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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2010(2010), No. 72, pp. 1--12.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2010 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2010/72\hfil Almost periodic solutions]
{Almost periodic solutions of higher order differential equations
 on Hilbert spaces}

\author[L. T. Nguyen\hfil EJDE-2010/72\hfilneg]
{Lan Thanh Nguyen}

\address{Lan Thanh Nguyen \newline
Department of Mathematics, Western Kentucky University,
Bowling Green, KY 42101, USA}
\email{Lan.Nguyen@wku.edu}

\thanks{Submitted March 26, 2010. Published May 17, 2010.}
\subjclass[2000]{34G10, 34K06, 47D06}
\keywords{Almost periodic; higher order differential equations}

\begin{abstract}
 We find necessary and sufficient conditions for the differential
 equation
 $$
 u^{(n)}(t) = Au(t)+f(t), \quad t\in \mathbb{R}
 $$
 to have a unique almost periodic solution. Some applications are
 also given.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{proposition}[theorem]{Proposition}

\section{Introduction}

In this article, we study the almost periodicity of solutions to
the differential equation
\begin{eqnarray}\label{eq}
u^{(n)}(t)=Au(t)+f(t), \quad t\in \mathbb{R},
\end{eqnarray}
where $A$ is a linear, closed  operator on a Hilbert space $H$ and
$f$ is a  function from  $\mathbb{R}$ to $H$.  The asymptotic
behavior and, in particular, the almost periodicity of solutions
of \eqref{eq} has been a subject of intensive study for recent
decades, see e.g. \cite{arba,lezi,minh,lan2,phong,vusc,ruvu}
and references therein. A particular condition for almost periodicity
is the countability of the spectrum of the solution. In this paper we
investigate the almost periodicity of mild solutions of Equation
\eqref{eq}, when $A$ is a linear, unbounded operator on a Hilbert
space $H$. We use the Hilbert space $AP(\mathbb{R}, H)$ introduced
in \cite{ka}, defined by follows: Let $(, )$ be the inner product
of $H$ and let $AP_b(\mathbb{R},E)$ be the space of all almost
periodic functions from $\mathbb{R}$ to $H$. The completion of
$AP_b(\mathbb{R},E)$ is then a Hilbert space with the inner
product defined by:
$$
\langle f,g\rangle  := \lim_{T\to \infty}\frac{1}{2T}
\int_{-T}^T(f(s), g(s)) ds.
$$
First, we establish the relationship between the Bohr transforms
of the almost periodic solutions of \eqref{eq} and those of the
inhomogeneity $f$. We then give  a necessary and sufficient
condition so that \eqref{eq} admits a unique almost periodic
solution for each almost periodic inhomogeneity $f$.
As applications, in Section 4 we show a short proof of
the Gearhart's Theorem: If $A$ is generator of a strongly
continuous semigroup $T(t)$, then $1\in \varrho(T(1))$ if and
only if $2k\pi i\in \varrho(A)$ and
$\sup_{k\in \mathbb{Z}}\|(2k\pi i- A)^{-1}\|< \infty $.

\section{Hilbert space of almost periodic functions}

Let us fix some notation. Define $S(t)f$ as $(S(t)f)(s) =f(s+t)$.
Recall that a bounded,  uniformly continuous function $f$ from
$\mathbb{R}$ to a Banach space $H$ is almost periodic, if the set
$\{S(t)f:t\in \mathbb{R}\}$ is relatively compact in
$BUC(\mathbb{R}, H)$, the space of bounded uniformly continuous
functions with sup norm topology.  Let $H$ be now a complex
Hilbert space with $(, )$ and $\|\cdot \|$ be the inner product
and the norm  in $H$, respectively. Let $AP_b(\mathbb{R},H)$ be
the space of all almost periodic functions from $\mathbb{R}$ to
$H$. In $AP_b(\mathbb{R},H)$ the following expression
$$
\langle f,g\rangle  := \lim_{T\to \infty}
\frac{1}{2T}\int_{-T}^T(f(s), g(s)) ds
$$
exists and defines an inner product. Hence, $AP_b(\mathbb{R},H)$
is a pre-Hilbert space and its completion, denoted by
$AP(\mathbb{R},H)$, is a Hilbert space. The inner product and the
norm in $AP(\mathbb{R},H)$ are denoted by $\langle, \rangle$ and
$\| \cdot \|_{AP}$, respectively.

For each function $f\in AP(\mathbb{R},H)$, the Bohr transform is
defined by
$$
a(\lambda,f):=\lim_{T\to \infty}\frac{1}{2T}\int_{-T}^Tf(s)
e^{-i\lambda s}ds.
$$
The set
$$
\sigma(f):=\{\lambda \in \mathbb{R}:a(\lambda, f)\not= 0\}
$$
is called the Bohr spectrum of $f$. It is well known that
$\sigma(f)$ is countable for each function $f\in
AP(\mathbb{R},H)$. The Fourier-Bohr series of $f$ is
$$
\sum_{\lambda \in \sigma(f)}a(\lambda, f)e^{i\lambda t}
$$
and it converges to $f$ in the norm topology of
$AP(\mathbb{R},H)$. The following Parseval's equality also holds:
$$
\|f\|_{AP(\mathbb{R},H)}^2=\sum_{\lambda \in
\sigma(f)}\|a(\lambda, f)\|^2.
$$
For more information about the almost periodic functions and
properties of the Hilbert space $AP(\mathbb{R},H)$, we refer
readers to \cite{ka,lezi,phong}.

