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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2012 (2012), No. 212, pp. 1--11.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2012 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2012/212\hfil Almost automorphic mild solutions]
{Almost automorphic mild solutions of hyperbolic evolution equations
 with stepanov-like almost automorphic forcing term}

\author[I. Mishra, D. Bahuguna \hfil EJDE-2012/212\hfilneg]
{Indira Mishra, Dhirendra Bahuguna}  % in alphabetical order

\address{Indira Mishra \newline
Department of Mathematics \& Statistics\\
Indian Institute of Technology-Kanpur,
Kanpur - 208016, India}
\email{indiram@iitk.ac.in}


\address{Dhirendra Bahuguna \newline
Department of Mathematics \& Statistics\\
Indian Institute of Technology-Kanpur,
Kanpur - 208016, India}
\email{dhiren@iitk.ac.in}

\thanks{Submitted December 1, 2011. Published November 27, 2012.}
\subjclass[2000]{34K06, 34A12, 37L05}
\keywords{Almost automorphic; evolution equation;
hyperbolic semigroups; \hfill\break\indent 
 extrapolation spaces; interpolation spaces;
 neutral differential equation; mild solution}

\begin{abstract}
 This article concerns the existence and uniqueness of almost automorphic
 solutions to the semilinear parabolic boundary differential equations
 \begin{gather*}
 x'(t)=A_mx(t)+f(t,x(t)), \quad t\in \mathbb{R}, \\
 Lx(t)=\phi(t,x(t)), \quad t\in \mathbb{R},
 \end{gather*}
 where $A:=A_m|_{\ker L}$ generates a hyperbolic analytic semigroup on a Banach
 space $X$, with Stepanov-like almost automorphic nonlinear term, defined
 on some extrapolated space $X_{\alpha-1}$, for $0<\alpha<1$ and $\phi$
 takes values in the boundary space $\partial X$. 
\end{abstract}

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


\section{Introduction}

In this article, we prove  existence and uniqueness results 
of almost automorphic solutions to the following semilinear parabolic boundary differential equations, with Stepanov-like
almost automorphic nonlinear term using the techniques initiated by Diagana and
N'Gu\`er\`ekata in \cite{dn}.
\begin{equation} \label{SBDE}
\begin{gathered}
x'(t)=A_mx(t)+h(t,x(t)), \quad t\in \mathbb{R}, \\
Lx(t)=\phi(t,x(t)), \quad t\in \mathbb{R},
\end{gathered}
\end{equation}
where the first equation stands in the complex Banach space $X$,
called the state space and the second equation lies in a boundary
 space $\partial X;  (A_m, D(A_m))$
is a densely defined linear operator on $X$
and $L:D(A_m)\to \partial X$ is a bounded linear operator.

Motivation for this paper come basically from the following three sources.

The first one is a nice paper by Boulite et al \cite{bol}.
They have established the existence and uniqueness of almost automorphic 
solutions to the semilinear boundary differential equation \eqref{SBDE}
 using extrapolation methods.

The second source of motivation is a recent paper by Baroun et al \cite{brn}, 
where the authors have considered the same equation as \eqref{SBDE}
and proved the existence of almost periodic (almost automorphic) solutions, when
the nonlinear term $h$ is almost periodic (almost automorphic), whereas we prove the
assertion by taking $h$ to be Stepanov-like almost automorphic function.
The functions $h$ and $\phi$ are defined on some continuous interpolation space
$X_\beta$, $0\le \beta<1$, with respect to the sectorial operator 
$A:=A_m|_{\ker L}$.

To prove our results, we make use of the techniques initiated by Diagana 
and N'Gu\`er\`ekata \cite{dn}, which is also our third source of motivation.

Likewise \cite{bol, brn} we solve the \eqref{SBDE} by transforming 
the semilinear boundary differential equation \eqref{SBDE}
into an equivalent semilinear evolution equation,
\begin{equation}
x'(t)=A_{\alpha-1}x(t)+h(t,x(t))+(\lambda-A_{\alpha-1})L_\lambda\phi(t,x(t)),
 \quad t\in \mathbb{R},
\end{equation}
where $A_{\alpha-1} \ 0\le\beta<\alpha<1$, is the continuous extension
of $A:=A_m|_{ker L}$
to the extrapolated Banach space $X_{\alpha-1}$ of $X_\alpha$ with respect
to $A$ and the semilinear term
$h(t,x)+(\lambda-A_{\alpha-1})L_\lambda\phi(t,x):=f(t,x)$ is an
$X_{\alpha-1}$ valued function. As in \cite{bol, brn} we also assume
 Greiner's assumption introduced by Greiner \cite{gre}, which is stated
in Section \ref{s4}. Under Greiner's assumption on $L$, the operator
 $L_\lambda:=(L|_{\ker(\lambda-A_m)})^{-1}$, called
the Drichilet map of $A_m$, is a bounded linear map from $\partial X$ to $X$,
 where $X_{\alpha-1}$ is a larger Banach space than $X$.
The extrapolation theory was introduced by
Da Prato, Grisvard \cite{da} and Nagel \cite{ngel} and is used for
various purposes.
One can see Section \ref{s2} for the mentioned notion
(cf. \cite{ngel,lun} for more details).

These days people have increasing interest in showing almost automorphy of the
solutions of the functional differential equations see for e.g. 
\cite{bol, brn, dn,
td, gurek, im, gp}. We refer \cite{gurek}, for the more details on the topic.


Our results generalize the existing ones in \cite{bol}, 
in the sense that the function
$h$ is assumed to be Stepanov-like almost automorphic functions.

