\documentclass[reqno]{amsart}

\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2004(2004), No. 79, 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  (login: ftp)}
\thanks{\copyright 2004 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}

\title[\hfilneg EJDE-2004/79\hfil Periodicity of mild solutions]
{Periodicity of mild solutions to  higher order
differential equations in Banach spaces}

\author[T. L. Nguyen\hfil EJDE-2004/79\hfilneg]
{Thanh Lan Nguyen}

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

\date{}
\thanks{Submitted February 18, 2004. Published June 4, 2004.}
\subjclass[2000]{34G10, 34K06, 47D06}
\keywords{Abstract Cauchy problems, Fourier series,
 periodic mild solutions, \hfill\break\indent semigroups and cosine families}

\begin{abstract}
 We give necessary and sufficient conditions for the periodicity
 of mild solutions to the the higher order differential equation
 $u^{(n)}(t)=Au(t)+f(t)$, $0\le t \le T$,
  in a Banach space $E$.
 Applications are made to the cases, when $A$ generates a $C_0$-semigroup
 or a cosine family, and when $E$ is a Hilbert space.
\end{abstract}

\maketitle

\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lem}[theorem]{Lemma}
\newtheorem{defi}[theorem]{Definition}
\newtheorem{prop}[theorem]{Proposition}
\newtheorem{corollary}[theorem]{Corollary}

\section{Introduction}

This paper concerns the periodicity of solutions to the higher
order Cauchy problem
\begin{equation}\label{high}
\begin{gathered}
u^{(n)}(t)=Au(t)+f(t), \quad 0 \le t\le T\\
u^{(i)}(0)=x_i, \quad i=0,1,\dots , n-1,
\end{gathered}
\end{equation}
where $A$ is a linear and closed  operator on a Banach space $E$, and
$f$ is a  function from  $[0,T]$ to $E$.  The asymptotic behavior
and, in particular, the periodicity of solutions of \eqref{high}
has been subject to intensive study in recent decades. It is
well-known \cite{hale} that, if $A$ is an $n\times n$ matrix on
$\mathbb{C}^n$, then the first order Cauchy problem
\begin{equation}\label{first}
\begin{gathered}
u'(t)=Au(t)+f(t), \quad 0 \le t\le T ,\\
u(0)=x
\end{gathered}
\end{equation}
in $E=\mathbb{C}^n$ admits a unique $T$-periodic solution for each
continuous $T$-periodic forcing term $f$ if and only if
$\lambda_k=2k\pi t/T$, $k\in\mathbb{Z}$, are not eigenvalues of $A$. This
result was extended by Krein and Dalecki \cite{dakr, kr} to the
Cauchy problem in an abstract Banach space. In
\cite[Theorem II 4.3]{dakr} it was claimed that, if $A$ is a linear 
  bounded operator on $E$, then  \eqref{first} admits a unique $T$-periodic
solution  for each $f\in C[0,T]$ if and only if $2k\pi i/T \in
\varrho(A)$, $k \in \mathbb{Z}$. Here $\varrho(A)$ denotes the resolvent
set of $A$.  Unfortunately, the above result does not hold any
more when $A$ is an unbounded operator (see \cite{gvw}). For the
case, when $A$ generates a strongly continuous semigroup,
periodicity of solutions of \eqref{first} was studied  in
\cite{haraux, pruss}. Corresponding results on the periodic
solutions of the second order Cauchy problem were obtained in
\cite{cili, schuler}, when $A$ is generator  of a cosine family.
Related results can also be found in \cite{eiti, hakr, lang, lm,
lizama, vusc} and the references therein.

In this paper we investigate the periodicity of mild solutions of the higher
order Cauchy problem \eqref{high} when $A$ is a linear, unbounded operator.
The main tool we use here is the Fourier series method. For an integrable
function $f(t)$ from $[0,T]$ to  $E$, the Fourier coefficient of $f(t)$ is
defined as
$$f_k = \frac{1}{T}\int_0^Tf(s)e^{-2k\pi is/T} ds, \quad k \in \mathbb{Z}.
$$
Then $f(t)$ can be represented by Fourier series
\begin{eqnarray*}
f(t) \approx \sum_{k = -\infty}^{\infty}e^{2k\pi it/T} f_k.
\end{eqnarray*}
First, we establish the relationship between the Fourier
coefficients of the periodic solutions of \eqref{high} and those
of the inhomogeneity $f$. We then give different equivalent
conditions so that \eqref{high} admits a unique periodic solution
for each  inhomogeneity $f$ in a certain function space. As
applications, in Section 3 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}}\|R(2k\pi i, A)\|< \infty
$. Corresponding result for the spectrum of a cosine family is
also presented.