Let $W^k(AP)$ be the space consisting of all almost periodic
functions $f$, such that $f'$, $f''$, \dots , $f^{(k)}$ are in
$AP(\mathbb{R},H)$. $W^k(AP)$ is then a Hilbert space with the
norm
$$
\|f\|^2_{W^k(AP)}:=\sum_{i=0}^k\|f^{(i)}\|^2_{AP(\mathbb{R}, H)}.
$$
Note   that, for $k\ge 0$, the $W^{k+1}(AP)$-topology is stronger
than the sup-norm topology in $C_b^{k}(\mathbb{R}, H)$, the space
of $k$-times continuously differentiable functions with all
derivatives until order $k$ inclusively bounded (see
\cite{triebel}). We will use the following lemma in the sequel.
(See also \cite[Lemma 2.1]{lan2}).

\begin{lemma}\label{lem2}
If $F$ is a function in $W^1(AP)$ and $f= F'$, then we have
\begin{equation}\label{der}
a(\lambda,f)=\lambda i \cdot a(\lambda,F).
\end{equation}
\end{lemma}

\begin{proof}
 If $\lambda \neq 0$, using  integration by part we have
\begin{align*}
\frac{1}{2T}\int_{-T}^Te^{-i\lambda s}f(s) ds
&= \frac{1}{2T}F(t)e^{-i\lambda t}|_{-T}^T
   +\frac{i\lambda }{2T}\int_{-T}^TF(s)e^{-i\lambda s}ds\\
&= \frac{F(T)e^{-i\lambda T}-F(-T)e^{i\lambda T}}{2T}
   +i\lambda \frac{1}{2T}\int_{-T}^TF(s)e^{-i\lambda s}ds.
\end{align*}
Let $T\to \infty$, and note that $F(t)$ is bounded, we have \eqref{der}.
If $\lambda =0$, then
$$
a(0,f)=\lim_{T\to \infty}\frac{1}{2T}\int_{-T}^Tf(s) ds
= \lim_{T\to \infty}\frac{F(T)-F(-T)}{2T}=0,
$$
which also satisfies \eqref{der}.
\end{proof}

Finally, for a linear and closed operator $A$ in a Hilbert space $H$,
we denote the domain, the range, the spectrum and the resolvent
set of $A$ by $D(A)$, $Range(A)$, $\sigma (A)$ and $\varrho(A)$,
respectively.

\section{Almost periodic mild solutions of  differential equations}

We now turn to the  differential equation
\begin{equation} \label{high}
u^{(n)}(t)=Au(t)+f(t), \quad t\in \mathbb{R},
\end{equation}
where $n\in \mathbb{N}^+$ and $A$ is a linear and closed operator on $H$.
First we define two types of solutions to Equation \eqref{high}.
Let $I: C(\mathbb{R}, H) \to C(\mathbb{R}, H)$ be the
operator defined by $If(t): = \int_0^tf(s)ds$ and $I^{n} f:=
I(I^{n-1}f)$.

\begin{definition}\label{def3.1} \rm
(a) We say that $u: \mathbb{R} \to H$ is a classical
solution of  \eqref{high}, if $u $ is $n$-times continuously
differentiable , $u(t)\in D(A)$  and \eqref{high} is
satisfied for all $t\in \mathbb{R}$.

(b) For $f\in C(\mathbb{R}, H)$, a continuous function $u$ is
called a mild solution of  \eqref{high}, if $I^{n}u(t)\in
D(A)$  and there exist $n$ points $ v_0$, $v_1$, \dots , $v_{n-1}$ in
$H$ such that
\begin{equation}\label{mild}
u(t) = \sum_{j=0}^{n-1}\frac{t^j}{j!}v_j + AI^{n}u(t) + I^{n}f(t)
\end{equation}
for all $t\in \mathbb{R}$.
\end{definition}

\noindent{\bf Remark.}
 Using the standard argument, we can prove the following:
\begin{itemize}
\item[(i)] If a mild solution $u$ is $m$ times differentiable, $0\leq m < n$ , then $v_i$, ($i= 0,1,\dots , m$), are the initial values, i.e. $u(0)= v_0$, $u'(0)=v_1$, \dots , and $u^{(m)}(0)=v_{m}$.
    \item[(ii)] If $n=1$ and $A$ is the generator of a $C_0$ semigroup $T(t)$, then a continuous function $u:\mathbb{R}\to E$ is a mild solution of \eqref{high} if and only if  it has the form
$$u(t) =T(t-s)u(s)+\int_s^tT(t-r)f(r)dr$$
for $t\ge s$.
\item[(iii)]  If $u$ is a bounded mild solution of \eqref{high} corresponding to a bounded inhomogeneity $f$ and $\phi \in L^1(\mathbb{R}, E)$ then $u*\phi$ is a mild solution of \eqref{high} corresponding to $f*\phi$.
\end{itemize}
The mild solution to \eqref{high} defined by \eqref{mild} is really an extension of
 classical solution in the sense that every classical solution is a mild solution and conversely, if a mild solution is $n$-times continuously differentiable, then it is a classical solution. That statement is actually contained in the following lemma (see also \cite{lan1}).