\section{Preliminaries} \label{s2}

In this section, we begin with fixing some notation and recalling 
the definitions and
basic results on generators of interpolation and extrapolation spaces.
Let $X$ be a complex Banach space and $(A,D(A))$ be a sectorial operator
 on $X$; that is, there
exist the constants $\omega\in \mathbb{R}$, $\phi\in (\frac{\pi}{2},\pi)$ and 
$M>0$ such that
\begin{gather*}
\|R(\lambda,A-\omega)\|_{\mathcal{L}(X)}\le \frac{M}{|\lambda-\omega|}, \quad
\forall \lambda\in \Sigma_{\omega,\phi}, \\
\text{where  } \Sigma_{\omega,\phi}:=\{\lambda\in \mathbb{C}: \lambda\neq \omega, 
\, |\arg(\lambda-\omega)|\le \phi\}\subset \rho(A).
\end{gather*}
The real \textit{interpolation space} $X_\alpha$ for $\alpha\in (0,1)$,
 is a Banach space endowedwith the norm,
\begin{equation}
\|x\|_\alpha:=\sup_{\lambda>0}\|\lambda^\alpha(A-\omega)R(\lambda, A-\omega)x\|.
\end{equation}
Here we denote by, $X_0:=X$, $X_1:=D(A)$, $\|x\|_0=\|x\|$, and
$\|x\|_1=\|(A-\omega)x\|$.
The \textit{extrapolation space} $X_{-1}$ associated with $A$,
is defined to be the completion of $(\widehat{X},\|\cdot\|_{-1})$, where
 $\widehat{X}:=\overline{D(A)}$, endowed with the
norm $\|\cdot\|_{-1}$ given by
$$
\|x\|_{-1}:=\|(\omega-A)^{-1}x\|, \quad x\in X.
$$
In a similar fashion, we can define the space
$X_{\alpha-1}:=(X_{-1})_\alpha=\overline{\widehat{X}}^{\|.\|_{\alpha-1}}$, with
$\|x\|_{\alpha-1}=\sup_{\lambda>0}\|\lambda^\alpha R(\lambda,A_{-1}-\omega)x\|$.
The restriction $A_{\alpha-1}: X_\alpha\to X_{\alpha-1}$ of $A_{-1}$ generates the
analytic semigroup $(T_{\alpha-1}(t))_{t\ge 0}$ on $X_{\alpha-1}$
which is the extension of $T(t)$ to $X_{\alpha-1}$.
Observe that $\omega-A_{\alpha-1}: X_\alpha\to X_{\alpha-1}$
is an isometric isomorphism.

We have the following continuous embedding of the spaces, which
will be frequently used here.
\begin{gather*}
D(A)\hookrightarrow X_\beta\hookrightarrow D((\omega-A)^\alpha)\hookrightarrow
X_\alpha\hookrightarrow X,\\
 X\hookrightarrow X_{\beta-1}\hookrightarrow D((\omega-A_{-1})^\alpha)
\hookrightarrow X_{\alpha-1}\hookrightarrow X_{-1},
\end{gather*}
for all $0<\alpha <\beta<1$.

Now we state certain propositions for the proofs of which one can see \cite{brn}.

\begin{proposition}  \label{p2.1}
Assume that $0<\alpha\le 1$ and $0\le \beta\le 1$. 
Then the following assertions hold for 
$0<t\le t_0$, $t_0>0$ and $\tilde{\epsilon}>0$ such that
 $0<\alpha-\tilde{\epsilon}<1$ with
constants possibly depending on $t_0$.
\begin{itemize}
\item[(i)]  The operator $T(t)$ has continuous extensions 
$T_{\alpha-1}(t):X_{\alpha-1}\to X$ satisfying
\begin{equation}
\|T_{\alpha-1}(t)\|_{\mathcal{L}(X_{\alpha-1},X)}
\le ct^{\alpha-1-\tilde{\epsilon}}, \label{2.2}
\end{equation}
\item[(ii)] For $x\in X_{\alpha-1}$ we have
\begin{equation}
\|T_{\alpha-1}(t)\|_\beta\le ct^{\alpha-\beta-1-\tilde{\epsilon}}
\|x\|_{\alpha-1}. \label{2.3}
\end{equation}
\end{itemize}
\end{proposition}

\begin{remark} \rm
We can remove $\tilde{\epsilon}$ in Proposition \ref{p2.1} by extending
 $T(t)$ to operators from $D(\omega-A_{-1})^{\alpha\pm \tilde{\epsilon}}$ 
to $X$, with norms bounded by $t^{\alpha-1\pm\tilde{\epsilon}}$, where
 $0<\alpha\pm\tilde{\epsilon}<1$, and therefore by
employing the reiteration theorem and the interpolation property, 
the inequality in the assertion (i) can be obtained without 
$\tilde{\epsilon}$. For a more general situation see \cite{man}.
\end{remark}

\begin{definition} \rm
An analytic semigroup $(T(t))_{t\ge 0}$ is said to be hyperbolic if 
it satisfies the following three conditions.
\begin{itemize}
\item[(i)] there exist two subspaces $X_s$ (the stable space) and
 $X_u$ (the unstable space) of $X$ such that $X=X_s\oplus X_u$;

\item[(ii)] $T(t)$ is defined on $X_u, \ T(t)X_u\subset X_u$, and 
$T(t)X_s\subset X_s$ for all $t\ge 0$;

