Well-posedness of second-order integro-differential equations

Shangquan Bu; Gang Cai

doi:10.4153/S0008439526102136

Well-posedness of second-order integro-differential equations

Part of: Volterra integral equations Groups and semigroups of linear operators, their generalizations and applications Abstract integral equations, integral equations in abstract spaces Abstract harmonic analysis

Published online by Cambridge University Press: 08 June 2026

Shangquan Bu and

Gang Cai

Show author details

Shangquan Bu: Affiliation:
Tsinghua University, China e-mail: bushangquan@tsinghua.edu.cn
Gang Cai*: Affiliation:
Chongqing Normal University , China
*: e-mail: caigang-aaaa@163.com

Article contents

Abstract
Introduction
Preliminaries
Lp$L^p$-well-posedness of (P5)$(P_5)$
Bp,qs$B_{p,q}^s$-well-posedness and Fp,qs$F_{p,q}^s$-well-posedness of (P5)$(P_5)$
Applications
Footnotes
References

Rights & Permissions

Abstract

In this article, we study well-posedness of the second-order integro-differential equations ($P_5$): $ \mu * (Mu)"(t) + \nu * {(Mu)}'(t) + \eta *(Mu)(t) = \tau * Au(t) + f(t), \ (t\in {\mathbb T}:=[0,2\pi ])$ with periodic boundary conditions $ Mu(0)=Mu(2\pi ), (Mu)'(0)=(Mu)'(2\pi )$, where $A,M$ are closed operators in a Banach space X such that $D(A)\subset D(M)$, $\mu , \nu , \eta ,$ and $\tau $ are finite scalar measures on ${\mathbb R}$. Using known results on Fourier multipliers, we find necessary and sufficient conditions on the measures $\mu , \nu , \eta , \tau $ and the operators $A,M$ to ensure well-posedness of ($P_5$) in $L^p({\mathbb T}; X)$, periodic Besov spaces $B_{p,q}^s({\mathbb T}; X),$ and Triebel–Lizorkin spaces $F_{p,q}^s({\mathbb T}; X)$.

Keywords

Fourier multiplier well-posedness integro-differential equation Besov spaces Triebel–Lizorkin spaces

MSC classification

Primary: 45N05: Abstract integral equations, integral equations in abstract spaces 43A15: $L^p$-spaces and other function spaces on groups, semigroups, etc. 45D05: Volterra integral equations 47D06: One-parameter semigroups and linear evolution equations

Information

Type: Article
Information: Canadian Mathematical Bulletin , First View , pp. 1 - 23

DOI: https://doi.org/10.4153/S0008439526102136 [Opens in a new window]
Copyright: © The Author(s), 2026. Published by Cambridge University Press on behalf of Canadian Mathematical Society

1 Introduction

In a series of recent publications, operator-valued Fourier multipliers on vector-valued function spaces are studied (see, e.g., [Reference Alvarez and Lizama1–Reference Arendt, Batty and Bu3, Reference Arendt and Bu5, Reference Arendt and Bu6, Reference Bu and Kim10, Reference Keyantuo and Lizama14, Reference Ponce17–Reference Weis21]). They are needed to study the existence and uniqueness of differential equations on Banach spaces. In [Reference Arendt, Batty and Bu3, Reference Arendt and Bu5, Reference Arendt and Bu6, Reference Bu and Fang8–Reference Bu and Kim10], the authors study well-posedness of the classical second-order problem ( $P_{1}$ ) on $L^{p}$ spaces, Besov spaces, and Triebel–Lizorkin spaces using operator-valued Fourier multipliers, where

$$ \begin{align*} (P_{1}) \quad \left\{ \begin{array}{@{}ll} u"(t)= Au(t) + f(t), \quad{(t\in{\mathbb T})}\\ u(0)=u(2\pi), \ u'(0)=u'(2\pi),\\ \end{array} \right. \end{align*} $$

here X is a Banach space, A is a closed linear operator defined in $X,$ and f is an X-valued function defined on $[0,2\pi ]$ . ( $P_{1}$ ) is $L^{p}$ -well-posed if and only if $-k^{2}\in \rho (A)$ for all $k\in {\mathbb Z}$ and $(k^{2}R(-k^{2},A))_{k\in {\mathbb Z}}$ is Rademacher bounded whenever X is a UMD Banach space and $1<p<\infty $ [Reference Arendt and Bu5]. In Besov spaces $B_{p,q}^s({\mathbb T}; X)$ and Triebel–Lizorkin spaces $F_{p,q}^s({\mathbb T}; X)$ setting, well-posedness of ( $P_{1}$ ) is equivalent to the assertion that $-k^{2}\in \rho (A)$ for all $k\in {\mathbb Z}$ and $(k^{2}R(-k^{2},A))_{k\in {\mathbb Z}}$ is bounded [Reference Arendt and Bu6, Reference Bu and Kim10]. Recently, well-posedness of the following more general second-order integro-differential equations:

$$ \begin{align*} (P_{2}) \quad \left\{ \begin{array}{@{}ll} u"(t)+\alpha {u}'(t)+\frac{d}{dt}(\int^{t}_{-\infty}b(t-s)u(s)ds)\\\qquad\, =Au(t)-\int^{t}_{-\infty}a(t-s)Au(s)ds+f(t), &{(t\in{\mathbb T})}\\u(0)=u(2\pi), u'(0)=u'(2\pi)\\ \end{array} \right. \end{align*} $$

StartLayout 1st Row left parenthesis upper P 2 right parenthesis StartLayout Enlarged left brace 1st Row 1st Column u quotation mark left parenthesis t right parenthesis plus alpha u prime left parenthesis t right parenthesis plus StartFraction d Over d t EndFraction left parenthesis integral Subscript negative infinity Superscript t Baseline b left parenthesis t minus s right parenthesis u left parenthesis s right parenthesis d s right parenthesis 2nd Row 1st Column equals upper A u left parenthesis t right parenthesis minus integral Subscript negative infinity Superscript t Baseline a left parenthesis t minus s right parenthesis upper A u left parenthesis s right parenthesis d s plus f left parenthesis t right parenthesis comma 2nd Column left parenthesis t element of double struck upper T right parenthesis 3rd Row 1st Column u left parenthesis 0 right parenthesis equals u left parenthesis 2 pi right parenthesis comma u prime left parenthesis 0 right parenthesis equals u prime left parenthesis 2 pi right parenthesis EndLayout EndLayout

has been considered in [Reference Bu and Fang9], where $\alpha $ is a constant and $a, b\in L^1 ({\mathbb R}_+ )$ . Much literature has devoted to the similar first-order integro-differential equations $(P_{3})$ :

$$ \begin{align*} (P_{3}) \quad \left\{ \begin{array}{@{}ll} \gamma_{0}u'(t)+\frac{d}{dt}(\int^{t}_{-\infty}b(t-s)u(s)ds)+\gamma_{\infty}u(t)\\\qquad\, =c_{0}Au(t)-\int^{t}_{-\infty}a(t-s)Au(s)ds+f(t), &{(t\in{\mathbb T})}\\u(0)=u(2\pi),\\ \end{array} \right. \end{align*} $$

