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

\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2006(2006), No. 79, pp. 1--10.\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 2006 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2006/79\hfil Pseudo almost periodic solutions]
{Pseudo almost periodic solutions to some differential equations
with infinite delay}

\author[T. Diagana\hfil EJDE-2006/79\hfilneg]
{Toka Diagana}

\address{Toka Diagana \newline
Department of Mathematics, Howard University,
2441 6th Street N.W., Washington D.C. 20059, USA}
\email{tdiagana@howard.edu}

\date{}
\thanks{Submitted May 29, 2006. Published July 19, 2006.}
\subjclass[2000]{34G20, 47A55}
\keywords{Sectorial operator; analytic semigroup; infinite delay;
\hfill\break\indent
hyperbolic semigroup; almost periodic; pseudo almost periodic;
heat equation}

\begin{abstract}
 This paper studies some sufficient conditions for the existence and
 uniqueness of a pseudo almost periodic mild solution to
 abstract hyperbolic differential equations with infinite delay.
 Applications include the existence and uniqueness of a pseudo
 almost periodic mild solution to the heat equation.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{example}[theorem]{Example}
\numberwithin{equation}{section}

\section{Introduction}

Let $(\mathcal{M}, \|\cdot\|)$ be a Banach space. In Burton and
Zhang \cite{TAB}, the degree-theoretic by Granas \cite{G} was used
to obtain some sufficient conditions for the existence of
$T$-periodic solutions to the differential equations with infinite
delay given by
\begin{eqnarray}\label{delay}
\frac{du}{dt} + Lu = \int_{-\infty}^t C(t,s) u(s) ds + f(u) +
F(t),
\end{eqnarray}
where $-L$ is a sectorial linear operator on $\mathcal{M}$, $f: \mathcal{M} \mapsto
\mathcal{M}$, $F: \mathbb{R} \mapsto \mathcal{M}$ are some $\mathcal{M}$-valued smooth functions with
$F(t + T) = F(t)$, $(C(t,s))_{t \geq s}$ is a bounded linear
operator on $\mathcal{M}$ with $C(t+T, s+T) = C(t,s)$, for some $T > 0$.
Abstract results were then applied to some nonlinear heat
equations with memory.

In this paper we study some sufficient condition for the existence
and uniqueness of a pseudo almost periodic (mild) solution to
(\ref{delay}). One should point out that the Banach fixed-point
combined with techniques related to the theory of intermediate
spaces will be preferred to that of Granas' theory used in
\cite{TAB}. In particular, we will be studying the case when the
sectorial operator $-L$ is hyperbolic, equivalently,
$$
\sigma (-L) \cap i \mathbb{R} = \{\emptyset\}.
$$
Upon making some
assumptions, it will be shown that (\ref{delay}) has a unique
$\mathcal{M}_\alpha$-valued pseudo almost periodic mild solution
($\mathcal{M}_\alpha \subset \mathcal{M}$ being an abstract intermediate space).
Applications include the study of pseudo almost periodic mild
solutions to the Cauchy problem for the heat equation in $\mathcal{M} =
C[0,1]$ given by the partial differential equations
\begin{gather}
 u_t(t,x) = u_{xx}(t,x) + H u(t,x)+ f(u(t,x)) + F(t)
+ \int_{-\infty}^t C(t, s) u(s,x)ds \label{11}\\
u(t,0) = u(t, 1) = 0, \quad \forall \quad t \in \mathbb{R},\label{12}
\end{gather}
where $H \in \mathbb{R}$ is a constant, and $f, F$ are some appropriate
functions.

In fact, to study (\ref{delay}) we consider the problem, which
consists of studying pseudo almost periodic solutions to the
differential equation
\begin{eqnarray}\label{14}
\frac{du}{dt}  + Lu = Ku + f(u) + F(t), \quad \forall t \in \mathbb{R},
\end{eqnarray}
where $-L$ is a sectorial linear operator on $\mathcal{M}$, and $K: \mathcal{M}
\mapsto \mathcal{M}$ is the linear operator defined by
$$
 K u(t) := \int_{-\infty}^t C(t,s) u(s) ds.
$$

To deal with (\ref{14}), as in \cite{BMN, GT} we will make
extensive use of some abstract intermediate spaces$\mathcal{M}_\alpha$ for
$\alpha \in (0 , 1)$. In contrast with the fractional power spaces
considered in some recent papers of the author et al.~\cite{dmn,
DMNT}, the interpolation and H\"older spaces, for instance, depend
only on $D(L)$ and $\mathcal{M}$ and can be explicitly expressed in many
concrete examples. The literature on those intermediate spaces is
very extensive; thus we refer the reader only to the excellent
book by Lunardi~\cite{lunardi}, which contains a comprehensive
presentation on this topic and related issues.