\item[(iii)] there exist constants $M, \delta>0$ such that
\begin{equation}
\|T(t)P_s\|\le Me^{-\delta t}, \; t\ge 0, \quad
\|T(t)P_u\|\le Me^{\delta t}, \; t\le 0, \label{2.4}
\end{equation}
where $P_s$ and $P_u$ are the projections onto $X_s$ and $X_u$, respectively.
\end{itemize}
\end{definition}

Recall that an analytic semigroup $(T(t))_{t\ge 0}$ is hyperbolic 
if and only if
$\sigma(A)\cap i\mathbb{R}=\phi$, (cf. \cite[Prop. 1.15]{ngel}).
In the next proposition, we show the hyperbolicity of the extrapolated 
semigroup $(T_{\alpha-1}(t))_{t\ge 0}$. Before stating the proposition, 
we assume that the part of $A$, $A|_{P_u}:P_u(X)\to P_u(X)$ is bounded, 
which implies
$$
\|AP_u\|\le C,
$$
 where $C$ is some constant.

\begin{proposition}
Let $T(\cdot)$ be hyperbolic and $0<\alpha\le 1$. Then the operators
 $P_s$ and $P_u$
admit continuous extensions $P_{u,\alpha-1}:X_{\alpha-1}\to X$ and
$P_{s,\alpha-1}:X_{\alpha-1}\to X_{\alpha-1}$ respectively.
Moreover we have the following assertions.
\begin{itemize}
\item[(i)] $P_{u,\alpha-1}X_{\alpha-1}=P_uX$; 
\item[(ii)] $T_{\alpha-1}(t)P_{s,\alpha-1}=P_{s,\alpha-1}T_{\alpha-1}(t)$;
\item[(iii)] $T_{\alpha-1}(t): P_{u,\alpha-1}(X_{\alpha-1})\to
P_{u,\alpha-1}(X_{\alpha-1})$
is an invertible function with inverse $T_{\alpha-1}(-t)$;
\item[(iv)] for $0<\alpha-\tilde{\epsilon}<1$, we have
\begin{gather}
\|T_{\alpha-1}(t)P_{s,\alpha-1}x\|
\leq  mt^{\alpha-1-\tilde{\epsilon}}e^{-\gamma t}\|x\|_{\alpha-1}
\quad \text{for  } x\in X_{\alpha-1}  \text{ and }  t\ge 0, \label{2.5}\\
\|T_{\alpha-1}(t)P_{u,\alpha-1}x\| 
\leq  C e^{\delta t}\|x\|_{\alpha-1}
\quad \text{for  } x\in X_{\alpha-1} \text{ and }  t\le 0, \label{2.6}
\end{gather}
\end{itemize}
\end{proposition}

\begin{proposition} \label{p2.6}
For $x\in X_{\alpha-1}$ and $0\le \beta\le 1$, $0<\alpha<1$, we have the
following assertions.
\begin{itemize}
\item[(i)] there is a constant $c(\alpha,\beta)$, such that
\begin{equation}
\|T_{\alpha-1}(t)P_{u,\alpha-1}x\|_\beta\le c(\alpha,\beta)e^{\delta t}
\|x\|_{\alpha-1} \quad \text{for } t\le 0, \label{2.8}
\end{equation}
\item[(ii)] there is a constant $m(\alpha,\beta)$, such that for $t\ge 0$ and
$0<\alpha-\tilde{\epsilon}<1$.
\begin{equation}
\|T_{\alpha-1}(t)P_{s,\alpha-1}x\|_\beta\le m(\alpha,\beta)e^{-\gamma t}
t^{\alpha-\beta-\tilde{\epsilon}-1}\|x\|_{\alpha-1}. \label{2.9}
\end{equation}
\end{itemize}
\end{proposition}

\begin{definition} \rm
A continuous function $f:\mathbb{R}\to X$, is called almost automorphic, if for
every sequence $(\sigma_n)_{n\in \mathbb{N}}$ of real numbers,
 there is a subsequence
$(s_n)_{n\in \mathbb{N}}\subset (\sigma_n)_{n\in \mathbb{N}}$ such that
$$
\lim_{n,m\to \infty}f(t+s_n-s_m)=f(t), \quad\text{for  each  } 
t\in \mathbb{R}.
$$
This is equivalent to
$$
g(t)=\lim_{n\to \infty}f(t+s_n), \quad\text{and}\quad
f(t)=\lim_{n\to \infty}g(t-s_n),
$$
are well defined for each $t\in \mathbb{R}$. The function $g$ 
in the above definition measurable but not necessarily continuous.
\end{definition}

\begin{remark} \rm
An almost automorphic function is continuous but may not be uniformly 
continuous, for e.g.
let $p(t)=2+\cos (t)+\cos (\sqrt2 t)$ and
$f:\mathbb{R}\to \mathbb{R}$ defined as
$f:=\sin(1/p)$, then $f\in AA(X)$, but $f$ is not uniformly continuous
 on $\mathbb{R}$, so $f\notin AP(X)$.
\end{remark}

\begin{lemma} \label{l2.8}
We have the following properties of almost automorphic functions:
\begin{itemize}
\item[(a)] For $f\in AA(X)$, the range 
$\mathcal{R}_f:=\{f(t) : t\in \mathbb{R}\}$
is precompact in $X$, so that $f$ is bounded.

\item[(b)] For $f, g\in AA(X)$ then $f+g\in AA(X)$.

\item[(c)] Assume that $f_n\in AA(X)$ and $f_n\to g$ uniformly
 on $\mathbb{R}$, then $g\in AA(X)$.