StartLayout 1st Row left parenthesis upper P 3 right parenthesis StartLayout Enlarged left brace 1st Row 1st Column gamma 0 u prime left parenthesis t right parenthesis plus StartFraction d Over d t EndFraction left parenthesis integral Subscript negative infinity Superscript t Baseline b left parenthesis t minus s right parenthesis u left parenthesis s right parenthesis d s right parenthesis plus gamma Subscript infinity Baseline u left parenthesis t right parenthesis 2nd Row 1st Column equals c 0 upper A u left parenthesis t right parenthesis minus integral Subscript negative infinity Superscript t Baseline a left parenthesis t minus s right parenthesis upper A u left parenthesis s right parenthesis d s plus f left parenthesis t right parenthesis comma 2nd Column left parenthesis t element of double struck upper T right parenthesis 3rd Row 1st Column u left parenthesis 0 right parenthesis equals u left parenthesis 2 pi right parenthesis comma EndLayout EndLayout

where $\gamma _0, \ \gamma _\infty $ , and $c_0$ are constants, and $a, b\in L^1({\mathbb R}_+)$ . The class of equation of type $(P_{2})$ and $(P_{3})$ arises as models for nonlinear heat conduction in material of fading memory type and population dynamics. Such kind of integro-differential equations has been recently studied in an abstract way by Keyantuo, Lizama, and Poblete. More precisely, they have considered the following first-order problem:

$$ \begin{align*} (P_{4}) \quad \left\{ \begin{array}{@{}ll} \mu *u'(t)+ \nu *u(t) = \eta * Au(t) + f(t), &{(t\in{\mathbb T})}\\ u(0)=u(2\pi),\\ \end{array} \right. \end{align*} $$

here $\mu , \nu ,$ and $\eta $ are finite scalar measures on ${\mathbb R}$ . Under suitable conditions on the measures $\mu , \nu , \eta $ and the operator A, they completely characterized well-posedness of $(P_4)$ in Lebesgue–Bochner spaces $L^p({\mathbb T}; X)$ , periodic Besov spaces $B_{p, q}^s({\mathbb T}; X),$ and periodic Triebel–Lizorkin spaces $F_{p, q}^s({\mathbb T}; X)$ [Reference Keyantuo, Lizama and Poblete15].

In this article, we study similar second-order degenerate integro-differential equations:

$$ \begin{align*} (P_{5}) \quad \left\{ \begin{array}{@{}ll} \mu *(Mu)"(t)+ \nu *(Mu)'(t) + \eta * (Mu)(t) = \tau * Au(t) + f(t), &{(t\in{\mathbb T})}\\ Mu(0)=Mu(2\pi), (Mu)'(0) = (Mu)'(2\pi),\\ \end{array} \right. \end{align*} $$

where $\mu , \nu , \eta ,$ and $\tau $ are finite scalar measures on ${\mathbb R}$ , and A and M are closed linear operators in X such that $D(A)\subset D(M)$ . It is clear that if we take $M=I_X$ , $\mu = \delta _0, \ \nu = \alpha \delta _0 + b, \ \eta = 0,$ and $\tau = \delta _0 - a$ in $(P_5)$ , then we recover $(P_2)$ . If we take $M=I_X$ and $\mu =0$ , then $(P_5)$ becomes $(P_4)$ . Moreover, we take $M=I_X$ , $\mu = \nu = \delta _0$ , and $\tau = a$ , then $(P_5)$ becomes $(3)$ in [Reference Ponce18] (or $(1.3)$ in [Reference Ponce17]) when $b=0$ . Here, we identify a finite scalar measure with its density in $L^1({\mathbb R})$ with respect to Lebesgue measure.

We notice that ( $P_5$ ) has been studied by several authors in a more simple form and different boundary conditions. For instance, Chill and Srivastava [Reference Chill and Srivastava11] have considered $L^{p}$ -well-posedness on a finite interval [0, T) for the abstract second-order problem

$$ \begin{align*} (P_{6}) \quad \left\{ \begin{array}{@{}ll} u"(t)+Bu'(t)+Au(t)=f(t), \ \quad{(t\in{\mathbb T})}\\ u(0)=x_0, \ u'(0)=x_1.\\ \end{array} \right. \end{align*} $$

The semigroup theory and the trace spaces played important roles in their discussion. Under a suitable condition on the operators A and B, they gave a necessary and sufficient condition for $(P_6)$ to be $L^p$ -well-posed.

We will study well-posedness of $(P_{5})$ on three function spaces: $L^{p}({\mathbb T}; X)$ for ${1 < p < \infty }$ , periodic Besov spaces $B^{s}_{p,q}({\mathbb T}; X)$ for $1\leq p, q\leq \infty , \ s>0,$ and periodic Triebel–Lizorkin spaces $F^{s}_{p,q}({\mathbb T}; X)$ for $1\leq p < \infty , \ 1\leq q\leq \infty ,\ s>0$ , where ${{\mathbb T} = [0, 2\pi ]}$ . The main tools will be operator-valued Fourier multiplier theorems obtained in [Reference Arendt and Bu5, Theorem 1.3], [Reference Arendt and Bu6, Theorem 4.5], and [Reference Bu and Kim10, Theorem 3.2] on these function spaces. We completely characterize well-posedness of $(P_5)$ using the Fourier transform of the measures $\mu , \nu , \eta , \tau $ and the closed operators A and M. For instance, when X is a UMD space and $1 < p < \infty $ , $(-k^2\tilde \mu (k) + ik\tilde \nu (k) + \tilde \eta (k))_{k\in {\mathbb Z}}$ and $(\tilde \tau (k))_{k\in {\mathbb Z}}$ are $1$ -regular, then $(P_5)$ is $L^p$ -well-posed if and only if $\rho _{M}(A;\mu ,\nu ,\eta ,\tau )=\mathbb {Z}$ (the resolvent set of $(P_5)$ ) and $\{k^2MN_k: k\in {\mathbb Z}\}$ , $\{AN_k: k\in {\mathbb Z}\}$ , and $\{N_k: k\in {\mathbb Z}\}$ are R-bounded (Theorem 3.4), where

$$\begin{align*}N_k=[(-k^2\tilde \mu(k) + ik\tilde \nu(k) + \tilde\eta(k))M-\tilde\tau(k)A]^{-1}, (k\in{\mathbb Z}), \end{align*}$$

$\tilde \mu , \tilde \tau , \tilde \nu , \tilde \eta $ are the Fourier transforms of $\mu , \tau , \nu , \eta $ . Similar characterizations have also been obtained in $B_{p,q}^s$ -well-posedness and $F_{p,q}^s$ -well-posedness setting. The conditions that we impose on the measures $\mu , \tau , \nu , \eta $ are satisfied by a class of measures, which correspond to the most common kernels encountered in the applications. Furthermore, it is easy to see that our results are in accordance with the known results for $(P_{1})$ [Reference Arendt and Bu5, Reference Arendt and Bu6, Reference Bu and Kim10] and $(P_2)$ [Reference Bu and Fang9].

