonormal basis of $X$.

\item  For every $x \in X$, $T(t)x = \sum_{n=1}^{\infty}
 e^{-n^{2}t} \langle x, z_{n}\rangle z_{n}$. Moreover,
 the semigroup $(T(t))_{t\geq 0}$ is compact, analytic,
self-adjoint and    $  \| T(t)
\|\leq e^{-t}$ for every $t\geq 0$.

 \item For $f \in X$,
$(-A)^{-\theta}f = \sum_{n=1}^{\infty} n^{-2\theta}  \langle f, z_{n}\rangle
z_{n}$ and  the operator $(-A)^{\theta}$ is given by
   $ (-A)^{\theta} f =
   \sum_{n=1}^{\infty} n^{2\theta}
 \langle f, z_{n}\rangle z_{n}$ on
 $$ D((-A)^{\theta}) = \{f \in X:
\sum_{n=1}^{\infty} n^{2\theta}<f, z_{n}> z_{n}  \in X \}.
$$
Moreover, $\|(-A)^{-1/2} \| =1$ and $\|(-A)^{1/2}T(t)\| \leq
\frac{e^{\frac{-t}{2}}t^{-\theta}}{\sqrt{2}}$ for every  $ t>0$.
\end{enumerate}
\end{theorem}

This theorem follow  from
 \cite[Theorem 2.3.5]{Cu} and \cite[Theorem 4]{R2}.
\bigskip
\par By defining  the  functions $f(\cdot),g(\cdot):\mathbb{R}\times
X\to X$
\begin{gather*}
g(t,x)(\xi)= a(t)\int_{0}^{\pi} b(\eta,\xi) x(\eta)d \eta,\\
f(t,x)(\xi)= F(t, x(\xi)),
\end{gather*}
the system  \eqref{H2}-\eqref{H3} can be written
 as the abstract differential equation \eqref{eq2}-\eqref{eq3}.
 Moreover,  $f,g$ are  continuous
 function,  $g$ is $D(A)$-valued, $Ag:\mathbb{R}\times X\to X$ is
 continuous and
\begin{gather*}
\| (- A)^{\theta}g(t,\cdot )\|_{\mathcal{L}(X)}\leq | a(t) | L_{g},
\quad\theta=0,\frac{1}{2},1,  \\
\| f(t,x) -f(t,y)\| \leq \mu (t)\| x-y\|,
\end{gather*}
for every $t\in \mathbb{R}$ and every $x,y\in X$.
 Obviously, our results can be  applied in the case $Y=X$. In this
particular case, the next results  is consequence of  Theorem
\ref{29}.

 \begin{theorem}\label{32}
Under the above conditions,  the following properties are satisfied.
\begin{enumerate}

\item Assume that $f:[0,\infty)\times X\to X $ is pointwise
a.a.p. and that $a(\cdot)$ is asymptotically almost periodic. Then
there exists $\varepsilon>0$
    such that for every $ y_{0}\in B_{\varepsilon }(0,X)$ there exists
an
    asymptotically almost periodic S-classical solution, $u(\cdot,y_{0})$,  of
    \eqref{eq2}-\eqref{eq3} such that   $u(0,y_{0}))=y_{0}$.
\item  If  $f:[0,\infty)\times Y\to X $ is
   pointwise  almost periodic and $a(\cdot)$ is almost periodic,
 then there  exists an
 almost periodic S-classical solution of \eqref{eq2}-\eqref{eq3}.
\end{enumerate}
\end{theorem}

\begin{remark} \rm
By using the results in this paper, in a forthcoming paper  we will  study
the existence of almost periodic solutions for the Navier-Stokes
equation
\begin{equation}
u'(t,x)=Au(t)+(u(t)\cdot\nabla)u(t)+g'(t)
\end{equation}
where $g\in C(\mathbb{R}:V)$ and $V=\{u\in H_{0}^{1}:\mathop{\rm div} u=0\}$.
See \cite{bri1} for details about this matter.
\end{remark}


\subsection*{Acknowledgement} The author wishes to thank  the
referees for their comments and suggestions.

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\end{document}