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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2018 (2018), No. 79, pp. 1--31.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2018 Texas State University.}
\vspace{7mm}}

\begin{document}
\title[\hfilneg EJDE-2018/79\hfil Well-posedness of integro-differential equations]
{Well-posedness of degenerate integro-differential equations
in function spaces}

\author[R. Aparicio, V. Keyantuo \hfil EJDE-2018/79\hfilneg]
{Rafael Aparicio, Valentin Keyantuo}

\address{Rafael Aparicio \newline
University of Puerto Rico, R\'io Piedras Campus,
Statistical Institute and Computerized Information Systems,
Faculty of Business Administration,
15 AVE Unviversidad STE 1501, San Juan, PR 00925-2535, USA}
 \email{rafael.aparicio@upr.edu}

\address{Valentin Keyantuo \newline
University of Puerto Rico,
R\'io Piedras Campus, Department of Mathematics,
Faculty of Natural Sciences,
17 AVE Universidad STE 1701, San Juan, PR 00925-2537, USA}
\email{valentin.keyantuo1@upr.edu}

\dedicatory{Communicated by Jerome A. Goldstein}

\thanks{Submitted September 1, 2017. Published March 20, 2018.}
\subjclass[2010]{45N05, 45D05, 43A15, 47D99}
\keywords{Well-posedness; maximal regularity;  $R$-boundedness;
\hfill\break\indent operator-valued Fourier multiplier;
 Lebesgue-Bochner spaces; Besov spaces;
\hfill\break\indent Triebel-Lizorkin spaces; H\"older spaces}

\begin{abstract}
 We use operator-valued Fourier multipliers to obtain characterizations
 for well-posedness of a large class of degenerate integro-differential
 equations of second order in time in Banach spaces.
 We treat periodic vector-valued Lebesgue, Besov and Trieblel-Lizorkin spaces.
 We observe  that in the Besov space context, the results are applicable
 to the more familiar scale of periodic vector-valued H\"older spaces.
 The equation under consideration are important in several applied problems
 in physics and material science, in particular for phenomena where memory
 effects are important.  Several examples are presented to illustrate the results.
\end{abstract}

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

\section{Introduction}

 In this article, we consider the following problem  which consists
 in a second order degenerate integro-differential equation with infinite delay
in a Banach space:
\begin{equation}{\label{eq1}}
 \begin{aligned}
& (Mu')'(t)-\Lambda u'(t)
 -\frac{d}{dt}\int_{-\infty}^t c(t-s)u(s) ds\\
&=\gamma u(t)+ Au(t)
 +\int_{-\infty}^t b(t-s) Bu(s) ds
  +f(t),\quad  0\leq t\leq 2\pi,
  \end{aligned}
\end{equation}
and  periodic boundary conditions $u(0)=u(2\pi)$,
 $(Mu')(0)=(Mu')(2\pi)$.
Here,  $A, B, \Lambda$ and $M$ are closed linear operators in a Banach
space $X$ satisfying  the assumption
 $D(A)\cap D(B)\subset D(\Lambda)\cap D(M)$,   $b, c\in L^1(\mathbb{R}_+)$,
$f$ is an $X$-valued function defined on $[0, 2\pi]$, and
$\gamma$ is a constant.
 In case $M=0$, the second boundary condition above becomes irrelevant and we
are in the presence of a first order degenerate equation.

Equations of the form \eqref{eq1} appear in a variety of applied
problems. The case where the memory effect is absent has been studied by many
authors.
The monograph \cite{FY} by Favini and Yagi is devoted to these problems
and contains meaningful applications  to concrete problems.
Recently applications to inverse problems and
in the context of multivalued operators have been investigated
(see e.g. \cite{FLT}).
 The book \cite{MF01} by Melnikova and Filinkov also treats abstract
degenerate equations.
Evolutionary integro-differential equations   arise typically
in mathematical physics by constitutive laws pertaining to materials
for which memory effects are important, when
combined with the usual conservation laws such as balance of
energy or balance of momentum. For details concerning the
underlying physical principles, we refer to Coleman-Gurtin \cite{CG}, Lunardi
\cite{Lu}, Nunziato \cite{Nu}, and the monograph Pr\"uss \cite{Pr}
(particularly Chapter II, Section 9) for work on the subject.
The latter reference contains a wealth of results on
general aspects of evolutionary integral equations and their relevance
in concrete models  from the physical sciences. Equations of first and
second order in time are of interest.
Typical examples for $\ b(\cdot)$ and $c(\cdot)$ are
the completely monotonic functions $Ke^{-\omega t}t^\mu$ where
$K\ge 0$, $\omega>0$ and $\mu>-1$, and linear combinations thereof.

Several authors have considered particular
cases of the above equation. Earlier papers: Lunardi \cite{Lu}, Da
Prato-Lunardi \cite{PL, PL1},    Clement-Da Prato \cite{CPr},
Pr\"uss \cite{Pr1}, Nunziato \cite{Nu}, Alabau-Boussouira-Cannarsa-Sforza
\cite{ACS08} and \cite{Sf} for example, use various
techniques for the solvability of problems of this type.
In the case of Hilbert spaces, the results obtained by these authors are complete.
 This is due to the fact that Plancherel's theorem is available in Hilbert space.
When $X$ is not a Hilbert space, this is no longer the case because of 
Kwapien's theorem which states that the validity of Plancherel's theorem 
for $X$-valued  functions requires  $X$ to be isomorphic to Hilbert
space (see for example  Arendt-Bu \cite{AB}). Beginning with the papers by Weis
\cite{W1,W2}, Arendt-Bu \cite{AB}, Arendt-Batty-Bu \cite{ABHN}, it became
possible to completely characterize well-posedness of the problem in periodic
vector-valued function spaces. Initially, Arendt and Bu \cite{AB} dealt with the
problem $u'(t)=Au(t)+f(t)$,
$u(0)=u(2\pi)$. Maximal regularity for the evolution problem in $L^p$ was
treated earlier by Weis \cite{W1,W2} (see also \cite{CPr} for a different proof of
the operator-valued Mikhlin multiplier theorem using a transference principle).
The study in the $L^p$ framework (when $1<p<\infty$) was made possible thanks to
the introduction of the concept of randomized boundedness (hereafter
$R$-boundedness, also known as Riesz-boundedness or Rademacher-boundedness).
 With this,   necessary conditions for operator-valued Fourier multipliers were
found in this context.
  In addition, the space $X$ must have the $UMD$ property.
This was done initially by L. Weis   \cite{W1,W2}
for the evolutionary problem and then by Arendt-Bu \cite{AB} for periodic
boundary conditions. For non-degenerate integro-differential equations both in
the periodic and non periodic cases, operator-valued Fourier multipliers have
been used by various authors to obtain well-posedness in various scales of
function spaces: see \cite{Bu1, BY1, CBu16, KL1, KL2, KL3, KL4, Pb}
and the corresponding references. The well-posedness or maximal regularity
results are important in that they allow for the treatment of nonlinear problems.
Earlier results on the application of operator-valued Fourier
 multiplier theorems to evolutionary integral equations can be found in \cite{CPr}.
More recent examples of second order integro-differential equations with
frictional damping and memory terms have been studied in the paper \cite{CDG15}

We use the operator-valued Fourier multiplier theorems
obtained by Arendt and Bu \cite{AB2} on   $B^s_{pq} (0,2\pi;X)$, and Bu
and Kim \cite{BK} on $F^s_{pq} (0,2\pi;X)$ to give a
characterization of well-posedness of \eqref{eq1} in these spaces in terms of
operator-valued Fourier multipliers and then we derive concrete conditions that
allow us to apply this characterization.

More recently, degenerate equations have attracted the attention of many
authors. Both first and second order equations have been considered. The first
order degenerate equation
\begin{equation}{\label{eq12}}
(Mu)'(t)= Au(t)+f(t),\quad  0\leq t\leq 2\pi,
 \end{equation}
with periodic boundary condition $Mu(0)=Mu(2\pi)$, has been studied
by  Lizama and  Ponce \cite{LP}; under suitable assumptions on the
 modified resolvent operator associated to \eqref{eq12},
they gave necessary and sufficient conditions to ensure the well-posedness of
\eqref{eq12} in Lebesgue-Bochner spaces
 $ L^p (0,2\pi; X)$, Besov spaces $B^s_{pq} (0,2\pi; X)$ and
Triebel-Lizorkin spaces $F^s_{pq} (0,2\pi; X)$.

Recently Bu \cite{Bu2} studied the following second order degenerate equation
\begin{equation}{\label{eq13}}
(Mu')'(t)= Au(t)+f(t),\quad  0\leq t\leq 2\pi,
\end{equation}
 with periodic boundary conditions $u(0)=u(2\pi)$, $(Mu')(0)=(Mu')(2\pi)$. He
also obtained necessary and sufficient conditions to ensure the well-posedness
of \eqref{eq13} in Lebesgue-Bochner spaces $ L^p (0,2\pi; X)$, Besov spaces
$B^s_{pq} (0,2\pi; X)$ and Triebel-Lizorkin spaces $F^s_{pq} (0,2\pi; X)$
under some suitable conditions on the  modified resolvent
operator associated to \eqref{eq13}.
Operator-valued Fourier multiplier techniques have been used recently,
most notably by  Bu and  Cai for
 handling degenerate problems in various classes of function spaces
(see e.g. \cite{BuCai2015, CBu16}.


  For more references on degenerate equations and their relevance in concrete
problems, we refer to the book \cite{FY} by Favini and Yagi. Other  references
are  Barbu and  Favini \cite{FY},  Favaron and  Favini \cite{FF} and
 Showalter \cite{Sh791, Sh792}.
The latter author has studied extensively the class of Sobolev type equations.

  When more than one unbounded operators are involved in \eqref{eq1}, a
strengthening of the definition of well-posedness is necessary. The resulting
definition (Definition \ref{DefWell} below) which we provide, seems to be
new in this context. In fact, our definition is parallel to the usual one for
partial differential equations, in the sense of Hadamard, namely existence,
 uniqueness and continuous dependence of the solution on the data of the problem. The
definition given is consistent with the previously adopted ones in the case
where only one unbounded operator appears in the equation.

We  study  equation \eqref{eq1} in the spaces of
$2\pi$-periodic vector-valued functions, namely: Lebesgue-Bochner spaces
$ L^p (0,2\pi;X)$, Besov spaces $B^s_{pq} (0,2\pi;X)$  and
 Triebel-Lizorkin spaces $F^s_{pq} (0,2\pi;X)$.

This article is organized as follows: in Section 2  we collect
some  preliminary results and definitions. In Section 3, we give  necessary
and sufficient conditions for well-posedness of the
 \eqref{eq1} in the Lebesgue
Bochner spaces   $ L^p  (0,2\pi;X) $,  Besov spaces
$B_{pq}^s (0,2\pi;X)$ and  Triebel-Lizorkin $F_{pq}^s (0,2\pi;X)$ spaces
in terms of operator-valued Fourier multipliers. In Section 4, we give
concrete conditions on the data ensuring applicability of the results
established in Section 3. We stress that in the $L^p$ case, the results use the
concept of $R$-boundedness  and require the space $X$ to be UMD (this is equivalent
 to the continuity of the Hilbert transform on $L^p( \mathbb{R}, X)$,
$1<p<\infty$). The the concept of $R$-boundedness
 first appeared in the context of evolution equations in the papers \cite{W1,W2}
of  Weis (see also the article \cite{GW2}).

   In the other cases (namely $B_{pq}^s (0,2\pi;X)$ and  $F_{pq}^s
(0,2\pi;X)$), these restrictions are no longer needed but
 one requires instead higher order boundedness conditions on the ``modified resolvents''
involved.

  In the final Section 5, we consider some
 examples where the results above apply. We single out the following modified
version of problem which is considered in Favini-Yagi \cite[Example 6.1]{FY}
\begin{equation}
\begin{gathered}
\begin{aligned}
&\frac{\partial}{\partial t}(m(x)\frac{\partial u(t,x)}{\partial
t})-\Delta\frac{\partial u(t,x)}{\partial t}\\
&= \Delta u(t,x)+\int_{-\infty}^t{b(t-s)\Delta u(s,x)}ds+f(t,x), \quad
(t,x)\in[0,2\pi]\times\Omega,
\end{aligned}\\
u(t,x)=\frac{\partial u(t,x)}{\partial t}=0, \quad
(t,x)\in[0,2\pi]\times\partial\Omega,\\
u(0,x)=u(2\pi,x), \quad m(x)\frac{\partial u(0,x)}{\partial t}=m(x)\frac{\partial
u(2\pi,x)}{\partial t}, \quad x\in\Omega,
\end{gathered}
\end{equation}
 where $\Omega\subset \mathbb{R}^n$ is an open subset and $\Delta$ is the
Laplace operator. We consider the problem in the space $X=L^r(\Omega)$,
$1<r<\infty$. This is a degenerate wave equation with memory and a damping term. We treat
the problem for periodic boundary conditions. The  authors of the cited papers
also study   the evolutionary problem as well, including asymptotic behavior of
solutions. They consider only the case when $a=0$, that is they do not
incorporate the memory term in the equation. They restrict their study to the
H\"older spaces. For periodic boundary conditions, we obtain complete
characterization of well-posedness in the three scales of spaces: $L^p$,
$B^s_{pq}$, and $F^s_{pq}$. We are also able to treat this problem replacing
$\Delta $ with $-\Delta$ in the right hand side. The latter equation is the
focus of the reference \cite{FY}.

\section{ Preliminaries}

 In this section, we collect some results and definitions that will be used in
the sequel. Let $X$ be a complex Banach space. We denote as usual by
$L^1(0,2\pi,X)$ the space of Bochner integrable functions with values in $X$.
 For a function $ f\in L^1(0,2\pi;X)$, we denote by
$ \hat f (k), k \in \mathbb{Z} $ the $k$th Fourier coefficient of $f$:
 $$
\hat f(k) = \frac{1}{2\pi} \int_0^{2\pi} e_{-k} (t) f(t) dt,
$$
where $e_k(t) = e^{ikt} , t \in \mathbb{R}$.

Let $ u\in L^1(0,2\pi; X)$.  We denote again by $u$ its
periodic extension   to $\mathbb{R} $. Let
$a\in L^1(\mathbb{R_+})$. We consider the the function
\begin{equation*}
F(t) = \int_{-\infty}^t a(t-s)u(s) ds, \quad t\in\mathbb{R}.
\end{equation*}
Since
\begin{equation}{\label{eq2.2}}
F(t) = \int_{-\infty}^t a(t-s)u(s) ds =\int_0^\infty a(s)u(t-s)ds,
\end{equation}
we have
 $\|F\|_{L^1}\leq\|a\|_1\|u\|_{L^1}=\Vert a\Vert_{L^1(\mathbb{R}_+)} \Vert u\Vert_{L^1(0,\, 2\pi; X)}$  and $F$ is periodic of period $T=2\pi$ as
$u$. Now
using Fubini's theorem and \eqref{eq2.2} we obtain, for $k\in\mathbb{Z}$, that
\begin{equation}{\label{eq3.3}}
\hat F(k) = \tilde a (ik) \hat u (k), k \in \mathbb{Z}
\end{equation}
 where
$\tilde a(\lambda) = \int_0^{\infty} e^{-\lambda t} a(t) dt $
denotes the Laplace transform of $a$. This identity plays a crucial role in the
paper.

Let $X,Y$ be Banach spaces. We denote by $ \mathcal{L}(X,Y)$  the
set of all bounded linear operators from $X$ to $Y$. When $X=Y$,
we write simply $ \mathcal{L}(X)$.

For results on operator-valued  Fourier multipliers and
$R$-boundedness (used in the next section), as well as some applications to
evolutionary partial differential equations,  we refer to
 Amann \cite{Am}, Bourgain \cite{Bo, Bo2}, Cl\'ement-de
Pagter-Sukochev-Witvliet \cite{CPSW}, Weis \cite{W1, W2},
Girardi-Weis \cite{GW1}, \cite{GW2}, Kunstmann-Weis \cite{KW},  Cl\'ement-
Pr\"uss \cite{CP}, Arendt \cite{A}
and Arendt-Bu \cite{AB}. The scalar case is presented for example in
Schmeisser-Triebel \cite[Chapter 3]{ST}. This reference
 also considers the case where $X$ is a Hilbert space (Chapter 6). Here, we will
merely present the
appropriate definitions.


We shall frequently identify the spaces of (vector or
operator-valued) functions defined on $ [0,2\pi]$ to their
periodic extensions to $\mathbb{R}$. Thus, in this section, we
consider the spaces:

\subsection*{Lebesgue-Bochner spaces}
For $1\leq p\le\infty$, we denote $L_{2\pi}^p(\mathbb{R};X)$ (denoted also
$L^p (0,2\pi; X)$, $1 \leq p \leq \infty $) of all $2\pi$-periodic
Bochner measurable $X$-valued functions $f$ such that the
restriction of $f$ to $[0,2\pi]$ is $p$-integrable, usual modification if
$p=\infty$. The space is equipped with the norm
\begin{equation}
\|f\|_p
=\|f\|_{L^p(0,2\pi,X)}
=\begin{cases}
\big( \frac{1}{2\pi}\int_0^{2\pi}\|f(t)\|_X^pdt\big)^{1/p}
 &\text{if }1\leq p<\infty,\\[4pt]
 \operatorname{ess\,sup}_{t\in[0, 2\pi]}\|f(t)\|_X &\text{if } p=\infty.
\end{cases}
\end{equation}


\subsection*{Besov spaces}
 We briefly recall the
 the definition of $2\pi$-periodic Besov space in the vector-valued case
  introduced in \cite{AB2}. Let
 $\mathcal{S}(\mathbb{R})$ be the Schwartz space of all rapidly decreasing
 smooth functions on $\mathbb{R}$. Let $\mathcal{D}(0,2\pi)$ be the space
 of all infinitely differentiable functions on $[0,2\pi]$ equipped with
 the locally convex topology given by the family of seminorms
 $$
\|f\|_\alpha=\sup_{x\in[0,2\pi]}|f^{(\alpha)}(x)|
$$
for  $\alpha\in\mathbb{N}_0:=\mathbb{N}\cup\{0\}$. Let
$\mathcal{D}'(0,2\pi, X): =\mathcal{L}( \mathcal{D}(0,2\pi),X)$
be the space of all bounded linear operators
 from $\mathcal{D}(0,2\pi)$ to $X$ ($X$-valued distributions).
In order to define the Besov spaces,
 we consider the dyadic-like subsets of $\mathbb{R}$:
$$
I_0=\{t\in\mathbb{R}:  |t|\leq2\} \text{, } I_k=\{t\in\mathbb{R}:
 2^{k-1}<|t|\leq2^{k}\}
$$
for $k\in\mathbb{N}$. Let $\Phi(\mathbb{R})$ be the set of all systems
$\phi=(\phi_k)_{k\in\mathbb{N}_0}\subset\mathcal{S}(\mathbb{R})$
satisfying $\text{supp}(\phi_k)\subset \overline{I}_k$ for each
$k\in\mathbb{N}_0$, $\sum_{k\in\mathbb{N}_0}\phi_k(x)=1$ for
$x\in\mathbb{R}$,  and for each $\alpha\in\mathbb{N}_0$,
 $\sup_{x\in\mathbb{R},k\in\mathbb{N}_0}2^{k\alpha}|\phi_k^{(\alpha)}(x)|<\infty$.
 Let $\phi=(\phi_k)_{k\in\mathbb{N}_0}\in\Phi(\mathbb{R})$ be fixed. For
$1\leq p,q\leq\infty$,   $s\in\mathbb{R}$, the $X$-valued $2\pi$-periodic
Besov space is denoted by
  $B_{pq}^s(0,2\pi,X)$ and defined by the set
$$
\Big\{f\in\mathcal{D}'(0,2\pi;X):\|f\|_{pq}^s:=
\Big(\sum_{j\geq0}2^{sjq}\|\sum_{k\in\mathbb{Z}}
e_k\otimes\phi_j(k)\hat{f}(k)\|_p^q\Big)^{1/q}<\infty\Big\}
$$
with the usual modification if $q=\infty$.