The article is organized as follows: in Section 2, we give some preliminaries concerning M-boundedness and R-boundedness for operator sequences. In Section 3, we study $L^{p}$ -well-posedness of $(P_5)$ . $B_{p,q}^s$ -well-posedness and $F_{p,q}^s$ -well-posedness of $(P_5)$ will be discussed in Section 4. Section 5 gives some examples to which our abstract results may be applied.

2 Preliminaries

Let X be a Banach space, we denote by ${\mathcal L}(X)$ the set of all bounded linear operators on X. If A is a closed operator in X, we will denote by $\rho (A)$ its resolvent set. When $\lambda \in \rho (A)$ , $R(\lambda , A) = (\lambda - A)^{-1}$ will be the resolvent operator. In what follows, we consider $D(A),D(M)$ as Banach spaces equipped with their respective graph norms. Now, we study the following problems:

where $\mu , \nu , \eta ,$ and $\tau $ are finite scalar measures on the real line, $A,M$ are closed linear operators in X such that $D(A)\subset D(M),$ and f is an X-valued function defined on ${\mathbb T} := [0, 2\pi ]$ . The solution u of $(P_5)$ will be an X-valued function defined on ${\mathbb T}$ (extended by periodicity to ${\mathbb R}$ ). The resolvent set of $(P_5)$ is defined by

$$ \begin{align*}\rho_{M}(A;\mu,\nu,\eta,\tau)\hspace{-1pt}:=\hspace{-1pt} \big\{k\in\mathbb{Z}: (-k^2\tilde \mu(k)\hspace{-1pt} +\hspace{-1pt} ik\tilde \nu(k)\hspace{-1pt} +\hspace{-1pt} \tilde\eta(k))M\hspace{-1pt}-\hspace{-1pt}\tilde\tau(k)A\text{ is invertible from} \end{align*} $$

$$ \begin{align*}D(A)\text{ onto }X \text{ and } [(-k^2\tilde \mu(k) + ik\tilde \nu(k) + \tilde\eta(k))M-\tilde\tau(k)A]^{-1} \in \mathcal{L}(X) \big\}. \end{align*} $$

For the sake of simplicity, when $k\in \rho _{M}(A;\mu ,\nu ,\eta ,\tau )$ , we will use the following notation:

(2.1) $$ \begin{align} N_k&=[(-k^2\tilde \mu(k) + ik\tilde \nu(k) + \tilde\eta(k))M-\tilde\tau(k)A]^{-1},\ (k\in\mathbb{Z})\nonumber\\ &:=[\alpha_kM - \beta_kA]^{-1},\ (k\in\mathbb{Z}). \end{align} $$

If $k\in \rho _{M}(A;\mu ,\nu ,\eta ,\tau )$ , then $MN_k$ and $AN_k$ make sense as $D(A)\subset D(M)$ by assumption, and they belong to $\mathcal {L}(X)$ by the Closed Graph Theorem and the closedness of A and M.

To study well-posedness of $(P_5)$ , we need to introduce some notions and notations. Let $f\in L^{1}({\mathbb T}; X)$ , we denote by

$$ \begin{align*}\hat{f}(k)=\frac{1}{2\pi}\int^{2\pi}_{0}e_{-k}(t)f(t)dt\end{align*} $$

the k-th Fourier coefficient of f, where $ k\in {\mathbb Z}$ and $e_{k}(t)=e^{ikt}$ for $t\in {\mathbb R}$ . For $x\in X$ , we let $e_k\otimes x$ be the X-valued function given by $t\rightarrow e_k(t)x$ . If $\mu $ is a finite scalar measure on ${\mathbb R}$ , we denote by $\tilde \mu $ its Fourier transform, that is,

$$ \begin{align*}\tilde\mu (\omega) = \int_{-\infty}^\infty e^{-i\omega t} \mu (dt), \ \ \ (\omega \in {\mathbb R}). \end{align*} $$

We notice that when $\mu $ has a density $\phi \in L^1({\mathbb R})$ , then

$$ \begin{align*}\tilde\mu (\omega) = \int_{-\infty}^\infty e^{-i\omega t}\phi (t) dt = {\mathcal F} \phi (\omega), \ \ \ (\omega \in {\mathbb R}) \end{align*} $$

coincides with the usual Fourier transform of $\phi $ .

Let $f\in L^1({\mathbb T}; X)$ (extended by periodicity to ${\mathbb R}$ ) and $\mu $ be a finite scalar measure on ${\mathbb R}$ . The convolution of $\mu $ and f is defined by

$$ \begin{align*}\mu * f(t) = \int_{-\infty}^{\infty} f(t-s) \mu (ds), \ \ \ (t\in {\mathbb R}). \end{align*} $$

It is clear that this definition makes sense and $\mu * f \in L^1({\mathbb T}; X)$ . By Fubini’s theorem, we have

(2.2) $$ \begin{align} \widehat{\mu *f}(k) = \tilde \mu(k) \hat f(k), \ \ \ (k\in{\mathbb Z}). \end{align} $$

Definition 2.1 [Reference Arendt and Bu5]

Let X be a Banach space and let $1 \leq p < \infty $ . We say that a sequence $(M_{k})_{k\in {\mathbb Z}}\subset \mathcal {L}(X)$ is an $L^p$ -multiplier, if for each $f\in L^p ({\mathbb T}; X)$ , there exists a unique $g\in L^p ({\mathbb T}; X)$ such that $\hat {g}(k)=M_{k}\hat {f}(k)$ for all $k\in {\mathbb Z}$ .

It follows from the Closed Graph Theorem that if $(M_{k})_{k\in {\mathbb Z}}$ is an $L^p$ -multiplier, then there exists a constant $C>0$ such that for $f\in L^p({\mathbb T}; X)$ , we have $\|\sum _{k\in {\mathbb Z}}e_{k}\otimes {M}_{k}\hat {f}(k)\|_{L^p}\leq {C}\|f\|_{L^p}$ . This implies that when $(M_{k})_{k\in {\mathbb Z}}$ is an $L^p$ -multiplier, then $(M_{k})_{k\in {\mathbb Z}}$ must be norm bounded. It is also clear from the definition that if $(M_{k})_{k\in {\mathbb Z}}$ and $(N_{k})_{k\in {\mathbb Z}}$ are $L^p$ -multipliers, then $(N_{k}M_{k})_{k\in {\mathbb Z}}$ is also an $L^p$ -multiplier.