\begin{lemma}\label{ntime}
Suppose $0\le m\le  n$ and $u$ is a mild solution of  {\rm
\eqref{high}}, which is $m$-times continuously differentiable.
Then for all $t\in \mathbb{R}$ we have $I^{n-m}u(t)\in D(A)$   and
\begin{equation} \label{deri}
u^{(m)}(t) = \sum_{j=m}^{n-1}\frac{t^{j-m}}{(j-m)!}v_j +AI^{n-m}u(t)
+I^{n-m}f(t),
\end{equation}
where $v_m, \dots , v_{n-1}$ are given in Definition \ref{def3.1}(b).
\end{lemma}

\begin{proof}
 If $m=0$, then \eqref{deri} coincides with \eqref{mild}. We prove for
 $m=1$: Let $v(t):= AI^nu(t)$. Then, by \eqref{mild}, $v$ is
continuously differentiable and
$$
v'(t)= u'(t)- \sum_{j=1}^{n-1}\frac{t^{j-1}}{(j-1)!}v_j-I^{n-1}f(t).
$$
Let  $h>0$ and put
$$
v_h :=\frac{1}{h}\int_t^{t+h}I^{n-1}u(s)ds.
$$
Then $v_h\to (I^{n-1}u)(t)$ for $h\to 0$ and
\begin{align*}
\lim_{h\to 0}Av_h
&=\lim_{h\to 0}\frac{1}{h} \Big(A\int_0^{t+h}I^{n-1}u(s)ds
-A\int_0^t I^{n-1}u(s)ds\Big)\\
&=  \frac{1}{h}(v(t+h)-v(t))\\
&= v'(t).
\end{align*}
Since $A$ is a closed operator, we obtain that $I^{n-1}u(t)\in D(A)$ and
$$
AI^{n-1}u(t) =u'(t)- \sum_{j=1}^{n-1}
\frac{t^{j-1}}{(j-1)!}v_j-I^{n-1}f(t),
$$
from which \eqref{deri} with $m=1$ follows.  If $m>1$,
 we obtain \eqref{deri} by repeating the above process $(m-1)$ times.
\end{proof}

In particular, if the mild solution $u$ is $n$-times continuously
differentiable, then \eqref{deri} becomes $u^{(n)}(t)=Au(t)+f(t)$;
 i.e. $u$ is a classical solution of \eqref{high}.

We now consider the mild solutions of \eqref{high}, which are
$(n-1)$ times continuously differentiable. The following proposition
describes the connection between the Bohr transforms of such
solutions and those of $f(t)$.

\begin{proposition}\label{prop1}
Suppose $A$ is a linear and closed operator on $H$,
$f\in AP(\mathbb{R},H)$ and $u$ is an almost periodic mild solution of
\eqref{high}, which belongs to $C_b^{n-1}(\mathbb{R}, H)$.
Then
\begin{equation}\label{fourier}
[(\lambda i)^n -A]a(\lambda,u) = a(\lambda,f)
\end{equation}
for every $\lambda\in \mathbb{R}$.
\end{proposition}