The concept of the pseudo almost periodicity, which is the central
question in this paper was introduced in the literature in the
early nineties by Zhang~\cite{z1, z2, z3} as a natural
generalization of the well-known Bohr almost periodicity. Thus
this new concept is welcome to implement another existing
generalization of almost periodicity, the concept of
asymptotically almost periodicity due to
Fr\'echet~\cite{cor,fink}. For more on these new and old concepts
and related topics, see, e.g.,
\cite{Ezzinbi1,Dads1,Maniar,Cuevas1,Diagana1,Diagana2,DH,z1}
and the references therein.

The existence of almost periodic, asymptotically almost periodic,
and pseudo almost periodic solutions is one of the most attracting
topics in the qualitative theory of differential equations due to
their significance and applications in physics, mathematical
biology, chemistry, control theory, and many others.

Some contributions on almost periodic, asymptotically almost
periodic, and pseudo almost periodic solutions to abstract
differential and partial differential equations have recently been
made in \cite{Maniar, Cuevas1, Diagana1, Diagana2, dmn, Li1}.
However, the existence of pseudo almost periodic to (\ref{delay})
in the case when $-L$ is hyperbolic is an untreated topic and this
is the main motivation of the present paper. Among other things,
some sufficient conditions for the existence and uniqueness of a
pseudo almost periodic (mild) solution to \eqref{delay} are
obtained whenever $-L$ generates a hyperbolic analytic semigroup
$(T(t))_{t\geq0}$, which is not necessarily strongly continuous at
$0$.

\section{preliminaries}

This section is devoted to some preliminary facts needed in the
sequel. We basically use a similar setting as in \cite{BMN, GT}.
Throughout the rest of this paper, $(\mathcal{M}, \|\cdot\|)$ stands for a
Banach space, $-L$ is a sectorial linear operator
(Definition~\ref{sect}) which is not necessarily densely defined.

If $L$ is a linear operator on $\mathcal{M}$, then $\rho(L)$, $\sigma(L)$,
$D(L)$, $N(L)$, $R(L)$ stand for the resolvent, spectrum, domain,
kernel, and range of $L$.

If $\mathcal{M}_0, \mathcal{M}_1$ are Banach spaces, then the space $(B(\mathcal{M}_0 ,
\mathcal{M}_1), \|\cdot\|_{B(\mathcal{M}_0, \mathcal{M}_1)})$ denotes the Banach space of all
bounded linear operators from $\mathcal{M}_0$ into $\mathcal{M}_1$ equipped with its
operator topology. If $\mathcal{M}_0 = \mathcal{M}_1$, then this is simply denoted
$B(\mathcal{M}_0)$.

\subsection{Sectorial Linear Operators and Their Semigroups}

\begin{definition} \label{def2.1} \rm
A linear operator $L: D(L) \subset \mathcal{M} \mapsto \mathcal{M}$ (not necessarily
densely defined) is said to be sectorial if the following hold:
There exist constants $\omega\in \mathbb{R}$, $\theta\in (\frac\pi2,\pi)$,
and $M>0$ such that
\begin{equation} \label{sect}
\begin{gathered}
\rho(L)\supset S_{\theta,\omega}:=\{\lambda\in \mathbb{C}:\lambda\neq\omega,
 |\mathop{\rm arg}(\lambda-\omega)|<\theta\},  \\
 \|R(\lambda, L)\|\leq \frac{M}{|\lambda-\omega|},  \quad
\lambda\in S_{\theta,\omega},
\end{gathered}
\end{equation}
where $R(\lambda, L) = (\lambda I - L)^{-1}$ for each $\lambda\in
\rho(L).$
\end{definition}

\begin{remark} \label{rmk2.2} \rm
If $-L$ is sectorial, it then generates an analytic semigroup
$(T(t))_{t\geq0}$, which maps $(0,\infty)$ into $B(\mathcal{M})$ such that
there exist $M_0, M_1 > 0$ with
\begin{equation}
\begin{gathered}
\|T(t)\|\leq M_0 e^{\omega t}, \quad t> 0,\\
\|t(-L-\omega I)T(t)\|\leq M_1 e^{\omega t}, \quad t> 0.
\end{gathered}\label{analy}
\end{equation}
\end{remark}