In the study of $L^p$ -well-posedness of $(P_5)$ , we need the notion of Rademacher boundedness for sets of bounded linear operators. We refer to Bourgain [Reference Bourgain7], Weis [Reference Weis20, Reference Weis21], and Arendt and Bu [Reference Arendt and Bu5] for more information about this property. We merely recall the definition and some basic properties.

Let $r_{j}$ be the j-th Rademacher function on [0,1] given by $r_j(t) = \mathrm{sgn} (\sin (2^{j-1}\pi t))$ . For $x\in X$ , we denote by $r_{j}\otimes x$ the vector-valued function $t\rightarrow r_{j}(t)x$ .

Definition 2.2 Let X be a Banach space. A family $\textbf {T}\subset \mathcal {L}(X)$ is called R-bounded (R stands for Rademacher) if there exists $C \geq 0$ such that

(2.3) $$ \begin{align} \Big\Vert \sum^{n}_{j=1}r_{j}\otimes T_{j}x_{j}\Big\Vert _{L^{1}(0,1; X)}\leq C\Big\Vert \sum^{n}_{j=1}r_{j}\otimes x_{j}\Big\Vert _{L^{1}(0,1; X)} \end{align} $$

for all $T_{1}, \ldots , T_{n}\in \textbf {T}$ , $x_{1}, \ldots , x_{n}\in X$ and $n\in {\mathbb N}.$

Remark 2.3

(a) Let $\textbf {S,T}\subset {\mathcal L}(X)$ be R-bounded sets. Then it is clear from the definition that $\textbf {ST}:=\{ST: S\in \textbf {S}, T\in \textbf {T}\}$ and $\textbf {S}+\textbf {T}:=\{S + T: S\in \textbf {S}, T\in \textbf {T}\}$ are R-bounded.
(b) Each subset $\textbf {M}\subset \mathcal {L}(X)$ of the form $\textbf {M}=\{\lambda I: \lambda \in \Omega \}$ is R-bounded whenever $\Omega \subset {\mathbb C}$ is bounded. This follows from Kahane’s contraction principle [Reference Pietsch and Wenzel16, Section 3.5.4].

Let X be a Banach space and $(M_k)_{k\in {\mathbb Z}} \subset {\mathcal L}(X)$ . We let

$$ \begin{align*}\Delta^0M_k = M_k, \ \ \ \Delta M_k = \Delta^1M_k = M_{k+1} - M_k, \ \ \ (k\in{\mathbb Z}), \end{align*} $$

and

$$ \begin{align*}\Delta^nM_k = \Delta (\Delta^{n-1}M_k), \ \ \ \ (k\in{\mathbb Z}) \end{align*} $$

for $n = 2, 3, \ldots $ . The sequence $(\Delta ^nM_k)_{k\in {\mathbb Z}}$ is the n-th derivative of $(M_k)_{k\in {\mathbb Z}}$ . Following [Reference Keyantuo, Lizama and Poblete15], we say that $(M_k)_{k\in {\mathbb Z}}$ is M-bounded (M stands for Marcinkiewicz) of order n for some $n\in {\mathbb N}\cup \{0\}$ , if

(2.4) $$ \begin{align} \sup_{0\leq l\leq n} \sup_{k\in{\mathbb Z}} \big\Vert k^l\Delta^lM_k\big\Vert < \infty. \end{align} $$

Analogously, we define M-boundedness of order n for sequences $(a_k)_{k\in {\mathbb Z}}$ of complex numbers. This notion is very convenient when studying operator-valued Fourier multipliers on vector-valued function spaces on ${\mathbb T}$ . We also need the notion of regularity for scalar sequences.

Definition 2.4 [Reference Keyantuo, Lizama and Poblete15]

Let $(a_k)_{k\in {\mathbb Z}}$ be a scalar sequence, we say that $(a_k)_{k\in {\mathbb Z}}$ is n-regular for some $n\geq 1$ , if there exists $k_0>0$ , such that $a_k\neq 0$ for all $|k|\geq k_0$ and

(2.5) $$ \begin{align} \sup_{0\leq l\leq n} \sup_{|k|\geq k_0} \big\vert k^l(\Delta^la_k)/a_k\big\vert < \infty. \end{align} $$

Note that if $(a_k)_{k\in {\mathbb Z}}$ is $1$ -regular, then $\lim _{|k|\rightarrow \infty }\frac {a_{k+1}}{a_k}=1$ . It is clear from the definitions that when $(a_k)_{k\in {\mathbb Z}}$ is bounded and n-regular, then it is M-bounded of order n. Therefore, by Remark 2.3(b), we obtain that $(a_kI_X)$ is $L^p$ -Fourier multiplier when $(a_k)_{k\in {\mathbb Z}}$ is bounded and $1$ -regular. It can be seen easily that $(k^2)_{k\in \mathbb {Z}}$ is $3$ -regular. We will use the following results established in [Reference Keyantuo, Lizama and Poblete15].

Theorem 2.5 [Reference Keyantuo, Lizama and Poblete15, Theorem 4.6]

Let $(a_k)_{k\in {\mathbb Z}}$ and $(b_k)_{k\in {\mathbb Z}}$ be given sequences of complex numbers and let $1\leq n \leq 3$ . Then

(i) $(a_k)_{k\in {\mathbb Z}}$ is n-regular if and only if $(1/a_k)_{k\in {\mathbb Z}}$ is n-regular;
(ii) if $(a_k)_{k\in {\mathbb Z}}$ and $(b_k)_{k\in {\mathbb Z}}$ are n-regular, then $(a_k/b_k)$ is n-regular.