It is known that  $B_{pq}^s(0,2\pi,X)$ is independent of the choice of $\phi$,
and different choices of $\phi$ in the class $\Phi(\mathbb{R})$ lead
to equivalent norms $\|\cdot\|_{pq}^s$. Equipped with the norm
$\|\cdot\|_{pq}^s$, $B_{pq}^s(0,2\pi,X)$ is a Banach space.

It is also known that  is $s_1\leq s_2$, then $B_{pq}^{s_2}(0,2\pi,X)\subset
B_{pq}^{s_1}(0,2\pi, X)$
 and  the embedding is continuous \cite{AB2}. When $s>0$, it is proved in
\cite{AB2} that $B_{pq}^{s}(0,2\pi,X)\subset L^p(0,2\pi,X)$
 and the embedding is continuous; moreover, $f\in B_{pq}^{s+1}(0,2\pi,X)$ if and
only if $f$ is differentiable a.e on $[0,2\pi]$
  and $f'\in B_{pq}^{s}(0,2\pi,X)$. In the case where $p = q = \infty$ and $0
< s < 1$ we have that $B^s_{\infty\infty}(0,2\pi,X)$ corresponds
to the space $C^s(0,2\pi, X)$ of H\"older continuous functions with equivalent
norm
$$
\|f\|_{C^s(0,2\pi;X)}=\sup_{t_1
\neq t_2}\dfrac{\|f(t_2)-f(t_1)\|_X}{|t_2-t_1|^s}+\|f\|_\infty.
$$

\subsection*{Triebel-Lizorkin spaces}  Let
$\phi=(\phi_k)_{k\in\mathbb{N}_0}\in\Phi(\mathbb{R})$ be fixed with $\phi$ and
$\Phi(\mathbb{R})$ as above. For $1\leq p<\infty$, $1\leq q\leq\infty$,
  $s\in\mathbb{R}$, the $X$-valued $2\pi$-periodic  Triebel-Lizorkin space with
parameters $s$, $p$ and $q$ is denoted by
  $F_{pq}^s(0,2\pi;X)$ and defined by the set
\[
  \Big\{f\in\mathcal{D}'(0,2\pi,X):
\|f\|_{pq}^s:= \big\|\big(\sum_{j\geq0}2^{sjq}\|\sum_{k\in\mathbb{Z}}
e_k\otimes\phi_j(k)\hat{f}(k)\|_X^q\big)^{1/q}\big\|_p<\infty\Big\}
\]
with the usual modification if $q=\infty$.

It is known that set $F_{pq}^s(0,2\pi,X)$ is independent of the choice of
$\phi$, and again,
different choices
of $\phi$ lead to equivalent norms $\|\cdot\|_{pq}^s$. Equipped with the norm
$\|\cdot\|_{pq}^s$, $F_{pq}^s(0,2\pi,X)$ is a Banach space.

It is also  known that  if $s_1\leq s_2$, then
$F_{pq}^{s_2}(0,2\pi,X)\subset F_{pq}^{s_1}(0,2\pi, X)$
and  the embedding is continuous \cite{BK}. When $s>0$, it is show in \cite{BK}
that $F_{pq}^{s}(0,2\pi,X)\subset L^p(0,2\pi,X)$
and the embedding is continuous; moreover, $f\in F_{pq}^{s+1}(0,2\pi,X)$ if and
only if $f$ is differentiable a.e on $[0,2\pi]$
and $f'\in F_{pq}^{s}(0,2\pi,X)$. The exceptional case $p=\infty$ will  not be
considered in this paper. We refer to Schmeisser-Triebel
\cite[ Section 3.4.2]{ST} for a discussion. Note that
$F^s_{pp}((0, 2\pi); X) = B^s_{pp}((0, 2\pi); X)$ by an inspection of   the definitions.

We give the definition of operator-valued Fourier multipliers in each of the
cases that will be of interest to us. First, in the case of Lebesgue spaces,
we have: (See \cite{AB,AB2, BK}).

\begin{definition} \label{def2.1}\rm
Let $X$ and $Y$ be Banach spaces.
For $ 1 \leq p \leq \infty, $ we say that a sequence
 $ ( M_k)_{k \in \mathbb{Z}} \subset \mathcal{L}(X,Y)$ is an
$L^p$-multiplier, if for each $ f \in L^p (0,2\pi; X)$ there
exists $ u  \in L^p (0,2 \pi ; Y) $ such that
$$
\hat u (k)  = M_k \hat f (k) \mbox{ for all } k \in \mathbb{Z}.
$$
\end{definition}

In the case of Besov spaces, we have the following concept.

\begin{definition} \label{def2.2} \rm
Let $X$ and $Y$ be Banach spaces. For  $1 \leq p , q\leq \infty$, $ s>0$, we
say that a sequence $( M_k )_{k \in \mathbb{Z}}\subset \mathcal{L}(X,Y)$  is an
$B^{s}_{pq}$-multiplier, if for each $ f \in B^{s}_{pq}(0,2\pi; X)$
there exists $ u  \in B^{s}_{pq}(0,2\pi ; Y) $ such that
$$
\hat u (k)  = M_k \hat f (k) \mbox{ for all } k \in \mathbb{Z}.
$$
\end{definition}

Finally, in the case of Triebel-Lizorkin spaces, we have the following concept.

\begin{definition} \label{def2.3} \rm
Let $X$ and $Y$ be Banach spaces. For $1\leq p<\infty$, $1\leq q\leq \infty$,
$s>0$, and let
$ ( M_k )_{k \in \mathbb{Z}} \subset \mathcal{L}(X,Y)$, we
say that a sequence $( M_k )_{k \in \mathbb{Z}}\subset \mathcal{L}(X,Y)$
is an $F^{s}_{pq}$-multiplier,  if for each $ f \in F^{s}_{pq}(0,2\pi; X)$
there exists $ u  \in F^{s}_{pq}(0,2\pi ; Y) $ such that
$$
\hat u (k)  = M_k \hat f (k) \mbox{ for all } k \in \mathbb{Z}.
$$
\end{definition}

From the uniqueness theorem of Fourier series, it follows that $u$ is
uniquely determined by $f$ in each of the above mentioned cases.

We denote by $\mathcal{Y}=\mathcal{Y}(X)$ any of the following spaces of
$X$-valued functions: $L^p (0,2\pi; X)$, $1 \leq p \leq \infty $;
$B^{s}_{pq}(0,2\pi; X)$, $1 \leq p , q\leq \infty$, $ s>0$;
 $F^{s}_{pq}(0,2\pi; X)$, $1\leq p<\infty$, $1\leq q\leq \infty$, $s>0$.  We
define the sets
\begin{gather*}
\mathcal{Y}^{[1]}=\{u \in \mathcal{Y}: u \text{ is almost everywhere
differentiable and }u'\in \mathcal{Y}\}, \\
\mathcal{Y}^{[1]}_{\rm per}=\{u \in \mathcal{Y}: \exists v\in\mathcal{Y},
\text{ such that }\hat v(k) = ik \hat u(k)\text{ for all }k
\in\mathbb{Z} \}
\end{gather*}
 In the case that $\mathcal{Y}=L^{p}(0,2\pi;X)$,
$\mathcal{Y}^{[1]}$ is denoted by $W^{1,p}(0,2\pi;X)$ and
$\mathcal{Y}^{[1]}_{\rm per}$ by $W^{1,p}_{\rm per}(0,2\pi;X)$.
   In the case that $\mathcal{Y}=B_{pq}^{s}(0,2\pi;X)$,
$\mathcal{Y}^{[1]}=B_{pq}^{s+1}(0,2\pi;X)$. In the case that
$\mathcal{Y}=F_{pq}^s(0,2\pi;X)$, $\mathcal{Y}^{[1]}=F_{pq}^{s+1}(0,2\pi;X)$.

\begin{remark} \label{RB} \rm
Using integration by parts,  the fact that $\mathcal{Y}\subset L^1(0,2\pi, X)$
and the uniqueness theorem of Fourier coefficients, we have
\begin{equation}\label{per}
\begin{gathered}
\mathcal{Y}^{[1]}_{\rm per}=\{u\in\mathcal{Y}^{[1]}:u(0)=u(2\pi)\},\\
\mathcal{Y}^{[1]}_{\rm per}=\{u\in\mathcal{Y}^{[1]}:\widehat{u}'(k) = ik \hat
u(k)\text{ for all }k
\in\mathbb{Z}\}.
\end{gathered}
\end{equation}
Therefore, if $u\in\mathcal{Y}^{[1]}_{\rm per}$, then $u$ has a unique continuous
representative such that $u(0)=u(2\pi)$. We always identify $u$ with this
continuous function.
\end{remark}

\begin{remark} \label{R4}  \rm
It is clear from the definitions that:
\begin{itemize}
\item[(a)] if $(M_k)_{k\in\mathbb{Z}},
(N_k)_{k\in\mathbb{Z}}\subset\mathcal{L}(X,Y)$ are $\mathcal{Y}$-Fourier
multipliers and $\alpha,\beta$ are constants, then
$(\alpha M_k+\beta N_k)_{k\in\mathbb{Z}}\subset\mathcal{L}(X,Y)$ is a
$\mathcal{Y}$-Fourier multiplier as well.

\item[(b)] if $(M_k)_{k\in\mathbb{Z}}\subset\mathcal{L}(X,Y)$ and
$(N_k)_{k\in\mathbb{Z}}\subset\mathcal{L}(Y,Z)$ are $\mathcal{Y}$-Fourier
multipliers, then $(N_kM_k)_{k\in\mathbb{Z}}\subset\mathcal{L}(X,Z)$ is a
$\mathcal{Y}$-Fourier multiplier as well. In particular, when $X=Y=Z$, if
$(M_k)_{k\in\mathbb{Z}}, \,
(N_k)_{k\in\mathbb{Z}}$ are $\mathcal{Y}$-Fourier
multipliers, then $(N_kM_k)_{k\in\mathbb{Z}}$ is a
$\mathcal{Y}$-Fourier multiplier as well.
\end{itemize}
\end{remark}

\begin{proposition}[{\cite[Fejer's Theorem]{AB}}]\label{Fejer's}
Let $f\in L^p(0,2\pi;X))$, then one has
$$
f=\lim_{n\to\infty}\frac{1}{n+1}\sum_{m=0}^n\sum_{k=-m}
^me_k\hat{f}(k)
$$
 with convergence in $L^p(0,2\pi;Y))$.
\end{proposition}

\begin{remark} \label{R5} \rm
(a)  If $(kM_k)_{k\in\mathbb{Z}}$ is a
$\mathcal{Y}$-Fourier multiplier, then $(M_k)_{k\in\mathbb{Z}}$ is also a
$\mathcal{Y}$-Fourier multiplier.

(b) If $(M_k)_{k\in\mathbb{Z}}\subset\mathcal{L}(X,Y)$ is a
$\mathcal{Y}$-Fourier
multiplier, then there exists a bounded linear operator
$T\in\mathcal{L}(\mathcal{Y}(X),\mathcal{Y}(Y))$ satisfying
$\widehat{(Tf)}(k)=M_k\hat{f}(k)$ for all $k\in\mathbb{Z}$. This implies in
particular that the sequence   $(M_k)_{k\in\mathbb{Z}}$ must be bounded.
\end{remark}

For $ j \in \mathbb{N}$, denote by $r_j$ the $j$-th Rademacher
function on $[0,1]$, i.e. $ r_j (t) = sgn(\sin(2^j \pi t))$. For
$x \in X$ we denote by $ r_j \otimes x $ the vector valued
function $ t \to r_j(t)x$.

The important concept of $R$-bounded  for a given family of
bounded linear operators is defined as follows.

\begin{definition} \rm
A family $ \mathbf{T} \subset \mathcal{L}(X,Y)$ is called
$R$-bounded if there exists $ c_q \geq 0$ such that
\begin{equation}{\label{eq4}}
\| \sum_{j=1}^n r_j \otimes T_j x_j \|_{L^q(0,1;X)}
 \leq c_q \| \sum_{j=1}^n r_j \otimes  x_j \|_{L^q(0,1;X)}
\end{equation}
for all $ T_1,\dots, T_n \in \mathbf{T}, x_1,\dots,x_n \in X$ and
 $ n \in \mathbb{N}, $ where $ 1 \leq q  < \infty$. We denote by
$R_q(\mathbf{T})$ the smallest constant $c_q$ such that \eqref{eq4}
holds.
\end{definition}


\begin{remark} {\label{R1}}  \rm
 Several useful properties of
$R$-bounded families can be found in the monograph of Denk-Hieber-Pr\"uss
\cite[Section 3]{DHP}, see also \cite{A, AB, CPSW, PW, KW}.
 We collect some of them here for later use.

\begin{itemize}
\item[(a)] Any finite subset of $\mathcal{L}(X)$ is is $R$-bounded.

\item[(b)] If $\mathbf{S}\subset \mathbf{T}\subset\mathcal{L}(X)$ and $\mathbf{T}$ is $R$-bounded,
then $\mathbf{S}$ is  $R$-bounded and $R_p(\mathbf{S})\leq R_p(\mathbf{T})$.

\item[(c)] Let $ \mathbf{S}, \mathbf{T}
\subset \mathcal{L}(X)$ be $R$-bounded sets. Then $ \mathbf{S}
\cdot \mathbf{T} := \{ S \cdot T : S \in \mathbf{S}, T \in
\mathbf{T} \}$ is $R$-bounded and $$ R_p (\mathbf{S} \cdot
\mathbf{T}) \leq R_p ( \mathbf{S}) \cdot R_p( \mathbf{T}).$$

\item[(d)] Let $ \mathbf{S}, \mathbf{T}
\subset \mathcal{L}(X)$ be $R$-bounded sets. Then $ \mathbf{S}
+ \mathbf{T} := \{ S + T : S \in \mathbf{S}, T \in
\mathbf{T} \}$ is $R$- bounded and $$ R_p (\mathbf{S} +
\mathbf{T}) \leq R_p ( \mathbf{S}) + R_p( \mathbf{T}).$$

\item[(e)] If $\mathbf{T}\subset\mathcal{L}(X)$ is $R$- bounded, then $\mathbf{T}\cup \{0\}$
 is $R$-bounded and $R_p(\mathbf{T}\cup \{0\})= R_p(\mathbf{T})$.

\item[(f)] If $\mathbf{S},\mathbf{T}\subset\mathcal{L}(X)$ are $R$- bounded, then
$\mathbf{T}\cup \mathbf{S}$  is $R$-bounded and 
$$
R_p(\mathbf{T}\cup \mathbf{S})\leq R_p(\mathbf{S})+R_p(\mathbf{T}).
$$

\item[(g)] Also, each subset $M \subset \mathcal{L}(X) $ of the form $ M =
\{\lambda I  : \lambda \in \Omega \}$ is $R$-bounded whenever $
\Omega \subset \mathbb{C}$ is bounded ($I$ denotes the identity
operator on $X$).
\end{itemize}

The proofs of (a),  (e), (f), and (g) rely  on  Kahane's contraction principle.

We sketch a proof of (f). Since we assume that
$\mathbf{S},\mathbf{T}\subset\mathcal{L}(X)$ are $R$-bounded, it follows
from (e) (which is a consequence of Kahane's contraction principle) that
$\mathbf{S}\cup \{0\}$ and $\mathbf{T}\cup \{0\}$ are $R$-bounded.
We now observe that $\mathbf{S}\cup  \mathbf{T} \subset  \mathbf{S}\cup \{0\}
+\mathbf{T}\cup \{0\}$. Then using (d) and (b) we conclude that
$\mathbf{S}\cup  \mathbf{T}$ is $R$-bounded.

