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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2009(2009), No. 111, pp. 1--14.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2009 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2009/111\hfil Quadratic-mean almost periodic solutions]
{Existence of quadratic-mean almost periodic solutions to some
stochastic hyperbolic differential equations}

\author[P. H. Bezandry, T. Diagana\hfil EJDE-2009/111\hfilneg]
{Paul H. Bezandry, Toka Diagana}  % in alphabetical order

\address{Paul H. Bezandry \newline
Department of Mathematics, Howard University,
Washington, DC 20059, USA}
\email{pbezandry@howard.edu}

\address{Toka Diagana \newline
Department of Mathematics, Howard University, Washington,
DC 20059, USA}
\email{tdiagana@howard.edu}

\thanks{Submitted May 1, 2009. Published September 10, 2009.}
\subjclass[2000]{34K14, 60H10, 35B15, 34F05}
\keywords{Stochastic differential equation; stochastic processes;
\hfill\break\indent quadratic-mean almost periodicity; Wiener process}

\begin{abstract}
 In this paper we obtain the existence of quadratic-mean almost
 periodic solutions to some classes of partial hyperbolic
 stochastic differential equations. The main result of this paper
 generalizes in a natural fashion some recent results by authors.
 As an application, we consider the existence of quadratic-mean
 almost periodic solutions to the stochastic heat equation with
 divergence terms.
\end{abstract}

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


\section{Introduction}

Let $(\mathbb{H}, \|\cdot\|, \langle \cdot, \cdot\rangle)$ be a real
Hilbert space which is separable and let $(\Omega, \mathcal{F},
\mathbf{P})$ be a complete probability space equipped with a
normal filtration $\{\mathcal{F}_t: t\in\mathbb{R}\}$, that is, a
right-continuous, increasing family of sub $\sigma$-algebras of
$\mathcal{F}$.

For the rest of this article, if $\mathcal{A}: D(\mathcal{A}) \subset \mathbb{H} \mapsto
\mathbb{H}$ is a linear operator, we then define the operator $A:
D(A) \subset L^2 (\Omega, \mathbb{H}) \mapsto L^2 (\Omega, \mathbb{H})$ as
follows: $X \in D(A)$ and $A X = Y$ if and only if $X, Y \in L^2
(\Omega, \mathbb{H})$ and $\mathcal{A} X(\omega) = Y(\omega)$ for all $\omega \in
\Omega$.

Let $\mathcal{A}: D(\mathcal{A})\subset \mathbb{H} \mapsto
\mathbb{H}$ be a sectorial linear operator. For
$\alpha \in (0 , 1)$, let $\mathbb{H}_\alpha$ denote the
intermediate Banach space between $D(\mathcal{A})$ and
$\mathbb{H}$. Examples of those $\mathbb{H}_\alpha$ include, among
others, the fractional spaces $D((-\mathcal{A})^\alpha)$, the real
interpolation spaces $D_{\mathcal{A}}(\alpha,\infty)$ due to Lions
and Peetre, and the H\"older spaces $D_{\mathcal{A}}(\alpha),$
which coincide with the continuous interpolation spaces that both
Da Prato and Grisvard introduced in the literature.

In Bezandry and Diagana \cite{BD}, the concept of quadratic-mean
almost periodicity was introduced and studied. In particular, such
a concept was, subsequently, utilized to study the existence and
uniqueness of a quadratic-mean almost periodic solution to the
class of stochastic differential equations
\begin{equation}\label{xy}
 dX(t)= A X(t) dt + F(t, X(t)) dt + G(t, X(t)) dW(t), \quad t\in\mathbb{R},
\end{equation}
where $A: D(A) \subset L^2(\Omega; \mathbb{H}) \mapsto L^2(\Omega; \mathbb{H})$ is
a densely defined closed linear operator, and $F: \mathbb{R} \times
L^2(\Omega; \mathbb{H}) \mapsto L^2(\Omega; \mathbb{H})$,
$G: \mathbb{R} \times L^2(\Omega; \mathbb{H}) \mapsto L^2 (\Omega; \mathcal{L}_2^0)$
 are jointly continuous
functions satisfying some additional conditions.

Similarly, in \cite{BD1}, Bezandry and Diagana made extensive use
of the same very concept of quadratic-mean almost periodicity to
study the existence and uniqueness of a quadratic-mean almost
periodic solution to the class of nonautonomous semilinear
stochastic differential equations
\begin{equation}\label{I}
dX(t)= A(t) X(t) \,dt + F(t, X(t))\,dt + G(t, X(t))\,dW(t),
\quad t\in\mathbb{R},
\end{equation}
where $A(t)$ for $t \in \mathbb{R}$ is a family of densely defined closed
linear operators satisfying the so-called Acquistapace and Terreni
conditions \cite{AT},
$F: \mathbb{R} \times L^2 (\Omega, \mathbb{H}) \to L^2 (\Omega, \mathbb{H})$,
$G: \mathbb{R} \times L^2 (\Omega, \mathbb{H}) \to L^2 (\Omega, \mathcal{L}_2^0)$
are jointly continuous satisfying some additional
conditions, and $W(t)$ is a Wiener process.

The present paper is definitely inspired by \cite{BD, BD1, D1} and
consists of studying the existence of quadratic-mean almost
periodic solutions to the stochastic differential equation of the
form
\begin{equation}\label{TY}
\begin{aligned}
&d\Big(X(\omega, t)+f(t, \mathcal{B}X(\omega, t))\Big)\\
&= \big[\mathcal{A}X(\omega, t)+ g(t, \mathcal{C}X(\omega, t))\big]dt
 + h(t, \mathcal{L} X(\omega, t)) dW(\omega, t)
\end{aligned}
\end{equation}
for all $t \in \mathbb{R}$ and $\omega\in\Omega$, where $\mathcal{A}:
D(\mathcal{A})\subset \mathbb{H}\to \mathbb{H}$ is a sectorial
linear operator whose corresponding analytic semigroup is
hyperbolic, that is, $\sigma (\mathcal{A})\cap i\mathbb{R} =\emptyset $,
$\mathcal{B}$, $\mathcal{C}$, and $\mathcal{L}$ are (possibly
unbounded linear operators on $\mathbb{H}$) and $f: \mathbb{R}\times
\mathbb{H}\to \mathbb{H}_\beta    (0<\alpha <\frac{1}{2} <\beta
<1)$, $g: \mathbb{R}\times \mathbb{H}\to \mathbb{H}$, and $h: \mathbb{R}\times
\mathbb{H}\to \mathcal{L}_2^0$ are jointly continuous functions.

To analyze  \eqref{TY}, our strategy consists of studying the
existence of quadratic-mean almost periodic solutions to the
corresponding class of stochastic differential equations of the
form
\begin{equation}\label{A}
 d\Big(X(t)+F(t, BX(t))\Big)=\big[AX(t)+ G(t,
CX(t))\big]dt  + H(t, LX(t)) dW(t)
\end{equation}
for all $t \in \mathbb{R}$, where $A: D(A)\subset L^2(\Omega, \mathbb{H})\to
L^2(\Omega, \mathbb{H})$ is a sectorial linear operator whose
corresponding analytic semigroup is hyperbolic, that is, $\sigma
(A)\cap i\mathbb{R} =\emptyset $, $B$, $C$, and $L$ are (possibly
unbounded linear operators on $L^2(\Omega, \mathbb{H})$) and
$F: \mathbb{R}\times L^2(\Omega, \mathbb{H})\to L^2(\Omega, \mathbb{H}_\beta)$
$(0<\alpha <\frac{1}{2} <\beta <1)$,
$G: \mathbb{R}\times L^2(\Omega, \mathbb{H})\to L^2(\Omega, \mathbb{H})$, and
$H: \mathbb{R}\times L^2(\Omega, \mathbb{H})\to L^2(\Omega, \mathcal{L}_2^0)$
are jointly continuous functions satisfying some
additional assumptions.

It is worth mentioning that the main results of this paper
generalize those obtained in Bezandry and Diagana \cite{BD1}.