3 $L^p$ -well-posedness of $(P_5)$

In this section, we consider $L^p$ -well-posedness of $P_5$ when $1 < p < \infty $ . First, we introduce the periodic Sobolev spaces $W^{m,p}_{\mathrm{per}}({\mathbb T}; X)$ when $m = 1, 2$ . We let $W^{1,p}_{\mathrm{per}}({\mathbb T}; X)$ be the space of all $u\in L^p({\mathbb T}; X)$ such that u is a.e. differentiable, $u'\in L^p({\mathbb T}; X)$ and $u(0) = u(2\pi )$ . It is shown in [Reference Arendt and Bu5] that

(3.1) $$ \begin{align} W^{1,p}_{\mathrm{per}}({\mathbb T}; X) = \Big\{u\in L^p({\mathbb T}; X): & \text{ there exists }\ v\in L^p({\mathbb T}; X), \\ &\hspace{-1.4cm}\text{such that} \ \hat v(k) = ik\hat u(k) \ \text{for all} \ k\in {\mathbb Z} \Big\}. \nonumber \end{align} $$

StartLayout 1st Row 1st Column upper W Subscript per Superscript 1 comma p Baseline left parenthesis double struck upper T semicolon upper X right parenthesis equals left brace u element of upper L Superscript p Baseline left parenthesis double struck upper T semicolon upper X right parenthesis colon 2nd Column there exists v element of upper L Superscript p Baseline left parenthesis double struck upper T semicolon upper X right parenthesis comma 2nd Row 1st Column Blank 2nd Column such that ModifyingAbove v With caret left parenthesis k right parenthesis equals i k ModifyingAbove u With caret left parenthesis k right parenthesis for all k element of double struck upper Z right brace period EndLayout

If u and v are as in (3.1), then $v = u'$ a.e.and $u(0) = u(2\pi )$ . We define $W^{2,p}_{\mathrm{per}}({\mathbb T}; X)$ as the space of all $u\in W^{1,p}_{\mathrm{per}}({\mathbb T}; X)$ such that $u' \in W^{1,p}_{\mathrm{per}}({\mathbb T}; X)$ . Thus, by (3.1),

(3.2) $$ \begin{align} W^{2,p}_{\mathrm{per}}({\mathbb T}; X) = \Big\{u\in L^p({\mathbb T}; X): & \text{ there exists }\ w\in L^p({\mathbb T}; X), \\ &\hspace{-1.4cm}\text{such that} \ \hat w(k) = -k^2\hat u(k) \ \text{for all} \ k\in {\mathbb Z} \Big\}.\nonumber \end{align} $$

StartLayout 1st Row 1st Column upper W Subscript per Superscript 2 comma p Baseline left parenthesis double struck upper T semicolon upper X right parenthesis equals left brace u element of upper L Superscript p Baseline left parenthesis double struck upper T semicolon upper X right parenthesis colon 2nd Column there exists w element of upper L Superscript p Baseline left parenthesis double struck upper T semicolon upper X right parenthesis comma 2nd Row 1st Column Blank 2nd Column such that ModifyingAbove w With caret left parenthesis k right parenthesis equals minus k squared ModifyingAbove u With caret left parenthesis k right parenthesis for all k element of double struck upper Z right brace period EndLayout

If u and w are as in (3.2), then $w = u"$ a.e.and $u(0) = u(2\pi ), \ u'(0) = u'(2\pi )$ .

Let $1\leq p<\infty $ , the solution space of the $L^p$ -well-posedness of $(P_5)$ is defined by

$$ \begin{align*} S_p(A,M):=\big\{u\in L^p(\mathbb{T}; D(A)):\ Mu\in W_{\text{per}}^{2,p}(\mathbb{T}; X)\big\},\nonumber \end{align*} $$

here we consider $D(A)$ and $D(M)$ as Banach spaces equipped with their graph norms. $S_p(A,M)$ is a Banach space equipped with the norm

$$ \begin{align*} \|u\|_{S_p(A,M)}:=\|u\|_{L^p}+\|Au\|_{L^p}+\|(Mu)'\|_{L^p}+\|(Mu)"\|_{L^p}.\nonumber \end{align*} $$

If $u\in S_p(A,M)$ , then $Mu$ and $(Mu)'$ are X-valued continuous functions on $\mathbb {T}$ , and $Mu(0)=Mu(2\pi )$ , $(Mu)'(0)=(Mu)'(2\pi )$ by [Reference Arendt and Bu5, Lemma 2.1].

We say that $(P_5)$ is $L^p$ -well-posed, if for each $f\in L^p({\mathbb T}; X)$ , there exists a unique $u\in S_p(A,M)$ such that $(P_5)$ is satisfied a.e. on ${\mathbb T}$ . We notice that when $(P_5)$ is $L^p$ -well-posed, then each term appeared in $(P_5)$ is in $L^p({\mathbb T}; X)$ , this follows from Young’s inequality. The following result is fundamental in our investigation. For the concept of UMD Banach spaces, we refer to [Reference Bourgain7].

If $(P_5)$ is $L^p$ -well-posed and $u\in S_p(A,M)$ is the unique strong $L^p$ -solution of $(P_5)$ , then there exists a constant $C>0$ such that for each $f\in L^p(\mathbb {T}; X)$ , we have

(3.3) $$ \begin{align} \|u\|_{S_p(A,M)}\leq C\|f\|_{L^p}. \end{align} $$

This can be obtained easily from the Closed Graph Theorem. In order to prove our main result of this section, we need the following preparations.

Theorem 3.1 [Reference Arendt and Bu5, Theorem 1.3]

Let X be a UMD Banach space, $(M_k)_{k\in {\mathbb Z}}\subset {\mathcal L}(X),$ and let $1 < p < \infty $ . We assume that the sets $\{M_k: \ k\in {\mathbb Z}\}$ and $\{k\Delta M_k: \ k\in {\mathbb Z}\}$ are R-bounded, then $(M_k)$ is an $L^p$ -multiplier.

Proposition 3.2 [Reference Arendt and Bu5, Proposition 1.11]

Let $X,\ Y$ be Banach spaces, $1\leq p < \infty $ , and let $(M_k)_{k\in \mathbb {Z}}\subset \mathcal {L}(X,Y)$ be an $L^p$ -Fourier multiplier, then the set $\{M_k:\ k\in \mathbb {Z}\}$ is R-bounded.

We will use the following result due to Conejero, Lizama, Murillo-Arcila, and Seoane-Sepulveda.

Theorem 3.3 [Reference Conejero, Lizama, Murillo-Arcila and Seoane-Sepúlveda12, Theorem 1.1]

Let $T:D(T)\subset X\rightarrow X$ be a closed linear operator defined in Banach space X. For each $k\in {\mathbb Z}$ , let $H_k:X\rightarrow D(T)$ be a sequence of bounded and linear operators such that $0\in \rho (H_k)$ for all $k\in {\mathbb Z}$ . Suppose that ${(c_k)_{k\in {\mathbb Z}}\subset {\mathbb C}}$ is a 1-regular sequence and denote $M_k:=c_kTH_k$ and $L_k:=(H_k^{-1}-H_{k+1}^{-1})H_k$ . If ${\{M_k:k\in {\mathbb Z}\}}$ and $\{kL_k:k\in {\mathbb Z}\}$ are R-bounded sets, then $\{M_k:k\in {\mathbb Z}\}$ is MR-bounded of order 1. If, in addition, $(c_k)_{k\in {\mathbb Z}}$ is 2-regular and the set $\{k^2\Delta L_k:k\in {\mathbb Z}\}$ is R-bounded, then $\{M_k:k\in {\mathbb Z}\}$ is MR-bounded of order 2.