\begin{proof}
Suppose $u$ is an almost periodic mild solution of
 \eqref{high}, which belongs to $C_b^{n-1}(\mathbb{R}, H)$
and $\lambda $ is a real number. Using \eqref{deri} with $m=n-1$
we have
\begin{equation} \label{n-1}
u^{(n-1)}(t) = u^{(n-1)}(0)+AIu(t) +If(t).
\end{equation}
For $\lambda \neq  0$, multiplying each side of  \eqref{n-1}
with $e^{-i\lambda t}$ and taking definite integral from
$-T$ to $T$ on both sides, we have
\begin{equation} \label{f1}
\begin{aligned}
 \int_{-T}^Te^{-i\lambda t}u^{(n-1)}(t)dt
&=  \int_{-T}^Te^{-i\lambda t}u^{(n-1)}(0)dt
 +A\int_{-T}^Te^{-i\lambda t}\int_0^tu(s)\,ds\,dt\\
&\quad +\int_{-T}^Te^{-i\lambda t}\int_0^tf(s)\,ds\,dt.
\end{aligned}
\end{equation}
Here we used the fact that $\int_{a}^b Au(t)dt=A\int_a^bu(t)dt$
for a closed operator $A$. It is easy to see that
$$
\int_{-T}^Te^{-i\lambda t}u^{(n-1)}(0)dt
=-\frac{e^{-i\lambda T}u^{(n-1)}(0)
-e^{i\lambda T}u^{(n-1)}(0)}{i\lambda }
$$
and,  applying integration by part for any integrable function $g(t)$,
we have
\begin{equation} \label{f2}
\begin{aligned}
 \int_{-T}^Te^{-i\lambda t}\int_0^tg(s)\,ds\,dt
&=-\frac{1}{i\lambda}e^{-i\lambda t}\int_0^tg(s)ds|_{-T}^T
 +\frac{1}{i\lambda }\int_{-T}^Te^{-i\lambda t}g(t)dt\\
&= -\frac{1}{i\lambda}e^{-i\lambda T}\int_0^T g(t)dt
 +\frac{1}{i\lambda} e^{i\lambda T}\int_0^{-T}g(t)dt\\
&\quad +\frac{1}{i\lambda }\int_{-T}^Te^{-i\lambda t}g(t)dt.
\end{aligned}
\end{equation}
Using \eqref{f2} for $g(t)=u(t)$ and $g(t)=f(t)$ in \eqref{f1},
respectively, we have
\begin{equation} \label{d2}
\begin{aligned}
& \frac{1}{2T}\int_{-T}^Te^{-i\lambda t}u^{(n-1)}(t)dt\\
&= -\frac{e^{-i\lambda T}u^{(n-1)}(0)
 -e^{i\lambda T}u^{(n-1)}(0)}{2 i\lambda T}
 -\frac{e^{-i\lambda T}}{2 i\lambda T}\Big(A\int_0^Tu(t)dt
 +\int_0^Tf(t)dt\Big)\\
&\quad +\frac{e^{i\lambda T}}{2 i\lambda T}\Big(A\int_0^{-T}u(t)dt
 +\int_0^{-T}f(t)dt\Big)\\
&\quad + \frac{1}{i\lambda 2T}\Big(A\int_{-T}^Te^{-i\lambda t}u(t)dt
 +\int_{-T}^Te^{-i\lambda t}f(t)dt\Big) \\
&=  I_1 +I_2+I_3,
\end{aligned}
\end{equation}
where
$$
I_1=  -\frac{e^{-i\lambda T}u^{(n-1)}(0)
 -e^{i\lambda T}u^{(n-1)}(0)}{2 i\lambda T}\to 0
$$
as $T\to \infty$;
\begin{align*}
I_2&= - \frac{e^{-i\lambda T}}{2i\lambda T}\Big(A\int_0^Tu(t)dt
 +\int_0^Tf(t)dt\Big)
 +\frac{e^{i\lambda Ti\lambda}}{2 i\lambda T}
 \Big(A\int_0^{-T}u(t)dt
 +\int_0^{-T}f(t)dt\Big)\\
&= -\frac{e^{-i\lambda T}}{2i\lambda T}\Big(u^{(n-1)}(T)-u^{(n-1)}(0)\Big)
+\frac{e^{i\lambda T}}{2i\lambda T}\Big(u^{(n-1)}(-T)-u^{(n-1)}(0)\Big)\\
&\quad \to 0 \quad\text{as }T\to \infty,
\end{align*}
and
\begin{equation}\label{f3}
I_3= \frac{1}{i\lambda}\Big(\frac{1}{2T}A\int_{-T}^Te^{-i\lambda t}u(t)dt +\frac{1}{2T}\int_{-T}^Te^{-i\lambda t}f(t)dt\Big).
\end{equation}
Let $u_T:= \frac{1}{2T}\int_{-T}^Te^{-i\lambda t}u(t)dt $. It is clear that
\begin{equation}\label{close1}
\lim_{T\to \infty}u_T= a(\lambda, u)
\end{equation}
 and from \eqref{f3} we have
\begin{equation} \label{close2}
\begin{aligned}
  Au_T&= \frac{1}{2T}A\int_{-T}^Te^{-i\lambda t}u(t)dt\\
  &= i\lambda I_3-\frac{1}{2T}\int_{-T}^Te^{-i\lambda t}f(t)dt\\
 &=  i\lambda\Big(\frac{1}{2T}\int_{-T}^{T}u^{(n-1)}(t)dt-I_1-I_2\Big) -\frac{1}{2T}\int_{-T}^Te^{-i\lambda t}f(t)dt\\
 &\quad \to i\lambda a(\lambda , u^{(n-1)})-a(\lambda, f)
\quad\text{as } T\to \infty .
\end{aligned}
\end{equation}
Since $A$ is a closed operator, from \eqref{close1} and \eqref{close2},
we obtain
$a(\lambda, u) \in D(A)$ and
\[
 Aa(\lambda, u) = i\lambda a(\lambda, u^{(n-1)})-a(\lambda, f)
 = (i\lambda)^na(\lambda, u)-a(\lambda, f),
\]
 from which \eqref{fourier} follows.
Next, if $\lambda =0$, using Formula \eqref{n-1}, we have
\begin{gather*}
u^{(n-1)}(T)=v_{n-1}+A\int_0^Tu(t)dt +\int_0^T f(t)dt,\\
u^{(n-1)}(-T)=v_{n-1}+A\int_0^{-T}u(t)dt +\int_0^{-T} f(t)dt.
\end{gather*}
Hence,
\begin{equation} \label{sub}
\frac{u^{(n-1)}(T)-u^{(n-1)}(-T)}{2T}
=A\frac{1}{2T}\int_{-T}^Tu(t)dt +\frac{1}{2T}\int_{-T}^T f(t)dt.
\end{equation}
Let $u_T= \frac{1}{2T}\int_{-T}^Tu(s)ds$. Then
 $\lim_{t\to \infty}u_T=a(0, u)$,  and by \eqref{sub},
\begin{align*}
Au_T&=  \frac{1}{2T}A\int_{-T}^Tu(s)ds\\
&=  \frac{u^{(n-1)}(T) -u^{(n-1)}(-T)}{2T}
 - \frac{1}{2T}\int_{-T}^Tf(s)ds
 \to -a(0, f) \quad\text{as } T\to \infty .
\end{align*}
Again, since $A$ is a closed operator, it implies
$a(0, u) \in D(A)$ and $Aa(0, u) =-a(0, f)$, from
which \eqref{fourier} follows, and this completes the proof.
\end{proof}

Note that Proposition \ref{prop1} also holds in a Banach space.
We are now going to look for conditions that Equation \eqref{high}
has an almost periodic mild solution.