We make the following general observation which will be valid throughout the
paper, notably in Section 4. Whenever we wish to establish $R$-boundedness of a
family of operators $(M_k)_{k\in\mathbb{Z}}$, if at some point we make an
exception such as $(k\ne 0)$, $(k\notin \{-1,0\})$ and so on, then later we
recover the property for the entire family using items (a), (c) and (f)
of the foregoing remark. The corresponding observation for boundedness is
clear.
\end{remark}

 \begin{remark} \label{R2} \rm
 If $X=Y$ is a $UMD$ space and $ M_k
= m_k I $ with $ m_k \in \mathbb{C}$, then the Marcinkiewicz condition
$\sup_{k}{|m_k|} + \sup_k |k (m_{k+1} - m_k)|< \infty $
 implies that the set
$ \{ M_k \}_{k \in \mathbb{Z}} $ is an $L^p$-multiplier. (see
\cite{AB} or \cite[Theorem 4.4.3]{Am}).
\end{remark}

Another important notion in Banach space theory is that of Fourier type for a
Banach space. Conditions for Fourier multipliers are simplified when the Banach
spaces involved satisfy this condition. The Hausdorff-Young inequality states
that for $1\le p\le 2$, the Fourier transform maps
$L^p(\mathbb{R}):= L^p(\mathbb{R};\mathbb{C})$  continuously into
$L^{p'}  (\mathbb{R})$
where $ \frac{1}{p}+\frac{1}{p'}=1$, with the
convention that $p'=\infty$ when $p=1$. In particular, when
$p=2$, Plancherel's theorem holds. When $X$ is a Banach space and
one considers $L^p(\mathbb{R};X)$, the situation is no longer the
same. It is known that Plancherel's theorem (here we mean $L^2-$continuity of the $X-$valued Fourier transform)  holds if and only if
$X$ is isomorphic to a Hilbert space (see e.g. \cite{Am,ABHN,AB,GW2}). For every Banach space, the
Hausdorff-Young theorem holds with $p=1$. A Banach space is said
to have non-trivial Fourier type if the Hausdorff-Young theorem
holds true for some $p\in (1, 2]$. By a result of Bourgain \cite{Bo, Bo2},
$UMD$ spaces are examples of spaces with nontrivial Fourier type
(see \cite{GW2,AB1}). More generally, $B$-convex spaces,
in particular superreflexive Banach spaces have nontrivial
Fourier type (\cite[Proposition 3]{Bo2}). However,  there exist
non reflexive Banach spaces with nontrivial Fourier type. The implications
of the property of having non trivial Fourier type are studied in Giradi-Weis
\cite{GW2}.

For Banach spaces with non trivial Fourier type, in particular for $UMD$ spaces,
the conditions for the validity of operator-valued Fourier  multiplier
theorems are greatly simplified.

\section{Characterization in terms of Fourier multipliers}

In this section, we characterize the well-posedness of the problem
\begin{equation} \label{eP}
\begin{gathered}
\begin{aligned}
&(Mu')'(t)-\Lambda u'(t)
 -\frac{d}{dt}\int_{-\infty}^t c(t-s)u(s) ds\\
&=\gamma u(t)+ Au(t)+\int_{-\infty}^t b(t-s) Bu(s) ds
  +f(t),\,\,\, 0\leq t\leq 2\pi, \end{aligned} \\
  u(0)=u(2\pi)\quad\text{and}\quad (Mu)'(0)=(Mu)'(2\pi)
  \end{gathered}
\end{equation}
in the vector-valued Lebesgue, Besov, and
Triebel-Lizorkin  spaces. Here $A, B, \Lambda$ and $M$ are closed
linear operators in a Banach space $X$ satisfying
$D(A)\cap D(B)\subset D(\Lambda)\cap  D(M)$,
$b, c\in L^1(\mathbb{R}_+)$,  $f$ is an $X$-valued function defined on
 $[0, 2\pi]$, and $\gamma$ is a constant.  The results are in terms of
operator-valued Fourier multipliers.

 Let $b,c$ be complex valued functions and $\gamma$ a constant.
We define the $M,\Lambda$-resolvent set of $A$ and $B$,
$\rho_{\Lambda,M,\tilde{b},\tilde{c}}(A,B)$, associated to \eqref{eP} by
\[
\{\lambda\in\mathbb{C}
\vert\mathcal{M}(\lambda):D(A)\cap D(B)\to X\text{ is bijective and  }
[\mathcal{M}(\lambda)]^{-1}\in\mathcal{L}(X)\}
\]
where $\mathcal{M}(\lambda)=\lambda^2M-A-
\tilde{b}(\lambda)B-\lambda \Lambda-\lambda \tilde{c}(\lambda)I-\gamma I$. 
Thus,
$\lambda\in\rho_{\Lambda,M,\tilde{a},\tilde{b},\tilde{c}}(A,B)$ if and only if
$[\mathcal{M}(\lambda)]^{-1}$ is a linear continuous isomorphism from $X$ onto $D(A)\cap
D(B)$.
 Here  we consider $D(A)$, $D(B)$, $D(\Lambda)$ and $D(M)$ as normed spaces
equipped with their respective graph norms. These are Banach space since all the
operators are closed.
For $a\in L^1(\mathbb{R}_+)$, $u\in \mathcal{Y}$, we denote by $a*u$ the
function
\begin{equation}
(a*u)(t):=\int_{-\infty}^ta(t-s)u(s)ds
\end{equation}
Since $\mathcal{Y}\subset L^1(0,2\pi;X)$, it follows that
  $a*u\in L^1(0,2\pi;X)$
 and $(a*u)(0)=(a*u)(2\pi)$ by \eqref{eq2.2}.  
With this notation we may rewrite
\eqref{eq1} in the following way:
\begin{equation*}
(Mu')'(t)-\Lambda u'(t)-\frac{d}{dt}(c*u)(t)\\
=\gamma u(t)+ Au(t)+(b*Bu)(t)+f(t), \quad 0\leq t\leq 2\pi.
\end{equation*}

If  $b$, $c\in L^1(\mathbb{R}_+)$ and
$u \in L^1(0,2\pi;D(A))\cap L^1(0,2\pi;D(B))$, then  $c*u$,
$b*Bu\in L^1(0,2\pi;X)$ by \eqref{eq2.2} and
$\widehat{(c*u)}(k)=\tilde{c}(ik)\hat{u}(k)$,
$\widehat{(a*Au)}(k)=\tilde{a}(ik)A\hat{u}(k)$ and
$\widehat{(b*Bu)}(k)=\tilde{b}(ik)B\hat{u}(k)$  by \eqref{eq3.3}. If
additionally we have that $ \frac{d}{dt}(c*u)\in L^1(0,2\pi;X)$, then $c*u\in
W^{1,1}(0,2\pi;X)$ and $(c*u)(0)=(c*u)(2\pi)$. Then
 $\widehat{\frac{d}{dt}(c*u)}(k)=ik\tilde{c}(ik)\hat{u}(k)$  by  \eqref{per}.

In what follows, we adopt the following notation:
\begin{equation} \label{eq2}
 b_k:=\tilde{b}(ik), c_k:=\tilde{c}(ik)
\end{equation}

\begin{remark} \label{R3} \rm
 By the Riemann-Lebesgue lemma, the sequences
$(b_k)_{k\in\mathbb{Z}}$ and $(c_k)_{k\in\mathbb{Z}}$
so defined are bounded. In fact $\lim_{|k|\to\infty}b_k=0$, and similarly
for $(c_k)_{k\in\mathbb{Z}}$. Moreover,  $(b_kI)_{k\in\mathbb{Z}}$ and
$(c_k I)_{k\in\mathbb{Z}}$ define a $\mathcal{Y}$-Fourier multiplier.
\end{remark}

We now give the definition of solutions of \eqref{eP} in our relevant cases.

\begin{definition}\label{def3.2} \rm
 A function $u\in \mathcal{Y}$ is called a {\it strong $\mathcal{Y}$-solution}
of  \eqref{eP}
if $u\in \mathcal{Y}(D(A))\cap \mathcal{Y}(D(B))\cap \mathcal{Y}^{[1]}_{\rm per}$,
$u'\in \mathcal{Y}(D(\Lambda))\cap \mathcal{Y}(D(M))$,
 $Mu'\in \mathcal{Y}^{[1]}_{\rm per}$,
and equation \eqref{eq1} holds for almost all $t\in [0,2\pi]$.
\end{definition}


\begin{lemma}\label{Fourier}
 Let $X$ be a Banach space,
and $A$, $B$, $\Lambda$, $M$
be closed linear operators in $X$ such that 
$D(A)\cap D(B)\subset D(\Lambda)\cap D(M)$. 
Suppose that $\gamma$ is a constant, $b,c\in L^1(\mathbb{R}_+)$,   and consider  $b_k$,
$c_k$ as in \eqref{eq2}. Assume that $u$ is a strong $\mathcal{Y}$-solution
of  \eqref{eP}. Then
 $$
[-k^2M-A  -b_kB-ik\Lambda-ikc_kI-\gamma I]\hat{u}(k)=\hat{f}(k).
$$
for all $k\in\mathbb{Z}$.
\end{lemma}

\begin{proof}
Let $k\in\mathbb{Z}$. Since $u$ is a strong $\mathcal{Y}$-solution of 
\eqref{eP},
 $u\in \mathcal{Y}(D(A))\cap \mathcal{Y}(D(B))\cap \mathcal{Y}^{[1]}_{\rm per}$,
$ u'\in \mathcal{Y}(D(\Lambda))\cap \mathcal{Y}(D(M))$,
 $Mu'\in \mathcal{Y}^{[1]}_{\rm per}$ and
\begin{align*}
&(Mu')'(t)-\Lambda u'(t)-\frac{d}{dt}(c*u)(t)\\
&=\gamma u(t)+Au(t)+ (b*Bu)(t)+f(t), \quad \text{for a.e }
 t\in[0, 2\pi].
\end{align*}

Since $u\in \mathcal{Y}(D(A))\cap \mathcal{Y}(D(B))$, we have
$$
\hat{u}(k)\in D(A)\cap D(B)\quad\text{and}\quad 
\widehat{Au}(k)=A\hat{u}(k),\hat{Bu}(k)=B\hat{u}(k).
$$
by \cite[Lemma 3.1]{AB}. Since
$u\in\mathcal{Y}^{[1]}_{\rm per}$, we have $\widehat{u}'(k)=ik\hat{u}(k)$ by
\eqref{per}.
 Since $ u'\in \mathcal{Y}(D(\Lambda))\cap \mathcal{Y}(D(M))$, it follows that
$\widehat{(\Lambda u')}=\Lambda \widehat{u}'(k)=ik\Lambda\hat{u}(k)$,
$\widehat{M u'}=M\widehat{u}'(k)=ikM\hat{u}(k)$ by \cite[Lemma 3.1]{AB}.
 Since $Mu'\in \mathcal{Y}^{[1]}_{\rm per}$, it follows that
$\widehat{(Mu')'}=ik\widehat{M u'}(k)=-k^2M\hat{u}(k)$ by \eqref{per}.
Since $u\in\mathcal{Y}(D(A))\subset L^1(0,2\pi;D(A))$,
$u\in\mathcal{Y}(D(B))\subset L^1(0,2\pi;D(B))$   and  $b$, $c\in
L^1(\mathbb{R}_+)$, it follows that  $c*u$, $b*Bu\in L^1(0,2\pi;X)$,
$(c*u)(0)=(c*u)(2\pi)$ by \eqref{eq2.2} and
$\widehat{(c*u)}(k)=\tilde{c}(ik)\hat{u}(k)$,
$\widehat{(b*Bu)}(k)=\tilde{b}(ik)B\hat{u}(k)$ by  \eqref{eq3.3}.
Since $\mathcal{Y}\subset L^1(0,2\pi;X)$, we have
 $u$,  $\Lambda u'$, $(Mu')'$ and $f\in L^1(0,2\pi;X)$. So $u$, $Au$, $Bu$,
$b*Bu$, $\Lambda u'$, $(Mu')'$ and
$f$ all belong to $L^1(0,2\pi;X)$. Then $\frac{d}{dt}(c*u)$ must be in
$L^1(0,2\pi;X)$.
Therefore $c*u\in W^{1,1}_{\rm per}(0,2\pi;X)$ and
 $\widehat{\frac{d}{dt}(c*u)}(k)=ik\tilde{c}(ik)\hat{u}(k)$ by  \eqref{per}.

 Taking Fourier series on both sides of \eqref{eq1} we obtain
$$
[-k^2M-A  -b_kB-ik\Lambda-ikc_kI-\gamma I]\hat{u}(k)=\hat{f}(k), \quad
k\in\mathbb{Z}.
$$
\end{proof}

When \eqref{eP} is $\mathcal{Y}$ well-posed, the map
$\mathcal{S}:\mathcal{Y}\to\mathcal{Y}$, $f\mapsto u$ where $u$ is the unique
strong solution, is linear. We adopt the following definition of well-posedness.

\begin{definition}\label{DefWell} \rm
We say that  \eqref{eP} is $\mathcal{Y}$-well-posed, if for each
$f\in  \mathcal{Y}$, there exists a unique  strong $\mathcal{Y}$-solution
 $u$ of \eqref{eP} which depends continuously on
$f$ in the sense that the operator $\mathcal{S}:\mathcal{Y}\to \mathcal{Y}$
defined by $\mathcal{S}(f)=u$ where $u$ is the unique
strong $\mathcal{Y}$-solution of \eqref{eP} is continuous.
\end{definition}

\begin{remark} \label{WD} \rm
We note that, according to Section 2,  \cite{AB,AB2,BK}, all the spaces of
vector-valued functions
$\mathcal{Y}$ concerned in this paper are continuously embedded in
$L^1(0,2\pi,X)$.
It follows that: If $f_n \to f$ in $\mathcal{Y}$, then $f_n\to f$ in
$L^1(0,2\pi,X)$ and
consequently for each $k\in\mathbb{Z}$,
$\lim_{n\to\infty}\hat{f_n}(k)=f(k)$ in $X$.
\end{remark}

 Our definition imposes an additional condition to that given in the previous
works such as \cite{Bu2}, \cite{LP} that allows us to establish the following
characterization of well-posed of \eqref{eP} in terms of Fourier multipliers.
Actually, the above definition stems from the Hadamard concept of well-posedness
 in partial differential equations. We refer for example to  \cite{FAT} and
\cite{ABHN} for the presentation of this fundamental concept.

\begin{theorem}\label{t2}
 Let $X$ be a Banach space and $A$, $B$, $\Lambda$, $M$
be closed linear operators in $X$ such that $
D(A)\cap D(B)\subset D(\Lambda)\cap D(M)$.  Suppose that $\gamma$ is a constant, $b,c\in L^1(\mathbb{R}_+)$,
and  consider  $b_k$, $c_k$ as in \eqref{eq2}.
Then the following assertions are equivalent.
\begin{itemize}
 \item[(i)] \eqref{eP} is $\mathcal{Y}$-well-posed.

 \item[(ii)] $i\mathbb{Z}\subset \rho_{\Lambda, M,\tilde{b},\tilde{c}}(A,B)$
and $(k^2MN_k)_{k\in\mathbb{Z}}$, $(BN_k)_{k\in\mathbb{Z}}$,
$(k\Lambda N_k)_{k\in\mathbb{Z}}$,
  $(kN_k)_{k\in\mathbb{Z}}$ are $\mathcal{Y}$-Fourier multipliers, where
$$
N_k=[k^2M+A  +b_kB+ik\Lambda+ikc_kI+\gamma I]^{-1}
$$
\end{itemize}
In this case the following maximal regularity property
holds: The unique strong $\mathcal{Y}$-solution $u$ is such that  $Au$,
 $b*Bu$, $\Lambda u$, $\Lambda u'$, $c*u$, $\frac{d}{dt}(c*u)$, $Mu$,
$Mu'$ and $(Mu')'$ all belong to $\mathcal{Y}$ and there exists a constant $C>0$
 independent of $f\in \mathcal{Y}$ such that
\begin{align*}
&\|u\|_\mathcal{Y}
+\|Au\|_\mathcal{Y}+\|b*Bu\|_\mathcal{Y}
+\|\Lambda u\|_\mathcal{Y}+\|\Lambda
u'\|_\mathcal{Y}+\|c*u\|_\mathcal{Y}\\
&+\|\frac{d}{dt} (c*u)\|_\mathcal{Y}+\|Mu\|_\mathcal{Y}
 +\|Mu'\|_\mathcal{Y}+\|(Mu')'\|_\mathcal{Y}\leq C\|f\|_\mathcal{Y}
\end{align*}
\end{theorem}

\begin{proof}
 (i) $\Rightarrow$ (ii). Let $k\in\mathbb{Z}$ and $y\in X$. Define
$f(t)=e^{ikt}y$. Then $\hat{f}(k)=y$. By assumption, there exists a unique
strong $\mathcal{Y}$-solution $u$ of \eqref{eP}. By Lemma \ref{Fourier}, we have
that for all $k\in\mathbb{Z}$,
$$
[-k^2M-A -b_kB-i k\Lambda-ikc_kI-\gamma I]\hat{u}(k)=y
$$
It follows that $$[-k^2M-A-b_kB-i k\Lambda-ikc_kI-\gamma I]$$ is surjective for each
$k\in\mathbb{Z}$.
Next we prove that for each $k\in\mathbb{Z}$,
$$
[-k^2M-A  -b_kB-ik\Lambda-ikc_kI-\gamma I]
$$
is injective. Let $x\in D(A)\cap D(B)$ such that
\begin{equation}\label{injec}
[-k^2M-A  -b_kB-ik\Lambda-ikc_kI-\gamma I]x=0
\end{equation}
Define $u(t)=e^{ikt}x$ when $t\in[0,2\pi]$.
Then $\hat{u}(k)=x$ and $\hat{u}(n)=0$ for all
$n\in\mathbb{Z}$, $n\neq k$. By \eqref{injec} we have
 \begin{align*}
\widehat{(Mu')'}(n)-\widehat{\Lambda
u'}(n)-\widehat{\frac{d}{dt}(c*u)}(n)&=\gamma
\hat{u}(n)+ \widehat{Au}(n)+\widehat{(b*Bu)}(n),
\end{align*}
for all $n\in\mathbb{Z}$.  From uniqueness theorem of Fourier coefficients, we
conclude that $u$ satisfies
\[
(Mu')'(t)-\Lambda u'(t)-\frac{d}{dt}(c*u)(t)
 =\gamma u(t)+ Aw(t)+ (b*Bu)(t)
\]
 for almost all $t\in[0,2\pi]$. Thus $u$ is a strong
$\mathcal{Y}$-solution of \eqref{eP}
with    $f=0$. We obtain $x=0$ by the
uniqueness assumption. We have shown that
$$
[-k^2M-A  -b_kB-ik\Lambda-ikc_kI-\gamma I]
$$
is injective for each $k\in\mathbb{Z}$.  Now
we show that
$$
N_k=[k^2M+A  +b_kB+ik\Lambda+ikc_kI+\gamma I]^{-1}\in\mathcal{L}(X)
$$
Let $k\in\mathbb{Z}$ and $(x_n)_{n\in\mathbb{N}}$ be a sequence in $X$ such that
$x_n\to x$.
For each $n\in\mathbb{N}$ we define $f_n(t)=e^{ikt}x_n$ and $f(t)=e^{ikt}x$.
Then $f_n,f\in \mathcal{Y}$,
 for every $n\in\mathbb{N}$ and $f_n\to f$ in $\mathcal{Y}$. Since
\eqref{eP} is $\mathcal{Y}$-well-posed,
 for each $f_n, f\in\mathcal{Y}$ there exists a unique strong
$\mathcal{Y}$-solution $\mathcal{S}(f_n)=u_n$, $\mathcal{S}(f)=u$.
Since $f_n\to f$ in $\mathcal{Y}$, we have
  $u_n\to u$ in $\mathcal{Y}$ by continuity of
$\mathcal{S}$. Therefore $\hat{u}_n(k)\to \hat{u}(k)$  by Remark \ref{WD}.
 Since
$$
-k^2M-A  -b_kB-ik\Lambda-ikc_kI-\gamma I
$$
is bijective, we obtain
  $\hat{u}_n(k)=-N_kx_n, \hat{u}(k)=-N_kx$ by Lemma \ref{Fourier}; then
$N_kx_n\to N_kx$.
  Thus by the Closed Graph Theorem, $N_k\in\mathcal{L}(X)$. Thus
$i\mathbb{Z}\subset \rho_{\Lambda,M,\tilde{b},\tilde{c}}(A,B)$.