Let us fix some notation. A continuous function on $[0,T]$ is said
to be $T$-periodic if $u(0)= u(T)$. For the sake of simplicity
(and without loss of generality) we assume  $T = 1$ and put
$J:=[0,1]$. For $p\ge 1$,  $L_p(J)$ denotes the space of
$E$-valued functions on $J$ with $\int_0^1\|f(t)\|^pdt <\infty$
and $C(J)$ the space of  functions on $J$ with and $\|f\|=
\sup_{J}\|f(t)\|<\infty$. Moreover, for $m>0$ we define the
following function spaces

\noindent(1) $W_p^m(J): =\{f\in L_p(J): f', f'', \dots ,
f^{(m)}\in L_p(J)\}$. $W_p^m(J)$ is then a Banach space with the
norm
$$ \|f\|_{W_p^m}:=\sum_{k=0}^m\|f^{(k)}\|_{L_p(J)}.
$$
(2) $ P^m(J):=\{f\in C(J): f, f', \dots , f^{(m)} \mbox{ are in }
P(J)\}$. That means $P^m(J)$ is the space of all functions on $J$,
which  can be extended to 1-periodic, $m$-times continuously
differentiable functions on $\mathbb{R}$. $P^m(J)$ is a Banach space with
the norm
$$
\|f\|_{P^m(J)}:=\sum_{k=0}^m\|f^{(k)}\|_{C(J)}.
$$
(3) $WP_p^m(J):= P^{m-1}(J)\cap W_p^m(J)$. It is easy to see that
$WP_p^m(J)$ is a Banach space with $W_p^m(J)$-norm.

We will use the following simple lemma.

\begin{lem}\label{2}
If $F$ is a continuous function on $J$ such that $f= F' \in L_p(J)$, then
for $k\neq 0$ we have
$$
F_k=\frac{1}{2k\pi i}f_k +\frac{F(0)-F(1)}{2k\pi i},
$$
where $f_k$ and $F_k$ are the Fourier series of $f$ and $F$, respectively.
\end{lem}

\section{Periodic Mild Solutions of Higher Order Differential Equations}
 Let $J$ be the interval $[0,1]$ and $p\ge 1$. For each function $f\in L_p(J)$
 we define the function $If$ by $If(t): = \int_0^tf(s)ds$ and, for $n\ge 2$,
the function $I^nf$ by $I^{n} f(t):= I(I^{n-1}f)(t)$.
\begin{defi} \label{def2.1} \rm
\begin{itemize}
\item[(1)] A continuous function $u$  is called a mild solution of \eqref{high}
 on $J$, if $I^nu(t)\in D(A)$  and, for all $t\in J$,
\begin{equation}\label{mild}
u(t) = \sum_{i=0}^{n-1}\frac{t^i}{i!}x_i + AI^{n}u(t) + I^{n}f(t)\,.
\end{equation}
\item[(2)] A function $u$ is a classical solution of {\rm \eqref{high}} on $J$,
if $u(t)\in D(A)$, $u$ is $n$-times continuously differentiable, and
\eqref{high} holds for $t\in J$.
\end{itemize}
\end{defi}

\subsection*{Remarks}
\begin{itemize}
\item[(i)] If $n=1$ and $A$ is the generator of a $C_0$ semigroup $T(t)$, then  a continuous function $u: J\to E$ is a mild solution of \eqref{high} if and only if  it has the form
$$u(t) =T(t)x_0+\int_0^tT(t-r)f(r)dr, \hspace{.2cm} t\in J.$$
(See \cite{abhn}).
\item[(ii)]
Similarly, if  $n=2$ and $A$  generates a cosine family $(C(t))$ on $E$, then any continuously differentiable function $u$ on $E$ of the form
$$u(t) = C(t)x_0 + S(t)x_1 +\int_0^tS(t-\tau)f(\tau)d\tau,\hspace{.2cm} t\in J, $$
where $(S(t))$ is the associated sine family, is a mild solution of \eqref{high} (see Section 3 for more details).
\end{itemize}
%
The mild solution to \eqref{high} defined by (\ref{mild}) is really an extension
of a 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.

\begin{lem}\label{ntime}
Suppose $0\le m\le  n$ and $u$ is a mild solution of  \eqref{high}, which is $m$-times continuously differentiable. Then 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)!}x_j +AI^{n-m}u(t) +I^{n-m}f(t).
\end{equation}
\end{lem}

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

In particular, if the mild solution $u$ is $n$-times continuously differentiable,
then (\ref{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 Fourier coefficients of such solutions and those of $f(t)$.