\begin{theorem}\label{theorem1}
Suppose $A$ is a linear and closed operator and  $f$ is a function
in $AP(\mathbb{R}, H)$. Then the following statements are
equivalent
\begin{itemize}
\item[(i)] Equation  \eqref{high} has an almost periodic mild solution,
which is in $W^n(AP)$;
\item[(ii)] For every $\lambda\in \sigma(f)$,
 $a(\lambda, f)\in \mathop{\rm Range}((i\lambda)^n-A)$ and there
exists a set $\{x_{\lambda}\}_{\lambda\in \sigma(f)}$ in $H$ satisfying
$((i\lambda)^n-A)x_{\lambda}=a(\lambda,f)$, for which the following
inequalities
\begin{equation}\label{ab}
  \sum_{\lambda\in \sigma(f)} |\lambda |^{2k}\|x_{\lambda}\|^2 < \infty
\end{equation}
hold for  $k=0, 1, 2, \dots , n$.
\end{itemize}
\end{theorem}

\begin{proof}
(i) $\Rightarrow$ (ii): Let $u(t)$ be an almost periodic solution
to \eqref{high}, which is in $W^n(AP)$. By Proposition \ref{prop1},
$((i\lambda)^n -A)a(\lambda, u)= a(\lambda, f)$.
Hence $a(\lambda, f)\in Range((i\lambda)^n-A)$ for all
$\lambda \in \sigma(f)$.
Put now $x_{\lambda}:= a(\lambda, u)$ for $\lambda \in \sigma(f)$.
Then it satisfies $((i\lambda)^n -A)x_{\lambda}= a(\lambda, f)$.
Moreover, by Lemma \ref{der},
$(i\lambda)^k x_{\lambda}=a(\lambda, u^{(k)})$.
Hence, for $0\le k\le n$ we have
\begin{align*}
\sum_{\lambda\in \sigma(f)} |\lambda | ^{2k}\|x_{\lambda}\|^2
&= \sum_{\lambda\in \sigma(f)} |a(\lambda, u^{(k)})|^2\\
&\le  \sum_{\lambda\in \sigma(f)\cup \sigma(u^{(k)})} |a(\lambda, u^{(k)})|^2\\
& =\sum_{\lambda\in \sigma(u^{(k)})} |a(\lambda, u^{(k)})|^2\\
&= \|u^{(k)}\|_{AP}^2 ,
\end{align*}
from which \eqref{ab} follows.

(ii) $\Rightarrow$ (i):
Let $\{x_{\lambda}\}_{\lambda\in \sigma(f)}$ be a set in $H$
satisfying $((i\lambda)^n-A)x_{\lambda}=a(\lambda,f)$, for which
\eqref{ab} holds. Put
\[
f_N(t):= \sum_{\lambda\in \sigma(f), |\lambda|<N}e^{i\lambda t}
 a(\lambda, f), \quad
u_N(t):= \sum_{\lambda\in \sigma(f), |\lambda|<N}
e^{i\lambda t}x_{\lambda}.
\]
It is then easy to find their norms
$$
\|u_N^{(k)}\|_{AP}^2= \sum_{\lambda\in \sigma(f), |\lambda|<N}
|\lambda |^{2k}\| x_{\lambda}\|^2 .
$$
 From \eqref{ab} it implies that $u_N^{(k)}\to U_k$ as $N\to
\infty$ for some functions $U_k$ ($k=0, 1, 2, \dots , n$) in the
topology of $AP(\mathbb{R},H)$. Since the differential operator is
closed, we obtain  $U_k'=U_{k-1}$ and $\lim_{N\to
\infty}u_N=U_0$ in the topology of $W^n(AP)$. It remains to show
that $U_0$ is a mild solution of \eqref{high}. In order to do
that, note  $u_N$ is a classical, and hence, a mild  solution of
\eqref{high} corresponding to $f_N$; i.e.,
\begin{equation}\label{n}
u_N(t) = \sum_{i=0}^{n-1}\frac{t^i}{i!}u_N^{(i)}(0) + AI^{n}u_N(t)
+ I^{n}f_N(t) .
\end{equation}
For each $t\in \mathbb{R}$ we have
$$
\lim_{N \to \infty} \int_0^tf_N(s)ds =\int_0^tf(s)ds, \quad
\lim_{N \to \infty} \int_0^tu_N(s)ds =\int_0^tU_0(s)ds.
$$
 Hence,
$$
\lim_{N \to \infty}I^ku_N(t) = I^kU_0(t), \quad
\lim_{N \to \infty}I^kf_N(t) = I^kf(t)
$$
for $k=0, 1, 2,\dots , n$. Using Equation \eqref{n}, we have
\begin{align*}
\lim_{N\to \infty} A(I^nu_N(t))
&= \lim_{N\to \infty} \Big( u_N(t) - \sum_{i=0}^{n-1}\frac{t^i}{i!}u_N^{(i)}(0) - I^{n}f_N(t)  \Big) \\
&=  U_0(t) - \sum_{i=0}^{n-1}\frac{t^i}{i!}U_0^{(i)}(0) - I^{n}f(t).
\end{align*}
Since $A$ is a closed operator, we obtain $ I^nU_0(t) \in D(A)$ and
$$
 A(I^nU_0(t))=  u(t) - \sum_{i=0}^{n-1}\frac{t^i}{i!}U_0^{(i)}(0)
- I^{n}f(t),
$$
which shows that $U_0$ is a mild solution of \eqref{high} and the
proof is complete.
\end{proof}

Note that if condition (ii) in Theorem \ref{theorem1} holds,
Equation \eqref{high} may have two or more almost periodic mild
solutions. We are  going to find conditions such that for each
almost periodic function $f$, Equation \eqref{high} has a unique
almost periodic mild solution. We are now in the position to
state the main result.