We  now set for  each $k\in\mathbb{Z}$:
\begin{gather*}
M_k=k^2MN_k\quad
B_k=AN_k\\
S_k=BN_k\quad
H_k=kN_k.
\end{gather*}
  Next we show that $(M_k)_{k\in\mathbb{Z}}$, $(B_k)_{k\in\mathbb{Z}}$,
$(S_k)_{k\in\mathbb{Z}}$,   and $(H_k)_{k\in\mathbb{Z}}$ are
$\mathcal{Y}$-Fourier multipliers.
  Since $N_k\in\mathcal{L}(X)$, $B$, $\Lambda$, $M$ are closed,
$M_k$, $B_k$, $H_k$ and $S_k$
  are bounded for all $k\in\mathbb{Z}$. Now let $f\in\mathcal{Y}$, then there
exists a strong $\mathcal{Y}$-solution $u$ of \eqref{eP}. Then
$\hat{u}(k)=-N_k\hat{f}(k)$ for all $k\in\mathbb{Z}$ by Lemma \ref{Fourier}.
Therefore
  $$
\hat{u}(k)\in D(A)\cap D(B)\subset D(\Lambda)\cap D(M),
$$
for all $k\in\mathbb{Z}$.   Since $B$ is closed,
 \begin{equation*}
\widehat{Bu}(k)=B\hat{u}(k)=-BN_k\hat{f}(k)=-B_k\hat{f}(k)
\end{equation*}
for all $k\in\mathbb{Z}$ by \cite[Lemma 3.1]{AB}.
Since $\Lambda$, $M$ are closed, $u\in\mathcal{Y}^{[1]}_{\rm per}$,
$u'\in \mathcal{Y}(D(\Lambda))\cap\mathcal{Y}(D(M))$, and $Mu'\in
\mathcal{Y}^{[1]}_{\rm per}$, we have
\begin{gather*}
\widehat{u}'(k)=ik\hat{u}(k)=-ikN_k\hat{f}(k)=-iH_k\hat{f}(k), \\
\widehat{\Lambda u'}(k)=\Lambda \widehat{u}'(k)
=ik\Lambda\hat{u}(k)=-ik\Lambda N_k\hat{f}(k)=-iS_k\hat{f}(k), \\
\widehat{(Mu')'}(k)=ik\widehat{Mu'}(k)=ikM\widehat{u}'(k)=-k^2M\hat{u}(k)
=k^2MN_k\hat{f}(k)=M_k\hat{f}(k)
\end{gather*}
for all $k\in\mathbb{Z}$ by \eqref{per} and \cite[Lemma 3.1]{AB}.
It follows that  $(M_k)_{k\in\mathbb{Z}}$, $(B_k)_{k\in\mathbb{Z}}$,
$(S_k)_{k\in\mathbb{Z}}$,
 and $(H_k)_{k\in\mathbb{Z}}$ are $\mathcal{Y}$-Fourier multipliers.
Therefore the implication (i) $\Rightarrow$ (ii) is true.
\smallskip


 (ii) $\Rightarrow$ (i). Since
$$
k^2MN_k+AN_k+b_kBN_k+ik\Lambda N_k+ikc_kN_k+\gamma N_k =I,
$$
we have
$$
AN_k=I-\left(k^2MN_k+AN_k +b_kBN_k+ikc_kN_k+\gamma N_k\right)
$$
for each $k\in\mathbb{Z}$.  Therefore,  $(AN_k)_{k\in\mathbb{Z}}$ is a
$\mathcal{Y}$-Fourier multiplier by Remarks \ref{R4}, \ref{R5}, and \ref{R3}.
 Since $(k^2MN_k)_{k\in\mathbb{Z}}$, $(k\Lambda N_k)_{k\in\mathbb{Z}}$,
$(kN_k)_{k\in\mathbb{Z}}$, $(BN_k)_{k\in\mathbb{Z}}$, and
$(AN_k)_{k\in\mathbb{Z}}$
are  $\mathcal{Y}$-Fourier multipliers, it follows that
$(N_k)_{k\in\mathbb{Z}}$,
$(ikc_kN_k)_{k\in\mathbb{Z}}$,
$(c_kN_k)_{k\in\mathbb{Z}}$, $(ikN_k)_{k\in\mathbb{Z}}$,
$(ik\Lambda N_k)_{k\in\mathbb{Z}}$, $(\Lambda N_k)_{k\in\mathbb{Z}}$,
$(-k^2MN_k)_{k\in\mathbb{Z}}$ $(ikMN_k)_{k\in\mathbb{Z}}$, and
$(MN_k)_{k\in\mathbb{Z}}$ are also
$\mathcal{Y}$-Fourier
multipliers again by Remarks \ref{R4}, \ref{R5}, and \ref{R3}.
From the fact that $(AN_k)_{k\in\mathbb{Z}}$, $(BN_k)_{k\in\mathbb{Z}}$,
 $(\Lambda N_k)_{k\in\mathbb{Z}}$, $(MN_k)_{k\in\mathbb{Z}}$, and
$(c_kN_k)_{k\in\mathbb{Z}}$ are $\mathcal{Y}$-Fourier multipliers, then for all
$f\in \mathcal{Y}$,
 we conclude that exist $u$, $v_1$, $v_2$, $v_3$, $v_4$,  and $v_5\in
\mathcal{Y}$ such that
 \begin{equation}\label{D}
 \hat{u}(k)=N_k\hat{f}(k),
\end{equation}
and
\begin{equation}\label{Y}
\begin{gathered}
  \hat{v}_1(k)= AN_k\hat{f}(k)=A\hat{u}(k)=\widehat{Au}(k),\\
  \hat{v}_2(k)= BN_k\hat{f}(k)=B\hat{u}(k)=\widehat{Bu}(k),\\
 \hat{v}_3(k)= \Lambda N_k\hat{f}(k)=\Lambda\hat{u}(k)
=\widehat{\Lambda u}(k),\\
 \hat{v}_4(k)= MN_k\hat{f}(k)=M\hat{u}(k)=\widehat{Mu}(k),\\
 \hat{v}_5(k)= c_kN_k\hat{f}(k)=c_k\hat{u}(k)=\widehat{c*u}(k),
 \end{gathered}
\end{equation}
 for all $k\in\mathbb{Z}$ by the closedness of $A$, $B$,
$\Lambda$, $M$, and \eqref{eq3.3}.
Since $i\mathbb{Z}\subset\rho_{\Lambda,M,\tilde{b},\tilde{c}}(A,B)$,
it follows that
   $$
\hat{u}(k)\in D(A)\cap D(B)\subset D(\Lambda)\cap D(M),
$$
for all $k\in\mathbb{Z}$ by \eqref{D}.
  Since $A$, $B$, $\Lambda$, and $M$ are closed,
$$
u(t)\in D(A)\cap D(B)
$$
and $Au(t)=v_1(t)$, $Bu(t)=v_2(t)$, $\Lambda u(t)=v_3(t)$,
 $Mu(t)=v_4(t)$ and $(c*u)(t)=v_5(t)$ a.e.\
 $t\in[0,2\pi]$ by \eqref{Y} and
\cite[Lemma 3.1]{AB} (here we also use the fact  that $\mathcal{Y}\subset
L^p(0,2\pi,X)$). Therefore
$$
u\in \mathcal{Y}(D(A))\cap\mathcal{Y}(D(B)),
$$
and $c*u$, $\Lambda u$, $Mu\in\mathcal{Y}$. Since $(ikN_k)_{k\in\mathbb{Z}}$
is a $\mathcal{Y}$-Fourier multiplier,  there exists
$v_6\in\mathcal{Y}$ such that
 \begin{equation}\label{Y3}
  \hat{v}_6(k)=ikN_k\hat{f}(k)=ik\hat{u}(k)\in D(\Lambda)\cap D(M).
\end{equation}
for all $k\in\mathbb{Z}$. Therefore by \eqref{per} and
\eqref{Y3}, $u\in\mathcal{Y}^{[1]}_{\rm per}$, $\widehat{u}'(k)=ik\hat{u}(k)$ and
$$
\widehat{u}'(k)\in D(\Lambda)\cap D(M),
$$
for all $k\in\mathbb{Z}$.
Since $(ik\Lambda N_k)_{k\in\mathbb{Z}}$ and $(ikMN_k)_{k\in\mathbb{Z}}$ are
$\mathcal{Y}$-Fourier multipliers,  there exist $v_7$,
$v_9\in\mathcal{Y}$ such that
 \begin{equation}\label{Y1}
\begin{gathered}
 \hat{v_7}(k)=ik\Lambda N_k\hat{f}(k)=\Lambda(ik\hat{u}(k))
=\Lambda\widehat{u}'(k)=\widehat{\Lambda u'}(k),\\
\hat{v}_8(k)=ikMN_k\hat{f}(k)=M(ik\hat{u}(k))
=M\widehat{u}'(k)=\widehat{Mu'}(k),
 \end{gathered}
\end{equation}
for al $k\in\mathbb{Z}$.
  Since  $\Lambda$ and $M$ are closed,
$$
u'(t)\in D(\Lambda)\cap D(M)
$$
and $\Lambda u'(t)=v_7(t)$, $Mu'(t)=v_8(t)$ a.e.\
 $t\in[0,2\pi]$ by \eqref{Y1} and
\cite[Lemma 3.1]{AB} (here again, we also use the fact that
 $\mathcal{Y}\subset L^p(0,2\pi,X)$). Therefore
$$
u'\in\mathcal{Y}( D(\Lambda))\cap \mathcal{Y}(D(M)).
$$
Since  $(-k^2MN_k)_{k\in\mathbb{Z}}$ is a
$\mathcal{Y}$-Fourier multiplier, there exists $v_9\in\mathcal{Y}$ such that
 \begin{equation}\label{Y2}
\hat{v}_9(k)=-k^2kMN_k\hat{f}(k)=ik(ikM\hat{u}(k))
=ikM\widehat{u}'(k)=ik\widehat{Mu}'(k),
\end{equation}
for al $k\in\mathbb{Z}$ by  \eqref{Y1}.
Then $Mu'\in\mathcal{Y}^{[1]}_{\rm per}$. Since
$(ikc_kN_k)_{k\in\mathbb{Z}}$ is a $\mathcal{Y}$-Fourier multiplier, there
exists $v_{10}\in\mathcal{Y}$ such that
 \begin{equation}
  \hat{v}_{10}(k)=ikc_kN_k\hat{f}(k)=ikc_k\hat{u}(k)=ik\widehat{(c*u)}(k),
\end{equation}
for al $k\in\mathbb{Z}$ by \eqref{Y}. Then
$c*u\in\mathcal{Y}^{[1]}_{\rm per}$ by \eqref{per}. Since
$\hat{u}(k)=N_k\hat{f}(k)$, we have
$$
[-k^2M-A  -b_kB-i k\Lambda-ikc_kI-\gamma I](-\hat{u}(k))=\hat{f}(k),
$$
this means that
\[
(\widehat{M}w')'(k)-\widehat{\Lambda w'}(k)-\widehat{\frac{d}{dt}(c*w)}(k)
=\gamma \hat{w}(k)+\widehat{Aw}(k)+\widehat{(b*Bw)}(k)+\hat{f}(k),
\]
for all $k\in\mathbb{Z}$ where $w=-u$.
From the uniqueness theorem of Fourier coefficients,
we conclude that $w$ satisfies
 \[
(Mw')'(t)-\Lambda w'(t)-\frac{d}{dt}(c*w)(t)
 =\gamma w(t)+ Aw(t)+(b*Bw)(t) +f(t)
 \]
 for almost all $t\in[0,2\pi]$. Thus $w$ is a strong
$\mathcal{Y}$-solution of \eqref{eP}. To prove uniqueness, let $u$ be a strong
$\mathcal{Y}$-solution of \eqref{eP} with $f=0$.
  Then
 $$
[-k^2M-A -b_kB-i k\Lambda-ikc_kI-\gamma I]\hat{u}(k)=0
$$
for all  $k\in\mathbb{Z}$ by Lemma \ref{Fourier}.
Since $ik\in \rho_{\Lambda,M,\tilde{b},\tilde{c}}(A,B)$ for
all $k\in\mathbb{Z}$, it follows that $\hat{u}(k)=0$ for
all $k\in\mathbb{Z}$. From the uniqueness theorem of Fourier coefficients we
have that $u=0$. Now we show  the continuous dependence of $u$ on $f$.
Let $f\in \mathcal{Y}$, then the unique
strong $\mathcal{Y}$-solution of \eqref{eP}, $u$, is such that
$\hat{u}(k)=-N_k\hat{f}(k)$ for
all $k\in\mathbb{Z}$ by Lemma \ref{Fourier} and
$i\mathbb{Z}\subset \rho_{\Lambda, M,\tilde{b},\tilde{c}}(A,B)$.
Since $N_k$ is a
$\mathcal{Y}$-Fourier multiplier,  there exists a bounded linear operator
$T\in\mathcal{L}(\mathcal{Y},\mathcal{Y})$ such that
$\widehat{Tf}(k)=\hat{u}(k)$ for all  $k\in\mathbb{Z}$ by Remark \ref{R5}. Then
$Tf=u$, so $u$
depends continuously on $f$.

The last assertion of the theorem is a direct consequence of the fact that
$Au$, $b*Bu$, $\Lambda u$, $\Lambda u'$, $c*u$, $\frac{d}{dt}(c*u)$, $Mu$,
$Mu'$ and $(Mu')'\in \mathcal{Y}$ are defined through  the following operator
valued Fourier multipliers
$(-AN_k)_{k\in\mathbb{Z}}$,  $(-b_kBN_k)_{k\in\mathbb{Z}}$,
$(-\Lambda N_k)_{k\in\mathbb{Z}}$, $(-k\Lambda N_k)_{k\in\mathbb{Z}}$,
$(-c_kN_k)_{k\in\mathbb{Z}}$, $(-kc_kN_k)_{k\in\mathbb{Z}}$,
$(-MN_k)_{k\in\mathbb{Z}}$, $(kMN_k)_{k\in\mathbb{Z}}$,
$(k^2MN_k)_{k\in\mathbb{Z}}$ (here we use the Remarks \ref{R4},
\ref{R5}, and \ref{R3}).
\end{proof}

The last assertion of the previous theorem is known
as the {\it maximal regularity} property for \eqref{eP}.

\begin{remark}\label{rmk3.7} \rm
 We can construct the solution $ u(\cdot)$ given by the above theorems
using Proposition \ref{Fejer's} and
the fact that $\mathcal{Y}$ is continuously embedded in  $L^p(0,2\pi; X) $.
More precisely,
\begin{equation}
u(\cdot) = -\lim_{n\to\infty} \frac{1}{n+1} \sum_{m=0}^n
\sum_{k= -m}^m  e_k (\cdot) N_k \hat f(k),
\end{equation}
 with convergence in $L^p(0,2\pi;X)$.
\end{remark}

 \begin{remark}\label{Auto} \rm
 If at most one operator of those that appear in \eqref{eq1} is unbounded,
then the additional condition in our definition of well-posedness is obtained
automatically. In that case the operators
$$
-k^2M-A-b_kB-ik\Lambda-ikc_kI-\gamma I
$$
are closed for all $k\in\mathbb{Z}$ and once we show that they are bijective,
continuity
follows from the Closed Graph Theorem.
\end{remark}

\section{Concrete characterization on periodic Lebesgue, Besov
 and Triebel-Lizorkin spaces}

In this section, we give concrete conditions that allow us to apply
Theorem \ref{t2}. Specifically we obtain conditions under which the sequences
$(k^2MN_k)_{k\in\mathbb{Z}}$,
$(BN_k)_{k\in\mathbb{Z}}$, $(k\Lambda N_k)_{k\in\mathbb{Z}}$,
 and $(kN_k)_{k\in\mathbb{Z}}$  are Fourier multipliers in the scale of spaces
under consideration  by use of  the operator valued multiplier
theorems established in  \cite{AB1,AB,AB2,BK}.
Versions of the multiplier theorems on the real line can be found
in \cite{Am1,GW1,GW2} (the reference  \cite{GW2} contains concrete
criteria for $R$-boundedness of operator families),
\cite{W1,W2}. The $L^p$-case is much different from the other scales of
spaces in that it involves the notion of $R$-boundedness and one has to restrict
consideration to $UMD$ Banach spaces. Fortunately, many Banach spaces, for
example $L^p(\Omega,\mu)$, $1<p<\infty$ are $UMD$ spaces. In addition, the
$R$-boundedness condition holds for resolvents of many classical operators in
the analysis of partial differential equations of evolution type (see for example
 Kunstmann-Weis \cite{KW} and Girardi-Weis \cite{GW2}).

Let $\{a_k:k\in\mathbb{Z}\}\subset\mathbb{C}$ be a scalar sequence, we
denote by $\Delta a_k= a_{k+1}-a_k$. It is obvious that $\Delta$ is linear:
$\Delta(a_k+b_k)=\Delta a_k+\Delta b_k$;
$\Delta(\lambda a_k)=\lambda \Delta a_k$. Another property used frequently is
$\Delta( a_k b_k)=a_k\Delta b_k+(\Delta a_k)b_k$. Define
$\Delta^{n+1} \alpha_k=\Delta\Delta^n a_k$ for all
$n\in\mathbb{N}$, $k\in\mathbb{Z}$.
$\Delta^n$ is the $n^{th}$ order difference operator:
$$
\Delta^n a_k=\sum_{j=0}^n(-1)^{n-j}\binom{n}{j}a_{k+j}.
$$
We will use the following hypotheses:
\begin{itemize}
\item[(H0)] $\{a_k:k\in\mathbb{Z}\}$ is bounded.