\begin{prop}\label{for}
Suppose $f\in L_p(J)$ and $u$ is a mild solution of {\rm \eqref{high}},
which is $(n-1)$ times continuously differentiable. Then
\begin{equation}\label{formula}
 \frac{((2k\pi i)^n-A)u_k-f_k}{(2k\pi i)^n }= \sum_{j=0}^{n-1}\frac{u^{(j)}(0)-u^{(j)}(1)}{(2k\pi i)^{j+1}}
\end{equation}
for $k\not=0$.
\end{prop}

\begin{proof}
Let $u^{(j)}_k$ be the $k^{th}$ Fourier coefficient of $u^{(j)}$. Using the
identity
\begin{equation}\label{a1}
u^{(j)}_k=\frac{u^{(j)}(0)-u^{(j)}(1)}{2k\pi i} + \frac{1}{2k\pi i}u^{(j+1)}_k
\end{equation}
for $j= 0,1, 2,\dots , n-2$ (by Lemma \ref{2}), we obtain
\begin{equation}\label{d1}
u_k=\sum_{j=0}^{n-2}\frac{u^{(j)}(0)-u^{(j)}(1)}{(2k\pi i)^{j+1}}
+\frac{1}{(2k\pi i)^{n-1}}u^{(n-1)}_k.
\end{equation}
Since $u$ is $(n-1)$ times continuously differentiable, by Lemma \ref{ntime},
\begin{equation}\label{b1}
u^{(n-1)}(t)=u^{(n-1)}(0)+AIu(t) +If(t).
\end{equation}
Taking the $k^{th}$ Fourier coefficient on both sides of (\ref{b1}) and
using (\ref{a1}), we have
\begin{equation}\label{d2}
\begin{aligned}
 u^{(n-1)}_k
&= A(Iu)_k+(If)_k\\
&= A\Biggr(\frac{Iu(0)-Iu(1)}{2k\pi i} + \frac{1}{2k\pi i}(Iu)'_k\Bigr)
  +\Bigr(\frac{If(0)-If(1)}{2k\pi i} + \frac{1}{2k\pi i}(If)'_k\Bigr)\\
&= \frac{-(AIu(1)+If(1))}{2k\pi i} +\frac{Au_k+f_k }{2k\pi i}\\
&= \frac{u^{(n-1)}(0) -u^{(n-1)}(1)}{2k\pi i} +\frac{Au_k+f_k}{2k\pi i}.
\end{aligned}
\end{equation}
Here we have also used $Iu(0)=If(0)=0$, $(Iu)'_k=u_k$ and $(If)'_k=f_k$.
Combining (\ref{d1}) and (\ref{d2}), we obtain
\[
u_k=\sum_{j=0}^{n-1}\frac{u^{(j)}(0)-u^{(j)}(1)}{(2k\pi i)^{j+1}}
+ \frac{Au_k+f_k}{(2k\pi i)^n},
\]
from which (\ref{formula}) follows.\end{proof}


The interesting point of Proposition \ref{for} is  that the Fourier coefficients
of the mild solution $u$ depend not only on $u$ but also on its derivatives.
If $u$ is a mild solution in $P^{(n-1)}(J)$, then we have a nice relationship
between  Fourier coefficients of $u$ and those of $f$, as the following
proposition shows.

\begin{prop}\label{key}
Suppose $f\in L_p(J)$ and $u$ is a mild solution of  \eqref{high},
which is $(n-1)$ times continuously differentiable. Then $u\in P^{(n-1)}(J)$
if and only if
\begin{equation}\label{fourier}
((2k\pi i)^n -A)u_k = f_k
\end{equation}
for every $k\in \mathbb{Z}$.
\end{prop}

\begin{proof}
 Suppose $u$ is a mild 1-periodic solution of \eqref{high} in $P^{n-1}(J)$.
If $k\not= 0$, then (\ref{fourier}) follows directly from (\ref{formula}).
If $k=0$, using (\ref{deri}) with $m=n-1$ and $t=1$ we obtain
\begin{align*}
u^{(n-1)}(1)&= u^{(n-1)}(0)+ A \int_0^1u(s)ds+\int_0^1f(s)ds\\
&= u^{(n-1)}(0)+Au_0+f_0.
\end{align*}
Due to the 1-periodicity of $u^{(n-1)}$ we obtain $Au_0+f_0=0$, from
which (\ref{fourier})  holds for $k=0$.
Conversely, suppose (\ref{fourier}) holds for all $k\in \mathbb{Z}$.
Then, by (\ref{formula}),
\begin{equation}\label{mat}
\sum_{j=0}^{n-1}\frac{u^{(j)}(0)-u^{(j)}(1)}{(2k\pi i)^j}=0
\end{equation}
 all $k\not=0$. That means that for any positive integer $K$, the vector
$$
X=\Big(u(0)-u(1),u'(0)-u'(1), \dots , u^{(n-1)}(0)-u^{(n-1)}(1)\Big)^T
$$
is a solution of the  system of linear equations
$$
\begin{pmatrix}
1 & \frac{1}{2\pi i}&  \cdots &\frac{1}{(2\pi i)^{n-1}}\\
1 & \frac{1}{2\cdot 2\pi i}&  \cdots &\frac{1}{(2\cdot 2\pi i)^{n-1}}\\
\vdots & &  \ddots  & \vdots \\
1 & \frac{1}{2K\pi i}&  \cdots &\frac{1}{(2K\pi i)^{n-1}}
\end{pmatrix}_{n\times K}
\begin{pmatrix}
x_1\\
x_2\\
\vdots\\
x_n
\end{pmatrix}
=0 .
$$
 This can only happen if $X=0$, i.e. $u^{(j)}(0)-u^{(j)}(1)=0$ for
 $j=0,1, 2, \dots , (n-1)$. Hence, $u \in P^{(n-1)}(J)$, and the proposition
is proved. \end{proof}