The existence of almost periodic (respectively, periodic)
solutions to autonomous stochastic differential equations has been
studied by many authors, see, e.g., \cite{AT,BD,D,t3}
and the references therein. In particular,
Da Prato and Tudor \cite{da}, have studied the existence of almost
periodic solutions to  (\ref{I}) in the case when $A(t)$ is
periodic. Though the existence and uniqueness of quadratic-mean
almost periodic solutions to  \eqref{A} in the case when $A$ is
sectorial is an important topic with some interesting
applications, which is still an untreated question and constitutes
the main motivation of the present paper. Among other things, we
will make extensive use of the method of analytic semigroups
associated with sectorial operators and the Banach's fixed-point
principle to derive sufficient conditions for the existence and
uniqueness of a quadratic-mean almost periodic solution to
\eqref{A}. To illustrate our abstract results, we study the
existence of quadratic-mean almost periodic solutions to the
stochastic heat equation with divergence coefficients.

\section{Preliminaries}

For details on this section, we refer the reader to \cite{BD, C}
and the references therein. Throughout the rest of this paper, we
assume that $(\mathbb{K}, \|\cdot\|_{\mathbb{K}})$ and $(\mathbb{H}, \|\cdot\|)$ are real
separable Hilbert spaces, and $(\Omega, \mathcal{F}, {\mathbb{P}})$ is a
probability space. The space $L_2(\mathbb{K}, \mathbb{H})$ stands for the space of
all Hilbert-Schmidt operators acting from $\mathbb{K}$ into $\mathbb{H}$, equipped
with the Hilbert-Schmidt norm $\|\cdot\|_2$.

For a symmetric nonnegative operator $Q\in L_2(\mathbb{K}, \mathbb{H})$ with
finite trace we assume that $\{W(t), \; t\in\mathbb{R}\}$ is a $Q$-Wiener
process defined on $(\Omega, \mathcal{F}, {\mathbb{P}})$ with values in
$\mathbb{K}$. It is worth mentioning that the Wiener process $W$ can
obtained as follows: let $\{W_i(t), \; t\in\mathbb{R}\}, \; i=1, 2$, be
independent $\mathbb{K}$-valued $Q$-Wiener processes, then
\[
W(t)= \begin{cases}
  W_1(t) & \text{if }t\geq 0 , \\
  W_2(-t) & \text{if }t\leq 0,
 \end{cases}
\]
is $Q$-Wiener process with the real number line as time parameter.
We then let $\mathcal{F}_t=\sigma\{W(s), s\leq t\}$.

 The collection of all strongly measurable, square-integrable
 $\mathbb{H}$-valued random variables, will be denoted
$L^2(\Omega, \mathbb{H})$. Of course, this is a
Banach space when it is equipped with norm
$$
\|X\|_{L^2(\Omega, \mathbb{H})}=\Big(\mathrm{E}\|X\|^2\Big)^{1/2},
$$
where the expectation $\mathrm{E}$ is defined by
 $$
\mathrm{E}[g]=\int_{\Omega}g(\omega)d\mathbb{P}(\omega).
$$
Let $\mathbb{K}_0=Q^{1/2}\mathbb{K}$ and let $\mathcal{L}^0_2=L_2(\mathbb{K}_0, \mathbb{H})$ with
 respect to the norm
$$
\|\Phi\|^2_{\mathcal{L}^0_2}=\|\Phi  Q^{1/2}\|_2^2
=\mathop{\rm Trace}  (\Phi Q \Phi^{*})\,.
$$
Let $(\mathbb{B}, \|\cdot\|)$ be a Banach space. This setting requires the
following preliminary definitions.

\begin{definition} \label{def2.1} \rm
A stochastic process $X: \mathbb{R} \to L^2(\Omega; \mathbb{B})$ is said to be
continuous whenever
\[
\lim_{t\to s}\mathrm{E}\|X(t)-X(s)\|^2=0.
\]
\end{definition}

\begin{definition} \label{def2.2} \rm
A continuous stochastic process $X: \mathbb{R} \to L^2(\Omega; \mathbb{B})$
is said to be quadratic mean almost periodic if for each
$\varepsilon >0$ there exists
$l(\varepsilon) >0$ such that any interval of length
$l(\varepsilon)$ contains at least a number $\tau$ for which
$$
\sup_{t\in \mathbb{R}}\mathrm{E}\|X(t+\tau) - X(t)\|^2 <\varepsilon.$$
\end{definition}

The collection of all stochastic processes $X: \mathbb{R} \to L^2(\Omega ;
\mathbb{B})$ which are quadratic mean almost periodic is then denoted by
$AP({\mathbb{R}};L^2(\Omega ; \mathbb{B}))$.

The next lemma provides some properties of quadratic mean almost
periodic processes.

\begin{lemma} \label{lem2.3}
If $X$ belongs to $AP({\mathbb{R}};L^2(\Omega ; \mathbb{B}))$, then
\begin{itemize}
\item[(i)]  the mapping $t\to \mathrm{E}\|X(t)\|^2$ is
uniformly continuous; \item[(ii)]  there exists a constant $M > 0$
such that $\mathrm{E}\|X(t)\|^2\le M$, for all $t\in \mathbb{R}$.
\end{itemize}
\end{lemma}

Let $\mathop{\rm CUB}(\mathbb{R};  L^2(\Omega ; \mathbb{B}))$ denote the
collection of all stochastic processes
$X: \mathbb{R} \mapsto L^2(\Omega ; \mathbb{B}),$ which are continuous and
uniformly bounded. It is then easy
to check that $\mathop{\rm CUB}(\mathbb{R};  L^2(\Omega ; \mathbb{B}))$  is a
Banach space when it is equipped with the norm:
$$
\|X\|_{\infty}=\sup_{t \in \mathbb{R}}\Big(\mathrm{E}\|X(t)\|^2\Big)^{1/2}.
$$

\begin{lemma} \label{lem2.4}
$AP(\mathbb{R};L^2(\Omega ; \mathbb{B}))\subset \mathop{\rm CUB} (\mathbb{R};L^2(\Omega ;
\mathbb{B}))$ is a closed subspace.
\end{lemma}

In view of the above, the space $AP(\mathbb{R};L^2(\Omega ; \mathbb{B}))$ of
quadratic mean almost periodic processes equipped with the norm
$\|\cdot\|_\infty$ is a Banach space.

Let $(\mathbb{B}_1, \|\cdot\|_{\mathbb{B}_1})$ and $(\mathbb{B}_2, \|\cdot\|_{\mathbb{B}_2})$ be
Banach spaces and let $L^2(\Omega ; \mathbb{B}_1)$ and $L^2(\Omega ;
\mathbb{B}_2)$ be their corresponding $L^2$-spaces, respectively.

\begin{definition} \label{def2.5} \rm
A function $F: \mathbb{R} \times L^2(\Omega;
\mathbb{B}_1) \to L^2(\Omega; \mathbb{B}_2))$,
 $(t, Y) \mapsto F(t, Y)$, which is
jointly continuous, is said to be quadratic mean almost periodic
in $t \in \mathbb{R}$ uniformly in $Y\in\mathbb{K}$ where
$\mathbb{K} \subset L^2(\Omega ; \mathbb{B}_1)$ is a compact
if for any $\varepsilon >0$, there exists
$l(\varepsilon, \mathbb{K}) >0$ such that any interval of
length $l(\varepsilon, \mathbb{K})$ contains at least a
number $\tau$ for which
$$
\sup_{t\in \mathbb{R}}\mathrm{E}\|F(t+\tau, Y) - F(t, Y)\|^2_{\mathbb{B}_2}
<\varepsilon
$$
for each stochastic process $Y: \mathbb{R} \to \mathbb{K}$.
\end{definition}

\begin{theorem}\label{U}
Let $F: \mathbb{R} \times L^2(\Omega ; \mathbb{B}_1) \to L^2(\Omega ; \mathbb{B}_2)$, $(t,
Y) \mapsto F(t, Y)$ be a quadratic mean almost periodic process in
$t \in \mathbb{R}$ uniformly in $Y\in\mathbb{K}$, where $\mathbb{K} \subset L^2(\Omega ;
\mathbb{B}_1)$ is compact. Suppose that $F$ is Lipschitz in the following
sense:
$$
\mathrm{E} \|F(t, Y) - F(t, Z)\|_{\mathbb{B}_2}^2
\leq M \mathrm{E}\|Y - Z\|_{\mathbb{B}_1}^2
$$
for all $Y, Z\in L^2(\Omega ; \mathbb{B}_1)$ and for each
$t \in \mathbb{R}$, where $M >0$. Then for any quadratic mean almost
periodic process $\Phi: \mathbb{R} \to L^2(\Omega ; \mathbb{B}_1)$, the stochastic
process $t \mapsto F(t, \Phi(t))$ is quadratic mean almost
periodic.
\end{theorem}