\item[(H1)] $\{a_k:k\in\mathbb{Z}\}$, $\{k\Delta a_k:k\in\mathbb{Z}\}$ are bounded.

\item[(H2)] $\{a_k:k\in\mathbb{Z}\}$, $\{k\Delta a_k:k\in\mathbb{Z}\}$,
$\{k^2\Delta^2 a_k:k\in\mathbb{Z}\}$ are bounded.

\item[(H3)] $\{a_k:k\in\mathbb{Z}\}$, $\{k\Delta a_k:k\in\mathbb{Z}\}$,
$\{k^2\Delta^2 a_k:k\in\mathbb{Z}\}$,
$\{k^3\Delta^3 a_k:k\in\mathbb{Z}\}$
are bounded.
\end{itemize}
Clearly (H0) is weaker than (H1) which in turn is weaker than (H2), and the
latter is weaker than    (H3). In our cases (H0) is obtained automatically from the
Riemann-Lebesgue Lemma. The condition (H1)
will be used for $L^p$ well-posedness, while (H2) and (H3) are needed for
Besov spaces and Triebel-Lizorkin spaces respectively. Some variations to this
rule will occur  when the Banach space $X$ satisfies a special geometric
property such as being $UMD$ or having nontrivial Fourier type.

Examples of functions $a(t)$ such that $a_k=\tilde{a}(ik)$ satisfies  (H3) are
$a(t)=Ce^{-\omega t}t^\nu$ where $\omega>0$, $\nu>-1$
and $C$ is a   constant. We give a class of functions which discriminate between
the above conditions in the following example.

\begin{example} \label{examp4.1} \rm
Let $\beta>0$, $\omega>0$,   $c\in\mathbb{R}$ and consider  the family of
functions
$$
b(t)=\begin{cases}
0 &\text{if }0<t\leq\beta,\\
Ce^{-\omega t}(t-\beta)^\nu &\text{if }t>\beta
\end{cases}
$$
$b_k=\tilde{b}(ik)$.  Then
\begin{itemize}
 \item[(a)] For   $-1<\nu<0$ and $\beta\notin 2\pi\mathbb{Z}$,  $b_k$
satisfies (H0) but not (H1).

\item[(b)] For  $0\leq \nu<1$ and $\beta\notin 2\pi\mathbb{Z}$,
 $b_k$ satisfies (H1) but not (H2).

\item[(c)] For  $1\leq \nu<2$ and $\beta\notin 2\pi\mathbb{Z}$,
$b_k$ satisfies (H2) but not (H3).

\item[(d)] For  $\nu\geq2$ or $\beta\in 2\pi\mathbb{Z}$,  $b_k$ satisfies
(H3).
\end{itemize}
\end{example}
In the following theorem, we characterize well-posedness in the
vector-valued $L^p$ spaces.

\begin{theorem}\label{tfmp}
 Let $X$ be a $UMD$ Banach space, $1< p<\infty$ and  $A$, $B$, $\Lambda$,
$M$ be closed linear operators in $X$ such that
$D(A)\cap D(B)\subset D(\Lambda)\cap D(M)$. Suppose that $\gamma$ is a
constant, $b,c\in L^1(\mathbb{R}_+)$, and consider
$b_k$, $c_k$  as in \eqref{eq2} such that $\{b_k:k\in\mathbb{Z}\}$ and
$\{c_k:k\in\mathbb{Z}\}$ satisfy    (H1).
Then the following assertions are equivalent.
\begin{itemize}
 \item[(i)] \eqref{eP} is $L^p$-well-posed.
 \item[(ii)] $i\mathbb{Z}\subset \rho_{\Lambda,M,\tilde{b},\tilde{c}}(A,B)$
and $\{k^2MN_k:k\in\mathbb{Z}\}$, $\{BN_k:k\in\mathbb{Z}\}$, $\{k\Lambda
N_k:k\in\mathbb{Z}\}$,
 and $\{kN_k:k\in\mathbb{Z}\}$ are $R$-bounded, where
$$
N_k=[k^2M+A  +b_kB+ik\Lambda+ikc_kI+\gamma I]^{-1}
$$
\end{itemize}
\end{theorem}


\begin{proof}
(i) $\Rightarrow$ (ii) Assume that
\eqref{eP} is $L^p$-well-posed. Then by Theorem  \ref{t2},
$i\mathbb{Z}\subset \rho_{\Lambda, M,\tilde{b},\tilde{c}}(A,B)$ and
$(k^2MN_k)_{k\in\mathbb{Z}}$, $(BN_k)_{k\in\mathbb{Z}}$,
 $(k\Lambda N_k)_{k\in\mathbb{Z}}$, and $(kN_k)_{k\in\mathbb{Z}}$ are
$L^p$-Fourier multipliers. The
$R$-boundedness of $\{k^2MN_k:k\in\mathbb{Z}\}$, $\{BN_k:k\in\mathbb{Z}\}$,
$(k\Lambda N_k)_{k\in\mathbb{Z}}$,
 and $\{kN_k:k\in\mathbb{Z}\}$  now
follows from \cite[Proposition 1.11]{AB}.
\smallskip

(ii) $\Rightarrow$ (i) In view of Theorem \ref{t2}, it suffices
to show that $(k^2MN_k)_{k\in\mathbb{Z}}$, $(BN_k)_{k\in\mathbb{Z}}$,
$(k\Lambda N_k)_{k\in\mathbb{Z}}$,
 and $(kN_k)_{k\in\mathbb{Z}}$ are $L^p$-Fourier multipliers.

 For each $k\in\mathbb{Z}$ we define $M_k=k^2MN_k$, $B_k=BN_k$,
$H_k=kN_k$ and $S_k=k\Lambda N_k$.
These operators are bounded because $i\mathbb{Z}\subset
\rho_{\Lambda, M,\tilde{b},\tilde{c}}(A,B)$. Since
$\{kN_k:k\in\mathbb{Z}\}$ is $R$-bounded,  $\{N_k:k\in\mathbb{Z}\}$ is
$R$-bounded by Remark \ref{R1}.
 We observe that
\begin{align*}
N_{k+1}^{-1}N_k
&=\left[(k+1)^2M+A+b_{k+1}B
 +i(k+1)\Lambda+i(k+1)c_{k+1}
I+\gamma I\right]N_k\\
&=[N_k^{-1}+(2k+1)M+\Delta b_kB+ik\Delta c_kI+ic_{k+1}
I+i\Lambda]N_k\\
&=I+(2k+1)MN_k+\Delta b_kBN_k+ik\Delta c_kN_k+ic_{k+1}
N_k+i\Lambda N_k\\
&=I+\frac{2k+1}{k^
2}M_k+\Delta b_kB_k+i\Delta
c_kH_k+\frac{ic_{k+1}}{k}H_k+\frac{i}{k}S_k
\end{align*}
for all $k\in\mathbb{Z}$, $k\neq0$. If we define
\begin{equation} \label{Tk}
T_k=\frac{2k+1}{k^2}M_k+\Delta b_kB_k+i\Delta c_kH_k+i\frac
{c_{k+1}}{k} H_k+\frac{i}{k}S_k,
\end{equation}
then $N_{k+1}^{-1}N_k=I+T_k$ for all $k\in\mathbb{Z}$, $k\neq0$. Define
\begin{align*}
Q_k&=-kT_k\\
&=-[\frac{2k+1}{k}M_k+k\Delta b_kB_k+ik\Delta c_kH_k+{i} c_ {k+1 }H_k+iS_k].
\end{align*}
for all $k\in\mathbb{Z}$, $k\neq0$. Since
$\{b_k:k\in\mathbb{Z}\}$   and $\{c_k:k\in\mathbb{Z}\}$ satisfy    (H1),
  $\{Q_k:k\in\mathbb{Z}\}$ is
$R$-bounded by  Remark \ref{R1} and \ref{R3}.
 We observe that
 \begin{align*}
 k\Delta N_k
&=k(N_{k+1}-N_k)
  =kN_{k+1}(I-N_{k+1}^ {-1}N_k)\\
&=kN_{k+1}[I-(I+T_k)]
 =kN_{k+1}[-T_k]
  =N_{k+1}Q_k
 \end{align*}
Thus, we have
 \begin{gather*}
 k\Delta B_k=k\Delta (BN_k)=B(k\Delta N_k)=BN_{k+1}Q_k=B_{k+1}Q_k, \\
\begin{aligned}
 k\Delta H_k
&=k[(k+1)N_{k+1}-k N_k]\\
&=k[(k+1) N_{k+1}-(k+1)N_{k}+(k+1) N_{k}-k\Lambda N_k]\\
&=k[(k+1)\Delta N_k+ N_k]
=(k+1)(k\Delta N_k)+k N_k\\
&=(k+1) N_{k+1}Q_k+kN_k
=H_{k+1}Q_k+H_k,
\end{aligned} \\
 \begin{aligned}
 k\Delta S_k&=\Lambda(k[(k+1) N_{k+1}-k N_k])\\
&=\Lambda[H_{k+1}Q_k+H_k]
=S_{k+1}Q_k+S_k,
\end{aligned} \\
\begin{aligned}
k\Delta M_k
&=k((k+1)^2MN_{k+1}-k^2MN_k)\\
&=k((k+1)^2MN_{k+1}-(k+1)^2MN_{k}+(k+1)^2MN_{k}-k^2MN_k)\\
&=k[(k+1)^2M\Delta N_k+(2k+1)MN_{k}\\
&=(k+1)^2M[k\Delta N_k]+k(2k+1)MN_{k}\\
&=(k+1)^2MN_{k+1}Q_k+k(2k+1)MN_{k}\\
&=M_{k+1}Q_k+\frac{2k+1}{k}M_{k}
\end{aligned}
\end{gather*}
for all $k\in\mathbb{Z}$, $k\neq0$. Then
$\{k\Delta B_k:k\in\mathbb{Z}\}$,
 $\{k\Delta H_k:k\in\mathbb{Z}\}$,
 $\{k\Delta S_k:k\in\mathbb{Z}\}$,
 and $\{k\Delta M_k:k\in\mathbb{Z}\}$ are $R$-bounded by  Remark \ref{R1}.
Therefore by \cite[Theorem 1.3]{AB} we
obtain that $(B_k)_{k\in\mathbb{Z}}$, $(H_k)_{k\in\mathbb{Z}}$,
 $(S_k)_{k\in\mathbb{Z}}$,
 and $(M_k)_{k\in\mathbb{Z}}$ are $L^p$-Fourier multipliers.
\end{proof}


From the proof of  Theorem \ref{tfmp}, we deduce the following result for
$B^s_{pq}$-solutions in case $X$ has nontrivial Fourier type.

\begin{theorem}\label{tfm}
Let $X$ be a Banach space with nontrivial Fourier type
and  $A$, $B$, $\Lambda$, $M$
be closed linear operators in $X$ such that
$D(A)\cap D(B)\subset D(\Lambda)\cap D(M)$. Suppose that $\gamma$ is constant,
$b,c\in L^1(\mathbb{R}_+)$, and consider $b_k$, $c_k$ as in \eqref{eq2} such that
$(b_k)_{k\in\mathbb{Z}}$ and $(c_k)_{k\in\mathbb{Z}}$  satisfy  (H1).
 Then for $s>0$ and $1\le p$, $q\le \infty$, the
following are equivalent.
\begin{itemize}
 \item[(i)] \eqref{eP} is $B^s_{p,q}$-well-posed.
 \item[(ii)] $i\mathbb{Z}\subset \rho_{\Lambda,M,\tilde{b},\tilde{c}}(A,B)$
and $\{k^2MN_k:k\in\mathbb{Z}\}$, $\{BN_k:k\in\mathbb{Z}\}$, $\{k\Lambda
N_k:k\in\mathbb{Z}\}$,
 and $\{kN_k:k\in\mathbb{Z}\}$ are bounded, where
$$
N_k=[k^2M+A  +b_kB+ik\Lambda+ikc_kI+\gamma I]^{-1}
$$
\end{itemize}
\end{theorem}

\begin{proof}
 (i) $\Rightarrow$ (ii).  Assume that
\eqref{eP} is $B_{pq}^s$-well-posed. Then by Theorem  \ref{t2},
$i\mathbb{Z}\subset \rho_{\Lambda, M,\tilde{b},\tilde{c}}(A,B)$ and
$(k^2MN_k)_{k\in\mathbb{Z}}$,  $(BN_k)_{k\in\mathbb{Z}}$,
$(k\Lambda N_k)_{k\in\mathbb{Z}}$
 and $(kN_k)_{k\in\mathbb{Z}}$ are $B_{pq}^s$-Fourier multipliers. The
boundedness of $(k^2MN_k)_{k\in\mathbb{Z}}$,  $(BN_k)_{k\in\mathbb{Z}}$,
$(k\Lambda N_k)_{k\in\mathbb{Z}}$,
 and $(kN_k)_{k\in\mathbb{Z}}$   now follows from Remark \ref{R5}.
\smallskip

(ii) $\Rightarrow$ (i).  In view of Theorem \ref{t2}, it suffices
to show that $(k^2MN_k)_{k\in\mathbb{Z}}$,  $(BN_k)_{k\in\mathbb{Z}}$,
$(k\Lambda N_k)_{k\in\mathbb{Z}}$,
 and $(kN_k)_{k\in\mathbb{Z}}$ are $B_{pq}^s$-Fourier multipliers.  By
   \cite[Theorem 4.5]{AB2} the proof follows the same lines as that of the
preceding theorem.
\end{proof}

 We now consider the problem of  well-posedness in Besov spaces
$B^s_{pq}(0,2\pi,X)$ for arbitrary Banach spaces $X$. For
 this, assumption (H0) and (H1) are no longer sufficient. It is
 proved in \cite[Theorem 4.2]{AB2} that for any sequence
 $(M_k)_{k\in\mathbb{Z}}\subset \mathcal{L}(X)$, the so-called variational
Marcinkiewicz  condition; that is,
 \begin{equation}
 \sup_{k\in\mathbb{Z}}\|M_k\|+\sup_{j\ge 0}
 \Big(\sum_{2^j\le | k|<2^{j+1}}\|  \Delta M_k\|\Big)<\infty
 \end{equation}
 implies that $(M_k)_{k\in\mathbb{Z}}$ is a $B^s_{pq}$-Fourier multiplier if
and only if
 $1<p<\infty$ and $X$ is a $UMD$ space.

For   Banach spaces with nontrivial Fourier type, a  condition
which implies that $(M_k)_{k\in\mathbb{Z}}$ is a Fourier multiplier for the
scale $B^s_{p,q}$, $s\in\mathbb{R}$, $1\le p, q\le\infty$ is the Marcinkiewicz
condition of order one:
\begin{equation}\label{Mar1}
 \sup_{k\in\mathbb{Z}}(\| M_k\|+\|
 k\Delta M_k\|)<\infty,
 \end{equation}
see \cite[Theorem 4.5]{AB2}, which is used in the proof of Theorem \ref{tfm}.

For arbitrary Banach spaces, a Marcinkiewicz condition of order
two is needed, namely,
\begin{equation}\label{Mar2}
 \sup_{k\in\mathbb{Z}}(\| M_k\|+\|
 k\Delta M_k\|+k^2\| \Delta^2 M_k\|)<\infty,
 \end{equation}
see \cite[Theorem 4.5]{AB2}.
Our next result uses this condition to obtain maximal regularity
of \eqref{eP} when $X$ does not necessarily have nontrivial Fourier  type.

\begin{theorem}\label{tfmg}
Let $X$ be a Banach space and  $A$, $B$, $\Lambda$, $M$
be closed linear operators in $X$ such that  $D(A)\cap D(B)\subset
D(\Lambda)\cap D(M)$. Suppose that $\gamma$ is a constant,
$b,c\in L^1(\mathbb{R}_+)$, and consider $b_k$, $c_k$ as in \eqref{eq2} such that
 $(b_k)_{k\in\mathbb{Z}}$,
 and $(c_k)_{k\in\mathbb{Z}}$  satisfy  (H2). Then for
$s>0$ and $1\le p, \, q\le \infty$, the following statements are equivalent.
\begin{itemize}
 \item[(i)] \eqref{eP} is $B^s_{pq}$-well-posed.
 \item[(ii)] $i\mathbb{Z}\subset \rho_{\Lambda,M,\tilde{b},\tilde{c}}(A,B)$
and $\{k^2MN_k:k\in\mathbb{Z}\}$, $\{BN_k:k\in\mathbb{Z}\}$, $\{k\Lambda
N_k:k\in\mathbb{Z}\}$,
 and $\{kN_k:k\in\mathbb{Z}\}$ are bounded, where
$$N_k=[k^2M+A
 +b_kB+i k\Lambda+ikc_kI+\gamma I]^{-1}$$
\end{itemize}
\end{theorem}

\begin{proof} (i) $\Rightarrow$ (ii). Assume that
\eqref{eP} is $B^s_{pq}$-well-posed. Then by Theorem  \ref{t2},
$i\mathbb{Z}\subset \rho_{\Lambda, M,\tilde{b},\tilde{c}}(A,B)$ and
$(k^2MN_k)_{k\in\mathbb{Z}}$, $(BN_k)_{k\in\mathbb{Z}}$,
$(k\Lambda N_k)_{k\in\mathbb{Z}}$
 and $(kN_k)_{k\in\mathbb{Z}}$ are $B^s_{pq}$-Fourier multipliers. The
boundedness of $\{k^2MN_k:k\in\mathbb{Z}\}$,
$\{BN_k:k\in\mathbb{Z}\}$, $\{kN_k:k\in\mathbb{Z}\}$,
 and $\{kN_k:k\in\mathbb{Z}\}$ now
follows from Remark \ref{R5}.
\smallskip

(ii) $\Rightarrow$ (i). By Theorem \ref{t2}, it suffices
to show that the families $(k^2MN_k)_{k\in\mathbb{Z}}$, $(BN_k)_{k\in\mathbb{Z}}$,
$(k\Lambda N_k)_{k\in\mathbb{Z}}$,
 and $(kN_k)_{k\in\mathbb{Z}}$ are $B_{pq}^s$-Fourier multipliers. Let
$M_k=k^2MN_k$,  $B_k=BN_k$, $H_k=kN_k$,
and $S_k=k\Lambda N_k$.   Since (H2) implies (H1),  the verification of the
Marcinkiewicz condition of order one is similar
 to what was done in the proof of Theorem \ref{tfmp}.
 It remains to prove that
 $\sup_{k\in\mathbb{Z}}\|k^2\Delta^2 M_k\|<\infty$,
$\sup_{k\in\mathbb{Z}}\|k^2\Delta^2 B_k\|<\infty$,
$\sup_{k\in\mathbb{Z}}\|k^2\Delta^2 S_k\|<\infty$,  and
$\sup_{k\in\mathbb{Z}}\|k^2\Delta^2 H_k\|<\infty$.