\item[(d)] $AA(X)$, equipped with the sup norm given by
\begin{equation}
\|f\|=\sup_{t\in \mathbb{R}}\|f(t)\|,
\end{equation}
turns out to be a Banach space.
\end{itemize}
\end{lemma}

\subsection{$S^p$-Almost automorphy}

\begin{definition}\cite{pnk} \rm
The Bochner transform $f^b(t,s)$, $t\in \mathbb{R}, \, s\in [0,1]$
of a function $f:\mathbb{R}\to X$ is defined by $f^b(t,s):=f(t+s)$.
\end{definition}

\begin{definition} \rm
The Bochner transform $f^b(t,s,u)$, $t\in \mathbb{R}$,  $s\in [0,1]$, 
$u\in X$ of a function $f(t,u)$ on $\mathbb{R}\times X$, with values
 in $X$, is defined by
 $$
f^b(t,s,u):=f(t+s,u)
$$ 
for each $x\in X$.
\end{definition}

\begin{definition} \rm
For $p\in (1,\infty)$, the space $BS^p(X)$ of all Stepanov bounded functions,
with the exponent $p$, consists of all measurable functions
$f:\mathbb{R}\to X$
such that $f^b$ belongs to $L^{\infty}(\mathbb{R};L^p((0,1),X))$. This is a
Banach space with the norm
\begin{equation}
\|f\|_{S^p}:= \|f^b\|_{L^\infty(\mathbb{R},L^p)}=\sup_{t\in \mathbb{R}}
\Big(\int_t^{t+1}\|f(\tau)\|^pd\tau\Big)^{1/p}.
\end{equation}
\end{definition}

\begin{definition}\cite{gp} \rm
 The space $AS^p(X)$ of Stepanov almost automorphic functions
(or $S^p$-almost automorphic) consists of all $f\in BS^p(X)$ such that
$f^b\in AA(L^p(0,1;X))$. That is, a function $f\in L^p_{\rm loc}(\mathbb{R},X)$
is said to be $S^p$-almost automorphic if its Bochner transform
$f^b:\mathbb{R}\to L^p(0,1;X)$ is almost automorphic in the sense that,
for every sequence $(s'_n)_{n\in \mathbb{N}}$ of real numbers,
there exists a subsequence $(s_n)_{n\in \mathbb{N}}$ and a function 
$g\in L^p_{\rm loc}(\mathbb{R},X)$ such that
\begin{gather*}
\Big[\int_t^{t+1}\|f(s_n+s)-g(s)\|^pds \Big]^{1/p}\to 0,\\
\Big[\int_t^{t+1}\|g(s-s_n)-f(s)\|^pds \Big]^{1/p}\to 0,
\end{gather*}
as $n\to \infty$ pointwise on $\mathbb{R}$.
\end{definition}

\begin{remark} \rm
$AS^p(X_{\alpha-1})$ is the extrapolated space of $AS^p(X_\alpha)$
 equipped with norm $\|\cdot\|_{S_{\alpha-1}^p}$, given by
$$
\|f\|_{S_{\alpha-1}^p}:=\sup_{t\in \mathbb{R}}\Big(\int_t^{t+1}\|f(\tau)\|
_{\alpha-1}^pd\tau\Big)^{1/p}.
$$
\end{remark}

\begin{remark} \rm
It is clear that if $1\le p<q<\infty$ and $f\in L^q_{\rm loc}(\mathbb{R};X)$ is 
$S^q$-almost automorphic, then $f$ is $S^p$-almost automorphic. 
Also if $f\in AA(X)$,
then $f$ is $S^p$-almost automorphic for any $1\le p<\infty$.
\end{remark}

Let $(Y, \|\cdot\|_Y)$ be an abstract Banach space.

\begin{definition}\rm
A function $F:\mathbb{R}\times Y\to X$, $(t,u)\mapsto F(t,u)$ with
$F(\cdot,u)\in L^p_{\rm loc}(\mathbb{R};X)$ for each $u\in Y$, 
is said to be $S^p$-almost automorphic in $t\in \mathbb{R}$ uniformly 
in $u\in Y$ if $t\mapsto F(t,u)$ is
$S^p$-almost automorphic for each $u\in Y$, that is for every sequence
of real numbers $(s'_n)_{n\in \mathbb{N}}$, there exists a subsequence
$(s_n)_{n\in \mathbb{N}}$ and a function
 $G(\cdot,u)\in L^p_{\rm loc}(\mathbb{R}, X)$
such that following statements hold
\begin{gather*}
\Big[\int_t^{t+1}\|F(s_n+s)-G(s)\|^pds \Big]^{1/p}\to 0, \\
\Big[\int_t^{t+1}\|G(s-s_n)-F(s)\|^pds \Big]^{1/p}\to 0,
\end{gather*}
as $n\to \infty$ pointwise on $\mathbb{R}$ for each $u\in Y$.
\end{definition}

The collection of all $S^p$-almost automorphic functions from 
$f:\mathbb{R}\times Y\mapsto X$
will be denoted by $AS^p(\mathbb{R}\times Y)$.
Now we have the following composition theorem due 
to Diagana \cite{td}.

\begin{theorem}\cite{td} \label{thm2.16}
Assume that $\phi\in AS^p(Y)$ such that 
$K:=\overline{\{\phi(t) :  t\in \mathbb{R}\}}\subset Y$ is a relatively compact
subset of $X$. Let $F\in AS^p(\mathbb{R}\times Y)$
and let the function $(t,u) \mapsto F(t,u)$ be Lipschitz continuous 
that is there exists a constant $L>0$ such
that 
$$
\|F(t,u)-F(t,v)\|\le L\|u-v\|_Y,
$$
for all $t\in \mathbb{R}, (u,v) \in Y\times Y$.
Then the function $\Gamma:\mathbb{R}\to X$
defined by $\Gamma(\cdot):=F(\cdot,\phi(\cdot))$ belongs to $AS^p(X)$.
\end{theorem}

\section{Main results}

In this section we discuss the existence and uniqueness of almost
automorphic solutions of the following semilinear evolution equation,
\begin{equation}
x'(t)=A_{\alpha-1}x(t)+f(t,x(t)), \quad t\in \mathbb{R}, \label{see}
\end{equation}
with the following assumptions;
\begin{itemize}
\item[(A1)] $A$ is the sectorial operator and the generator of a hyperbolic
analytic semigroup $(T(t))_{t\ge 0}$.