\section{Sectorial Operators on $\mathbb{H}$}

In this section, we introduce some notations and collect some
preliminary results from Diagana \cite{toka1} that will be used
later. If $\mathcal{A}$ is a linear operator on $\mathbb{H}$, then
$\rho (\mathcal{A})$, $\sigma (\mathcal{A})$, $D(\mathcal{A})$,
$\ker  (\mathcal{A})$, $R(\mathcal{A})$ stand
for the resolvent set, spectrum, domain, kernel, and range of
$\mathcal{A}$.
If $\mathbb{B}_1, \mathbb{B}_2$ are Banach spaces, then the notation
$B(\mathbb{B}_1, \mathbb{B}_2)$ stands for the Banach space of bounded linear
operators from $\mathbb{B}_1$ into $\mathbb{B}_2$. When $\mathbb{B}_1 = \mathbb{B}_2$, this is
simply denoted $B(\mathbb{B}_1)$.

\begin{definition} \label{def3.1} \rm
A linear operator
$\mathcal{A}: D(\mathcal{A})\subset \mathbb{H}\to \mathbb{H}$
(not necessarily densely defined) is said to be sectorial if
the following hold: there exist constants $\zeta\in\mathbb{R}$,
$\theta\in (\frac{\pi}{2}, \pi)$, and $M>0$ such
that $S_{\theta, \zeta}\subset\rho (\mathcal{A})$,
\begin{gather*}
S_{\theta, \zeta}:= \{\lambda\in \mathbb{C} : \lambda\ne\zeta,];
 |\arg (\lambda -\zeta)|<\theta\}, \\
 \|R(\lambda, \mathcal{A})\|\leq\frac{M}{|\lambda - \zeta|}, \quad
\lambda\in S_{\theta,
\zeta}
\end{gather*}
where $R(\lambda, \mathcal{A})=(\lambda I - \mathcal{A})^{-1}$ for each
$\lambda\in\rho (\mathcal{A})$.
\end{definition}

\begin{remark} \label{rmk3.2} \rm
If the operator $\mathcal{A}$ is sectorial, then it generates
an analytic semigroup $(T(t))_{t\ge 0}$, which maps $(0, \infty)$
into $B(\mathbb{H})$ and such that there exist constants $M_0$,
$M_1>0$ such that
\begin{gather}\label{IR}
\|T(t)\|\leq M_0 e^{\zeta t}, \quad t>0\\
\|t(\mathcal{A}-\zeta  I) T(t)\|\leq M_1 e^{\zeta t}, \quad t>0\label{JR}
\end{gather}
\end{remark}

\begin{definition} \label{def3.3} \rm
A semigroup $(T(t))_{t\geq 0}$ is hyperbolic; that is, there exist
a projection $P$ and constants $M$, $\delta >0$ such that $T(t)$
commutes with $P$, $\ker (P)$ is invariant with respect
$T(t)$, $T(t): R(S)\to R(S)$ is invertible, and
\begin{gather}\label{B}
\|T(t)Px\|\leq M e^{-\delta t}\|x\|, \quad   t>0,\\
\|T(t)Sx\|\leq M e^{\delta t}\|x\|, \quad   t\leq 0,\label{C}
\end{gather}
where $S:=I - P$ and, for $t\leq 0$, $T(t):= (T(-t))^{-1}$.
\end{definition}

Recall that the analytic semigroup $(T(t))_{t\geq 0}$ associated
with the linear operator $\mathcal{A}$ is hyperbolic if and if
 $\sigma (\mathcal{A})\cap i\mathbb{R} =\emptyset$.

\begin{definition} \label{def3.4} \rm
Let $\alpha\in (0, 1)$. A Banach space $(\mathbb{H}_\alpha,
\|\cdot\|_{\alpha})$ is said to be an intermediate space between
$D(\mathcal{A})$ and $\mathbb{H}$, or a space of class
$\mathcal{J}_\alpha$, if $D(\mathcal{A})\subset \mathbb{H}_\alpha\subset
\mathbb{H}$ and there is a constant $c>0$ such that
\begin{equation}\label{D}
\|x\|_\alpha\leq c\|x\|^{1-\alpha}\|x\|^\alpha_{[D(\mathcal{A})]}, \quad
 x\in D(\mathcal{A}), \end{equation}
where $\|\cdot\|_{[D(\mathcal{A})]}$ is the graph norm of $\mathcal{A}$.
 Here, $\|u\|_{[D(\mathcal{A})]}=\|u\|+\|\mathcal{A}u\|$,
for each $u\in D(\mathcal{A})$.
\end{definition}

Concrete examples of $\mathbb{H}_\alpha$ include
$D((-\mathcal{A})^\alpha)$
for $\alpha\in (0, 1)$, the domains of the fractional powers of
$\mathcal{A}$, the real interpolation spaces
$D_\mathcal{A}(\alpha, \infty)$,
$\alpha\in (0, 1)$, defined as the space of all
$x\in \mathbb{H}$ such that
$$
[x]_\alpha=\sup_{0\leq t\leq 1}\|t^{1-\alpha}
(\mathcal{A}-\zeta I) e^{-\zeta t}T(t)x\|<\infty,
$$
with the norm
$$
\|x\|_\alpha=\|x\|+ [x]_\alpha,
$$
and the abstract Holder spaces
$D_\mathcal{A}(\alpha):=\overline{D(\mathcal{A})}^{ \|\cdot\|_\alpha}$.

\begin{lemma}[\cite{D1, toka1}]\label{KL}
 For the hyperbolic analytic semigroup
$(T(t))_{t\geq 0}$, there exist constants $C(\alpha)>0$,
$\delta >0$, $M(\alpha)>0$, and $\gamma >0$ such that
\begin{gather}\label{Y}
\|T(t)S x\|_{\alpha}\leq c(\alpha) e^{\delta t}\|x\|\quad
 \text{for } t\leq 0,\\
\|T(t)Px\|_\alpha\leq M(\alpha)t^{-\alpha} e^{-\gamma
t}\|x\|\quad\text{for } t>0.\label{Z}
\end{gather}
\end{lemma}

The next Lemma is crucial for the rest of the paper. A version of
it in a general Banach space can be found in Diagana
\cite{D1,toka1}.

\begin{lemma}[\cite{D1, toka1}] \label{BC}
 Let $0<\alpha <\beta <1$. For the hyperbolic analytic
semigroup $(T(t))_{t\geq 0}$, there exist constants
$c>0$, $\delta >0$, and $\gamma >0$ such that
\begin{gather}\label{HI}
\|\mathcal{A}T(t)Q x\|_{\alpha}\leq n(\alpha, \beta)
e^{\delta t}\|x\|\leq n'(\alpha, \beta)
e^{\delta t}\|x\|_{\beta},\quad\text{for } t\leq 0\\
\|\mathcal{A}T(t)P x\|_\alpha\leq M(\alpha) t^{-\alpha} e^{-\gamma
t}\|x\|\leq M'(\alpha) t^{-\alpha} e^{-\gamma
t}\|x\|_{\beta}, \quad\text{for }  t>0.\label{HJ}
\end{gather}
Also, for $\Xi\in \mathcal{L}_2^0$,
\begin{gather}\label{VW}
\|\mathcal{A}T(t)Q  \Xi\|_{\mathcal{L}_2^0}
 \leq n_1(\alpha, \beta) e^{\delta t}\|\Xi\|_{\mathcal{L}_2^0},
\quad\text{for }  t\leq 0\\
\|\mathcal{A}T(t)P  \Xi\|_{\mathcal{L}_2^0}\leq M_1(\alpha) t^{-\alpha}
e^{-\gamma t}\|\Xi\|_{\mathcal{L}_2^0}, \quad\text{for }   t>0.\label{WY}
\end{gather}
\end{lemma}