We recall from the proof of Theorem \ref{tfmp} that the family
$(T_k)_{k\in\mathbb{Z}}$ defined through
$$
T_k=\frac{2k+1}{k^2}M_k+\Delta b_kB_k+i\Delta c_kH_k+i\frac
{c_{k+1 }}{k}H_k+i\frac{1}{k}S_k, \, k\neq0
$$
is such that
$N_{k+1}^{-1}N_k=I+T_k$, $Q_k=-kT_k$, $k\Delta
N_k=N_{k+1}Q_k$  for all
$k\in\mathbb{Z}$, $k\neq0$,  and $\{kT_k:k\in\mathbb{Z}\}$  is bounded.

We observe that
\begin{align*}
\Delta T_k&=\Delta(\frac{2k+1}{k^2}M_k)+\Delta(\Delta
b_k)B_k+i\Delta(\Delta(c_k)H_k)\\
&\quad +i\Delta(\frac {c_{k+1}}{k}H_k)+i\Delta(\frac{1}{k}S_k))
\end{align*}
However,
\begin{align*}
\Delta(\frac{2k+1}{k^2}M_k)
&=\frac{2k+3}{(k+1)^2}M_{k+1}-\frac{2k+1}{k^2}M_k\\
&=\frac{2k+3}{(k+1)^2}M_{k+1}-\frac{2k+3}{(k+1)^2}M_{k}+\frac{2k+3}{(k+1)^2}M_{
k}-\frac{2k+1}{k^2}M_k\\
&=\frac{2k+3}{(k+1)^2}\Delta M_k-\frac{2k^2+4k+1}{k^2(k+1)^2}M_k\\
&=\frac{2k+3}{k(k+1)^2}(k\Delta M_k)-\frac{2k^2+4k+1}{k^2(k+1)^2}M_k,
\end{align*}

\begin{align*}
\Delta(\frac{1}{k}S_k)&=\frac{1}{k+1}S_{k+1}-\frac{1}{k}S_k\\
 &=\frac{1}{k+1}S_{k+1}-\frac{1}{k+1}S_{k}+\frac{1}{k+1}S_{k}-\frac{1}{k}S_k\\
 &=\frac{1}{k+1}\Delta S_k-\frac{1}{k(k+1)}S_k\\
 &=\frac{1}{k(k+1)}(k\Delta S_k)-\frac{1}{k(k+1)}S_k,
 \end{align*}

 \begin{align*}
\Delta(\frac {c_{k+1}}{k} H_k)
&=\frac {c_{k+2}}{k+1}H_{k+1}-\frac {c_{k+1}}{k}H_k\\
 &=\frac {c_{k+2}}{k+1}H_{k+1}-\frac {c_{k+2}}{k+1}H_{k}+\frac
{c_{k+2}}{k+1}H_{k}-\frac {c_{k+2}}{k}H_{k}+\frac {c_{k+2}}{k}H_{k}-\frac
{c_{k+1 }}{k}H_k\\
 &=\frac {c_{k+2}}{k+1}\Delta H_k+\frac{\Delta
c_{k+1}}{k}H_k-\frac{c_{k+2}}{k(k+1)}H_k\\
 &=\frac {c_{k+2}}{k(k+1)}(k\Delta H_k)+\frac{(k+1)\Delta
c_{k+1}}{k(k+1)}H_k-\frac{c_{k+2}}{k(k+1)}H_k,
 \end{align*}

\begin{align*}
\Delta[k(\Delta b_k)B_k]
&=(\Delta b_{k+1})B_{k+1}-(\Delta b_k)B_k\\
&=(\Delta b_{k+1})B_{k+1}-(\Delta b_{k+1})B_{k}+(\Delta b_{k+1})B_{k}-(\Delta
b_k)B_k\\
&=(\Delta b_{k+1})\Delta B_{k}+(\Delta^2 b_{k})B_k\\
&=\frac{1}{k(k+1)}((k+1)\Delta b_{k+1})(k\Delta B_{k})+\frac{1}{k^2}(k^2\Delta^2
b_{k})B_k,
\end{align*}
and
\begin{align*}
\Delta((\Delta c_k)H_k))
&=(\Delta c_{k+1})H_{k+1}-(\Delta c_k)H_k\\
&=(\Delta c_{k+1})H_{k+1}-(\Delta c_{k+1})H_{k}+(\Delta c_{k+1})H_{k}-(\Delta
c_k)H_k\\
&=(\Delta c_{k+1})\Delta H_{k}+(\Delta^2 c_k)H_k\\
&=\frac{1}{k(k+1)}((k+1)\Delta c_{k+1})(k\Delta H_{k})+\frac{1}{k^2}(k^2\Delta^2
c_k)H_k
\end{align*}
for all $k\in\mathbb{Z}$, $k\neq 0,-1$. Since
$\{b_k:k\in\mathbb{Z}\}$ and
 $\{c_k:k\in\mathbb{Z}\}$ satisfy (H2), we have $(M_k)_{k\in\mathbb{Z}}$,
$(B_k)_{k\in\mathbb{Z}}$,  $(S_k)_{k\in\mathbb{Z}}$, $(H_k)_{k\in\mathbb{Z}}$
 satisfy the Marcinkiewicz condition of order one, and  $\{c_k:k\in\mathbb{Z}\}$
is bounded by Remark \ref{R3}.
  It follows that $\sup_{k\in\mathbb{Z}}\{k^2\|\Delta T_k\|\}<\infty$.

We observe that from \eqref{Tk} we have
\begin{align*}
k^2\Delta^2 N_k
&=k^2[\Delta N_{k+1}-\Delta N_k]\\
 &=k^2[-N_{k+2}T_{k+1}+N_{k+1}T_k]\\
 &=-k^2N_{k+2}[T_{k+1}-N_{k+2}^{-1}N_{k+1}T_k]\\
 &=-k^2N_{k+2}[T_{k+1}-(I+T_{k+1})T_k]\\
 &=-k^2N_{k+2}[T_{k+1}-T_k-T_{k+1}T_k]\\
  &=-k^2N_{k+2}[\Delta T_k-T_{k+1}T_k]\\
  &=-N_{k+2}[k^2\Delta T_k-\frac{k}{k+1}Q_{k+1}Q_k]
  =N_{k+2}R_k
\end{align*}
where we have set $R_k=-[k^2\Delta T_k-\frac{k}{k+1}Q_{k+1}Q_k]$ for all
$k\in\mathbb{Z}$, $k\neq0,-1$. Since
$\{Q_k:k\in\mathbb{Z}\}$ and $\{k^2\Delta T_k:k\in\mathbb{Z}\}$ are
bounded,  $\{R_k:k\in\mathbb{Z}\}$ is bounded.

Now, we have
\[
 k^2\Delta^2 B_k=k^2\Delta^2( BN_k)=B(k^2\Delta^2 N_k)=BN_{k+2}R_k=B_{k+2}R_k,
\]
\begin{align*}
k^2\Delta^2 H_k
&=k^2\Delta^2( kN_k)\\
 &=k^2[(k+2)N_{k+2}-2(k+1)N_{k+1}+kN_k]\\
 &=k^2[kN_{k+2}-2kN_{k+1}+kN_k]+2k^2N_{k+2}-2k^2N_{k+1}\\
 &=k^3\Delta^2N_k+2k^2\Delta N_{k+1}\\
 &=k(k^2\Delta^2N_k)+\frac{2k^2}{k+1}[(k+1)\Delta N_{k+1}]\\
 &=kN_{k+2}R_k+\frac{2k^2}{k+1}N_{k+2}Q_{k+1}\\
&=\frac{k}{k+2}H_{k+2}R_k+\frac{2k^2}{(k+1)(k+2)}H_{k+2}Q_{k+1},
\end{align*}

\begin{align*}
k^2\Delta^2 S_k
&=k^2\Delta^2(k\Lambda N_k)=k^2\Lambda\Delta^2
(kN_k)=\Lambda(k^2\Delta^2 H_k)\\
&=\Lambda\Big(\frac{k}{k+2}H_{k+2}R_k+\frac{2k^2}{(k+1)(k+2)}H_{k+2}Q_{
k+1}\Big)\\
&=\frac{k}{k+2}S_{k+2}R_k+\frac{2k^2}{(k+1)(k+2)}S_{k+2}Q_{k+1}.
\end{align*}
Finally,
\begin{align*}
k^2\Delta^2 M_k
&=k^2\Delta^2( k^2MN_k)\\
 &=k^2[(k+2)^2MN_{k+2}-2(k+1)^2MN_{k+1}+k^2MN_k]\\
&=k^2[k^2MN_{k+2}-2k^2MN_{k+1}+k^2MN_k]+k^2(4k+4)MN_{k+2}\\&\hspace{.5cm}-2k^2(2k+1)MN_{k+1}
\\
 &=k^2M(k^2\Delta^2N_k)+\frac{2k^2(2k+1)}{k+1}M[(k+1)\Delta
N_{k+1}]+2k^2MN_{k+2}\\
 &=k^2MN_{k+2}R_k+\frac{2k^2(2k+1)}{k+1}MN_{k+2}Q_{k+1}+2k^2MN_{k+2}\\
&=\frac{k^2}{(k+2)^2}M_{k+2}R_k+\frac{2k^2(2k+1)}{(k+1)(k+2)^2}M_{k+2}Q_{
k+1}+\frac { 2k^2 }{(k+2)^2}M_{k+2}
 \end{align*}
for all $k\in\mathbb{Z}$, $k\neq0,-1,-2$. Since
$\{B_k:k\in\mathbb{Z}\}$,
$\{S_k:k\in\mathbb{Z}\}$, $\{H_k:k\in\mathbb{Z}\}$, $\{M_k:k\in\mathbb{Z}\}$,
$\{Q_k:k\in\mathbb{Z}\}$, and
$\{R_k:k\in\mathbb{Z}\}$ are bounded,   $\{k^2\Delta^2
B_k:k\in\mathbb{Z}\}$, $\{k^2\Delta^2
H_k:k\in\mathbb{Z}\}$, $\{k^2\Delta^2
S_k:k\in\mathbb{Z}\}$ and $\{k^2\Delta^2
M_k:k\in\mathbb{Z}\}$ are bounded. This completes the proof.
\end{proof}

From the proof of  Theorem \ref{tfmg} and using \cite[Theorem 3.2]{BK}, we
deduce the following result for
$F^s_{pq}$-solutions in the case that $1<p<\infty$, $1<q\leq\infty$ and $s>0$.

\begin{theorem}\label{tfm1}
Let $X$ be a Banach space and  $A$, $B$, $\Lambda$, $M$
be closed linear operators in $X$ such that
$D(A)\cap D(B)\subset D(\Lambda)\cap D(M)$. Suppose that $\gamma$ is a constant,
$b,c\in L^1(\mathbb{R}_+)$, and consider $b_k$, $c_k$ as in \eqref{eq2} such that
 $(b_k)_{k\in\mathbb{Z}}$
 and $(c_k)_{k\in\mathbb{Z}}$  satisfy {\rm (H2)}. Then for
$s>0$ and $1<p<\infty, \, 1<q\le \infty$, the following are equivalent.
\begin{itemize}
 \item[(i)] \eqref{eP} is $F^s_{p,q}$-well-posed.
 \item[(ii)] $i\mathbb{Z}\subset \rho_{\Lambda,M,\tilde{b},\tilde{c}}(A,B)$
and $\{k^2MN_k:k\in\mathbb{Z}\}$, $\{BN_k:k\in\mathbb{Z}\}$, $\{k\Lambda
N_k:k\in\mathbb{Z}\}$,
 and $\{kN_k:k\in\mathbb{Z}\}$ are bounded, where
$$
N_k=[k^2M+A  +b_kB+i k\Lambda+ikc_kI+\gamma I]^{-1}
$$
\end{itemize}
\end{theorem}

\begin{proof} (i) $\Rightarrow$ (ii). Follows from Theorem \ref{t2} and
Remark \ref{R5}.

(ii) $\Rightarrow$ (i). Follows from  \cite[Theorem 3.2]{BK} using the same lines
as the proof of the preceding theorem.
\end{proof}

We now consider the problem of well-posedness in the vector-valued
Triebel-Lizorkin spaces $F_{pq}^s(0,2\pi, X)$ with parameters
$1\leq p<\infty$, $1\leq q\leq\infty$ and
$s>0$. For this, assumption (H2) is no longer sufficient.

A condition which implies that $(M_k)_{k\in\mathbb{Z}}$ is a Fourier multiplier
for the scale $F_{pq}^s$, $s\in\mathbb{R}$, $1<p<\infty$, $1<q\leq\infty$ is
the Marcinkiewicz condition of order two which is used in the proof of Theorem
\ref{tfm1}.

For $1\leq p<\infty$, $1\leq q\leq\infty$ and
$s\in\mathbb{R}$, a Marcinkiewicz condition of order three is needed, namely,
\begin{equation}\label{Mar3}
 \sup_{k\in\mathbb{Z}}(\| M_k\|+\|
 k\Delta M_k\|+k^2\| \Delta^2 M_k\|+|k|^3\|\Delta^3M_k\|)<\infty.
 \end{equation}
Our next result uses this condition to obtain characterization of
$F_{pq}^s$-well-posedness of the Problem \eqref{eP}.

\begin{theorem}\label{tfmf}
Let $X$ be a Banach space and let $A$, $B$, $\Lambda$, $M$
be closed linear operators in $X$ such that  $D(A)\cap D(B)\subset
D(\Lambda)\cap D(M)$. Suppose that $\gamma$ is a constant,
$b,c\in L^1(\mathbb{R}_+)$, and consider $b_k$, $c_k$ as in \eqref{eq2}
 such that  $(b_k)_{k\in\mathbb{Z}}$
 and $(c_k)_{k\in\mathbb{Z}}$  satisfy {\rm (H3)}. Then for
$s>0$ and $1\leq p<\infty$, $1\leq q\le \infty$, the following assertion are
equivalent.
\begin{itemize}
 \item[(i)] \eqref{eP} is $F^s_{p,q}$-well-posed.
 \item[(i)] $i\mathbb{Z}\subset \rho_{\Lambda,M,\tilde{b},\tilde{c}}(A,B)$
and $\{k^2MN_k:k\in\mathbb{Z}\}$, $\{BN_k:k\in\mathbb{Z}\}$,
$\{k\Lambda N_k:k\in\mathbb{Z}\}$,
 and $\{kN_k:k\in\mathbb{Z}\}$ are bounded, where
$$
N_k=[k^2M+A  +b_kB+i k\Lambda+ikc_kI+\gamma I]^{-1}
$$
\end{itemize}
\end{theorem}

\begin{proof} (i) $\Rightarrow$ (ii). Assume that
\eqref{eP} is $F^s_{pq}$-well-posed. Then by Theorem  \ref{t2},
$i\mathbb{Z}\subset \rho_{\Lambda,M,\tilde{b},\tilde{c}}(A,B)$ and
$(k^2MN_k)_{k\in\mathbb{Z}}$, $(BN_k)_{k\in\mathbb{Z}}$,
 $(k\Lambda N_k)_{k\in\mathbb{Z}}$,
 and $(kN_k)_{k\in\mathbb{Z}}$ are $F^s_{pq}$-Fourier multipliers. The
boundedness of $\{k^2MN_k:k\in\mathbb{Z}\}$, $\{BN_k:k\in\mathbb{Z}\}$,
$(k\Lambda N_k)_{k\in\mathbb{Z}}$,
 and $\{kN_k:k\in\mathbb{Z}\}$  follows of Remark  \ref{R5}.
\smallskip

(ii) $\Rightarrow$ (i). In view of Theorem \ref{t2}, it suffices
to show that the families $(k^2MN_k)_{k\in\mathbb{Z}}$,
$(BN_k)_{k\in\mathbb{Z}}$, $(k\Lambda N_k)_{k\in\mathbb{Z}}$,
 and $(kN_k)_{k\in\mathbb{Z}}$ are $F_{pq}^s$-Fourier multipliers. Let
$M_k=k^2MN_k$,  $B_k=BN_k$, $H_k=kN_k$
and $S_k=k\Lambda N_k$.   Since (H3) implies (H2) and (H2) implies (H1),
the verification of the Marcinkiewicz condition of order two and one is equal
 to what was done in the proof of Theorem \ref{tfmg}.

 It remains to prove the following inequalities:
\begin{gather*}
\sup_{k\in\mathbb{Z}}\|k^3\Delta^3 M_k\|<\infty,\quad
\sup_{k\in\mathbb{Z}}\|k^3\Delta^3 B_k\|<\infty,\\
\sup_{k\in\mathbb{Z}}\|k^3\Delta^3 S_k\|<\infty, \quad
\sup_{k\in\mathbb{Z}}\|k^3\Delta^3 H_k\|<\infty.
\end{gather*}
 But we obtain this using the same technique as used in the proof
 of the previous theorems.
\end{proof}

The following remark concerns the independence on the parameters
regarding the results of Section 4.

\begin{remark} \rm
$\bullet$ In Theorem \ref{tfmp}, if the problem is well-posed for
 some $p\in (1,\infty)$, then it well-posed for  all $p\in (1,\,\infty)$.

$\bullet$ Likewise, in Theorems \ref{tfm}, \ref{tfmg},  \ref{tfm1},
and  \ref{tfmf},  if the problem under consideration is well-posed for
one set of parameters in the range afforded by the corresponding
theorem then it is well-posed for  any set of parameters in that range.