For the rest of the paper, we suppose that the analytic
semigroup $(T(t))_{t\geq0}$ associated with $-L$ is hyperbolic,
that is, there exist a projection $P$ and constants $M, \delta>0$
such that each $T(t)$ commutes with $P$, $N(P)$ is invariant with
respect to $T(t)$, $T(t): R(Q)\longrightarrow R(Q)$ is invertible
and
\begin{gather}\label{hyP}
\|T(t)Px\|\leq Me^{-\delta t}\|x\| \quad \mbox{for } t\geq0, \\
\label{hyQ}
  \|T(t)Qx\|\leq Me^{\delta t}\|x\| \quad\mbox{for } t\leq0,
 \end{gather}
 where $Q:=I-P$ and,  for $ t\leq0$,  $T(t):= (T(-t))^{-1} $.

Recall that if a semigroup $(T(t))_{t\geq0}$ is analytic,  then
$(T(t))_{t\geq0}$ is hyperbolic if and only if
$$
\sigma(-L)\cap i\mathbb{R}=\emptyset,
$$
see for instance \cite[Prop 1.15, p.305]{nag}.


\subsection{Intermediate Spaces}

\begin{definition} \label{def2.3} \rm
Let $\alpha\in(0,1)$. A Banach space $(\mathcal{M}_\alpha,
\|\cdot\|_\alpha)$ is said to be an intermediate
 space between $D(-L)$ ($ = D(L)$) and $\mathcal{M}$, or a space of class $\mathcal{J}_\alpha$,
 if $D(L)\subset \mathcal{M}_\alpha\subset \mathcal{M}$ and there is a constant  $C>0$
 such that
 \begin{equation}\label{extrap}
 \|x\|_{\alpha}\leq
C.\|x\|^{1-\alpha}\|x\|_{L}^{\alpha},\quad x\in D(L),
\end{equation}
where $\|\cdot\|_L$ is the graph norm of $L$.
\end{definition}

Concrete examples of $\mathcal{M}_\alpha$ include $D((-L)^\alpha)$ for
$\alpha\in(0,1)$, the domains
 of the fractional powers of $-L$,  the real interpolation
 spaces $D_L(\alpha,\infty)$,  $\alpha\in(0,1)$,  defined as follows
\begin{gather*}
    D_L(\alpha,\infty):=\{x\in \mathcal{M}:  [x]_\alpha
=\sup_{0< t\leq1}\|t^{1-\alpha}(-L-\omega I)e^{-\omega
    t}T(t)x\|<\infty\}\\
    \|x\|_{\alpha}=\|x\|+[x]_{\alpha},
\end{gather*}
 and the abstract H\"older spaces
$D_L(\alpha):=\overline{D(L)}^{\|.\|_\alpha}$.

For the hyperbolic analytic  semigroup $(T(t))_{t\geq0}$, we can
easily check that   estimations
similar to \eqref{hyP} and \eqref{hyQ} hold also with norms
  $\|\cdot\|_\alpha$. In fact,  as the part of $A$ in $R(Q)$ is
  bounded, it follows from the inequality \eqref{hyQ} that
  \begin{equation*}
  \|AT(t)Qx\|\leq
C'e^{\delta t}\|x\| \quad\mbox{for } t\leq0.
\end{equation*}
Hence, from \eqref{extrap} there exists a constant $C(\alpha)>0$
such that
   \begin{equation}\label{hyQ*}
  \|T(t)Qx\|_{\alpha}\leq
C(\alpha)e^{\delta t}\|x\| \quad\mbox{for } t\leq0.
\end{equation}
We have also
  \begin{equation*}
\|T(t)Px\|_{\alpha}\leq \|T(1)\|_{B(\mathcal{M},\mathcal{M}_\alpha)}\|T(t-1)Px\|
\quad\mbox{for } t\geq1,
\end{equation*}
 and then from \eqref{hyP}, we obtain
\begin{equation*}
\|T(t)Px\|_{\alpha}\leq M'e^{-\delta t}\|x\|,  \quad
t\geq1,\end{equation*} where $M'$ depends on $\alpha$. For
$t\in(0,1]$, by \eqref{analy} and \eqref{extrap}
$$
\|T(t)Px\|_{\alpha}\leq M'' t^{-\alpha}\|x\|.
$$
Hence, there exist constants $M(\alpha)>0$ and $\gamma>0$ such
that
  \begin{equation}\label{hyP*}
\|T(t)Px\|_{\alpha}\leq M(\alpha)t^{-\alpha}e^{-\gamma t}\|x\|
\quad\mbox{for } t>0.
\end{equation}