\section{Existence of Quadratic-Mean Almost Periodic Solutions}

This section is devoted to the existence and uniqueness of a
quadratic-mean almost periodic solution to the stochastic
hyperbolic differential equation  \eqref{A}

\begin{definition} \label{def4.1} \rm
Let $\alpha\in (0, 1)$. A continuous random function, $X : \mathbb{R}\to
L^2(\Omega; \mathbb{H}_\alpha)$ is said to be a bounded solution
of \eqref{A} provided that the function $s\to A T(t-s)P F(s,
BX(s))$ is integrable on $(-\infty, t)$,   $s\to A T(t-s)Q F(s,
BX(s))$ is integrable on $(t, \infty)$ for each $t\in\mathbb{R}$, and
\begin{align*}
X(t)
&=-F(t, BX(t))-\int_{-\infty}^t A T(t-s)P F(s, BX(s))\,ds\\
&\quad +\int_t^\infty A T(t-s)S F(s, B X(s))\,ds\\
&\quad +\int_{-\infty}^t T(t-s)P G(s, C X(s))\,ds
-\int_t^\infty T(t-s)S G(s, C X(s))\,ds\\
&\quad +\int_{-\infty}^t T(t-s)P H(s, L X(s))\,dW(s) -\int_t^\infty
T(t-s)S H(s, L X(s))\,dW(s)
\end{align*}
for each $t\in\mathbb{R}$.
\end{definition}

In the rest of this article, we denote by $\Gamma_1$,
$\Gamma_2$, $\Gamma_3$, $\Gamma_4$, $\Gamma_5$, and $\Gamma_6$ the
nonlinear integral operators defined by
\begin{gather*}
(\Gamma_1X)(t):=\int_{-\infty}^t A T(t-s)P F(s, BX(s))\,ds,\\
(\Gamma_2X)(t):=\int_t^\infty A T(t-s)S F(s, B X(s))\,ds, \\
(\Gamma_3X)(t):=\int_{-\infty}^t T(t-s)P G(s, C X(s))\,ds\\
(\Gamma_4X)(t):=\int_t^\infty T(t-s)S G(s, C X(s))\,ds,\\
(\Gamma_5X)(t):=\int_{-\infty}^t T(t-s)P H(s, L X(s))\,dW(s), \\
(\Gamma_6X)(t):=\int_t^\infty T(t-s)S H(s, L X(s))\,dW(s).
\end{gather*}

To discuss the existence of quadratic-mean almost periodic
solution to  \eqref{A} we need to set some assumptions on
$A$, $B$, $C$, $L$, $F$, $G$, and $H$. First of all, note
that for $0<\alpha <\beta <1$, then
$$
L^2(\Omega, \mathbb{H}_\beta)
\hookrightarrow L^2(\Omega, \mathbb{H}_\alpha)\hookrightarrow L^2(\Omega;
\mathbb{H})
$$
are continuously embedded and hence there exist constants
$k_1>0$, $k(\alpha)>0$ such that
\begin{gather*}
\mathrm{E}\|X\|^2\leq k_1\mathrm{E}\|X\|^2_\alpha \quad
 \text{for each }  X\in L^2(\Omega, \mathbb{H}_\alpha),\\
\mathrm{E}\|X\|^2_\alpha\leq k(\alpha)\mathrm{E}\|X\|^2_\beta \quad
\text{for each }  X\in L^2(\Omega, \mathbb{H}_\beta).
\end{gather*}

\begin{enumerate}


\item[(H1)] The operator $\mathcal{A}$ is sectorial and generates a
hyperbolic (analytic) semigroup $(T(t))_{t\geq 0}$.

\item [(H2)] Let $\alpha\in (0, \frac{1}{2}) $. Then
$\mathbb{H}_\alpha=D((-\mathcal{A})^\alpha)$, or
$\mathbb{H}_\alpha=D_\mathcal{A}(\alpha, p),   1\leq p\leq \infty$, or
$\mathbb{H}_\alpha=D_\mathcal{A}(\alpha)$, or
$\mathbb{H}_\alpha=[\mathbb{H}, D(\mathcal{A})]_\alpha$. We also assume
that $B,  C,  L: L^2(\Omega, \mathbb{H}_\alpha)\to L^2(\Omega;\mathbb{H})$ are
bounded linear operators and set
$$
\varpi := \max \Big(\|B\|_{B(L^2(\Omega, \mathbb{H}_\alpha),
L^2(\Omega; \mathbb{H}))},  \|C\|_{B(L^2(\Omega, \mathbb{H}_\alpha),
L^2(\Omega; \mathbb{H}))},  \|L\|_{B(L^2(\Omega, \mathbb{H}_\alpha),
L^2(\Omega; \mathbb{H}))}\Big).
$$

\item [(H3)] Let $\alpha\in (0, \frac{1}{2})$ and
$\alpha <\beta <1$. Let
$F: \mathbb{R}\times L^2(\Omega; \mathbb{H})\to L^2(\Omega, \mathbb{H}_\beta)$,
$G:\mathbb{R}\times L^2(\Omega; \mathbb{H})\to L^2(\Omega; \mathbb{H})$ and
$H:\mathbb{R}\times L^2(\Omega;\mathbb{H})\to L^2(\Omega; \mathcal{L}_2^0)$ are
quadratic-mean almost periodic. Moreover, the functions
$F$, $G$, and $H$ are uniformly Lipschitz with respect to the
second argument in the following sense: there exist positive
constants $K_F$, $K_G$, and $K_H$ such that
\begin{gather*}
\mathrm{E}\|F(t, \psi_1)-F(t, \psi_2)\|_{\beta}^2
\leq K_F  \mathrm{E}\|\psi_1-\psi_2\|^2,\\
\mathrm{E}\|G(t, \psi_1)-G(t, \psi_2)\|^2\leq  K_G
\mathrm{E}\|\psi_1-\psi_2\|^2, \\
\mathrm{E}\|H(t, \psi_1)-H(t, \psi_2)\|_{\mathcal{L}_2^0}^2\leq  K_H
\mathrm{E}\|\psi_1-\psi_2\|^2,
\end{gather*}
for all stochastic processes $\psi_1, \psi_2 \in L^2(\Omega;\mathbb{H})$ and
$t\in\mathbb{R}$.
\end{enumerate}

\begin{theorem}\label{AB}
Under assumptions {\rm (H1)--(H3)}, the evolution equation \eqref{A}
has a unique quadratic-mean almost periodic mild solution whenever
$\Theta <1$, where
\begin{align*}
\Theta &:= \varpi \Big[k'(\alpha) K'_F\Big\{1+c
\Big(\frac{\Gamma (1-\alpha)}{\gamma^{1 -\alpha}}
 +\frac{1}{\delta}\Big)\Big\}+ k_1'\cdot K'_G
 \Big( M'(\alpha) \frac{\Gamma (1-\alpha)}{\gamma^{1-\alpha}}
 +\frac{ C'(\alpha)}{\delta}\Big)\\
&\quad +c\sqrt{\mathop{\rm Tr} Q}\cdot K'_H\cdot k'_1\cdot
\Big\{\frac{K'(\alpha, \beta)}{\sqrt{\delta}}
+2 K'(\alpha, \gamma, \delta, \Gamma)\Big\}  \Big].
\end{align*}
\end{theorem}

To prove this Theorem \ref{AB}, we will need the following
lemmas, which will be proven under our initial assumptions.

\begin{lemma}\label{CD}
Under assumptions {\rm (H1)--(H3)}, the integral operators
$\Gamma_1$ and $\Gamma_2$ defined above map
 $AP(\mathbb{R}; L^2(\Omega, \mathbb{H}_\alpha))$ into itself.
\end{lemma}

\begin{proof}
The proof for the quadratic-mean almost periodicity of $\Gamma_2 X$
is similar to that of $\Gamma_1X$ and hence will be omitted.
 Let $X\in AP(\mathbb{R}; L^2(\Omega; \mathbb{H}_\alpha))$.
Since $B\in B(L^2(\Omega; \mathbb{H}_\alpha), L^2(\Omega;\mathbb{H}))$
it follows that the function $t\to BX(t)$ belongs to
$AP(\mathbb{R}; L^2(\Omega;\mathbb{H}))$. Using Theorem \ref{U} it follows
that $\Psi (\cdot)= F (\cdot, BX(\cdot))$ is in
$AP(\mathbb{R}; L^2(\Omega; \mathbb{H}_\beta))$ whenever
$X\in AP(\mathbb{R}; L^2(\Omega; \mathbb{H}_\alpha))$. We can now
show that $\Gamma_1X\in AP (\mathbb{R}; L^2(\Omega; \mathbb{H}_\alpha))$.
Indeed, since $X\in AP(\mathbb{R}; L^2(\Omega; \mathbb{H}_\beta))$,
for every $\varepsilon >0$ there exists
$l(\varepsilon)>0$ such that for all $\xi$ there is
$t\in [\xi, \xi + l(\varepsilon)]$ with the property:
$$
\mathrm{E}\|\Psi X(t+\tau)-\Psi X(t)\|^2_\beta
<\nu^2\varepsilon\quad\text{for each }  t\in\mathbb{R},
$$
where $ \nu = \frac{\gamma^{1-\alpha}}{M'(\alpha)\Gamma
(1-\alpha)}$ with $\Gamma (\cdot)$
being the classical gamma function.