From Proposition \ref{key} we obtain

\begin{corollary}\label{coro}
Suppose $f\in L_p(J)$. Then
\begin{itemize}
\item[(i)] If $((2k\pi i)^n  -A)$ is injective for $k\in \mathbb{Z}$,  then Equation {\rm \eqref{high}} has at most one  1-periodic mild solution, which belongs tp $P^{n-1}(J)$.
\item[(ii)] If there exists a number $k\in \mathbb{Z}$ such that $f_k\not\in Range((2k\pi i)^n -A)$, then Equation {\rm \eqref{high}} has no periodic mild solution which belongs to $P^{n-1}(J)$.
\item[(iii)] Let $u$ be a mild solution   of $u^{(n)}=Au$, which is $(n-1)$ times continuously differentiable. Then $u$ belongs to $P^{n-1}$ if and only if
$$(2k\pi i)^n u_k =Au_k,$$
i.e.,  $u_k$ is an {\it eigen-vector} of $A$ corresponding to $(2k\pi i)^n$, $k\in\mathbb{Z}$.
\end{itemize}
\end{corollary}

We are now in a position to state the main results.

\begin{theorem} \label{p1}
Let $A$ be a closed operator on $E$ and $0\le m\le n$. The  following
statements are equivalent.
\begin{itemize}
\item[(i)] For each function $f\in WP_p^m(J)$, Equation {\rm \eqref{high}}
admits a unique  mild solution in $WP_p^n(J)$
\item[(ii)] For each  $k\in \mathbb{Z}$, $2k\pi i\in \varrho(A)$  and there exists a
constant $C>0$ such that
\begin{equation}\label{ness}
\|\sum_k((2k\pi i)^n- A)^{-1}e^{2k\pi i\cdot }x_k)\|_{W_{p}^n(J)}\le C\cdot \|\sum_ke^{2k\pi i \cdot} x_k
\|_{W_{p}^m(J)}
\end{equation}
 for any finite sequence $\{x_k\}\subset E$
\end{itemize}
If $E$ is a Hilbert space, and $p=2$, then (i) and (ii) are equivalent to
\begin{itemize}
\item[(iii)] For every  $k\in \mathbb{Z}$, $(2k\pi i)^n\in \varrho(A)$  and
\begin{equation}\label{necess2}
\sup_{k\in \mathbb{Z}}\|k^{n-m}((2k\pi i)^n- A)^{-1}\|< \infty
\end{equation}
\end{itemize}
\end{theorem}
We will need the following lemma.

\begin{lem}\label{dense}
Let $F_1:= WP_p^m(J)$ and $F_2:= WP_p^n(J)$. Then the following are equivalent:
\\
({\it 1}) For each function $f\in F_1$, \eqref{high} admits a unique mild
solution $u$ in $F_2$.\\
({\it 2}) There exists a dense subset $D$ in $F_1$ such that:
\begin{itemize}
\item[(i)] For each function $f\in D$, {\rm \eqref{high}} admits a unique mild solution $u$ in $F_2$;
\item[(ii)] There exists a constant $C>0$ such that for all $f\in D$,
\begin{equation}\label{ine}
\|u\|_{F_2}\le C\|f\|_{F_1}\,.
\end{equation}
\end{itemize}
\end{lem}

\begin{proof}
(1)$\Rightarrow $ (2): We will prove (2) with $D=F_1$. It is easy to see
that (i)  is automatically satisfied. To show (ii), we define the operator
$G:F_1 \mapsto F_2$ by $Gf:=u$, where $u$ is the unique mild solution
of \eqref{high} in $F_2$. Then $G$ is a linear, everywhere defined operator.
We will prove  the boundedness of $G$ by showing that $G$ is a closed operator.
 To this end, let $\{f_j\}\subset F_1$ a sequence such that $f_j\to f$ in $F_1$
and $Gf_j\to u$ in $F_2$ for $j\to \infty$. For each
$t\in J$, let $v_j:=I^{n}(Gf_j)(t)$, then

$$
\lim_{j\to \infty}v_j= I^{n}u(t).
$$
Moreover,  from the identity
$$
(Gf_j)(t)=\sum_{i=0}^{n-1}\frac{t^j}{j!}(Gf_j)(0) + AI^{n}(Gf_j)(t) + I^{n}f_j(t)
$$
we have
\begin{align*}
Av_j&=AI^{n}(Gf_j)(t) \\
&=(Gf_j)(t)-\sum_{i=0}^{n-1}\frac{t^i}{i!}(Gf_j)(0)- I^{n}f_j(t)
\to u(t)-\sum_{i=0}^{n-1}\frac{t^i}{i!}u(0)- I^{n}f(t)
\end{align*}
as $j\to \infty$. Since $A$ is a closed operator, $I^{n}u(t)\in D(A)$ and
$$AI^{n}u(t)= u(t)-\sum_{i=0}^{n-1}\frac{t^i}{i!}u(0)- I^{n}f(t),
$$
i.e., $u$ is a mild solution of \eqref{high} and consequently, $Gf=u$.
 So, $G$ is a bounded operator from $F_1$ to $F_2$, from which (\ref{ine})
 follows with $C=\|G\|$.