\item[(A2)]  $f:\mathbb{R}\times X_\beta\to X_{\alpha-1}$, is Stepanov-like
almost automorphic in $t$, for each $x\in X_\beta$.

\item[(A3)] $f$ is uniformly Lipschitz with respect to the second argument,
 that is
\begin{equation}
\|f(t,x)-f(t,y)\|_{\alpha-1}\le k\|x-y\|_\beta,  \label{3.2}
\end{equation}
for all $t\in \mathbb{R}, \ x,y\in X_\beta$, and some constant $k>0$.
\end{itemize}

\begin{definition}\rm
A continuous function $x:\mathbb{R}\to X_\beta$, is said to be a mild solution
of \eqref{see}, if it satisfies following variation of constants formula
\begin{equation}
x(t)=T(t-s)x(s)+\int_s^t T_{\alpha-1}(t-\sigma)f(\sigma,x(\sigma))d\sigma
\end{equation}
for all $t\ge s, \ t,s\in \mathbb{R}$.
\end{definition}

\begin{definition} \rm
A function $u:\mathbb{R}\to X_\beta$, is said to be a bounded solution of
\eqref{see} provided that
\begin{equation}
u(t)=\int_{-\infty}^t T_{\alpha-1}(t-\sigma)P_{s,\alpha-1}f(\sigma,u(\sigma))
d\sigma
-\int_t^\infty T_{\alpha-1}(t-\sigma)P_{u,\alpha-1}f(\sigma,u(\sigma))d\sigma,
\label{3.4}
\end{equation}
$t\in \mathbb{R}$.
\end{definition}

Throughout the rest of this paper, we assume
$\mathcal{H} u(t):=H_1u(t)+H_2u(t)$,
where
\begin{gather*}
H_1u(t):=\int_{-\infty}^tT_{\alpha-1}(t-\sigma)
P_{s,\alpha-1}f(\sigma,u(\sigma))d\sigma,\\
H_2u(t):=\int _t^\infty T_{\alpha-1}(t-\sigma)
P_{u,\alpha-1}f(\sigma,u(\sigma))d\sigma,
\end{gather*}
for all $t\in \mathbb{R}$.

\begin{lemma} \label{l3.3}
Assume that  assumptions {\rm (A1)--(A3)} are satisfied. If
\begin{equation}
M(\alpha,\beta,q,\gamma)
:=\sum_{n=1}^\infty\Big[\int_{n-1}^ne^{-\gamma q \sigma}
\sigma^{-q(\beta+1+
\tilde{\epsilon}-\alpha)}d\sigma\Big]^{1/q} <\infty, \label{3.5}
\end{equation}
then the operator $\mathcal{H}$ maps $AA(X_\beta)\mapsto AA(X_\beta)$.
\end{lemma}

\begin{proof}
Let $u$ be in $AA(X_\beta)$. Then $u\in AS^p(X_\beta)$ and by Lemma \ref{l2.8}
the set $\overline{\{u(t) : t\in \mathbb{R}\}}$ is compact in $X_\beta$.
Since $f$ is Lipschitz, then it follows from
Theorem \ref{thm2.16} (also see \cite[Theorem 2.21]{td1}) that the function
$\phi(t):=f(t,u(t))$ belongs to
$AS^p(X_\beta)$. Now we show that $\mathcal{H}u\in AA(X_\beta)$.