 Now, the estimate in (\ref{HJ}) yields
\begin{align*}
&\mathrm{E}\|\Gamma_1X(t+\tau)- \Gamma_1X(t)\|^2_\alpha \\
&\leq \mathrm{E}\Big(\int_0^\infty \|AT(s) P
 [\Psi (t-s+\tau)-\Psi (t-s)]\|_\alpha ds\Big)^2\\
&\leq M'(\alpha)^2 \Big(\int_0^\infty s^{-\alpha}e^{-\gamma s} ds \Big)
\Big(\int_0^\infty s^{-\alpha }e^{-\gamma s}\mathrm{E}\|\Psi
(t-s+\tau)-\Psi (t-s)\|^2_\beta ds\Big)\\
&\leq \Big(\frac{M'(\alpha) \Gamma (1-\alpha)}{\gamma^{1-\alpha}}
 \Big)^2\sup_{t\in\mathbb{R}}\mathrm{E}\|\Psi (t+\tau) -\Psi (t)\|^2_\beta ds
<\varepsilon
\end{align*}
for each $t\in\mathbb{R}$, and hence
$\Gamma_1 X\in AP(\mathbb{R}; L^2(\Omega; \mathbb{H}_\alpha))$.
\end{proof}

\begin{lemma}\label{DE}
Under assumptions {\rm (H1)--(H3)}, the integral operators $\Gamma_3$
and $\Gamma_4$ defined above map $AP(\mathbb{R}; L^2(\Omega; \mathbb{H}_\alpha))$
into itself.
\end{lemma}

\begin{proof}
The proof for the quadratic-mean almost periodicity of
$\Gamma_4X$ is similar to that of $\Gamma_3X$ and hence will be
omitted. Note, however, that for $\Gamma_4X$, we make use of
\eqref{Y} rather than  \eqref{Z}.

Let $X\in AP(\mathbb{R}; L^2(\Omega, \mathbb{H}_\alpha))$.
Since $C\in B(L^2(\Omega; \mathbb{H}_\alpha), L^2(\Omega;\mathbb{H}))$,
it follows that $C X\in AP(\mathbb{R}, L^2(\Omega;\mathbb{H})))$.
Setting $\Phi (t)=G(t, CX(t))$ and
using Theorem \ref{U} it follows that
$\Phi\in AP(\mathbb{R}; L^2(\Omega, \mathbb{H})))$. We can now show that
$\Gamma_3X\in AP(\mathbb{R}; L^2(\Omega, \mathbb{H}_\alpha))$.
Indeed, since $\Phi\in AP(\mathbb{R}; L^2(\Omega, \mathbb{H})))$, for every
$\varepsilon >0$ there exists $l(\varepsilon)>0$
such that for all $\xi$ there is $\tau\in [\xi, \xi +l(\varepsilon)]$
with
\[
\mathrm{E}\|\Phi (t+\tau)-\Phi (t)\|^2<\mu^2\cdot\varepsilon \quad
\text{for each }   t\in\mathbb{R},
\]
where $ \mu = \frac{\gamma^{1-\alpha}}{M(\alpha)\Gamma (1-\alpha)}$.
Now using the expression
$$
(\Gamma_3X)(t+\tau)-(\Gamma_3X)(t)
=\int_0^{\infty} T(s)P [\Phi (t-s+\tau) - \Phi (t-s)]\,ds
$$
and \eqref{Z} it easily follows that
\[
\mathrm{E}\|(\Gamma_3X)(t+\tau)-(\Gamma_3X)(t)\|^2_\alpha
<\varepsilon \quad  \text{for each }   t\in\mathbb{R},
\]
and hence, $\Gamma_3X\in AP (\mathbb{R}; L^2(\Omega; \mathbb{H}_\alpha))$.
\end{proof}

\begin{lemma}\label{EF}
Under assumptions {\rm (H1)--(H3)}, the integral operators
$\Gamma_5$ and $\Gamma_6$  map
$AP(\mathbb{R}; L^2(\Omega; \mathbb{H}_\alpha))$ into itself.
\end{lemma}

\begin{proof}
Let $X\in AP(\mathbb{R}; L^2(\Omega; \mathbb{H}_\alpha))$.
Since $L\in B(L^2(\Omega; \mathbb{H}_\alpha), L^2(\Omega;\mathbb{H}))$,
it follows that $L X\in AP(\mathbb{R}, L^2(\Omega; \mathbb{H})))$.
Setting $\Lambda (t)=H(t, LX(t))$ and
using Theorem \ref{U} it follows that
$\Lambda\in AP(\mathbb{R}; L^2(\Omega; \mathcal{L}_2^0))$.
We claim that $\Gamma_5X\in AP(\mathbb{R}; L^2(\Omega; \mathbb{H}_\alpha))$.
Indeed, since $\Lambda\in AP(\mathbb{R}; L^2(\Omega; \mathcal{L}_2^0))$, for every
$\varepsilon >0$ there exists $l(\varepsilon)>0$ such that for
all $\xi$ there is
$\tau\in [\xi, \xi + l(\varepsilon)]$ with
\begin{equation}\label{MN}
\mathrm{E}\|\Lambda (t+\tau)-\Lambda
(t)\|^2_{\mathcal{L}_2^0}<\zeta\cdot\varepsilon  \quad \text{for each }
t\in\mathbb{R},
\end{equation}
where
$$
\zeta =\frac{1}{2c^2\mathop{\rm Tr} Q \cdot K(\alpha, \gamma, \delta, \Gamma)}.
$$
Now using the expression
$$
(\Gamma_5X)(t+\tau)-(\Gamma_5X)(t)=\int_0^{\infty} T(s)P
[\Lambda (t-s+\tau) - \Lambda (t-s)]\,dW(s),
$$
Equation (\ref{D}), the arithmetic-geometric inequality, and
Ito isometry we have
\begin{align*}
&\mathrm{E}\|(\Gamma_5X)(t+\tau)-(\Gamma_5X)(t)\|^2_\alpha \\
&= \big\|\int_0^{\infty} T(s)P [\Lambda (t-s+\tau)
 - \Lambda (t-s)]\,dW(s)\big\|^2_\alpha\\
&\leq  c^2 \mathrm{E}\Big\{(1-\alpha) \big\|\int_0^{\infty}
 T(s)P [\Lambda (t-s+\tau) - \Lambda (t-s)]\,dW(s)\big\|\\
&\quad + \alpha  \big\|\int_0^{\infty} T(s)P [\Lambda (t-s+\tau)
 - \Lambda (t-s)]\,dW(s)\big\|_{[D(A)]}\Big\}^2\\
&\leq  c^2 \mathrm{E}\Big\{\big\|\int_0^{\infty} T(s)
 P [\Lambda (t-s+\tau) - \Lambda (t-s)]\,dW(s)\big\|\\
&\quad +  \big\|A\int_0^{\infty} T(s)P [\Lambda (t-s+\tau)
 - \Lambda (t-s)]\,dW(s)\big\|\Big\}^2\\
&\leq  2c^2 \mathop{\rm Tr} Q \Big\{\int_0^{\infty}\mathrm{E}\|T(s)
 P [\Lambda (t-s+\tau) - \Lambda (t-s)]\|^2_{\mathcal{L}_2^0} ds\\
&\quad + \int_0^{\infty} \mathrm{E}\|AT(s)P [\Lambda (t-s+\tau) -
\Lambda (t-s)]\|^2_{\mathcal{L}_2^0} ds\Big\}.
\end{align*}
Now
\[
\mathrm{E}\|T(s)P [\Lambda (t-s+\tau)
- \Lambda (t-s)]\|^2_{\mathcal{L}_2^0}\leq M^2 e^{-2\delta s}\mathrm{E}
\|\Lambda (t-s+\tau) - \Lambda (t-s)\|^2_{\mathcal{L}_2^0},
\]
and
\begin{align*}
&\mathrm{E}\|AT(s)P [\Lambda (t-s+\tau)
- \Lambda (t-s)]\|^2_{\mathcal{L}_2^0}\\
&\leq M_1^2(\alpha) s^{-2\alpha}
 e^{-2\gamma s}\mathrm{E}\|\Lambda (t-s+\tau)
- \Lambda (t-s)\|^2_{\mathcal{L}_2^0}.
\end{align*}
Hence,
$$
\mathrm{E}\|(\Gamma_5X)(t+\tau)-(\Gamma_5X)(t)\|^2_\alpha
\leq   2c^2\mathop{\rm Tr} Q \cdot K(\alpha, \gamma, \delta, \Gamma)
 \sup_{t\in\mathbb{R}}\mathrm{E}\|\Lambda (t+\tau) - \Lambda (t)\|^2_{\mathcal{L}_2^0}.
$$
where
$$
K(\alpha, \gamma, \delta,
\Gamma)=\frac{M^2}{2\delta} + \frac{M_1^2(\alpha)
\Gamma (1-2\alpha)}{\gamma^{1-2\alpha}},
$$
and it follows from (\ref{MN}) that $\Gamma_5X\in AP (\mathbb{R}; L^2(\Omega;
\mathbb{H}_\alpha)$.