\begin{theorem}\label{theo2}
Suppose $A$ is a linear and closed operator on a Hilbert space $H$
and $M$ is a closed subset of $\mathbb{R}$. For $0\le k \le n$,
the following statements  are equivalent
\begin{itemize}
\item[(i)] For each function $f\in W^k(AP)$ with $\sigma(f)\subset M$,
 Equation \eqref{high} has a unique  almost periodic mild solution $u$
 in $W^n(AP)$ with $\sigma(u)\subseteq M$.
\item[(ii)] For each  $\lambda \in M$, $ (i \lambda)^n \in \varrho(A)$
and
\begin{equation}\label{ness2}
\sup_{\lambda \in M}|\lambda|^{m}\|( (i\lambda)^n -A)^{-1}\| < \infty
\end{equation}
for all $m=0, 1, 2, \dots , n-k$.
\end{itemize}
\end{theorem}

\begin{proof}
$(i) \Rightarrow (ii)$:
Let $W^k(AP)_{|M}$ be the subspace of all functions $f$ in $W^k(AP)$
with $\sigma(f)\subset M$. Then $W^k(AP)_{|M}$ is a Hilbert space
by nature. Let $x$ be any vector in $H$, $\lambda $ be a number in $M$
and let $f(t)= e^{i\lambda t}x$. Then $f\in W^k(AP)_{|M}$ and hence,
Equation \eqref{high} has a unique almost periodic solution $u$
in $W^n(AP)_{|M}$. By Theorem  \ref{theorem1},
$x= a(\lambda, f) \in \mathop{\rm Range}( (i\lambda)^n -A)$, hence
$((i\lambda)^n-A)$ is surjective for all $\lambda \in M$.
If   $((i\lambda)^n-A)$ were not injective, i.e., there exists a
nonzero vector $y\in H$ such that $((i\lambda )^n-A)y=0$, we show
that $u_2(t)=u(t)+ e^{i\lambda t}y$,  would be an other almost
periodic mild solution to \eqref{high} with
$\sigma(u_2)=\sigma(u)\subseteq M$. In deed, since $u$ is
$(n-1)$ times differentiable, we can use formula \eqref{deri} to
obtain
\begin{align*}
u_2^{(n-1)}(t)
&= u^{(n-1)}(0)+A\int_0^tu(s)ds+\int_0^tf(s)ds
 +(i\lambda)^{(n-1)}e^{i\lambda t}y\\
&= u^{(n-1)}(0)+A\int_0^tu(s)ds+\int_0^tf(s)ds
 +\frac{e^{i\lambda t}}{i\lambda}Ay\\
&= (u^{(n-1)}(0)+\frac{Ay}{i\lambda})+A\int_0^t(u(s)
 +e^{i\lambda s}y)ds+\int_0^tf(s)ds \\
&= u_2^{(n-1)}(0)+A\int_0^tu_2(s)ds+\int_0^tf(s)ds,
\end{align*}
 which means $u_2$ is another mild solution of \eqref{high}
corresponding to $f$, contradicting to the uniqueness of the solution.
Therefore, $((i\lambda)^n-A)$ is  bijective and
$(i\lambda)^n \in \varrho(A)$ for all $\lambda \in M$.

We now define the operator $L: W^k(AP)_{|M}\to W^n(AP)_{|M}$ by follows:
For each $f \in W^k(AP)_{|M}$, $Lf$ is the unique almost periodic
 mild solution to \eqref{high} corresponding to $f$.
By the assumption, $L$ is everywhere defined. We will prove that $L$
is a bounded operator by showing  $L$ is closed. Let $f_n \to f$ in
$W^k(AP)_{|M}$ and $Lf_n \to u$ in $W^n(AP)_{|M}$, where
\begin{equation}\label{lim}
(Lf_n)^{(n-1)}(t)= (Lf_n)^{(n-1)}(0)+A\int_0^t(Lf_n)(s)ds
+\int_0^tf_n(s)ds.
\end{equation}
For each $t\in \mathbb{R}$, we have $\lim_{n\to \infty}
(Lf_n)^{(n-1)}(t)=u^{(n-1)}(t)$, $\lim_{N \to \infty}
\int_0^tf_n(s)ds =\int_0^tf(s)ds$  and $ \lim_{n \to
\infty} \int_0^tLf_n(s)ds =\int_0^tu(s)ds$. Moreover, from
\eqref{lim} we have
\begin{align*}
A\int_0^t(Lf_n)(s)ds
&=  (Lf_n)^{(n-1)}(t)-(Lf_n)^{(n-1)}(0)-\int_0^tf_n(s)ds\\
&\quad \to u^{(n-1)}(t)-u^{(n-1)}(0)-\int_0^tf(s)ds,
\quad\text{as } n \to \infty,
\end{align*}
for each $t\in \mathbb{R}$. Since $A$ is a closed operator,
$\int_0^tu(s)ds \in D(A)$ and
$$
A\int_0^tu(s)ds =u(t)-u(0)-\int_0^tf(s)ds,
$$
which means $u$ is a mild solution to \eqref{high} corresponding to $f$.
Thus, $f\in D(L)$, $Lf=u$ and hence, $L$ is closed.