The following is the main result of this section.

Theorem 3.4 Let X be a UMD Banach space and let $1 < p < \infty $ . Assume that A and M are closed operators in X such that $D(A)\subset D(M)$ , $\mu , \nu , \eta ,$ and $\tau $ are bounded scalar measures on ${\mathbb R}$ such that $(\alpha _k)_{k\in {\mathbb Z}}$ and $(\beta _k)_{k\in {\mathbb Z}}$ are $1$ -regular, where $\alpha _k$ and $\beta _k$ are defined by (2.1). Then the following assertions are equivalent:

(i) $(P_5)$ is $L^p$ -well-posed;
(ii) $\rho _{M}(A;\mu ,\nu ,\eta ,\tau )=\mathbb {Z}$ and $\{k^2MN_k:k\in {\mathbb Z}\}$ , $\{AN_k:k\in {\mathbb Z}\}$ , $\{N_k:k\in {\mathbb Z}\}$ are R-bounded, where $N_k$ is defined by (2.1).

Proof Now we assume that (ii) is true and we are going to show that $(P_5)$ is $L^p$ -well-posed for $1 < p < \infty $ . We first show the uniqueness. Let $u\in S_p(A,M)$ such that

$$ \begin{align*}\mu * (Mu)"(t)+ \nu * (Mu)'(t) + \eta * (Mu)(t) - \tau * Au(t) = 0 \end{align*} $$

a.e. on ${\mathbb T}$ and $ Mu(0) = Mu(2\pi ), \ (Mu)'(0) = (Mu)'(2\pi )$ . Taking Fourier coefficient on both sides, we find that $\big [(-k^2\tilde \mu (k) + ik\tilde \nu (k) + \tilde \eta (k))M - \tilde \tau (k)A\big ]\hat u(k) = 0$ for $k\in {\mathbb Z}$ . Here, we have used (2.2) and the fact that when $h\in L^p({\mathbb T}; D(A))$ , then $\hat h(k) \in D(A)$ for all $k\in {\mathbb Z}$ and $A\hat h(k) = \widehat {Ah}(k)$ [Reference Arendt and Bu5, Lemma 3.1]. It follows that $\hat u(k) = 0$ for $k\in {\mathbb Z}$ as $\rho _{M}(A;\mu ,\nu ,\eta ,\tau )={\mathbb Z}$ by assumption. Thus, $u = 0$ by the Uniqueness Theorem [Reference Arendt and Bu5, p. 314]. We have shown the uniqueness.

Now we show the existence. Let $L_k=(N_k^{-1}-N_{k+1}^{-1})N_k$ , we observe

$$ \begin{align*} L_k&=[\alpha_kM - \beta_kA - \alpha_{k+1}M + \beta_{k+1}A]N_k\\&=-(\Delta\alpha_k)MN_k + (\Delta\beta_k)AN_k\\&=\frac{\Delta \alpha_k}{\alpha_k}(\alpha_kMN_k) + \frac{\Delta\beta_k}{\beta_k}(\beta_kAN_k) \end{align*} $$

StartLayout 1st Row 1st Column upper L Subscript k 2nd Column equals left bracket alpha Subscript k Baseline upper M minus beta Subscript k Baseline upper A minus alpha Subscript k plus 1 Baseline upper M plus beta Subscript k plus 1 Baseline upper A right bracket upper N Subscript k Baseline 2nd Row 1st Column Blank 2nd Column equals minus left parenthesis normal upper Delta alpha Subscript k Baseline right parenthesis upper M upper N Subscript k Baseline plus left parenthesis normal upper Delta beta Subscript k Baseline right parenthesis upper A upper N Subscript k Baseline 3rd Row 1st Column Blank 2nd Column equals StartFraction normal upper Delta alpha Subscript k Baseline Over alpha Subscript k Baseline EndFraction left parenthesis alpha Subscript k Baseline upper M upper N Subscript k Baseline right parenthesis plus StartFraction normal upper Delta beta Subscript k Baseline Over beta Subscript k Baseline EndFraction left parenthesis beta Subscript k Baseline upper A upper N Subscript k Baseline right parenthesis EndLayout

when $|k|\geq k_0$ . We also notice

(3.4) $$ \begin{align} \beta_kAN_k = \alpha_kMN_k - I_X. \end{align} $$

Since $\{k^2MN_k:k\in {\mathbb Z}\}$ is R-bounded and $\mu , \nu , \eta ,$ and $\tau $ are bounded scalar measures on ${\mathbb R}$ , we have that $\{\alpha _kMN_k:k\in {\mathbb Z}\}$ , $\{kMN_k:k\in {\mathbb Z}\},$ and $\{MN_k:k\in {\mathbb Z}\}$ are also R-bounded. Here, we have used the fact that the sums and products of R-bounded sets are also still R-bounded. We deduce by (3.4) that $\{\beta _kAN_k:k\in {\mathbb Z}\}$ is also R-bounded. Therefore, we obtain that $\{kL_k:k\in {\mathbb Z}\}$ is R-bounded, here we have use the fact that $(\alpha _k)_{k\in {\mathbb Z}}$ and $(\beta _k)_{k\in {\mathbb Z}}$ are $1$ -regular. If we put $c_k=1$ , $T=M$ (or $A,I_X$ ), and $H_k=N_k$ in Theorem 3.3, then by Theorem 3.1, we deduce that $(MN_k)_{k\in {\mathbb Z}} $ , $(AN_k)_{k\in {\mathbb Z}} $ , and $(N_k)_{k\in {\mathbb Z}} $ are $L^p$ -multipliers. Similarly, let $c_k=k^2$ (or $\beta _k$ ), $T=M$ (or A), and $H_k=N_k$ in Theorem 3.3, we obtain by Theorem 3.1 that $(k^2MN_k)_{k\in {\mathbb Z}} $ and $(\beta _kAN_k)_{k\in {\mathbb Z}} $ are $L^p$ -multipliers, here we have used the fact that $(k^2)_{k\in \mathbb {Z}}$ and $(\beta _k)_{k\in {\mathbb Z}}$ are $1$ -regular. Let $f\in L^p({\mathbb T}; X)$ be fixed. Then there exists $u_1, u_2, u_3, u\in L^p({\mathbb T}; X)$ such that