As for $\Gamma_6X\in AP(\mathbb{R}; L^2(\Omega, \mathbb{H}_\alpha))$,
since $\Lambda\in AP(\mathbb{R}; L^2(\Omega; \mathcal{L}^2_0))$, for every
$\varepsilon >0$ there exists
$l(\varepsilon)>0$ such that for all $\xi$ there is
$\tau\in [\xi, \xi + l(\varepsilon)]$ with
\begin{equation}\label{MQ}
\mathrm{E}\|\Lambda (t+\tau)-\Lambda
(t)\|^2_{\mathcal{L}^2_0}<\kappa\cdot\varepsilon  \quad \text{for each }
t\in\mathbb{R},
\end{equation}
where $\kappa =\frac{\delta}{c^2\cdot\mathop{\rm Tr} Q \cdot K(\alpha, \beta)}$.
Now using the expression
$$
(\Gamma_6X)(t+\tau)-(\Gamma_6X)(t)=\int_{-\infty}^0 T(s)S [\Lambda (t-s+\tau) - \Lambda (t-s)]\,dW(s)$$
Equation (\ref{D} ), the arithmetic-geometric inequality,
and Ito isometry we have
\begin{align*}
&\mathrm{E}\|(\Gamma_6X)(t+\tau)-(\Gamma_6X)(t)\|^2_\alpha \\
&= \big\|\int_{-\infty}^0 T(s)S [\Lambda (t-s+\tau)
 - \Lambda (t-s)]\,dW(s)\big\|^2_\alpha\\
&\leq  2c^2 \mathop{\rm Tr} Q \Big\{\int_{-\infty}^0 \mathrm{E}\|T(s)S
 [\Lambda (t-s+\tau) - \Lambda (t-s)]\|^2_{\mathcal{L}_2^0} ds\\
&\quad + \int_{-\infty}^0 \mathrm{E}\|AT(s)S [\Lambda (t-s+\tau) -
\Lambda (t-s)]\|^2_{\mathcal{L}_2^0} ds\Big\}
\end{align*}
However,
\begin{gather*}
\mathrm{E}\|T(s)S [\Lambda (t-s+\tau)
 - \Lambda (t-s)]\|^2_{\mathcal{L}_2^0}\leq M^2 e^{2\delta s}
\mathrm{E}\|\Lambda (t-s+\tau) - \Lambda (t-s)\|^2_{\mathcal{L}_2^0},
\\
\mathrm{E}\|AT(s)S [\Lambda (t-s+\tau)
- \Lambda (t-s)]\|^2_{\mathcal{L}_2^0}\leq n^2_1(\alpha, \beta)
 e^{2\delta s}\mathrm{E}\|\Lambda (t-s+\tau)
- \Lambda (t-s)\|^2_{\mathcal{L}_2^0}
\end{gather*}
Thus,
$$
\mathrm{E}\|(\Gamma_6X)(t+\tau)-(\Gamma_6X)(t)\|^2_\alpha
 \leq c^2\cdot\mathop{\rm Tr} Q \cdot \frac{K(\alpha, \beta)}{\delta}
\sup_{t\in\mathbb{R}}\mathrm{E}\|\Lambda (t+\tau) - \Lambda (t)\|^2_{\mathcal{L}_2^0} ds,
$$
where $K(\alpha, \beta))=M^2+n_1^2(\alpha, \beta)$ is a constant
depending on $\alpha$ and $\beta$ and it
follows from (\ref{MQ}) that
$\Gamma_6X\in AP (\mathbb{R};L^2(\Omega; \mathbb{H}_\alpha))$.
\end{proof}

We are ready for the proof of Theorem \ref{AB}.

\begin{proof}
Consider the nonlinear operator $\mathbb{M}$ on the space
$AP(\mathbb{R}; L^2(\Omega; \mathbb{H}_\alpha))$ equipped with the
$\alpha$-sup norm
 $\|X\|_{\infty, \alpha}=\sup_{t\in\mathbb{R}}(\mathrm{E}\|X(t)\|^2_\alpha)^{1/2}$
and defined by
\begin{align*}
\mathbb{M}X(t)
&=-F(t, BX(t))-\int_{-\infty}^t A T(t-s)P F(s,
BX(s))\,ds \\
&\quad +\int_t^\infty A T(t-s)S F(s, B X(s))\,ds\\
&\quad +\int_{-\infty}^t T(t-s)P G(s, C X(s))\,ds
-\int_t^\infty T(t-s)S G(s, C X(s))\,ds\\
&\quad +\int_{-\infty}^t T(t-s)P H(s, L X(s))\,dW(s)
-\int_t^\infty \! T(t-s)S H(s, L X(s))\,dW(s)
\end{align*}
for each $t\in\mathbb{R}$.

As we have previously seen, for every
$X\in AP(\mathbb{R}; L^2(\Omega; \mathbb{H}_\alpha))$,
$f(\cdot, BX(\cdot))\in AP(\mathbb{R}; L^2(\Omega;
\mathbb{H}_\beta))\subset AP(\mathbb{R}; L^2(\Omega; \mathbb{H}_\alpha))$.
In view of Lemmas \ref{CD}, \ref{DE}, and \ref{EF}, it follows
that $\mathbb{M}$ maps $AP(\mathbb{R}; L^2(\Omega; \mathbb{H}_\alpha))$ into
itself. To complete the proof one has to show that $\mathbb{M}$
has a unique fixed point.