This is a direct consequence of statement (ii) in each of the mentioned
theorems.
\end{remark}

\section{Examples and applications}

A large number of partial differential equations arising in physics and in
applied sciences can be written in the form of equation \eqref{eq1}; among them
there are some famous examples, such as the pseudo-parabolic  equations and the
Sobolev type equations. Sobolev type equations have  the form
\begin{equation}\label{Sob}
\Lambda u'=Au+f,
\end{equation}
generally denoting equations or systems in which spatial derivatives are
mixed with the time derivative of highest
order.  Showalter  \cite{Sh791, Sh792}  studied Sobolev type equations of
the  first and second order in time.
 Specifically, Equation \ref{Sob} is called {\it strongly regular} if
$\Lambda^{-1}A$ is continuous, {\it weakly regular } if $\Lambda$ is invertible
but does not dominate $A$ and { \it degenerate } if $\Lambda$ is not invertible.
Strongly regular Sobolev type equations
are also widely known as pseudoparabolic. The Sobolev type equations are of
interest not only for the sake of generalizations but also because they arise
naturally in a  variety
of applications (e.g. in acoustics, electromagnetics, viscoelasticity, heat
conduction etc., see e.g. \cite{LSY}).
 A general theory in the context of generalized semigroups is developed
in the monograph \cite{MF01}.

For the periodic case initially, Arendt and Bu \cite{AB} deal with the
problem $u'(t)=Au(t)+f(t)$, $u(0)=u(2\pi)$.
This problem corresponds to \eqref{eP} with   $M=B=0$, $\Lambda=-I$,
$c=0$, and $\gamma=0$. The additional  condition of our definition of
well-posedness is obtained automatically by Remark \ref{Auto}. In this case
their result are equivalent to our result by Remarks \ref{R4} and \ref{R5}.

 Arendt and Bu \cite{AB} (see also the review paper \cite{A})   consider  the
problem $u''(t)=Au(t)+f(t)$, $u(0)=u(2\pi)$, $u'(0)=u'(2\pi)$. This problem
corresponds to \eqref{eP} with   $M=I$, $\Lambda=B=0$
$c=0$, and $\gamma=0$. Here again the additional  condition of our
definition of well-posedness is obtained automatically by Remark \ref{Auto}. In
this case their result are equivalent to our result by Remarks  \ref{R5}.

 Keyantuo and  Lizama \cite{KL1, KL2}  considered well-posedness of \eqref{eP}
when $B=M=0$ and $\Lambda$ is a scalar operator. As noted earlier, this problem is
relevant for viscoelasticity and was previously studied in the framework of
periodic solutions by Da Prato-Lunardi \cite{PL} among other references,
 and on the real line by \cite{ CP, PL1}.
Second order equations are considered in this context in \cite{KL3, Pb}

The additional  condition of our definition of well-posedness is obtained
automatically by Remark \ref{Auto}. Their results can be deduced from ours.
Some additional papers on the subject are Bu \cite{Bu1, Bu2, BY1}.
 Delay equations are considered in \cite{BY2, PP}
with the method of operator-valued Fourier multipliers.

 Bu \cite{Bu2} considered the well-posedness of \eqref{eP} when $B=\Lambda=0$,
$c=0$, and $\gamma$. His results follow  from ours. With our
definition of well-posedness we do not need the a priori the estimate
\cite[(2.2)]{Bu2}. Thus, in the reference \cite{Bu2}, the author considers the problem
\begin{gather*}
 (Mu')'(t)= Au(t)+f(t),\quad 0\leq t\leq 2\pi,\\
u(0)=u(2\pi), \quad (Mu')(0)=(Mu')(2\pi).
 \end{gather*}

It follows from Theorem \ref{tfmp} that this problem is $L^p$-well-posed if and
only if $i\mathbb{Z}\subset
\rho_{0,M,\tilde{0},\tilde{0}}(A,0)=\rho_{M}(A)$
and $\{k^2MN_k:k\in\mathbb{Z}\}$
 and $\{kN_k:k\in\mathbb{Z}\}$ are $R$-bounded, where
$N_k=(k^2M+A)^{-1}$.
In a similar way, we deduce the results in $B^s_{p,q}$ and $F^s_{p,q}$ using
Theorem \ref{tfmg} and Theorem \ref{tfmf} respectively.

We introduce some facts on   uniformly elliptic operators on domains of
$\mathbb{R}^n$ to discuss the examples that follow.    Let
$\Omega\subset\mathbb{R}^n$ be open, $n\geq1$.
We consider measurable functions
$\alpha_{ik}$, $\beta_{k}$, $ \gamma_k$, and $\alpha_0$
$(1\leq j,k\leq n)$ on $\Omega$. We assume that the following
uniform ellipticity condition holds: The functions $\alpha_{kj}$, $\beta_{k}$,
 $ \gamma_k$,  $\alpha_0$ are bounded on $\Omega$, i.e., $\alpha_{kj}$, $\beta_{k}$,
$ \gamma_k$, $\alpha_0\in L^{\infty}(\Omega,\mathbb{C})$ for
$1\leq j,k\leq n$ and the principal part is
elliptic; i.e., there exists a constant $\eta>0$ such that
\begin{equation}\label{EL}
\operatorname{Re}(\sum_{j,k=1}^n\alpha_{kj}
(x)\xi_j\overline{\xi_k})\geq\eta|\xi|^2\quad \text{for all }\xi\in\mathbb{C}^n,
\text{ a.e. }  x\in \Omega.
\end{equation}
The largest possible $\eta$ in \eqref{EL} is called the ellipticity constant
of the matrix $(\alpha_{jk})_{1\leq j,k\leq n}$.
Then we consider the elliptic operator
$L:W_{\rm loc}^{1,2}(\Omega) \to \mathcal{D}(\Omega)'$ given by
$$
Lu=-\sum_{k,j=1}^nD_j(\alpha_{kj}D_ku)+\sum_{k=1}
^n(\beta_kD_ku-D_k(\gamma_ku))+\alpha_0u.
$$

With the help of bilinear forms we will define various realizations of
$L \in L^2(\Omega)$ corresponding to
diverse boundary conditions. Let $V$ be a closed subspace of
$W^{1,2}(\Omega)$ containing $W_0^{1,2}(\Omega)$. We define the form
$\alpha_V:V\times V\to\mathbb{C}$ by
$$
\alpha_V(u,v)=\int_{\Omega}\Big[\sum_{k,j=1}^n\alpha_{kj}D_ku\overline
{(D_jv)}+ \sum_{k=1}^n(\beta_k\overline{v}D_ku+\gamma_ku\overline{D_kv}
)+\alpha_0u\overline{v}\Big]dx.
$$
Then $\alpha_V$ is densely defined,
accretive,  and closed sesquilinear form on $L^2(\Omega)$
(see \cite[Chapter 4 p. 100-101]{Ouh}). Denote by $A_V$ the operator
on $L^2(\Omega)$ associated with $\alpha_V$. Then $-A_V$ generates a
$C_0$-semigroup $T_V$ on $L^2(\Omega)$ (see \cite[Proposition 1.51]{Ouh}).
It follows from the definition of the associated
operator that $A_V u = Lu$ for all $u \in D(A_V )$.
We will say that we have:
\begin{itemize}
 \item {Dirichlet boundary conditions} if $V=W_0^{1,2}(\Omega)$;
 \item {Neumann boundary conditions} if $V=W^{1,2}(\Omega)$;
 \end{itemize}
 We consider Dirichlet boundary conditions with $\Omega$ bounded
 and we assume the following additional conditions:  $\alpha_{kj}$ is real-valued
with $\alpha_{kj}=\alpha_{jk}$, $\beta_k=\gamma_k=0$, $\alpha_0\geq0$. Then,
in this case the  semigroup $T_V$ is positive,
$\|T_V(t)\|_{\mathcal{L}(L^2(\Omega))}\leq 1$ for all $t\geq0$, and $T_V$ is
given by an integral kernel $p_V(t,x,y)$ such that there exist constants $C>0$,
$b>0$, and $\delta>0$ such that
\begin{equation}\label{Ker}
|p_V(t,x,y)|\leq Ct^{-n/2}e^{-\delta t}e^{-\frac{|x-y|^2}{4bt}}
\end{equation}
 for every $t>0$ and a.e. $x,y\in\Omega$, see
\cite[Theorem 4.2, Corollary 6.14 and Theorem 4.28]{Ouh}
and \cite{Dav}.  For every $r\in(1,\infty)$, the
$C_0$-semigroup $T_V$ extends to a bounded $C_0$-semigroup $T_r$  on
$L^r(\Omega)$ with $\|T_r(t)\|_{\mathcal{L}(L^r(\Omega))}\leq 1$ for all
$t\geq0$, by \cite[Theorem 4.28]{Ouh}. By \eqref{Ker} there exist $M_r>0$, and
$\delta_r>0 $ depending only on $r$ such that
$\|T_r(t)\|_{\mathcal{L}(L^r(\Omega))}\leq M_re^{-\delta_r t}$ for all $t>0$ and
$r\in(1,\infty)$. Denote now by $-A_r$ the corresponding infinitesimal generator
on $L^r(\Omega)$.  If $\lambda\in\mathbb{C}$, $\operatorname{Re}\lambda>-\delta$,
then $\lambda\in \rho(-A_r)$ and
\begin{equation}\label{RB2}
R(\lambda,-A_r)u=\int_0^\infty e^{-\lambda t}T_r(t)udt\text{ for all }u\in
L^r(\Omega),
\end{equation}
by \cite[Theorem 3.1.7]{ABHN}.

 Let $r\in(1,\infty)$. The  $C_0$-semigroup $T_r$ extends to a bounded
holomorphic semigroup on the sector $\Sigma_{\pi/2}$, where $\Sigma_\theta$ is
the sector in the complex right half plane of angle $\theta\in(0,\pi]$.  By
\cite[Theorem 3.7.11]{ABHN} we have that $\Sigma_\pi\subset\rho(-A_r)$ and
$\sup_{\lambda\in\Sigma_{\pi-\varepsilon}}\|\lambda
R(\lambda,-A_r)\|<\infty$ for all $\varepsilon>0$.
Denote  by  $\sigma(A_r)$ the spectrum of the operator $A_r$ on
$L^r(\Omega)$. By  \cite[Theorem 7.10]{Ouh}, we have that
$\sigma(A_r)=\sigma(A_2)\subset(0,\infty)$ for all
$r\in(1,\infty)$.
 By \cite[Section 7.2.6]{A} we have that $\lambda R(\lambda,-A_r)$  is
$R$-bounded for all $\lambda\in \Sigma_{\pi/2+\theta_r}$ with
$0<\theta_r\leq\pi/2$.
Since $\lambda\to R(\lambda,-A_r)$ is analytic on
$\Sigma_\pi\cup\{\lambda\in\mathbb{C}:\operatorname{Re}\lambda>-\delta_r\}$,
it follows that $R(\lambda,-A_r)$ is $R$-bounded on every compact subset of
$\Sigma_\pi\cup\{\lambda\in\mathbb{C}:\operatorname{Re}\lambda>-\delta_r\}$ by
\cite[Proposition 3.10]{DHP}. By Remark \ref{R1}, we have that $R(\lambda,-A_r)$
and $\lambda R(\lambda,-A_r)$ are $R$-bounded on
$\Sigma_{\pi/2+\theta_r}\cup\{\lambda\in\mathbb{C}:\operatorname{Re}
\lambda\geq-\delta_r/2\}$. Using Kahane's principle, we obtain that
$R(\lambda,A_r)$ and $\lambda R(\lambda,A_r)$ are $R$-bounded on
$\mathbb{C}\setminus\Sigma_{\theta_r}\cup \{\lambda\in\mathbb{C}:
\operatorname{Re}\lambda\leq\delta_r/2\}$.

We conclude, with some examples using uniformly elliptic operators in
$L^r(\Omega)$ just discussed. General references on uniformly elliptic operators
in $L^p-$spaces and the associated heat kernel estimates are \cite{Dav} and \cite{Ouh}.

\begin{example}\label{E} \rm
Let us consider the boundary value problem (in which $L$ is a uniformly elliptic operator as defined above)
\begin{equation}\label{E1}
\begin{gathered}
\begin{aligned}
&\frac{\partial}{\partial t}(m(x)\frac{\partial u(t,x)}{\partial
t})+L\frac{\partial u(t,x)}{\partial t}\\
&=-Lu(t,x)-\int_{-\infty}^t{b(t-s)L
u(s,x)}ds+f(t,x), \quad (t,x)\in[0,2\pi]\times\Omega,
\end{aligned}\\
u(t,x)=\frac{\partial u(t,x)}{\partial t}=0, \quad
(t,x)\in[0,2\pi]\times\partial\Omega,\\
u(0,x)=u(2\pi,x), \quad m(x)\frac{\partial u(0,x)}{\partial t}=m(x)\frac{\partial
u(2\pi,x)}{\partial t}, \quad x\in\Omega,
\end{gathered}
\end{equation}
where  $f\in L^p(0,2\pi;L^r(\Omega))$ for $1<p, r<\infty$, $m$ is a real-valued
measurable function on $\Omega$ such that $m\in L^\infty(\Omega)$.
This is the degenerate wave equation with fading memory. The non-degenerate
equation is studied in \cite{ACS08}, and the reference list of this paper
contains additional works on that topic. Maximal regularity for the damped wave
equation in the absence of memory effects has been studied in \cite{ChS} and
\cite{KL3}.  The problem \eqref{E1} can also  be considered as  a modified
version of a problem which is considered in Favini-Yagi \cite[Example 6.24 p.
197]{FY}. They do not incorporate the delay aspect of the equation. They
restrict their study to the H\"older spaces. The authors are considered with the
evolutionary problem as well. For periodic boundary conditions, we obtain
complete characterization of well-posedness in the three scales of spaces:
$L^p$, $B^s_{pq}$, and $F^s_{pq}$.

We can rewrite  problem \eqref{E1} in as follows (where $A_r$ was defined above):
\begin{equation}\label{E11}
\begin{gathered}
\begin{aligned}
&\frac{\partial}{\partial t}(m(x)\frac{\partial u(t,x)}{\partial
t})+A_r\frac{\partial u(t,x)}{\partial t}\\
&=-A_ru(t,x)-\int_{-\infty}^t{b(t-s)A_r u(s,x)}ds+f(t,x), \quad
(t,x)\in[0,2\pi]\times\Omega,
\end{aligned}\\
u(0,x)=u(2\pi,x), \quad m(x)\frac{\partial u(0,x)}{\partial t}=m(x)\frac{\partial
u(2\pi,x)}{\partial t}, \quad x\in\Omega.
\end{gathered}
\end{equation}
If we suppose that $b_k$ defined by \eqref{eq2} satisfies (H1) and the
additional condition $|\operatorname{Im}b_k|<1$ for all $k\in\mathbb{Z}$,
it follows that
$\frac{k^2m(x)}{1+b_k+ik}\notin(0,\infty)$ for all $x\in\Omega$ and all
$k\in\mathbb{Z}$. Therefore $i\mathbb{Z}\subset \rho_{-A_r,
M,\tilde{b},\tilde{0}}(-A_r,-A_r)$, where $M$ is the multiplication
operator by $m$.    By Remark \ref{R3}, we have that there exists
$N\in\mathbb{N}$ such that
$\frac{k^2m(x)}{ik+1+b_k}\in\mathbb{C}\setminus\Sigma_{\theta_r}\cup\{
\lambda\in\mathbb{C}:\operatorname{Re}\lambda\leq\delta_r/2\}$
for all $x\in\Omega$ and
all $k\in\mathbb{Z}$ whit $|k|\geq N$. Then
$\{(\frac{k^2}{ik+1+b_k}M-A_r)^{-1}:k\in\mathbb{Z}, |k|\geq N\}$  and
$\{\frac{k^2}{ik+1+b_k}M(\frac{k^2}{ik+1+b_k}M-A_r)^{-1}:k\in\mathbb{Z}, |k|\geq N\}$
are $R$-bounded. Since
\begin{align*}
&\frac{k}{ik+1+b_k}A_r(\frac{k^2}{ik+1+b_k}M-A_r)^{-1}\\
&=-\frac{k}{ik+1+b_k}I
+\frac{k}{ik+1+b_k}\frac{k^2}{ik+1+b_k}M(\frac{k^2}{ik+1+b_k}M-A_r)^{-1},
 \end{align*}
it follows that
$\{\frac{k}{ik+1+b_k}A_r(\frac{k^2}{ik+1+b_k}M-A_r)^{-1}:k\in\mathbb{Z},|k|\geq
N\}$ is $R$-bounded as well by Remark  \ref{R1}. Since
$N_k=\frac{1}{ik+1+b_k}(\frac{k^2}{ik+1+b_k}M-A_r)^{-1}$,  we have shown that
$\{k^2MN_k:k\in\mathbb{Z}, |k|\geq N\}$,
$\{kA_rN_k:k\in\mathbb{Z}, |k|\geq N\}$ and
$\{kN_k:k\in\mathbb{Z}, |k|\geq N\}$ are $R$-bounded. Also by Remark
\ref{R1}, we have that $\{kN_k:k\in\mathbb{Z}\}$,
$\{kA_rN_k:k\in\mathbb{Z}\}$ and $\{k^2MN_k:k\in\mathbb{Z}\}$ are $R$-bounded.
Therefore, by Theorem \ref{tfmp}, it follows that \eqref{E11} is
$L^p(0,2\pi;L^r(\Omega))$-well-posed for all $1<p<\infty$.  Since $R$-boundedness
implies uniformly boundedness,  if we suppose that
$f\in B^s_{pq}(0,2\pi;L^r(\Omega))$ and $b_k$  satisfies (H2) with
$|\operatorname{Im}b_k|<1$ for all $k\in\mathbb{Z}$, then we have that \eqref{E11}
is $B^s_{pq}(0,2\pi;L^r(\Omega))$-well-posed for all $s>0$, $1\leq p,q\leq\infty$
by Theorem \ref{tfmg}. Observe that here we include the scale of vector-valued
H\"older spaces $C^s$, $0<s<1$. In the $F^s_{pq}$ case if
$f\in F^s_{pq}(0,2\pi;L^r(\Omega))$ and $b_k$  satisfies   (H3) with
$|\operatorname{Im}b_k|<1$
for all $k\in\mathbb{Z}$, then we have that \eqref{E11} is
$F^s_{pq}(0,2\pi;L^r(\Omega))$-well-posed for all $s>0$, $1\leq p<\infty$,
$1\leq q\leq\infty$, by Theorem \ref{tfmf}. Observe that  if $s>$, $1<p<\infty$
and $1<q\leq\infty$ we only need the (H2) condition for this scale. As a
particular example we have that
$b(t)=e^{-\varepsilon t}\frac{t^{\nu-1}}{\Gamma(\nu)}$ with $\varepsilon>0$
and $\nu>0$ satisfies the
required conditions for $b_k$ for all the cases.
\end{example}

\begin{example}\label{EII} \rm
Let us consider the boundary value problem
\begin{equation}\label{EII1}
\begin{gathered}
\begin{aligned}
&\frac{\partial}{\partial t}(m(x)\frac{\partial u(t,x)}{\partial
t})+L\frac{\partial u(t,x)}{\partial t}\\
&=Lu(t,x)+\int_{-\infty}^t{b(t-s)L
u(s,x)}ds+f(t,x), \, (t,x)\in[0,2\pi]\times\Omega,
\end{aligned} \\
u(t,x)=\frac{\partial u(t,x)}{\partial t}=0, \quad
(t,x)\in[0,2\pi]\times\partial\Omega,\\
u(0,x)=u(2\pi,x), \quad
 m(x)\frac{\partial u(0,x)}{\partial t}=m(x)\frac{\partial u(2\pi,x)}{\partial t},
\quad x\in\Omega,
\end{gathered}
\end{equation}
where  $f\in L^p(0,2\pi;L^r(\Omega))$ for $1<p, r<\infty$, $m$ is a
complex-valued measurable function on $\Omega$ such that $m\in
L^\infty(\Omega)$, $m(x)\in\sum_{\theta_r}\cup\{0\}$ for all $x\in\Omega$.
 Here, as in the previous example (and similarly in Example \ref{Ex21} below), $L$ is a uniformly elliptic operator.