\noindent (2)$\Rightarrow $ (1). For any $f\in F_1$ there exists a sequence
$\{f_j\}\subset D$ such that $f_j\to f$ for $j\to \infty$. Let $u_j$ be the
mild solution in $F_2$ corresponding to $f_j$, then, by (\ref{ine}), $u_j\to u$
for some $u\in F_2$. With the same manner as in the previous part, we can prove
that $u$ is a mild solution of \eqref{high} corresponding to $f$. The uniqueness
of this solution comes directly from (\ref{ine}).
\end{proof}

\begin{proof}[Proof of Theorem \ref{p1}]
(i) $\to$ (ii):   We first show that $(2k\pi i)^n\in \varrho(A)$ for $k\in \mathbb{Z}$.
To this end, let $f(t)=e^{2k\pi i t}x$, $x\in E$ and $u(t)$ be the unique mild
solution to \eqref{first} corresponding to $f$. By Lemma \ref{key} we have
$((2k\pi i)^n -A)u_k=x$. Hence $((2k\pi i)^n-A)$ is surjective.  On the other side,
if $((2k\pi i)^n-A)$ is not injective, i.e. there is a non-zero vector $x_0\in E$
such that $((2k\pi i)^n-A)x_0=0$, then  it is not hard to check that $u_1:\equiv 0$
and $u_2(t):= e^{2k\pi it}x_0$ are two distinct 1-periodic mild (classical) solution
f $u^{(n)}(t)=Au(t)$. It is contradicting to the uniqueness of $u$.
So $((2k\pi i)^n-A)$ is injective and hence bijective, i.e.
$(2k\pi i)^n\in \varrho(A)$.
Let now  $f(t):= \sum_ke^{2k\pi it}x_k$, where  $\{x_k\}$ is any finite sequence
in $E$. Then, by Lemma \ref{key},
$u(t) = \sum_k((2k\pi i)^n -A)^{-1}e^{2k\pi i t}x_k$ is the unique 1-periodic mild
solution to \eqref{high} corresponding to $f$. Thus, (\ref{ness}) is obtained by
inequality (\ref{ine}).

\noindent (ii) $\to$ (i):  Put
$$
\mathcal{M}:= \{f(t)=\sum_ke^{2k\pi it}x_k : \{x_k\} \mbox{ is a finite sequence in } E \}.
$$
Observe that $\mathcal{M}$ is dense in $WP_p^m(J)$. Moreover, if $f$ is a
function in $\mathcal{M}$, i.e.,  if $f(t)=\sum_ke^{2k\pi t}x_k$, then it is
easy to check that $u(t)= \sum_k((2k\pi i)^n -A )^{-1}e^{2k\pi it }x_k $ is a
unique 1-periodic mild solution of \eqref{high} corresponding to $f$ and, by
Corollary \ref{coro}{\it (i)}, it is the unique one. From (\ref{ine}) it follows
that $\|u\|_{W_p^n(J)}\le C \|f\|_{W_p^m(J)}$ for all $f\in \mathcal{M}$.
By Lemma \ref{dense}, that implies {\it (i)}.

Finally, if $E$ is a Hilbert space, then $WP_2^m(J)$ is a Hilbert space for
any $0\le m\le n$. Moreover, for $f(t)=\sum_ke^{2k\pi it}x_k$ and $u(t)= \sum_k((2k\pi i)^n-A)^{-1}e^{2k\pi it }x_k $ we have
 \begin{equation}\label{pr1}
\|f\|_{W_2^m(J)} = \sum_{j=0}^m\Big(\sum_k(2k\pi)^{2j}\|x_k\|^2
\Big)^{1/2}
\end{equation}
and
\begin{equation}\label{pr2}
 \|u\|_{W_2^n(J, E)}= \sum_{j=0}^{n} \Big(\sum_k (2k\pi )^{2j} \|((2k\pi i)^n-A)^{-1} x_k\|^2\Big)^{1/2}.
\end{equation}
Suppose {\it (ii)} holds, i.e., $\|u\|_{W_2^n(J)}\le C  \|f\|_{W_2^m(J)}$
for $f\in \mathcal{M}$. For any $k\in \mathbb{Z}$, take $f(t):= e^{2k\pi i t}x$.
From  (\ref{pr1}) and \eqref{pr2}, we have
\begin{equation}\label{pr3}
\|f\|_{W_2^m(J)} = \sum_{j=0}^m \|(2k\pi)^jx\| \le (2\pi)^m(m+1) \|k^mx\|
\end{equation}
and
\begin{equation}\label{pr4}
 \|u\|_{W_2^n(J)} = \sum_{j=0}^{n}  \|(2k\pi )^{j} ((2k\pi i)^n-A)^{-1} x\|
\ge  (2\pi)^n \|k^n((2k\pi i)^n-A)^{-1} x\|.
\end{equation}
Combining (\ref{ness}), \eqref{pr3} and (\ref{pr4}) we obtain
$$(2\pi)^n\|k^n((2k\pi i)^n-A)^{-1} x\|\le C\cdot (2\pi)^m(m+1) \|k^mx\|,
$$
from which (\ref{necess2}) follows.