Let $X,  Y\in AP(\mathbb{R}; L^2(\Omega; \mathbb{H}_\alpha))$. By (H1),
(H2), and (H3), we obtain
\begin{align*}
\mathrm{E}\|F(t, BX(t))-F(t, BY(t))\|^2_\alpha
&\leq  k(\alpha)K_F\mathrm{E}\|BX(t)-BY(t)\|^2\\
&\leq  k(\alpha)\cdot K_F \varpi^2 \|X-Y\|^2_{\infty, \alpha},
\end{align*}
which implies
$$
\|F(\cdot, BX(\cdot))-F(\cdot, BY(\cdot))\|_{\infty, \alpha}
 \leq k'(\alpha)\cdot K'_F \varpi \|X-Y\|_{\infty, \alpha}.
$$
Now for $\Gamma_1$ and $\Gamma_2$, we have the following
evaluations
\begin{align*}
&\mathrm{E}\|(\Gamma_1X)(t)-(\Gamma_1Y)(t)\|^2_\alpha \\
&\leq  \mathrm{E}\Big(\int_{-\infty}^t\|AT(t-s)
P [F(s, BX(s))-F(s, BY(s))]\|_\alpha ds\Big)^2\\
&\leq  c^2 \Big(\int_{-\infty}^t(t-s)^{-\alpha}e^{-\gamma (t-s)} ds\Big)\\
&\quad \times\Big(\int_{-\infty}^t(t-s)^{-\alpha}
 e^{-\gamma (t-s)}\mathrm{E}\|[F(s, BX(s))
 -F(s, BY(s))]\|^2_\alpha ds\Big)\\
&\leq  c^2 k(\alpha) K_F\varpi^2\|X-Y\|^2_{\infty, \alpha}
\Big(\int_{-\infty}^t(t-s)^{-\alpha} e^{-\gamma (t-s)} ds\Big)^2\\
&=  c^2 k(\alpha) K_F \Big(\frac{\Gamma (1
-\alpha)}{\gamma^{1- \alpha}}\Big)^2 \varpi^2
\|X-Y\|^2_{\infty, \alpha},
\end{align*}
which implies
$$
\|\Gamma_1 X - \Gamma_1 Y\|_{\infty, \alpha}
\leq c\cdot k'(\alpha)\cdot K'_F
 \frac{\Gamma (1-\alpha))}{\gamma^{1-\alpha}} \varpi
\|X-Y\|_{\infty, \alpha}.
$$
Similarly,
\begin{align*}
&\mathrm{E}\|(\Gamma_2X)(t)-(\Gamma_2Y)(t)\|^2_\alpha \\
&\leq \mathrm{E}\Big(\int_t^{\infty}\|AT(t-s) S [F(s, BX(s))
 -F(s, BY(s))]\|_\alpha ds\Big)^2\\
&\leq \frac{c^2 k(\alpha) K_F}{\delta^2} \varpi^2
 \|X-Y\|^2_{\infty, \alpha},
\end{align*}
which implies
$$
\|\Gamma_2 X - \Gamma_2 Y\|_{\infty, \alpha}
 \leq \frac{c\cdot k'(\alpha)\cdot K'_F}{\delta}
\varpi \|X-Y\|_{\infty, \alpha}.
$$
As to $\Gamma_3$ and $\Gamma_4$, we have the following evaluations
\begin{align*}
&\mathrm{E}\|(\Gamma_3X)(t)-(\Gamma_3Y)(t)\|^2_\alpha \\
&\leq  \mathrm{E}\Big(\int_{-\infty}^t\|T(t-s) P [G(s, C X(s))
 -G(s, C Y(s))]\|_\alpha ds\Big)^2\\
&\leq  k_1\cdot M^2(\alpha)\Big(\int_{-\infty}^t(t-s)^{-\alpha}
 e^{-\gamma (t-s)} ds\Big)\\
&\quad \times \Big(\int_{-\infty}^t(t-s)^{-\alpha}
 e^{-\gamma (t-s)}\mathrm{E}\|G(s, C X(s))-G(s,
 C Y(s))\|^2_\alpha ds\Big)\\
&\leq  k_1\cdot K_G\cdot M^2(\alpha)
 \Big(\frac{\Gamma (1-\alpha)}{\gamma^{1-\alpha}}\Big)^2 \varpi^2
\|X-Y\|^2_{\infty, \alpha},
\end{align*}
which implies
$$
\|\Gamma_3X - \Gamma_3Y\|_{\infty, \alpha}\leq k'_1\cdot K'_G\cdot
 M'(\alpha) \frac{\Gamma (1-\alpha))}{\gamma^{1-\alpha}}
 \varpi \|X-Y\|_{\infty, \alpha}.
$$
Similarly,
\begin{align*}
&\mathrm{E}\|(\Gamma_4X)(t)-(\Gamma_4Y)(t)\|^2_\alpha \\
&\leq  \mathrm{E}\Big(\int_t^{\infty}\|T(t-s) S [G(s, C X(s))
 -G(s, C Y(s))]\|_\alpha ds\Big)^2\\
&\leq  \frac{k_1 K_G C(\alpha)}{\delta^2}  \varpi^2 \|X-Y\|^2_{\infty,
\alpha},
\end{align*}
which implies
$$
\|\Gamma_4X - \Gamma_4Y\|_{\infty, \alpha}
 \leq \frac{k'_1\cdot K'_G\cdot C'(\alpha)}{\delta}
\varpi \|X-Y\|_{\infty, \alpha}.
$$
Finally for $\Gamma_5$ and $\Gamma_6$, we have the following
evaluations
\begin{align*}
&\mathrm{E}\|(\Gamma_5X)(t)-(\Gamma_5Y)(t)\|_\alpha^2\\
&\leq  2c^2 \mathop{\rm Tr} Q \{\int_0^{\infty}\mathrm{E}\|T(s)
 P [H(t, L X(t)) - H(t, L Y(t))\|^2_{\mathcal{L}_2^0} ds\\
&\leq  2c^2\cdot\mathop{\rm Tr} Q\cdot k_1\cdot K(\alpha, \gamma,
 \delta, \Gamma)\cdot K_H\cdot\varpi^2\|X-Y\|^2_{\infty, \alpha},
\end{align*}
which implies
$$
\|\Gamma X_5 - \Gamma_5 Y\|_{\infty, \alpha}
 \leq 2c\cdot\sqrt{\mathop{\rm Tr} Q}\cdot k'_1\cdot
 K'(\alpha, \gamma, \delta, \Gamma)\cdot K'_H\cdot
 \varpi \|X-Y\|_{\infty, \alpha}.
$$
Similarly,
\[
\mathrm{E}\|(\Gamma_6X)(t)-(\Gamma_6Y)(t)\|^2_\alpha\leq
c^2\cdot\mathop{\rm Tr} Q\cdot k_1\cdot K_H\cdot \frac{K(\alpha,
\beta)}{\delta} \varpi^2\|X-Y\|^2_{\infty, \alpha},
\]
which implies
$$
\|\Gamma X_6 - \Gamma_6 Y\|_{\infty, \alpha}
\leq c\cdot\sqrt{\mathop{\rm Tr} Q}\cdot k'_1\cdot K'_H\cdot
\frac{K'(\alpha, \beta)}{\sqrt{\delta}}\cdot\varpi
\|X-Y\|_{\infty, \alpha}.
$$
Consequently,
$$
\|\mathbb{M}X - \mathbb{M}Y\|_{\infty, \alpha}
\leq\Theta\cdot\|X-Y\|_{\infty, \alpha} .
$$
Clearly, if $\Theta <1$, then  \eqref{A} has a unique fixed-point
by Banach fixed point theorem, which is obviously the only
quadratic-mean almost periodic solution to it.
\end{proof}

\section{Example}

Let $\Gamma \subset \mathbb{R}^N$ ($N \geq 1$) be a open bounded subset
with $C^2$ boundary $\partial \Gamma$. To illustrate our abstract
results, we study the existence of quadratic mean almost periodic
solutions to the stochastic heat equation in divergence given by
\begin{equation}\label{H}
\begin{gathered}
\partial \Big[\Phi + F(t, \widehat {\mathop{\rm div}} \Phi)\Big] =
\Big[\Delta \Phi + G(t, \widehat {\mathop{\rm div}} \Phi)\Big]\partial_t 
+ H(t, \Phi) \partial W(t), \quad \text{in } \Gamma\\
\Phi =0, \quad \text{on }  \partial \Gamma
\end{gathered}
\end{equation}
where the unknown $\Phi$ is a function of $\omega \in \Omega$, $t
\in \mathbb{R}$, and $x \in \Gamma$, the symbols $\widehat {\mbox div}$
and $\Delta$ stand respectively for the first and second-order
differential operators defined by
$$
\widehat {\mbox div} := \sum_{j=1}^N \frac{\partial}{\partial x_j}
,\quad \Delta = \sum_{j=1}^N \frac{\partial^2}{\partial x_j^2},
$$
and the coefficients
$F, G: \mathbb{R} \times L^2(\Omega, \mathbb{H}_0^\alpha(\Gamma) \cap
\mathbb{H}^{2\alpha}(\Gamma)) \mapsto L^2(\Omega, L^2(\Gamma))$ and
$H: \mathbb{R}\times L^2(\Omega, \mathbb{H}_0^\alpha(\Gamma) \cap
\mathbb{H}^{2\alpha}(\Gamma))\to L^2(\Omega, \mathcal{L}_2^0)$ are quadratic-mean
almost periodic.