\subsection{Pseudo Almost Periodic Functions}

Let $C(\mathbb{R}, \mathcal{M})$ denote the collection of continuous functions from
$\mathbb{R}$ into $\mathcal{M}$. Let $(BC(\mathbb{R} , \mathcal{M}), \|\cdot\|_\infty)$ denote the
Banach space of all $\mathcal{M}$-valued bounded continuous functions
equipped with the sup norm defined by
${\|u\|_{\infty} := \sup_{t \in \mathbb{R}} \|u(t)\|}$ for
each $u \in B(\mathbb{R} , \mathcal{M})$. (If $\mathcal{M}_\alpha$ for $\alpha \in (0 , 1)$
is an intermediate space, then $BC(\mathbb{R} , \mathcal{M}_\alpha)$ will be
equipped with the $\alpha$- sup norm:
$\|u\|_{\infty, \alpha} = \sup_{t \in \mathbb{R}} \|u(t)\|_\alpha$ for each
$u \in BC(\mathbb{R} , \mathcal{M}_\alpha)$.) Similarly, $BC(\mathbb{R} \times \Omega)$,
where $\Omega \subset \mathcal{M}$ is an open subset, denotes the
collection of all bounded continuous functions $G: \mathbb{R} \times
\Omega \mapsto \mathcal{M}$.

Let $f \in BC(\mathbb{R}, \mathcal{M})$. Define the linear shift operator
$\sigma_\tau$ for some $\tau \in \mathbb{R}$ by $(\sigma_\tau f)(t) := f(t
+\tau)$ for each $t \in \mathbb{R}$. Similarly, if $G \in B(\mathbb{R} \times
\Omega)$, one defines the function $\sigma_\tau G( \cdot , x)$ for
each $x \in \Omega$ by $\sigma_\tau G (t ,x) := G( t + \tau , x)$
for each $t \in \mathbb{R}$.

\begin{definition}\label{D} \rm
A function $f \in BC(\mathbb{R} , \mathcal{M})$ is called (Bohr) almost periodic if
for each $\varepsilon > 0$ there exists $l_{\varepsilon}
> 0$ such that every interval of length  $l_{\varepsilon}
$ contains a number $\tau$ with the property: $\|\sigma_\tau f - f
\|_\infty < \varepsilon$.
\end{definition}
The number $\tau$ above is called an {\it
$\varepsilon$-translation} number of $f$, and the collection of
all such functions will be denoted $AP(\mathcal{M})$. The space $(AP(\mathcal{M}) ,
\|\cdot\|_\infty)$ is a Banach space.

\begin{definition}\label{DD} \rm
A function $F \in BC(\mathbb{R} \times \Omega)$ is called almost periodic
in $t \in \mathbb{R}$ uniformly in $x \in \Gamma \subset \Omega$ ($\Gamma$
being a compact subset of $\Omega$), if for each $\varepsilon > 0$
there exists $l_{\varepsilon}> 0$ such that every interval of
length  $l_{\varepsilon}> 0$ contains a number $\tau$ with the
property:
$\|\sigma_\tau F(\cdot, x) - F(\cdot, x) \|_\infty < \varepsilon$
for each $x \in \Gamma$.
\end{definition}
Here again, the number $\tau$ above is called an
$\varepsilon$-translation number of $F$ and the class of such
functions will be denoted $AP(\mathbb{R} \times \Omega)$.

Throughout the rest of the paper, we suppose $\Omega = \mathcal{M}$ and set
$$
AP_0(\mathcal{M}) := \{ f \in BC(\mathbb{R} , \mathcal{M}): \lim_{r \to \infty}
{\frac{1}{2r}} \int_{-r}^r \| f(s)\| ds = 0\},
$$
and define $AP_0(\mathbb{R} \times \mathcal{M})$ as the collection of functions
$ F \in BC(\mathbb{R} \times \mathcal{M})$ such that
$$
\lim_{r \to \infty}{\frac{1}{2r}} \int_{-r}^r \| F(t,u)\| dt = 0
$$
uniformly in $u \in \mathcal{M}$.


\begin{definition}\label{DDD} \rm
A function $f \in BC(\mathbb{R} , \mathcal{M})$ is called pseudo almost periodic if
it can be expressed as $f = g + \phi$, where $g \in AP(\mathcal{M})$ and
$\phi \in AP_0(\mathcal{M})$. The collection of such functions will be
denoted by $PAP(\mathcal{M})$.
\end{definition}


\begin{remark} \label{rmk2.&}\rm
\begin{enumerate}
\item The functions $g$ and $\phi$ in Definition \ref{DDD}
are respectively called the {\it almost periodic} and the {\it
ergodic perturbation} components of $f$.
\item The
decomposition given in Definition \ref{DDD} is unique.
\end{enumerate}
\end{remark}