Conversely, suppose (iii) holds, i.e., there is a positive constant $C$ such that
$\|(2k\pi i)^n-A)^{-1}\|\le C|k|^{m-n}$ for $k\in \mathbb{Z}$. Using that inequality
for the right hand side of \eqref{pr2} we obtain
\begin{align*}
 \|\sum_k((2k\pi i)^n- A)^{-1}e^{2k\pi i \cdot }x_k\|_{W_2^n(J)}
&\le C\sum_{j=0}^{n} \Big(\sum_k (2k\pi )^{2j}k^{2m-2n} \|x_k\|^2\Big)^{1/2}\\
&\le C_1 \sum_{j=0}^{n} \Big(\sum_k (2k\pi )^{2j+2m-2n} \|x_k\|^2\Big)^{1/2}\\
&\le  C_1(n+1)\Big(\sum_k (2k\pi )^{2m} \|x_k\|^2\Big)^{1/2}\\
&\le  C_1(n+1)\sum_{j=0}^{m} \Big(\sum_k (2k\pi )^{2j}\|x_k\|^2\Big)^{1/2}\\
&= C_1(n+1)\|\sum_ke^{2k\pi i \cdot} x_k
\|_{W_2^m(J)},
\end{align*}
where $C_1=C(2\pi)^{n-m}$. Thus, (\ref{ness}) holds and the theorem is
proved. \end{proof}

The next theorem shows the relationship between the regularity of the
inhomogeneity and that of the corresponding mild solution.
%
\begin{theorem} \label{p2}
If $A$ is a closed operator on $E$, then the  following  statements are equivalent.
\begin{itemize}
\item[(i)] For each $f\in L_p(J)$ Eq. \eqref{high} admits a unique mild solution in $P^{n-1}(J)$  .
\item[(ii)] $0\in \varrho(A)$ and for each $f\in L_p(J)$ with $\int_0^1f(s)ds =0$, Equation {\rm \eqref{high}} admits a unique mild solution in $P^{n-1}(J)$ .
 \item[(iii)] For each  $f\in WP_p^1(J)$, Equation \eqref{high} admits a unique
 1-periodic classical solution.
\end{itemize}
\end{theorem}

\begin{proof}
 If  (i) or (iii) holds, then, by the same reasoning as in the proof of
 Theorem \ref{p1}, we can prove that  $2k\pi i\in \varrho(A)$ for $k\in \mathbb{Z}$.

\noindent (i) $\to$ (iii):  Let  $F$  be any function in $WP_p^1(J)$.
Then $F$ can be written as by $F(t)= \int_0^tf(s)ds + x_0$, where $f\in L_p(J)$
 and $x_0$ is a vector in $E$. Since $F$ is 1-periodic we have $\int_0^1f(s)ds =0$.
Let $u$ be the mild solution to \eqref{high} corresponding to $f$, which is in
$P^{n-1}(J)$, and put
$$
U(t)= \int_0^tu(s)ds +A^{-1}u^{n-1}(0) -A^{-1}x_0.
$$
From identity (\ref{deri}) with $m=n-1$ we have
\begin{equation}\label{them}
u^{(n-1)}(1)=u^{n-1}(0) +A\int_0^1u(s)ds +\int_0^1f(s)ds.
\end{equation}
Note that $u^{(n-1)}(1)= u^{(n-1)}(0)$ and $\int_0^1f(s)ds=0$.
 Thus, from (\ref{them}) we obtain $A\int_0^1u(s)ds=0$,
which implies, due to $0\in \varrho(A)$,  $\int_0^1u(s)ds =0$.
Hence, $U$ is a 1-periodic function. Moreover,
\begin{align*}
U^{(n)}(t)&= u^{(n-1)}(t)\\
&= u^{n-1}(0)+A\int_0^tu(s)ds +\int_0^tf(s)ds\\
&= u^{n-1}(0)+A[U(t)-A^{-1}u_{n-1}+A^{-1}x_0] +(F(t)-x_0)\\
&= AU(t)+F(t).
\end{align*}
So, $U$ is an 1-periodic classical solution. The uniqueness of this
solution follows from the fact that $u\equiv 0$ is the unique
1-periodic mild solution to the homogeneous equation $u^{(n)}(t)=Au(t)$, which,
in turn, follows from (i).