For that we first define a sequence of integral operators
$\{\phi_n\}$ as follows
\begin{equation}
\phi_n(t):=\int_{n-1}^n T_{\alpha-1}(t-\sigma)P_{s,\alpha-1}g(\sigma)d\sigma,
\quad t\in \mathbb{R} \text{ \ and \ }  n=1,2,3\dots
\end{equation}
Putting $r=t-\sigma$,
\begin{equation}
\phi_n(t):=\int_{t-n}^{t-n+1} T_{\alpha-1}(r)P_{s,\alpha-1}g(t-r)dr.
\end{equation}
Let $0<\tilde{\epsilon}+\beta<\alpha$, $0<\alpha-\tilde{\epsilon}<1$ and
using Proposition \ref{p2.6} we have
\begin{align*}
\|\phi_n(t)\|_\beta
&\le  \int_{t-n}^{t-n+1}m(\alpha,\beta) r^{\alpha-1-\beta-\tilde{\epsilon}}
e^{-\gamma r}\|g(t-r)\|_{S^p_{\alpha-1}}dr\\
&\text{ now, $r\to (t-r)$,}\\
&\leq  \int_{n-1}^{n}m(\alpha,\beta)(t-r)^{\alpha-1-\beta-\tilde{\epsilon}}
e^{-\gamma (t-r)}\|g(r)\|_{S^p_{\alpha-1}}dr, \\
&\leq  \int_{n-1}^n m(\alpha,\beta)\sigma^{\alpha-\beta-1-\tilde{\epsilon}}
e^{-\gamma \sigma}\|g\|_{S^p_{\alpha-1}}d\sigma,\\
&\leq q(\alpha,\beta)\Big[\int_{n-1}^ne^{-\gamma q\sigma}
\sigma^{q(\alpha-\beta-1-\tilde{\epsilon})}d\sigma\Big]^{1/q}
\|g\|_{S_{\alpha-1}^p}.
\end{align*}
By Weierstrass theorem and \eqref{3.5}, it follows that the series
$$
\Phi(t):=\sum_{n=1}^\infty \phi_n(t)
$$
is uniformly convergent on $\mathbb{R}$.
Moreover $\Phi\in C(\mathbb{R},X_\beta)$;
\begin{equation}
\|\Phi(t)\|_\beta\le \sum_{n=1}^\infty \|\phi_n(t)\|_\beta\le
q(\alpha,\beta)M(\alpha,\beta,q,\gamma)\|\phi\|_{S_{\alpha-1}^p.}
\end{equation}
We show that for all $n=1,2,3$, $\phi_n\in AA(X_\beta)$.
Since $g\in AS^p(X_{\alpha-1})$, which implies that for every sequence
$(s'_n)_{n\in \mathbb{N}}$ of real numbers, there exist a subsequence
$(s_n)_{n\in \mathbb{N}}$ and a function $g'$ such that
\begin{equation}
\int_t^{t+1}\|g(\sigma+s_n)-g'(\sigma)\|^p_{\alpha-1}d\sigma \to 0.
\end{equation}
Let us define another sequence of integral operators
\begin{equation}
\widehat{\phi_n}(t)=\int_{n-1}^n T_{\alpha-1}(t-\sigma)P_{s,\alpha-1}g'(\sigma)d\sigma
 \quad \text{for }  n=1,2,3,\dots .
\end{equation}
Now we show for $n=1,2,3,\dots$ that $\phi_n\in AA(X_\beta)$.
Since $g\in AS^p(X_{\alpha-1})$, for every sequence
$(s'_n)_{n\in \mathbb{N}}$ of real
numbers, there exists a subsequence $(s_n)_{n\in \mathbb{N}}$
and a function $g'$ such that
\begin{equation}
\int_t^{t+1}\|g(\sigma+s_n)-g'(\sigma)\|_{\alpha-1}^pd\sigma\to 0.
\end{equation}
Define for all $n=1,2,3,\dots $ another sequence of integral operators
\begin{equation}
\widehat{\phi_n}(t)=\int_{n-1}^n T_{\alpha-1}(t-\sigma)
 P_{s,\alpha-1}g'(\sigma)d\sigma,
\end{equation}
for all $t\in \mathbb{R}$. Consider
\begin{align*}
&\phi_n(t+s_{n_k})-\widehat{\phi_n}(t)\\
&=\int_{n-1}^n T_{\alpha-1}(t+s_{n_k}-\sigma)
P_{s,\alpha-1}g(\sigma)d\sigma-\int_{n-1}^n T_{\alpha-1}(t-\sigma)
P_{s,\alpha-1}g'(\sigma)d\sigma,\\
&=\int_{n-1}^n T_{\alpha-1}(t-\sigma)
P_{s,\alpha-1}g(\sigma+s_{n_k})d\sigma
-\int_{n-1}^n T_{\alpha-1}(t-\sigma)
P_{s,\alpha-1}g'(\sigma)d\sigma,\\
&=\int_{n-1}^nT_{\alpha-1}(t-\sigma)P_{s,\alpha-1}
\big[g(\sigma+s_{n_k})-g'(\sigma)\big]d\sigma.
\end{align*}
Using Proposition \ref{p2.6}, we have
\begin{align*}
&\|\phi_n(t+s_{n_k})-\widehat{\phi_n}(t)\|_\beta\\
&\le \int_{n-1}^n m(\alpha,\beta)e^{-\gamma(t-\sigma)}
(t-\sigma)^{-(\beta-\alpha+\tilde{\epsilon}+1)}
\|g(\sigma+s_{n_k})-g'(\sigma)\|_{S^p_{\alpha-1}}d\sigma\\
&\to 0, \quad \text{as  } k\to \infty, \; t\in \mathbb{R},
 \quad (\text{since } g\in AS^p(X_{\alpha-1})).
\end{align*}
This implies that $\widehat{\phi_n}(t)=\lim_{k\to \infty}\phi_n(t+s_{n_k})$,
$n=1,2,3,\dots$ and $t\in \mathbb{R}$.

In a similar way, one can show that
 $\phi_n(t)=\lim_{k\to \infty} \widehat\phi_n(t-s_{n_k})$, for all 
$t\in \mathbb{R}$ and $n=1,2,3,\dots$.
Therefore for each $n=1,2,3,\dots$, the sequence $\phi_n\in AA(X_{\beta})$.
\end{proof}