From now on we equip $BC(\mathbb{R}, \mathcal{M})$, $AP(\mathcal{M})$, and $PAP(\mathcal{M})$ with
the sup norm $\| \cdot\|_{\infty}$ previously defined. In view of
the above, they all constitute Banach spaces.

\begin{definition}\label{DDDD} \rm
A function $f \in BC(\mathbb{R} \times \mathcal{M})$ for is called pseudo almost
periodic in $t \in \mathbb{R}$ uniformly in $x \in \mathcal{M}$ if it can be
expressed as $f = g + \phi$, where $g \in AP(\mathbb{R} \times \mathcal{M})$ and
$\phi \in AP_0(\mathbb{R} \times \mathcal{M})$. The collection of such functions
will be denoted by $PAP({\mathbb R \times \mathcal{M}})$.
\end{definition}

\section{Main results}

To study (\ref{14}) we first study pseudo almost periodic
solutions to the inhomogeneous differential equation
\begin{equation}\label{31}
u'(t) + Lu(t)= h(t),\quad t\in \mathbb{R},
\end{equation}
where $-L$ is sectorial on $\mathcal{M}$ and $h: \mathbb{R} \mapsto \mathcal{M}$ is pseudo
almost periodic.

Let $(T(t))_{t \geq 0}$ denote the analytic semigroup defined on
$\mathcal{M}$ whose infinitesimal generator is $-L$.

\begin{definition} \label{def3.1} \rm
A mild solution to \eqref{31} is a continuous  function
$u:\mathbb{R}\rightarrow \mathcal{M}_\alpha$
    satisfying
  \begin{equation}\label{vcf}
    u(t)= T(t-s)u(s)+\int_s^tT(t-\sigma)h(\sigma)\,d\sigma
    \end{equation}
for all $ t\geq s$ and all $ s\in \mathbb{R}$.
\end{definition}


\begin{proposition}\label{homg}
Let $-L$ be a sectorial linear operator on $\mathcal{M}$, which is hyperbolic,
i.e., $\sigma(-L) \cap i\mathbb{R} = \{\emptyset\}$. If $h\in PAP(\mathcal{M})$,
then (\ref{31}) has a unique $\mathcal{M}_\alpha$-valued pseudo almost
periodic mild solution $u(\cdot)$, which can be expressed as
\begin{equation}\label{inho}
    u(t)= \int_{-\infty}^tT(t-s)Ph(s)ds -\int_t^{+\infty}T(t-s)Qh(s)ds,\quad
    t\in\mathbb{R}.
    \end{equation}
\end{proposition}

\begin{proof} It is not hard to see that
 $$
  u(t)= \int_{-\infty}^tT(t-s)Ph(s)ds
 -\int_t^{+\infty}T(t-s)Qh(s)ds,\quad t\in\mathbb{R},
$$
is well defined for each $t\in \mathbb{R}$, and satisfies
 \begin{equation}\label{34}
    u(t)=  T(t-s)u(s)+\int_s^tT(t-\sigma)h(\sigma)\,d\sigma
\end{equation}
for all $ t\geq s$ and all $ s\in \mathbb{R}$, and hence $u$ is a mild
solution to \eqref{31}.

For the uniqueness, let $v$ be another bounded continuous mild
solution, i.e., $v$ satisfies \eqref{34}. Then, using both the
projection operator $P$ and the estimates
\eqref{hyQ*}-\eqref{hyP*} defined previously, it is not hard to
see that
\begin{gather*}
Pv(t) =\int_{-\infty }^tT(t-s)Ph(s)ds,\quad t\in \mathbb{R}, \\
 Qv(t)=\int_{+\infty }^tT(t-s)Qh(s)ds,\quad t\in \mathbb{R}.
\end{gather*}
Thus from the decomposition of the space $\mathcal{M}$ it follows that
$v(t)=u(t)$.