Define the linear operator appearing in  (\ref{H}) as follows:
$$
A X  = \Delta X \quad \text{for all }
 u \in D(A) = L^2(\Omega; \mathbb{H}_0^1(\Gamma) \cap
\mathbb{H}^2(\Gamma)).
$$
Using the fact that the operator $\mathcal{A}$, defined in
$L^2(\Gamma)$ by
$$
\mathcal{A} u = \Delta u \quad \text{for all } u \in
D(\mathcal{A}) = \mathbb{H}_0^1(\Gamma) \cap \mathbb{H}^2(\Gamma),
$$
is sectorial and whose corresponding analytic semigroup
is hyperbolic, one easily sees
that the operator $A$ defined above is sectorial and hence is the
infinitesimal generator of an analytic semigroup
$(T(t))_{t \geq 0}$. Moreover, the semigroup $(T(t))_{t \geq 0}$
is hyperbolic as
$$
\sigma(A) \cap i \mathbb{R} = \emptyset.
$$
For each $\mu \in (0, 1)$, we take $\mathbb{H}_\mu = D((-\Delta)^\mu) =
L^2(\Omega, \mathbb{H}_0^\mu(\Gamma) \cap \mathbb{H}^{2\mu}(\Gamma))$ equipped
with its $\mu$-norm $\|\cdot\|_\mu$. Moreover, since
$\alpha \in (0, \frac{1}{2})$, we suppose that
$\frac{1}{2} < \beta < 1$.
Letting $L= I$, and $B X = C X = \mathop{\rm div}\,X$ for all
$X \in L^2(\Omega, \mathbb{H}_\alpha) = L^2(\Omega, D((-\Delta)^\alpha)) =
L^2(\Omega, \mathbb{H}_0^\alpha(\Gamma) \cap \mathbb{H}^{2\alpha}(\Gamma))$, one
easily see that both $B$ and $C$ are bounded from
$L^2(\Omega, \mathbb{H}_0^\alpha(\Gamma) \cap \mathbb{H}^{2\alpha}(\Gamma))$ in
$L^2(\Omega, L^2(\Gamma))$ with $\varpi = 1$.

We require the following assumption:
\begin{itemize}

\item[(H4)] Let $\frac{1}{2} < \beta < 1$, and
$$
F: \mathbb{R}\times L^2(\Omega, \mathbb{H}_0^\alpha(\Gamma)
\cap \mathbb{H}^{2\alpha}(\Gamma)) \mapsto L^2(\Omega,
\mathbb{H}_0^\beta(\Gamma) \cap \mathbb{H}^{2\beta}(\Gamma))
$$
be quadratic-mean almost periodic in $t \in \mathbb{R}$ uniformly in
$X \in L^2(\Omega, \mathbb{H}_0^\alpha(\Gamma) \cap \mathbb{H}^{2\alpha}(\Gamma))$,
$G: \mathbb{R} \times L^2(\Omega, \mathbb{H}_0^\alpha(\Gamma) \cap \mathbb{H}^{2\alpha}(\Gamma))
 \mapsto L^2(\Omega, L^2(\Gamma))$ be quadratic-mean almost
periodic in $t \in \mathbb{R}$ uniformly in
$X \in L^2(\Omega, \mathbb{H}_0^\alpha(\Gamma) \cap
\mathbb{H}^{2\alpha}(\Gamma))$. Moreover, the functions $F, G$ are
uniformly Lipschitz with respect to the second argument in the
following sense: there exists $K'> 0$ such that
\begin{gather*}
\mathrm{E} \|F(t,\Phi_1)-F(t,\Phi_2)\|_\beta
\leq K' \mathrm{E} \|\Phi_1-\Phi_2\|_{L^2(\Gamma)}, \\
\mathrm{E} \|G(t,\Phi_1)-G(t,\Phi_2)\|_{L^2(\Gamma)}
\leq K' \mathrm{E} \|\Phi_1-\Phi_2\|_{L^2(\Gamma)},\\
\mathrm{E}\|H(t, \psi_1)-H(t, \psi_2)\|_{\mathcal{L}_2^0}^2
\leq  K' \mathrm{E}\|\psi_1-\psi_2\|_{L^2(\Gamma)}^2
\end{gather*}
for all $\Phi_1, \Phi_2, \psi_1, \psi_2\in L^2(\Omega; L^2(\Gamma))$
and $t\in \mathbb{R}$.
\end{itemize}

As a final result, we have the following theorem.

\begin{theorem} \label{thm5.1}
Under the above assumptions including {\rm (H4)},
 the $N$-dimensional stochastic heat equation \eqref{H} has a
unique quadratic-mean almost periodic solution
$\Phi \in L^2(\Omega, \mathbb{H}_0^1(\Gamma) \cap \mathbb{H}^2(\Gamma))$ whenever
$K'$ is small enough.
\end{theorem}


\begin{thebibliography}{10}

\bibitem{AT} P. Acquistapace and B. Terreni;
\emph{A Unified Approach to Abstract Linear Parabolic Equations},
 Tend. Sem. Mat. Univ. Padova 78 (1987) 47-107.

\bibitem{BD} P. Bezandry and T. Diagana;
\emph{Existence of Almost Periodic Solutions
to Some Stochastic Differential Equations}.  Applicable
Analysis. \textbf{86} (2007), no. 7,  pages 819-827.

\bibitem{BD1} P. Bezandry and T. Diagana;
\emph{Square-mean almost periodic solutions
nonautonomous stochastic differential equations}. Electron. J.
Diff. Equ. Vol. 2007(2007), No. 117, pp. 1-10.

\bibitem{C} C. Corduneanu;
\emph{Almost Periodic Functions}, 2nd Edition.
Chelsea-New York, 1989.

\bibitem{da} G. Da Prato and C. Tudor;
\emph{Periodic and Almost Periodic Solutions
for Semilinear Stochastic Evolution Equations},  Stoch. Anal.
Appl. \textbf{13}(1) (1995), 13--33.

\bibitem{D1} T. Diagana;
\emph{Existence of Weighted Pseudo Almost Periodic Solutions
to Some Classes of Hyperbolic Evolution Equations},  J. Math.
Anal. Appl., \textbf{350}(2009), No. 1, pp. 18-28.

\bibitem{toka1} T. Diagana;
\emph{Existence of pseudo almost periodic solutions to some
classes of partial hyperbolic evolution equations}. E. J.
Qualitative Theory of Diff. Equ., No. 3. (2007), pp. 1-12.

\bibitem{TT} T. Diagana;
\emph{Pseudo almost periodic functions in Banach
spaces}. Nova Science Publishers, Inc., New York, 2007.

\bibitem{D} A. Ya. Dorogovtsev and O. A. Ortega;
\emph{On the Existence of Periodic
Solutions of a Stochastic Equation in a Hilbert Space}.  Visnik
Kiiv. Univ. Ser. Mat. Mekh., No. \textbf{30} (1988), 21-30, 115

\bibitem{I} A. Ichikawa;
\emph{Stability of Semilinear Stochastic Evolution
Equations}.  J. Math. Anal. Appl., \textbf{90} (1982), no.1,
12-44.

\bibitem{Alessandra2}  A. Lunardi;
\emph{Analytic Semigroups and Optimal Regularity
in Parabolic Problems}, PNLDE Vol. \textbf{16}, Birkh\"{a}auser
Verlag, Basel, 1995.

\bibitem{KB} D. Kannan and A.T. Bharucha-Reid;
\emph{On a Stochastic Integro-differential Evolution of Volterra Type}.
J. Integral Equations, \textbf{10} (1985), 351-379.

\bibitem{K} T. Kawata;
\emph{Almost Periodic Weakly Stationary Processes.
Statistics and probability: essays in honor of C. R. Rao}, pp.
383--396, North-Holland, Amsterdam-New York, 1982.

\bibitem{KM1} D. Keck and M. McKibben;
\emph{Functional Integro-differential
Stochastic Evolution Equations in Hilbert Space}.  J. Appl.
Math. Stochastic Analy. \textbf{16}, no.2 (2003), 141-161.

\bibitem{MR} L. Maniar and R. Schnaubelt;
\emph{Almost Periodicity of Inhomogeneous
Parabolic Evolution Equations}, Lecture Notes in Pure and Appl.
Math. Vol. 234, Dekker, New York, 2003, pp. 299-318.

\bibitem{t3} C. Tudor;
\emph{Almost Periodic Solutions of Affine Stochastic
Evolutions Equations},  Stochastics and Stochastics Reports
\textbf{38} (1992), 251-266.

\end{thebibliography}


\end{document}