Now we state the main result of this Section.
\begin{theorem} \label{mthm}
Let $0\le \beta<\alpha$, $\tilde{\epsilon}>0$ such that
$0<\alpha-\tilde{\epsilon}<1$ and $0<\beta+\tilde{\epsilon}<\alpha$, moreover
assume that the constant
\[
K:=k.m(\alpha,\beta)\gamma^{\beta-\alpha+\tilde{\epsilon}}
\Gamma(\alpha-\beta-\tilde{\epsilon})+c(\alpha,\beta)\delta^{-1}<1
\]
and  equation \eqref{3.5} hold. Then under assumptions {\rm (A1)--(A3)}
and for $f\in AS^p(\mathbb{R}\times X_\beta, X_{\alpha-1})$,
equation \eqref{see} has unique almost automorphic solution $u\in AA(X_\beta)$,
satisfying the following variation of constants formula.
\[
u(t)=\int_{-\infty}^t T_{\alpha-1}(t-\sigma)P_{s,\alpha-1}f(\sigma,u(\sigma))d\sigma
-\int_t^\infty T_{\alpha-1}(t-\sigma)P_{u,\alpha-1}f(\sigma,u(\sigma))d\sigma,
\]
$t\in \mathbb{R}$.
\end{theorem}

\begin{proof}
We first show that $\mathcal{H}$ is a contraction. Let
$v, w\in AA(X_\beta)$ and consider the following
\begin{align*}
&\|H_1v(t)-H_1w(t)\|_\beta\\
&\le \int_{-\infty}^t m(\alpha,\beta)
(t-s)^{\alpha-\beta-1-\tilde{\epsilon}}e^{-\gamma(t-s)}
\|f(s,v(s))-f(s,w(s))\|_{\alpha-1}ds\\
&\le\int_{-\infty}^tkm(\alpha,\beta)(t-s)^{\alpha-\beta-1-
\tilde{\epsilon}}e^{-\gamma(t-s)}\|v(s)-w(s)\|_\beta ds\\
&\le k.m(\alpha,\beta)\gamma^{\beta-\alpha+\tilde{\epsilon}}
\Gamma(\alpha-\beta-\tilde{\epsilon})\|v-w\|_\beta,
\end{align*}
where $\Gamma(\alpha):=\int_0^\infty t^{\alpha-1} e^{-t}dt$.
Similarly we have
\begin{align*}
\|H_2v(t)-H_2w(t)\|_\beta
&\leq  \int_t^\infty c(\alpha,\beta)
e^{-\delta(t-s)}\|v(s)-w(s)\|_\beta ds\\
&\leq  c(\alpha,\beta)\delta^{-1}\|v-w\|_\beta.
\end{align*}
Consequently,
\begin{align*}
\|\mathcal{H}v(t)-\mathcal{H}v(t)\|_\beta
&\leq \Big(k.m(\alpha,\beta)\gamma^{\beta-\alpha+\tilde{\epsilon}}
\Gamma(\alpha-\beta-\tilde{\epsilon})+c(\alpha,\beta)\delta^{-1}\Big)
\|v-w\|_\beta\\
&< \|v-w\|_\beta.
\end{align*}
Hence by the well-known Banach contraction principle, $\mathcal{H}$
has unique fixed point $u$ in $AA(X_\beta)$ satisfying $\mathcal{H}u=u$
(cf. Lemma \ref{l3.3} for almost automorphy of solution).
\end{proof}

\section{Semilinear boundary differential equations} \label{s4}

Consider the  semilinear boundary differential equation
\begin{equation}\label{SBDE2}
\begin{gathered}
x'(t)=A_mx(t)+h(t,x(t)), \quad t\in \mathbb{R}, \\
Lx(t)=\phi(t,x(t)), \quad t\in \mathbb{R},
\end{gathered}
\end{equation}
where $(A_m,D(A_m))$ is a densely defined linear operator on a
Banach space $X$ and $L: D(A_m)\to \partial X$, the boundary Banach
space and the functions
$h:\mathbb{R}\times X_m\to \partial X$ and
$\phi:\mathbb{R}\times X_m\to \partial X$ are continuous.

Likewise \cite{bol, brn} here we assume the assumptions introduced by 
Greiner \cite{gre} which are given as follows
\begin{itemize}
\item[(H1)] There exists a new norm $|\cdot|$ which makes the domain $D(A_m)$ complete
and then denoted by $X_m$. The space $X_m$ is continuously embedded in $X$ and
$A_m\in \mathcal{L}(X_m,X)$.

\item[(H2)] The restriction operator $A:=A_m|_{ker(L)}$ is a sectorial operator
such that $\sigma(A)\cap i\mathbb{R}=\phi$.

\item[(H3)] The operator $L: X_m\to \partial X$ is bounded and surjective.

\item[(H4)] $X_m\hookrightarrow X_\alpha$ for some $0<\alpha<1$.

\item[(H5)] $h:\mathbb{R}\times X_\beta\to X$ and $\phi: \mathbb{R}\times
X_\beta\to \partial X$ are continuous for $0\le \beta<\alpha$.

\end{itemize}
 A function $x:\mathbb{R}\to X_\beta$ is a mild solution of
\eqref{SBDE} if we have the following
\begin{itemize}
\item[(i)] $\int_s^t x(\tau)d\tau\in X_m$,
\item[(ii)] $x(t)-x(s)=A_m\int_s^t x(\tau)d\tau
+\int_s^t h(\tau,x(\tau))d\tau$,
\item[(iii)] $L\int_s^tx(\tau)d\tau=\int_s^t\phi(\tau,x(\tau))d\tau$,
\end{itemize}
for all $t\ge s, \ t,s \in \mathbb{R}$.