 It remains to prove that $u\in PAP(\mathcal{M}_\alpha)$. Since $h \in PAP(\mathcal{M})$,
 write $h = \phi + \zeta$ where $\phi \in AP(\mathcal{M})$ and $\zeta \in AP_0(\mathcal{M})$.
Thus $Pu(t)$ and $Qu(t)$ can be rewritten as
\begin{gather*}
Pu(t) =\int_{-\infty }^tT(t-s)P\phi(s)ds + \int_{-\infty
}^tT(t-s)P\zeta(s)ds \\
 Qu(t) =\int_{+\infty }^tT(t-s)Q\phi(s)ds
+ \int_{+\infty }^tT(t-s)Q\zeta(s)ds.
\end{gather*}

Set $ \Phi(t) = \int_{-\infty }^tT(t-s)P\phi(s)ds$,
and $ \Psi(t)= \int_{-\infty }^tT(t-s)P\zeta(s)ds$
for each $t \in \mathbb{R}$. The next step consists of showing that $\Phi
\in AP(\mathcal{M}_\alpha)$ and $\Psi \in AP_0(\mathcal{M}_\alpha)$.

Clearly, $\Phi \in AP(\mathcal{M}_\alpha)$. Indeed, since
$\phi \in AP(\mathcal{M})$, therefore for every $\varepsilon > 0$ there exists
$\theta_{\varepsilon}> 0$ such that for all $\xi$ there is
$\tau \in [ \xi , \xi + \theta_{\varepsilon}]$ with
$$
\| \sigma_\tau \Phi - \Phi \|_{\infty} < \mu .  \varepsilon
$$
where $\mu = {\frac{\gamma^{1-\alpha}}{M(\alpha) \Gamma
(1-\alpha)}}$ with $\Gamma$ being the classical gamma
function.

Now
$$
\Phi(t + \tau) - \Phi(t)
= \int_{-\infty}^{t} T(t -s) P\left(\phi(s + \tau) - \phi(s)\right) ds.
$$
Thus, using \eqref{hyP*} it easily follows that $\| \Phi(t + \tau)
- \Phi(t) \|_\alpha < \varepsilon$ for each $t \in \mathbb{R}$, and hence
$\|\sigma_\tau \Phi - \Phi \|_{\infty, \alpha} < \varepsilon$,
that is, $\Phi \in AP(\mathcal{M}_\alpha)$.

Next, we show that $t \mapsto \Psi(t)$ is in $AP_0(\mathcal{M}_\alpha)$.
First, note that $t \mapsto \Psi(t)$ is a bounded continuous
function. It remains to show that
$$
\lim_{r\to \infty} {\frac{1}{2r}}  \int_{-r}^r
\| \Psi(t) \|_\alpha \, dt = 0.
$$
Once again using \eqref{hyP*} it follows that
$$
\lim_{r \to \infty} {\frac{1}{2r}}
 \int_{-r}^r \|\Psi(t) \|_\alpha \ dt \leq I + J
$$
where
$$
I := \lim_{r \to \infty} {\frac{M(\alpha)}{2r}} \int_{-r}^r
dt \mathcal{M}ig( \int_{-r}^t (t-s)^{-\alpha} e^{-\gamma(t-s)}  \|
\zeta(s) \| ds \mathcal{M}ig),
$$
and
$$
J: = \lim_{r \to \infty} {\frac{M(\alpha)}{2r}}  \int_{-r}^r
dt  \int_{-\infty}^{-r}  (t-s)^{-\alpha} e^{-\gamma(t-s)}
\|\zeta(s) \| ds.
$$
Now
\begin{align*}
I &= \lim_{r \to \infty} {\frac{M(\alpha)}{2r}} \int_{-r}^r
\| \zeta(t)\| \,dt \left( \int_{-r}^t (t-s)^{-\alpha} e^{-\gamma(t-s)} ds \right) \\
&=\lim_{r \to \infty} {\frac{M(\alpha)}{2r}} \int_{-r}^r
\| \zeta(t)\| \,dt \left( \int_{0}^{t+r} \sigma^{-\alpha} e^{-\gamma \sigma} d\sigma \right) \\
&\leq \lim_{r \to \infty} {\frac{M(\alpha)}{2r}} \int_{-r}^r
\| \zeta(t)\| \,dt \left( \int_{0}^{+\infty} \sigma^{-\alpha} e^{-\gamma \sigma} d\sigma \right) \\
 &= \lim_{r \to \infty} {\frac{M(\alpha) \Gamma (1-\alpha)}{2r \gamma^{1-\alpha}}} \int_{-r}^r
\| \zeta(t)\| \,dt \\
&= {\frac{M(\alpha)
\Gamma(1-\alpha)}{\gamma^{1-\alpha}}} \, . \, \lim_{r \to \infty}
{\frac{1}{2r}} \int_{-r}^r
\| \zeta(t)\| dt = 0.
\end{align*}
Similarly,
\begin{align*}
J &\leq \lim_{r \to \infty} {\frac{M(\alpha)
.\|\zeta\|_\infty}{2r}} \int_{-r}^r dt \int_{t+r}^{+\infty}
 \sigma^{-\alpha} e^{-\gamma \sigma}d\sigma\\
&\leq \lim_{r \to \infty}
{\frac{M(\alpha).\|\zeta\|_\infty}{2r}} \int_{-r}^r dt \int_{2r}^{+\infty}
 \sigma^{-\alpha}e^{-\gamma \sigma}d\sigma\\
&\leq \lim_{r \to \infty}
{\frac{M(\alpha).\|\zeta\|_\infty}{2r}} \int_{-r}^r dt \int_{2r}^{+\infty}
 (2r)^{-\alpha}e^{-\gamma \sigma}d\sigma\\
&= \lim_{r \to \infty}\left[\frac{M(\alpha) .\|\zeta\|_\infty
.e^{-2\gamma r}}{
(2r)^\alpha .\gamma} \right] = 0,
\end{align*}
and hence $\Psi$ belongs to $AP_0(\mathcal{M}_\alpha)$.