\noindent (iii) $\to$ (ii):
Let  $f$ be a function in $L_{p}(J)$ with $\int_0^1f(s)ds=0$.
Define $F(t):=\int_0^tf(s)ds$, then it is easy to see that $F\in WP_p^1(J)$.
Let $U$ be the unique 1-periodic classical solution of \eqref{first} corresponding
to $F$ and put $u:=U'$. Then $u\in P^{n-1}(J)$ and $U(t)=\int_0^tu(s)ds +U(0)$.
By the definition of $U$ and $F$, the equation
$U^{(n)}(t)=AU(t)+F(t)$ means
\[
u^{(n-1)}(t)=AU(0)+A\int_0^tu(s)ds+\int_0^tf(s)ds.
\]
Hence, by Lemma \ref{deri}, $u$ is a mild solution to \eqref{high} corresponding
to $f$.  The uniqueness of $u$ follows from Corollary \ref{coro}.

\noindent (ii) $\to$ (i):  Let $f$ be a function in $L_p(J)$.
Define $\tilde{f}(t):=f(t)-f_0$, where $f_0=\int_0^1f(s)ds$,
then $\int_0^1\tilde{f}(s)ds =0$. Let $\tilde{u}$ be the 1-periodic mild
solution to \eqref{high} corresponding to $\tilde{f}$ and put
$u(t):=\tilde{u}(t)-A^{-1}f_0$. Then $u$, as $\tilde{u}$, is in $P^{n-1}(J)$.
 Moreover,
\begin{align*}
u(t)&= \tilde{u}(t)-A^{-1}f_0\\
&= \Big(\sum_{k=0}^{n-1}\frac{t^k}{k!}\tilde{u}^{(k)}(0)+AI^n\tilde{u}(t) +I^n\tilde{f}(t)\Big ) -A^{-1}f_0\\
&= \Big(u(0)+A^{-1}f_0 +\sum_{k=1}^{n-1}\frac{t^k}{k!}u^{(k)}(0)\Big)
    +AI^n \Big(u(t)+ A^{-1}f_0\Big)\\
&\quad +I^n\Big(f(t)-f_0\Big)-A^{-1}f_0\\
&=\sum_{k=0}^{n-1}\frac{t^k}{k!}u^{(k)}(0))+AI^nu(t)+I^nf(t).
\end{align*}
Hence, $u$ is a mild solution to \eqref{high} corresponding to $f$.
The uniqueness of $u$  follows from Corollary \ref{coro}.
\end{proof}

\section{Applications}

\subsection*{A semigroup case}
Here, we consider the first order Cauchy problem
\begin{equation}\label{first2}
\begin{gathered}
u'(t)=Au(t)+f(t) \quad 0 \le t\le T\\
u(0)=x ,
\end{gathered}
\end{equation}
where  $A$ generates a $C_0$-semigroup $(T(t))_{t\ge 0}$. Recall that in
this case the mild solution is of the form
\begin{equation}\label{milda}
u(t)=T(t)x+\int_0^tT(t-s)f(s)ds.
\end{equation}
We have the following result, in which the equivalence between (i) and  (v)
 is the  Gearhart's Theorem \cite{ge}.

\begin{theorem}\label{cor1}
Let $A$ generate a $C_0$-semigroup $(T(t))_{t\ge 0}$.
Then the following statements are equivalent:
\begin{itemize}
\item[(i)] $1\in \varrho(T(1))$;

\item[(ii)] For every function $f\in L_p(J)$, Equation {\rm(\ref{first2})} admits a unique 1-periodic mild solution;

\item[(iii)] For every function $f\in WP_p^1(J)$, Equation {\rm(\ref{first2})} admits a unique mild solution in $WP_p^1(J)$;

\item[(iv)] For every function $f\in WP_p^1(J)$, Equation {\rm(\ref{first2})} admits a unique 1-periodic classical solution

\end{itemize}
%
If $E$ is a Hilbert space, all the above statements are equivalent to
\begin{itemize}
\item[(v)] $\{2k\pi i: k\in \mathbb{Z}\} \subset \varrho(A)$ and
$$
\sup_{k \in \mathbb{Z}}\|(2k\pi i-A)^{-1}\|< \infty .
$$
\end{itemize}
\end{theorem}

\begin{proof}
The equivalence (i) $\Leftrightarrow$ (ii) was proved in \cite{pruss}.
The equivalence (ii) $\Leftrightarrow$ (iv) follows from Theorem \ref{p2}
and, if $E$ is a Hilbert space, (iii) $\Leftrightarrow$  (v) follows from
Theorem \ref{p1}. The inclusion (iv) $\Rightarrow$ (iii) is obvious.
So, it remains to show  (iii) $\to$ (iv).