Now we transform \eqref{SBDE} to the equivalent semilinear evolution equation
\begin{equation}
x'(t)=A_{\alpha-1}x(t)+h(t,x(t))-A_{\alpha-1}L_0\phi(t,x(t)), \quad
 t\in \mathbb{R}, \label{4.1}
\end{equation}
where $L_0:=(L|_{}Ker(A_m))^{-1}$.

\begin{theorem} \label{mthm2}
Assume that functions $\phi\in AS^p(\mathbb{R}\times X_\beta,\partial X)$ 
and  $h\in AS^p(\mathbb{R}\times X_\beta, X)$, are globally Lipschitzian 
with small lipschitz constants.
Then under the assumptions (H1)-(H5), the semilinear boundary differential
equation \eqref{SBDE} has a unique mild solution $x\in AA(X_\beta)$, 
satisfying the following formula for all $t\in \mathbb{R}$.
\begin{equation} \label{4.2}
\begin{aligned}
x(t)&= \int_{-\infty}^t T(t-s)P_sh(s,x(s))ds
-\int_t^\infty T(t-s)P_uh(s,x(s))ds \\
&\quad -A\Big[\int_{-\infty}^t T(t-s)P_sL_0\phi(s,x(s))ds
-\int_t^\infty T(t-s)P_uL_0\phi(s,x(s))ds\Big].
\end{aligned}
\end{equation}
\end{theorem}

\begin{proof}
It is clear that $A_{\alpha-1}L_0$ is a bounded operator from
$\partial X\to X_{\alpha-1}$. Since $\phi\in AS^p(\mathbb{R}\times X_\beta,
\partial X)$ and 
$h\in AS^p(\mathbb{R}\times X_\beta,X)$ and from the injection
$X\hookrightarrow X_{\alpha-1}$, the function
 $f(t,x):=h(t,x)-A_{\alpha-1}L_0\phi(t,x)
\in AS^p(\mathbb{R}\times X_\beta,X_{\alpha-1})$. 
This function is also globally
Lipschitzian with a small constant. Hence by Theorem \ref{mthm},
 there is a unique mild solution $x\in AA(X_\beta)$ of \eqref{4.1}, satisfying
\[
x(t)=\int_{-\infty}^t P_{s,\alpha-1}T_{\alpha-1}(t-s)f(s,x(s))ds-
\int_t^\infty P_{u,\alpha-1}T_{\alpha-1}(t-s)f(s,x(s))ds,
\]
from which we deduce the variation of constants formula \eqref{4.2} and
$x\in AA(X_\beta)$ is the unique mild solution.
\end{proof}

\begin{example} \rm Consider the partial differential equation
\begin{equation} \label{4.3}
\begin{gathered}
\frac{\partial}{\partial t}u(t,x)=\Delta u(t,x)+au(t,x),
\quad t\in \mathbb{R}, \; x\in \Omega \\
\frac{\partial}{\partial n}u(t,x)=\Gamma(t,m(x)u(t,x)),
\quad t\in \mathbb{R}, \; x\in \partial \Omega.
\end{gathered}
\end{equation}
Where $a\in \mathbb{R}_+$ and $m$ is a $\mathbb{C}^1$ function
and $\Omega\subset \mathbb{R}^n$ is a bounded open subset of $\mathbb{R}^n$
with smooth boundary $\partial \Omega$.
Here we use the following notation/conventions:
$X=L^2(\Omega)$, $X_m=H^2(\Omega)$ and the boundary space
$\partial X=H^{1/2}(\partial \Omega)$. The operators $A_m: X_m\to X$,
given by $A_m \varphi=\Delta \varphi+a\phi$ and $L: X_m\to \partial X$, given by
$L\varphi:=\frac{\partial\varphi}{\partial n}$.
The operator $L$ is bounded and surjective,
follows from Sections \cite[4.3.3, 4.6.1]{tri}. It is also
known that the operator $A:=A_m|_{\ker L}$
generates an analytic semigroup, moreover we also have
$X_m\hookrightarrow X_\alpha$ for
$\alpha<3/4$ (cf. \cite[Sections 4.3.3, 4.6.1]{tri}).
The eigenvalues of Neumann Laplacian $A$ is a decreasing sequence
$(\lambda_n)$ with
$\lambda_0=0$, $\lambda_1<0$,
taking $a=-\lambda_1/2$, we have $\sigma(A)\cap i\mathbb{R}=\phi$. Hence the
analytic semigroup generated by $A$ is hyperbolic.
$$
\phi(t,\varphi)(x)=\Gamma(t,m(x)\varphi(x))=\frac{kb(t)}{1+|m(x)\varphi(x)|},
 \quad t\in \mathbb{R}, \; x\in \partial\Omega
$$
where $b(t)$ is $S^p$ Stepanov-like almost automorphic function and
$b(\cdot)$ has relatively compact range. It can be easily seen that
$\phi$ is continuous on $\mathbb{R}\times H^{2\beta'}(\Omega)$ for
some $\frac{1}{2}<\beta<\beta'<\frac{3}{4}$, which is embedded in
$\mathbb{R}\times X_\beta$ (cf. \cite{tri}). Using the definitions
of fractional Sobolev spaces, one can easily show that
$\phi(t,\varphi)(.)\in H^{1/2}(\partial\Omega)$ for all
$\varphi\in H^{2\beta'}\hookrightarrow H^1(\Omega)$. 
Moreover $\phi$ is globally Lipschitzian for each $\varphi\in X_\beta$.
Now for a small constant $k$, all assumptions of Theorem \ref{mthm2} are
satisfied. 
Hence \eqref{4.3} admits a unique almost automorphic mild solution $u$
with values in $X_\beta$.
\end{example}

\subsection*{Acknowledgements} 
Authors are thankful to the anonymous referee for his/her useful
comments/suggestions, which really helped us to improve our old manuscript. 
The first author would like to thank UGC-India for providing the 
financial support for this work.
The second author acknowledges the financial suupport from the Department 
of Science and Technology, New Delhi, under project No. SR/S4/MS:796/12.

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\end{document}