The proof for $Qu(\cdot)$ is similar to that of $Pu(\cdot)$.
However one makes use of \eqref{hyQ*} rather than \eqref{hyP*}.
\end{proof}

\begin{definition} \label{def3.3}
A mild solution of \eqref{14} is a continuous  function
$u:\mathbb{R}\rightarrow \mathcal{M}_\alpha$
    satisfying the integral equation
  \begin{equation}\label{vcf1}
    u(t)=    T(t-s)u(s)+\int_s^tT(t-\sigma) \left[ f(u(\sigma))
 + F(\sigma) + K u(\sigma)\right]\,d\sigma
    \end{equation}
for all $ t\geq s$ and all $ s\in \mathbb{R}$.
\end{definition}

To study \eqref{14} we require the following assumptions:

\begin{itemize}
\item[(H1)] The sectorial operator $-L$ is the generator of
a hyperbolic analytic semigroup $(T(t))_{t\geq0}$;

\item[(H2)] The operator $K: \mathcal{M}_\alpha \mapsto \mathcal{M}_\alpha$ is
bounded. We then set $\|K\|_{B(\mathcal{M}_\alpha)} = K_\alpha$.

\item[(H3)] The function $f: PAP(\mathcal{M}_\alpha) \mapsto
PAP(\mathcal{M}_\alpha)$ and $f$ is
    Lipschitz as follows: there exists $R > 0$ such that
    $$
    \|f(u(t))-f(v(t))\|\leq R\,.\,\|u(t)-v(t)\|_\alpha
    $$
    for all $t \in \mathbb R$ and $u,v\in PAP(\mathcal{M}_\alpha)$.
\item[(H4)] $F \in PAP(\mathcal{M}_\alpha)$.
\end{itemize}


\begin{theorem}\label{theo}
Under the  assumptions  (H1)-(H2)-(H3)-(H4),
the evolution equation \eqref{14} has
a unique pseudo almost periodic mild solution whenever
$$
\Theta_\alpha = (R + K_\alpha)
\left[ \frac{C(\alpha)}{\delta} + \frac{M(\alpha) \Gamma (1 -
\alpha)}{\gamma^{1- \alpha}}\right] < 1.
$$
\end{theorem}

    \begin{proof} Using similar arguments as in the proof of
Proposition \ref{homg}, it can be easily seen that each mild
solution $u$ to \eqref{14} is given by
\begin{equation} \label{fix}
\begin{aligned}
u(t)&= \int_{-\infty}^tT(t-s)P [ f(u(s)) + F(s) + K
u(s)]ds  \\
&\quad-\int_t^{+\infty}T(t-s) Q[ f(u(s)) + F(s) + K
u(s)]ds,\quad t\in\mathbb{R}.
\end{aligned}
\end{equation}
Now consider the nonlinear operator on $C(\mathbb{R},\mathcal{M}_\alpha)$ given by
\begin{equation} \label{fix1}
\begin{aligned}
\mathbb{D} y(t)&= \int_{-\infty}^tT(t-s)P [ f(y(s)) + F(s) + K
y(s)]ds  \\
&\quad -\int_t^{+\infty}T(t-s) Q[ f(y(s)) + F(s) + K
y(s)]ds,\quad t\in\mathbb{R},
\end{aligned}
\end{equation}
for each $y\in C(\mathbb{R},\mathcal{M}_\alpha)$.