To this end, let $u$ be the unique mild solution of (\ref{first2}),
which belong to $WP_p^1(J)$. Since $\int_0^tT(t-s)f(s)ds \in D(A)$ and
$t\rightarrow \int_0^tT(t-s)f(s)ds$ is continuously differentiable for
any $f\in W_p^1(J)$ (see e.g. \cite{ns}), we obtain that
$T(\cdot)u(0)\in W_p^1(J)$. It follows that $T(t)u(0)\in D(A)$ for $t> 0$
(since $t\mapsto T(t)x$ is differentiable at $t_0$ if and only if
$T(t_0)x\in D(A)$). Hence, $u(1)$, and thus, $x =u(1)$ belongs to $ D(A)$.
So $u$ is a classical solution. The uniqueness of the 1-periodic classical
solution is obvious.
\end{proof}

\subsection*{A cosine family case} We now consider the second order
Cauchy problem
\begin{equation}\label{second2}
\begin{gathered}
u''(t)=Au(t)+f(t) \quad 0 \le t\le T\\
u(0)=x , u'(0)=y,
\end{gathered}
\end{equation}
 where  $A$ is generator of a cosine family $(C(t))_{t\in \mathbb{R}}$ on $E$.
Recall (see u.g. \cite{abhn}) that in this case there exists a Banach space
$F$ such that $D(A)\hookrightarrow F\hookrightarrow E$ and  such that the operator
$$
\mathcal{A}:= \begin{pmatrix}
         0 & I \\
         A & 0 \\
\end{pmatrix}
$$
 with $D(\mathcal{A})= D(A)\times F$ generates the $C_0$-semigroup
$$\mathcal{T}(t):=\begin{pmatrix}
         C(t) & S(t) \\
         C'(t) & C(t) \\
\end{pmatrix}$$
on $F\times E$, where $S(t)$ is the associated sine family. Moreover,
it is not difficult to check that $u$ is a mild solution of (\ref{second2}),
which is continuously differentiable (a mild solution, which is in $WP_p^2(J)$,
or a classical solution of (\ref{second2}), respectively), if and only if
$\mathcal{U}=(u, u')^T$ is a mild solution (a mild solution,
which is in $WP_p^1(J)$, or a classical solution, respectively) of the first
order differential equation
\begin{equation}\label{first3}
\begin{gathered}
\mathcal{U}'(t)= \mathcal{A}\mathcal{U}(t) + (0, f(t))^{T}, \quad 0 \le t\le T,\\
\mathcal{U}(0)=(x, y)^T
\end{gathered}
\end{equation}
in the space $F\times E$. Using (\ref{milda}), we have the explicit form of $u$ by
$$ u(t)= C(t)x+S(t)y+\int_0^tS(s-\tau)f(\tau)d\tau.
$$
\begin{theorem}\label{cor2}
Let $A$ generate a cosine family $(C(t))_{t\in \mathbb{R}}$ in $E$. Then the following
statements are equivalent:
\begin{itemize}
\item[(i)] $1\in \varrho(C(1))$;

\item[(ii)] For each function $f\in L_p(J)$, Equation \eqref{second2}
has a unique 1-periodic mild solution, which is continuously differentiable;

\item[(iii)] For each function $f\in WP_p^1(J)$, Equation \eqref{second2}
admits a unique mild solution in $WP_p^2(J)$;

\item[(iv)] For each function $f\in WP_p^1(J)$, Equation \eqref{second2}
admits a unique 1-periodic classical solution;
\end{itemize}
%
If $E$ is a Hilbert space, all the above statements are equivalent to
\begin{itemize}
\item[(v)] $\{-4k^2 \pi^2: k\in \mathbb{Z}\} \subset \varrho(A)$ and
$\sup_{k \in \mathbb{Z}}\|k(4k^2\pi^2+A)^{-1}\|< \infty $.
\end{itemize}
\end{theorem}

\begin{proof} The equivalence  (i) $\Leftrightarrow$ (ii) is virtually
proved in \cite{schuler}. The equivalence  (ii) $\Leftrightarrow$  (iv)
 from Theorem \ref{p2} and, if $E$ is a Hilbert space, (iii) $\Leftrightarrow$  (v)
follows from Theorem \ref{p1}. The inclusion (iv) $\Rightarrow$ (iii) is obvious.
So, it remains to show  (iii) $\to$ (iv).
To this end, let $u$ be the 1-periodic mild solution of (\ref{second2}),
which is in $WP_p^2(J)$, then $\mathcal{U}=(u, u')^T$ is the 1-periodic mild
solution of (\ref{first3}), which is in $WP_p^1(J, F\times E)$. Since $\mathcal{A}$
is the generator of a $C_0$-semigroup, we can show (with the same manner as in the
proof of Theorem \ref{cor1}) that $\mathcal{U}$ is a 1-periodic classical
 solution of (\ref{first3}). It follows that $u$ is a 1-periodic classical
solution of (\ref{second2}).
\end{proof}

\subsection*{Acknowledgments}
The author would like to express his gratitude to the anonymous referee 
for his/her helpful remarks and suggestions.

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\end{document}