Next, for any $x\in H$ and $\lambda \in M$, put $f(t)= e^{i\lambda t}x$,
then $u(t)= e^{i\lambda t}( (i\lambda)^n-A)^{-1}x$ is the unique
almost periodic solution to \eqref{high}, i.e., $u=Lf$.
Using the boundedness of operator $L$, we obtain
$$
\sum_{j=0}^n|\lambda|^{2j}\|((i\lambda)^n -A)^{-1}x\|^2
= \|u\|_{W^n(AP)}^2 \le \|L\|^2\|f\|_{W^k(AP)}^2
= \|L\|^2\sum_{j=0}^k |\lambda|^{2j}\|x\|^2,
$$
which implies
\begin{equation}\label{so}
\|((i\lambda )^n-A)^{-1}x\|^2 \le
\|L\|^2\frac{\sum_{j=0}^k |\lambda|^{2j}}{\sum_{j=0}^n |\lambda|^{2j}}
\cdot \|x\|^2.
\end{equation}
for any $x\in H$ and any $\lambda \in M$.
For a real number $\lambda$ and an integer $m$
with $0\le m \le n-k$ it is easy to show the inequality
$$
\frac{\sum_{j=0}^k |\lambda|^{2j}}{\sum_{j=0}^n |\lambda|^{2j}}
\le \frac{1}{|\lambda|^{2m}}.
$$
Thus, from \eqref{so} we have
$$
\|((i\lambda )^n-A)^{-1}x\| \le \|L\|\frac{1}{|\lambda^m|}\cdot \|x\|,
$$
from which \eqref{ness2} follows.

$(ii) \Rightarrow (i)$:  Suppose $f$ is  in $W^k(AP)_{|M}$.
Put $x_{\lambda}:=((i\lambda)^n -A)^{-1}a(\lambda, f)$.
For any integer $m$ with $0\le m\le n$ we can write $m=m_1+m_2$
with $0\le m_1\le n-k$ and $0\le m_2 \le k$. We have
\begin{align*}
\sum_{\lambda \in \sigma(f)} \lambda^{2m}\|x_{\lambda}\|^2
&\le  \sum_{\lambda \in \sigma(f)} \Big( \Big(|\lambda|^{2m_1} \|
 ( (i\lambda)^n -A)^{-1}\|^2\Big)\cdot \Big( |\lambda |^{2m_2}\|a(\lambda, f)\|^2\Big)\Big)\\
&\le  \Big(\sup_{\lambda \in \sigma(f)} |\lambda|^{2m_1}\|
 ( (i\lambda)^n -A)^{-1}\|^2 \Big) \sum_{\lambda \in \sigma(f)}
 |\lambda |^{2m_2}\|a(\lambda, f)\|^2\\
&= \Big(\sup_{\lambda \in \sigma(f)} |\lambda|^{m_1}\|
 ( (i\lambda)^n -A)^{-1}\| \Big)^2 \|f^{(m_2)}\|_{AP}^2
 <  \infty .
\end{align*}
By Proposition \ref{theorem1}, Equation \eqref{high} has an almost
periodic mild solution in $W^n(AP)$. That solution is unique and is
in $W^n(AP)_{|M}$, since  its Bohr transforms are uniquely determined
by  $a(\lambda,u)=((i\lambda)^n-A)^{-1}a(\lambda, f)$ for all
$\lambda \in M$.
\end{proof}

We can apply Theorem \ref{theo2} to some particular sets for $M$.
First, if $M= \mathbb{R}$ we have

\begin{corollary}\label{cor1}
Suppose $A$ is a linear and closed operator on a Hilbert space $H$.
For $0 \le k\le n$, the following statements are equivalent
\begin{itemize}
\item[(i)] For each function $f\in W^k(AP)$, Equation  \eqref{high}
  has a unique  almost periodic mild solution in $W^n(AP)$.
\item[(ii)] $(i\mathbb{R})^n \subseteq \varrho(A)$ and
$$
\sup_{\lambda \in\mathbb{R}}|\lambda|^{n-k}\|( (i\lambda)^n
-A)^{-1}\| < \infty .
$$
\end{itemize}
\end{corollary}

Let  $L_2(0,1)$ be the Hilbert space of integrable functions $f$
from $(0,1)$ to $H$ with the norm
$$
\|f\|^2_{L_2(0,1)}= \int_0^1\|f(t)\|^2dt <\infty.
$$
If $M=\{2 p \pi : p\in \mathbb{Z}\}$, then the space $AP(\mathbb{R},
H)_{|M}$ becomes $L_2(0,1)$ and $W^k(AP)$ becomes $W^k(1)$, the
space of all periodic functions $f$ of period $1$ with $f^{(k)}\in
L_2(0,1)$. $W^k(1)$ is then a Hilbert space with the norm
$$
\|f\|^2_{W^k(1)}= \sum_{j=0}^k|f^{(k)}\|^2_{L_2(0,1)}.
$$
Note that, since $M=\{2 p \pi : p\in \mathbb{Z}\}$, Condition \eqref{ness2}
is satisfied for all $m$  from $0$ to $(n-k)$ if and only if
it is satisfied for only $m=n-k$. Hence, we obtain the following
corollary, which generalizes a result in \cite{lan1}.

\begin{corollary}[{\cite[Theorem 2.6]{lan1}}] \label{cor2}
Suppose $A$ is a linear and closed operator on a Hilbert space $H$.
For $0 \le k\le n$, the following statements are equivalent
\begin{itemize}
\item[(i)] For each function $f\in W^k(1)$,
Equation \eqref{high} has a unique  1-periodic mild solution
in $W^n(1)$.
\item[(ii)] For each $p\in \mathbb{Z}$, $ 2pi \pi  \in \varrho(A)$ and
\begin{equation}\label{ness3}
\sup_{p \in\mathbb{Z}}|2p\pi|^{(n-k)}\|( (2pi\pi)^n  -A)^{-1}\| < \infty .
\end{equation}
\end{itemize}
\end{corollary}