(3.5) $$ \begin{align} \hat u(k) = N_k\hat f(k),\ \widehat {u_1}(k) = AN_k\hat f(k) \end{align} $$

and

(3.6) $$ \begin{align} \widehat {u_2}(k) = MN_k\hat f(k),\ \widehat {u_3}(k) = -k^2MN_k\hat f(k) \end{align} $$

for $k\in {\mathbb Z}$ . We deduce from (3.4) that $\hat u(k) \in D(A)$ and $\widehat {u_1}(k) = AN_k\hat f(k) = A\hat u(k)$ for $k\in {\mathbb Z}$ . Hence, $u\in L^p({\mathbb T}; D(A))$ by [Reference Arendt and Bu5, Lemma 3.1]. From (4.15) that $\widehat {u_2}(k) = MN_k\hat f(k) = M\hat u(k)$ for $k\in {\mathbb Z}$ . Hence, $Mu\in L^p({\mathbb T}; X)$ by [Reference Arendt and Bu5, Lemma 3.1]. Again by (3.5), we have $\widehat {u_3}(k) = -k^2M\hat u(k)$ for $k\in {\mathbb Z}$ . This implies that $Mu\in W^{2,p}_{\mathrm{per}}({\mathbb T}; X)$ and $Mu(0) = Mu(2\pi ), \ (Mu)'(0) = (Mu)'(2\pi ), \ (Mu)" = u_3$ by (3.2).

From the equality $\hat u(k) = N_k\hat f(k)$ , we deduce that

$$ \begin{align*}\big[(-k^2\tilde \mu (k) + ik\tilde \nu(k) + \tilde\eta(k))M - \tilde\tau (k)A\big]\hat u(k) = \hat f(k) \end{align*} $$

for all $k\in {\mathbb Z}$ . This is equivalent to say that

$$ \begin{align*}\mu*(Mu)"(t)+ \nu *(Mu)'(t) + \eta * (Mu)(t) - \tau * Au(t) = f(t) \end{align*} $$

a.e. on ${\mathbb T}$ by the Uniqueness Theorem [Reference Arendt and Bu5, p. 314]. Here, we have used the fact that $\widehat {u'}(k) = ik\hat u(k)$ when $k\in {\mathbb Z}$ . This shows the existence. Thus, the implication (ii) $\Rightarrow $ (i) is true.

Now we give the proof for the implication (i) $\Rightarrow $ (ii). Assume that $(P_5)$ is $L^p$ -well-posed for $1 < p < \infty $ . Let $k\in \mathbb {Z}$ and $y\in X$ be fixed, we consider the function f defined by $f(t)=e^{ikt}y$ when $t\in \mathbb {T}$ . Then it is clear that $f\in L^p(\mathbb {T};X), \hat {f}(k)=y,$ and $\hat {f}(n)=0$ when $n\neq k$ . Since $(P_5)$ is $L^p$ -well-posed, there exists a unique $u\in S_p(A,M)$ such that

(3.7) $$ \begin{align} \mu*(Mu)"(t)+ \nu *(Mu)'(t) + \eta * (Mu)(t) - \tau * Au(t) = f(t) \end{align} $$

a.e. on $\mathbb {T}$ . We have $\hat {u}(n)\in D(A)$ when $n\in \mathbb {Z}$ by [Reference Arendt and Bu5, Lemma 3.1] as $u\in L^p (\mathbb {T}; D(A))$ . Taking Fourier transforms on both sides of (3.6), we obtain

(3.8) $$ \begin{align} \big[(-k^2\tilde \mu (k) + ik\tilde \nu(k) + \tilde\eta(k))M - \tilde\tau (k)A\big]\hat{u}(k)=y \end{align} $$

and $\big [(-n^2\tilde \mu (n) + in\tilde \nu (n) + \tilde \eta (n))M - \tilde \tau (n)A\big ]\hat {u}(n)=0$ when $n\neq k$ . This implies that the operator $\big [(-k^2\tilde \mu (k) + ik\tilde \nu (k) + \tilde \eta (k))M - \tilde \tau (k)A\big ]$ defined on $D(A)$ with values in X is surjective. To show that it is also injective, we let $x\in D(A)$ be such that

$$ \begin{align*}\big[(-k^2\tilde \mu (k) + ik\tilde \nu(k) + \tilde\eta(k))M - \tilde\tau(k)A\big]x=0. \end{align*} $$

Let u be the function given by $u(t)=e^{ikt}x$ when $t\in \mathbb {T}$ , then it is clear that $u\in S_p(A,M)$ and $(P_5)$ is satisfied a.e. on $\mathbb {T}$ when $f=0$ . Thus, u is a strong $L^p$ -solution of $(P_5)$ when taking $f=0$ . We get $x=0$ by the uniqueness assumption. We have shown that the operator $\big [(-k^2\tilde \mu (k) + ik\tilde \nu (k) + \tilde \eta (k))M - \tilde \tau (k)A\big ]$ from $D(A)$ into X is injective. Therefore, $\big [(-k^2\tilde \mu (k) + ik\tilde \nu (k) + \tilde \eta (k))M - \tilde \tau (k)A\big ]$ is bijective from $D(A)$ onto X.

Next, we show that $\big [(-k^2\tilde \mu (k) + ik\tilde \nu (k) + \tilde \eta (k))M - \tilde \tau (k)A\big ]^{-1}\in \mathcal {L}(X)$ . For $f(t)=e^{ikt}y$ , we let $u\in S_p(A,M)$ such that u is the unique strong $L^p$ -solution of $(P_5)$ . Then

(3.9) $$ \begin{align} \hat{u}(n)= \begin{cases}0&n\neq k,\\ \big[(-k^2\tilde \mu (k) + ik\tilde \nu(k) + \tilde\eta(k))M - \tilde\tau(k)A\big]^{-1}y&n=k, \end{cases} \end{align} $$

by (3.7). This implies that u is given by

$$ \begin{align*}u(t) = e^{ikt}\big[(-k^2\tilde \mu (k) + ik\tilde \nu(k) + \tilde\eta(k))M - \tilde\tau(k)A\big]^{-1}y \end{align*} $$

when $t\in \mathbb {T}$ . By (3.3), there exists a constant $C>0$ independent from y and k, such that $\|u\|_{S_p(A,M)}\leq C\|f\|_{L^p}$ . This implies that

$$ \begin{align*}\big\Vert \big[(-k^2\tilde \mu (k) + ik\tilde \nu(k) + \tilde\eta(k))M - \tilde\tau(k)A\big]^{-1}y\big\Vert\leq C\|y\| \end{align*} $$

when $y\in X$ , or equivalently,

$$ \begin{align*}\big\|\big[(-k^2\tilde \mu (k) + ik\tilde \nu(k) + \tilde\eta(k))M - \tilde\tau(k)A\big]^{-1}\big\|\leq C. \end{align*} $$

Therefore, we have shown that $k\in \rho _{M}(A;\mu ,\nu ,\eta ,\tau )$ for all $k\in \mathbb {Z}$ . Thus, $\rho _{M}(A;\mu ,\nu ,\eta ,\tau )=\mathbb {Z}$ .