Following   Example \ref{E}, we can rewrite the problem \eqref{EII1} in the form
\begin{equation}\label{E12}
\begin{gathered}
\begin{aligned}
&\frac{\partial}{\partial t}(m(x)\frac{\partial u(t,x)}{\partial
t})+A_r\frac{\partial u(t,x)}{\partial t}\\
&=A_ru(t,x)+\int_{-\infty}^t{b(t-s)A_r
u(s,x)}ds+f(t,x), \quad (t,x)\in[0,2\pi]\times\Omega,
\end{aligned}\\
u(0,x)=u(2\pi,x), \quad m(x)\frac{\partial u(0,x)}{\partial t}=m(x)\frac{\partial
u(2\pi,x)}{\partial t}, \quad x\in\Omega.
\end{gathered}
\end{equation}
If we suppose that $b_k$ defined by \eqref{eq2} satisfies   (H1) and the
additional condition $\operatorname{Re}b_k>-1$ for all $k\in\mathbb{Z}$, then
$\frac{k^2m(x)}{1+b_k-ik}\in\Sigma_{\pi/2+\theta_r}\cup\{0\}$ for all
$x\in\Omega$ and all $k\in\mathbb{Z}$. Therefore
$i\mathbb{Z}\subset \rho_{-A_r,M,\tilde{b},\tilde{0}}(A_r,A_r)$ and
$\{(\frac{k^2}{1+b_k-ik}M+A_r)^{-1}:k\in\mathbb{Z}\}$,
$\{\frac{k^2}{1+b_k-ik}M(\frac{k^2}{1+b_k-ik}M+A_r)^{-1}:k\in\mathbb{Z}\}$ are
$R$-bounded, here $M$ is the multiplication operator by $m$.     By Remarks
\ref{R1}  and  \ref{R3}, we have that
$\{\frac{k}{1+b_k-ik}(\frac{k^2}{1+b_k-ik}M+A_r)^{-1}:k\in\mathbb{Z}\}$ is also
$R$-bounded.  Since
\begin{align*}
&\frac{k}{1+b_k-ik}A_r(\frac{k^2}{1+b_k-ik}M+A_r)^{-1}\\
&=\frac{k}{1+b_k-ik}I
-\frac{k}{1+b_k-ik}\frac{k^2}{1+b_k-ik}M(\frac{k^2}{1+b_k-ik}M+A_r)^{-1},
 \end{align*}
it follows that
$\{\frac{k}{1+b_k-ik}A_r(\frac{k^2}{1+b_k-ik}M+A_r)^{-1}:k\in\mathbb{Z}\}$ is
$R$-bounded as well by Remark \eqref{R1}. Since
$N_k= \frac{1}{1+b_k-ik}(\frac{k^2}{1+b_k-ik}M+A_r)^{-1}$,  we have shown that
$\{k^2MN_k:k\in\mathbb{Z}\}$,  $\{kA_rN_k:k\in\mathbb{Z}\}$ and
$\{kN_k:k\in\mathbb{Z}\}$ are $R$-bounded. Therefore, by Theorem \ref{tfmp}, we
have that \eqref{E12} is
$L^p(0,2\pi;L^r(\Omega))$-well-posed for all $1<p<\infty$. Since $R$-boundedness
implies uniformly boundedness,  if we suppose that
$f\in B^s_{pq}(0,2\pi;L^r(\Omega))$ and $b_k$  satisfies (H2) with
$\operatorname{Re}b_k>-1$ for all $k\in\mathbb{Z}$, then we have that \eqref{E12} is
$B^s_{pq}(0,2\pi;L^r(\Omega))$-well-posed for all $s>0$, $1\leq p,q\leq\infty$
by Theorem \ref{tfmg}. Observe that here we include the scale of vector-valued
H\"older spaces $C^s$, $0<s<1$. In the $F^s_{pq}$ case if
$f\in F^s_{pq}(0,2\pi;L^r(\Omega))$ and $b_k$  satisfies (H3) with
$\operatorname{Re}b_k>-1$
for all $k\in\mathbb{Z}$, then we have that \eqref{E12} is
$F^s_{pq}(0,2\pi;L^r(\Omega))$-well-posed for all $s>0$, $1\leq p<\infty$,
$1\leq q\leq\infty$, by Theorem \ref{tfmf}.  Observe that  if $s>0$,
$1<p<\infty$ and $1<q\leq\infty$ we only need the (H2) condition for this scale.
As in the Example \ref{E}, a particular example of $b(t)$ we have
$b(t)=e^{-\varepsilon}\frac{t^{\nu-1}}{\Gamma(\nu)}$ with $\varepsilon>0$ and
$\nu>0$ that fulfills the required conditions for $b_k$ in all the cases.
\end{example}

\begin{example}\label{Ex21} \rm
Consider another initial-boundary value problem.
\begin{equation}\label{E2}
\begin{gathered}
\begin{aligned}
&\frac{\partial}{\partial t}(m_2(x)\frac{\partial u(t,x)}{\partial
t})-m_1(x)\frac{\partial u(t,x)}{\partial t}\\
&=Lu(t,x)+\int_{-\infty}^t{b(t-s)L u(s,x)}ds+f(t,x), \quad
 (t,x)\in[0,2\pi]\times\Omega,
\end{aligned}\\
u(t,x)=\frac{\partial u(t,x)}{\partial t}=0, \quad
(t,x)\in[0,2\pi]\times\partial\Omega,\\
u(0,x)=u(2\pi,x), \quad m_2(x)\frac{\partial u(0,x)}{\partial
t}=m_2(x)\frac{\partial u(2\pi,x)}{\partial t}, \quad x\in\Omega,
\end{gathered}
\end{equation}
where   $m_1$ and $m_2$ are real-valued measurable functions on $\Omega$ such
that $m\in L^\infty(\Omega)$, $m_2(x)\geq 0$, $\tau<|m_1(x)|\leq \mu$ for some
$\tau, \mu>0$, and $f\in L^p(0,2\pi;L^r(\Omega))$ for $1<p, r<\infty$.

Following the Example \ref{E}, we can rewrite the problem \eqref{E2} in the form
\begin{equation}\label{E21}
\begin{gathered}
\begin{aligned}
&\frac{\partial}{\partial t}(m_2(x)\frac{\partial u(t,x)}{\partial
t})-m_1\frac{\partial u(t,x)}{\partial t}\\
&=A_ru(t,x)+\int_{-\infty}^t{b(t-s)A_r
u(s,x)}ds+f(t,x), \quad (t,x)\in[0,2\pi]\times\Omega,
\end{aligned}\\
u(0,x)=u(2\pi,x), \quad m_2(x)\frac{\partial u(0,x)}{\partial
t}=m_2(x)\frac{\partial u(2\pi,x)}{\partial t}, \quad x\in\Omega.
\end{gathered}
\end{equation}
  If we  assume that  $\operatorname{Re}b_k>-1$ for all $k\in\mathbb{Z}$, then
$\frac{k^2m_2(x)+ikm_1(x)}{b_k+1}\notin(-\infty,0)$ for all $x\in\Omega$ and all
$k\in\mathbb{Z}$. Therefore
$i\mathbb{Z}\subset \rho_{\Lambda, M,\tilde{b},\tilde{0}}(A_r,A_r)$, where
$\Lambda$ and $M$ are the
multiplication operators by $m_1$ and $m_2$ respectively. By Remark \ref{R3}, we
have that there exists $N\in\mathbb{N}$ such that
$\frac{k^2m_2(x)+ikm_1(x)}{b_k+1}\in\Sigma_{\pi/2+\theta_r}$ for all
$x\in\Omega$ and all $k\in\mathbb{Z}$ whit $|k|\geq N$.
Then
$\{\frac{k^2m_2(x)+ikm_1(x)}{b_k+1}(\frac{k^2m_2(x)+ikm_1(x)}{b_k+1}+A_r)^{-1}
:k\in\mathbb{Z}, |k|\geq N, x\in\Omega\}$ are $R$-bounded. Since
$\{\frac{1}{km_2(x)+im_1(x)}:k\in\mathbb{Z}, x\in\Omega\}$ is bonded,
$\{\frac{k}{b_k+1}(\frac{k^2m_2(x)+ikm_1(x)}{b_k+1}+A_r)^{-1}:k\in\mathbb{Z},
|k|\geq N, x\in\Omega\}$ are $R$-bounded by Remark \ref{R1}. Since $m_1$  is
bounded, by Remark \ref{R1} we have that
$\{\frac{km_1(x)}{b_k+1}(\frac{k^2m_2(x)+ikm_1(x)}{b_k+1}+A_r)^{-1}:k\in\mathbb{
Z}, |k|\geq N, x\in\Omega\}$ are $R$-bounded.  Therefore,
$\{\frac{k^2m_2(x)}{b_k+1}(\frac{k^2m_2(x)+ikm_1(x)}{b_k+1}+A_r)^{-1}
:k\in\mathbb{Z}, |k|\geq N, x\in\Omega\}$ are $R$-bounded. Since
$N_k=\frac{1}{b_k+1}(\frac{k^2}{b_k+1}M+i\frac{k}{b_k+1}\Lambda+A_r)^{-1}$,
we have show that $\{kN_k:k\in\mathbb{Z}, |k|\geq N\}$, $\{k\Lambda
N_k:k\in\mathbb{Z}, |k|\geq N\}$ and $\{k^2MN_k:k\in\mathbb{Z}, |k|\geq N\}$ are
$R$-bounded. By Remark \ref{R1}, we have that $\{kN_k:k\in\mathbb{Z}\}$,
$\{k\Lambda N_k:k\in\mathbb{Z}\}$ and $\{k^2MN_k:k\in\mathbb{Z}\}$ are
$R$-bounded. Under the same conditions over $b_k$ in the Example \ref{EII}  and
$f\in\mathcal{Y}$ we can apply  Theorems \ref{tfmp}, \ref{tfmg} and \ref{tfmf}
to obtain that the \eqref{E21} is $\mathcal{Y}$-well-posed.
\end{example}
\newpage
\begin{example}\label{E31}\end{example}
Let us now consider the boundary-value problem
\begin{equation}\label{E3}
\begin{gathered}
\begin{aligned}
&\frac{\partial}{\partial t}(m_2(x)\frac{\partial u(t,x)}{\partial
t})-m_1(x)\frac{\partial u(t,x)}{\partial t}-\frac{\partial}{\partial
t}\int_{-\infty}^tc(t-s)u(s,x)ds\\
&=Lu(t,x)+\int_{-\infty}^t{b(t-s)L u(s,x)}ds \\
&\quad +\int_{-\infty}^tb(t-s)m_0(x)u(s,x)ds +f(t,x), \quad
(t,x)\in[0,2\pi]\times\Omega,
\end{aligned}\\
u(t,x)=\frac{\partial u(t,x)}{\partial t}=0, \quad
(t,x)\in[0,2\pi]\times\partial\Omega,\\
u(0,x)=u(2\pi,x), \quad m_2(x)\frac{\partial u(0,x)}{\partial
t}=m_2(x)\frac{\partial u(2\pi,x)}{\partial t}, \quad x\in\Omega,
\end{gathered}
\end{equation}
where  $m_0$, $m_1$, and $m_2$ are real-valued measurable functions on $\Omega$
such that $0\leq m_0(x)\leq \mu$, $\tau<| m_1(x)|\leq\mu$, $0\leq m_2(x)$, for
some $\mu, \tau>0$, all $x\in\Omega$,   and $f\in L^p(0,2\pi;L^r(\Omega))$ for
$1<p, r<\infty$.

Following the Example \ref{E}, we can rewrite the problem \eqref{E3} in the form
\begin{equation}\label{E32}
\begin{gathered}
\begin{aligned}
&\frac{\partial}{\partial t}(m_2(x)\frac{\partial u(t,x)}{\partial
t})-m_1(x)\frac{\partial u(t,x)}{\partial t}-\frac{\partial}{\partial
t}\int_{-\infty}^tc(t-s)u(s,x)ds\\
&=A_ru(t,x)+\int_{-\infty}^tb(t-s)m_0(x)u(s,x)ds
 +f(t,x), \quad (t,x)\in[0,2\pi]\times\Omega,
\end{aligned}\\
u(0,x)=u(2\pi,x), \quad m_2(x)\frac{\partial u(0,x)}{\partial
t}=m_2(x)\frac{\partial u(2\pi,x)}{\partial t}, \quad x\in\Omega.
\end{gathered}
\end{equation}
 If we suppose that  $\operatorname{Re}b_k\geq0$ and
$k\operatorname{Im}c_k\leq0$ for all $k\in\mathbb{Z}$, we then have that
$ {k^2m_2(x)+ikm_1(x)+b_km_0(x)+ikc_k}\notin(-\infty,0)$ for all
$x\in\Omega$ and all $k\in\mathbb{Z}$.
Therefore $i\mathbb{Z}\subset \rho_{\Lambda, M,\tilde{b},\tilde{c}}(A_r,B)$,
where $B$, $\Lambda$,
and $M$ are the multiplication operators by $m_0$, $m_1$, and $m_2$
respectively. In the similar way that in the Example \ref{Ex21} we can show that
$\{kN_k:k\in\mathbb{Z}\}$, $\{kBN_k:k\in\mathbb{Z}\}$,
$\{k\Lambda N_k:k\in\mathbb{Z}\}$ and $\{k^2MN_k:k\in\mathbb{Z}\}$ are
$R$-bounded where $N_k=(k^2M+ik\Lambda+{b_k}B+i{kc_k}I+A_r)^{-1}$.
Typical cases of functions $b$ and $c$ is the function
$e^{-\varepsilon t}$, $\varepsilon>0$. With $f\in\mathcal{Y}$ and the
appropriate  $b$, and $c$ we can obtain that the \eqref{E32} is
$\mathcal{Y}$-well-posed.

   In the case of Neumann boundary conditions, the
operator $A_r$ is not invertible. To apply the results to this case, we
  can add in the right side of each of the above equations the  term
$\eta u(t,x)$ for some $\eta>0$. Then the above conclusions hold in this case as
well.


\begin{example}\label{P} \rm
The following equation is a modification of
 the one studied by Chill and Srivastava \cite{ChS}. Here we have include
 memory term.
\begin{equation}
\begin{gathered}
u''(t)+\alpha A^{\frac{1}{2}} u'(t)=-Au(t)+\int_{-\infty}^t{b(t-s)A
u(s,x)}ds+f(t), \\
t\in[0,2\pi],\quad u(0)=u(2\pi),\quad u'(0)=u'(2\pi),
\end{gathered}
\end{equation}
where $A$ is a invertible sectorial operator in a Banach space $X$ which admits
a bounded $H^\infty$  functional calculus of angle $\beta$ (see for
example \cite{ChS}, \cite{DHP}) with
$\beta\in(0,\pi-2\tan^{-1}\frac{\sqrt{4-\alpha^2}}{\alpha})$ if
$0<\alpha<2$ or $\beta\in(0,\pi)$ if $\alpha\geq2$, $f\in B^s_{pq}(0,2\pi;X)$,
($1\leq p,q\leq\infty,\, s>0$), and  $b\in {L}^1(\mathbb{R}_+)$ is such that
$b_k=\tilde{b}(ik)$ satisfies $|CQb_k|<\frac{1}{2}$ where $Q$ is a constant
provided by \cite[Lemma 4.1]{ChS}
and $C$ is a constant provided by the $H^\infty$  functional calculus.

 In the same way as in the proof of theorem \cite[Theorem 4.1]{ChS} we have
that for $k\in\mathbb{Z}$,
$\|k^2(k^2-\alpha kiA^{\frac{1}{2}}-A)^{-1}\|\leq CP$,
$\|kA^{\frac{1}{2}}(k^2-\alpha kiA^{\frac{1}{2}}-A)^{-1}\|\leq CP$, and
$\|A(k^2-\alpha kiA^{\frac{1}{2}}-A)^{-1}\|\leq CP$. In  this case for
$k\in\mathbb{Z}$, we have that
$N_k=(k^2-\alpha kiA^{\frac{1}{2}}-A+b_kA)^{-1}$. Since
$\|b_kA(k^2-\alpha kiA^{\frac{1}{2}}-A)^{-1}\|\leq \frac{1}{2}$, we have
 $$
N_k=(k^2-\alpha kiA^{\frac{1}{2}}-A)^{-1}\sum_{n=0}^\infty (-1)^n
\Big(b_kA(k^2-\alpha kiA^{\frac{1}{2}}-A)^{-1}\Big)^n,
$$
which implies that  $\|k^2N_k\|\leq CP$ and $\|\alpha
kA^{\frac{1}{2}}N_k\|\leq CP$ for $k\in\mathbb{Z}$. Now if
$b_k$ satisfy (H2), then we have that the problem \ref{P} is
$B^s_{pq}$-well-posed. This gives in particular well-posedness in
the H\"older spaces $C^s(0,2\pi ; X)$, $0<s<1$.

In a similar way, one can handle the case of the vector-valued Triebel-Lizorkin
spaces $F^s_{pq}(0,2\pi ; X)$, $1\le p, q<\infty$, $s>0$.
\end{example}

\subsection*{Acknowledgments}
This work was partially supported by the Air Force Office
of Scientific Research (AFOSR) under the Award No: FA9550-15-1-0027.


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\end{document}