From the boundedness of $K$ as an operator of $\mathcal{M}_\alpha$
it is clear that $Ky(\cdot)$ is also pseudo almost periodic on
$\mathcal{M}_\alpha$ whenever $y$ does. Under (H.3), if $y\in
PAP(\mathcal{M}_\alpha)$, then $f(y(\cdot))\in
PAP(\mathcal{M}_\alpha)$. Considering Proposition \ref{homg}, for
$h(s)= f(y(s)) + F(s) + K y(s)$,
 it follows that the operator $\mathbb{D}$ maps $PAP(\mathcal{M}_\alpha)$
into itself.

Let $v,w\in PAP(\mathcal{M}_\alpha)$,
\begin{align*}
 \|\mathbb{D} v(t)-\mathbb{D} w(t)\|_\alpha
&\leq \|\int_{-\infty}^tT(t-s)P\left[f(v(s))-f(w(s))\right]ds\|_\alpha\\
&\quad+ \|\int_{-\infty}^tT(t-s)P\left[Kv(s))-Kw(s))\right]ds\|_\alpha\\
&\quad +\|\int_t^{+\infty}T(t-s)Q\left[f(v(s))-f(w(s))\right]ds\|_\alpha\\
&\quad +\|\int_t^{+\infty}T(t-s)Q\left[Kv(s)-Kw(s)\right]ds\|_\alpha.
\end{align*}
Using \eqref{hyP*} and \eqref{hyQ*} it follows that
\begin{align*}
  \|\mathbb{D} v(t)-\mathbb{D} w(t)\|_\alpha
&\leq  R. M(\alpha)\int_{-\infty}^t(t-s)^{-\alpha}e^{-\delta(t-s)}\|v(s)-
  w(s)\|_\alpha ds\\
&\quad + K_\alpha .
M(\alpha)\int_{-\infty}^t(t-s)^{-\alpha}e^{-\delta(t-s)}\|v(s)-
  w(s)\|_\alpha ds\\
&\quad + R . C(\alpha)\int^{+\infty}_te^{\delta(t-s)}\|v(s)-w(s)\|_\alpha ds\\
&\quad + K_\alpha .
C(\alpha)\int^{+\infty}_te^{\delta(t-s)}\|v(s)-w(s)\|_\alpha ds,
\end{align*}
and hence
$\|\mathbb{D} v - \mathbb{D} w\|_{\infty, \alpha} \leq \Theta_\alpha .\|v - w\|_{\infty,
\alpha}$, where
$$ \Theta_\alpha = (R + K_\alpha)
\left[ \frac{C(\alpha)}{\delta} + \frac{M(\alpha) \Gamma (1 -
\alpha)}{\gamma^{1- \alpha}}\right].
$$
Clearly, if $\Theta_\alpha < 1$, then \eqref{14}
 has a unique fixed-point by the Banach fixed point theorem, which
obviously is the only
pseudo almost periodic (mild) solution to \eqref{14}.
\end{proof}

\begin{example} \label{exa3.5} \rm
To deal with the system (\ref{11})-(\ref{12}), take
 $\mathcal{M}:=C[0 , 1]$, equipped with the sup norm. Define the operator
$-L$ by
$$
-L(\varphi):= \varphi'' + H.\varphi, \quad \forall \varphi \in D(-L),
$$
where $ D(-L):=\{ \varphi\in C^2[0,1], \varphi(0)= \varphi(1)= 0\}
\subset C[0 , 1]$ and $H \in \mathbb{R}$ is a constant.

 Clearly $-L$ is sectorial, and hence is the generator of an analytic
semigroup.  In addition to the above, the resolvent and spectrum
of $-L$ are respectively given by
$$
\rho(-L) = \mathbb{C} - \{ - n^2 \pi^2 + H: n \in \mathbb{N} \} \quad \mbox{and}
\quad \sigma(-L)= \{ - n^2 \pi^2 + H: n \in \mathbb{N} \}
$$
so that $\sigma(-L) \cap i \mathbb{R} = \{\emptyset\}$ whenever
$H \neq n^2 \pi^2$. In particular, if $H = m \pi^2$ where
$ m \in \mathbb{N}-\{0\}$
and $m$ is not a square, then $-L$ is hyperbolic.
\end{example}


\begin{theorem} \label{thm3.6}
Under assumptions (H2)--(H4), suppose that the constant $H \neq n^2 \pi^2$
for $n \in \mathbb{N}$. Then the heat equation \eqref{11}-\eqref{12} has a unique
$\mathcal{M}_{\alpha}$-valued pseudo almost periodic mild solution whenever
$R + K_{\alpha}$ is small enough.
\end{theorem}


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\end{document}