\section{Application: A $C_0$-semigroup case}

If $A$ generates a $C_0$-semigroup $(T(t))_{t\ge 0}$, then
  mild solutions of the first order differential equation
 \begin{equation}\label{first}
u'(t)=Au(t)+f(t) \, \hspace{1cm} t\in \mathbb{R},
\end{equation}
 can be expressed by
\begin{equation}\label{semi}
u(t)=T(t-s)u(s) +\int_s^tT(t-\tau)f(\tau)d\tau
\end{equation}
for $t\ge s$ (see \cite[Theorem 2.5]{abhn}).
We obtain the following result.

\begin{corollary}\label{cor11}
Let $A$ generate a $C_0$-semigroup $(T(t))$ on a Hilbert $H$ and
$M$ is a closed subset in $\mathbb{R}$. The following statements
are equivalent
\begin{itemize}
\item[(i)] For each function $f\in W^1(AP)_{|M}$,
 Equation  \eqref{first} has a unique solution  in $W^1(AP)_{|M}$.
\item[(ii)] For each function $f\in W^1(AP)_{|M}$, Equation
 \eqref{first} has a unique almost periodic classical solution $u$
 with $\sigma(u)\subset M$.
\item[(iii)] For each  $\lambda \in M$, $\lambda i\in \varrho(A)$ and
\begin{equation}
\sup_{\lambda \in M}\|(i \lambda -A)^{-1}\| < \infty .
\end{equation}
\end{itemize}
\end{corollary}

\begin{proof}
The equivalence $(i)\Leftrightarrow(iii) $ is shown in
Theorem \ref{theo2},
$(ii)\Rightarrow (i)$ is obvious. So, it remains to show the
implication $(i)\Rightarrow(ii)$.

Let $f$ be any function in $W^1(AP)$ and $u(t)$ be the unique mild
solution of \eqref{first}, which is in $W^1(AP)$. We will show
$u$ is a classical solution by showing $u(t_0)\in D(A)$ for every
point $t_0\in \mathbb{R}$.
Take any point $s_0\in \mathbb{R} $ with $s_0<t_0$. Since for each
almost everywhere differentiable function $f$, the function
$g(t):=\int_{s_0}^tT(t-s)f(s)ds$ is continuously differentiable and
$g(t)\in D(A)$ for  all $t\in [s_0,t_0]$ (see \cite{ns}).
So, from Formula \eqref{semi}, it suffices to show
$T(t_0-s_0)u(s_0)\in D(A)$.

 By the assumptions, function
$g(t):= T(t-s_0)u(s_0) =u(t)-\int_{s_0}^tT(t-s)f(s)ds$ is  almost
everywhere differentiable on $[s_0,t_0]$. It follows that
$g(t)\in D(A)$ for almost $t$ in $[s_0, t_0]$
(since $t\mapsto T(t)x$ is differentiable at right at $t_0$
if and only if $T(t_0)x\in D(A)$). Taking a point
$s_1 \in (s_0, t_0)$ such that $T(s_1-s_0)u(s_0)\in D(A)$,
then  $T(t_0-s_0)u(s_0)=T(t_0-s_1)T(s_1-s_0)u(s_0)\in D(A)$.
The uniqueness of this  classical solution is obvious and the
proof is complete.
\end{proof}

If  $M=\{2 k \pi : k\in \mathbb{Z}\}$ and $f$ is a 1-periodic function,
then it is easy to see that  solution $u$ is 1-periodic if and only
if $u(1)=u(0)$. Hence, to consider 1-periodic solution,
it suffices to consider $u$ in $[0,1]$ and in this interval we have
\begin{equation}\label{peri}
u(t)=T(t)u(0) +\int_0^tT(t-s)f(s)ds, \hspace{.3cm} 0\le t\le 1.
\end{equation}
 From Corollary \ref{cor11} we have a direct consequence below,
in which we show the Gearhart's Theorem
(the equivalence $(iv)\Leftrightarrow(v) $).
For the proof of that Corollary, note that the equivalence
$(i)\Leftrightarrow(ii) $ can be easily  proved by using
standard arguments and $(i)\Leftrightarrow(v) $ has been
shown in \cite{pruss}.

\begin{corollary}\label{cor22}
Let $A$ generate a $C_0$-semigroup $(T(t))$ on a Hilbert $H$,
then the following statements are equivalent
\begin{itemize}
\item[(i)] For each function $f\in L_2(0, 1)$, Equation  \eqref{first}
  has a unique 1-periodic mild solution.
\item[(ii)] For each function $f\in W^1(1)$, Equation  \eqref{first}
  has a unique 1-periodic classical solution.
\item[(iii)] For each function $f\in W^1(1)$, Equation \eqref{first}
  has a unique 1-periodic solution contained in $W^1(1)$.
\item[(iv)] For each  $k\in \mathbb{Z}$, $2k\pi i\in \varrho(A)$ and
$$
\sup_{k\in \mathbb{Z}}\|(2k\pi i -A)^{-1}\| < \infty .
$$
\item[(v)] $1\in \varrho(T(1))$.
\end{itemize}
\end{corollary}

\subsection*{Acknowledgments}
The author would like to express his gratitude to the anonymous
referee for pointing out several mistakes in the manuscript and
giving valuable suggestions.


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\end{document}