Finally, we prove that $(k^2MN_k)_{k\in \mathbb {Z}}$ and $(N_k)_{k\in \mathbb {Z}}$ define $L^p$ -Fourier multipliers. Let $f\in L^p(\mathbb {T};X)$ , then there exists $u\in S_p(A,,M)$ , a strong $L^p$ -solution of $(P_5)$ by assumption. Taking Fourier transforms on both sides of $(P_5)$ , we get that $\hat {u}(k)\in D(A)$ by [Reference Arendt and Bu5, Lemma 3.1] and

$$ \begin{align*}[(-k^2\tilde \mu (k) + ik\tilde \nu(k) + \tilde\eta(k))M - \tilde\tau(k)A]\hat{u}(k) = \hat{f}(k) \end{align*} $$

for $k\in \mathbb {Z}$ . Since $(-k^2\tilde \mu (k) + ik\tilde \nu (k) + \tilde \eta (k))M - \tilde \tau (k)A$ is invertible, we have

$$ \begin{align*}\hat{u}(k) = [(-k^2\tilde \mu (k) + ik\tilde \nu(k) + \tilde\eta(k))M - \tilde\tau(k)A]^{-1}\hat{f}(k) = N_k\hat f(k) \end{align*} $$

when $k\in \mathbb {Z}$ . We conclude from $u\in S_p(A,M)$ that $Mu\in W_{\text {per}}^{2,p}(\mathbb {T};X)$ . We have

$$ \begin{align*}\widehat{(Mu)"}(k)=-k^2M\hat{u}(k), \end{align*} $$

when $k\in \mathbb {Z}$ . We obtain that $(k^2MN_k)_{k\in \mathbb {Z}}$ and $(N_k)_{k\in \mathbb {Z}}$ define $L^p$ -Fourier multipliers. It follows from Proposition 3.2 that the sets $\{k^2MN_k:k\in \mathbb {Z}\}$ and $\{N_k:k\in \mathbb {Z}\}$ are R-bounded. We have shown that the implication (i) $\Rightarrow $ (ii) is true. This completes the proof.

We notice that, without the hypothesis that X is a UMD Banach space, the implication (i) $\Rightarrow $ (ii) remains valid under the same assumptions on $A,M$ and the measures $\mu , \nu , \eta ,$ and $\tau $ . Furthermore, since the second assertion in Theorem 3.4 does not depend on the parameter $1< p < \infty $ , we have the following easy but useful consequence.

Corollary 3.5 Let X be a UMD Banach space. Assume that A and M are closed operators in X such that $D(A)\subset D(M)$ , $\mu , \nu , \eta ,$ and $\tau $ are bounded scalar measures on ${\mathbb R}$ such that $(\alpha _k)_{k\in {\mathbb Z}}$ and $(\beta _k)_{k\in {\mathbb Z}}$ are $1$ -regular. Then if $(P_5)$ is $L^p$ -well-posed for one $1 < p < \infty $ , then it is $L^p$ -well-posed for all $1 < p < \infty $ .

4 $B_{p,q}^s$ -well-posedness and $F_{p,q}^s$ -well-posedness of $(P_5)$

In this section, we study $B_{p,q}^s$ -well-posedness and $F_{p,q}^s$ -well-posedness of $(P_5)$ . Firstly, we briefly recall the definition of periodic Besov spaces in the vector-valued case introduced in [Reference Arendt and Bu6]. Let ${\mathcal S}({\mathbb R})$ be the Schwartz space of all rapidly decreasing smooth functions on ${\mathbb R}$ . Let ${\mathcal D}({\mathbb T})$ be the space of all infinitely differentiable functions on ${\mathbb T}$ equipped with the locally convex topology given by the seminorms $\|f\|_{\alpha }=sup_{x\in {\mathbb T}}|f^{(\alpha )}(x)|$ for $\alpha \in {\mathbb N}_{0} ={\mathbb N}\cup \{0\}$ . Let ${\mathcal D}^{'}({\mathbb T}; X) =\mathcal {L}({\mathcal D}({\mathbb T}),X)$ be the space of all bounded linear operators from ${\mathcal D}({\mathbb T})$ to X. In order to define Besov spaces, we consider the dyadic-like subsets of ${\mathbb R}$ :

$$ \begin{align*}I_{0}=\bigl\{{t\in{\mathbb R}:\ |t|\leq2}\bigr\},{~}I_{k}=\bigl\{{t\in{\mathbb R}:\ 2^{k-1}<|t|\leq2^{k+1}}\bigr\} \end{align*} $$

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 $supp(\phi _{k})\subset \bar {I_{k}}$ for each $k\in {\mathbb N}_{0}$ ,

$$ \begin{align*} \sum_{k\in{\mathbb N}_{0}}\phi_{k}(x)=1 \qquad\text{for}\quad x\in{\mathbb R}, \end{align*} $$

and for each $\alpha \in {\mathbb N}_{0}$ ,

$$ \begin{align*}sup_{\begin{subarray}{l} x\in{\mathbb R}\\k\in{\mathbb N}_{0} \end{subarray}} 2^{k\alpha}|\phi_{k}^{(\alpha)}(x)|<\infty. \end{align*} $$

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 periodic Besov space is defined by

$$ \begin{align*} B^{s}_{p,q}({\mathbb T}; X)= &\left\{\vphantom{\left(\sum_{j\geq0}2^{sjq}\left\Vert\sum_{k\in{\mathbb Z}}e_{k}\otimes\phi_{j}(k)\hat{f}(k)\right\|^{q}_{p}\right)^{1/q}}f\in{\mathcal D}^{'}({\mathbb T}; X):\ \left\|f\right\|_{B^{s}_{p,q}}\right. \\& \quad \left.=\left(\sum_{j\geq0}2^{sjq}\left\Vert\sum_{k\in{\mathbb Z}}e_{k}\otimes\phi_{j}(k)\hat{f}(k)\right\|^{q}_{p}\right)^{1/q}<\infty\right\} \end{align*} $$

StartLayout 1st Row 1st Column upper B Subscript p comma q Superscript s Baseline left parenthesis double struck upper T semicolon upper X right parenthesis equals 2nd Column left brace f element of script upper D Superscript Super Superscript prime Superscript Baseline left parenthesis double struck upper T semicolon upper X right parenthesis colon StartMetric f EndMetric Subscript upper B Sub Subscript p comma q Sub Superscript s Subscript Baseline 2nd Row 1st Column Blank 2nd Column equals left parenthesis sigma summation Underscript j greater than or equals 0 Endscripts 2 Superscript s j q Baseline StartMetric sigma summation Underscript k element of double struck upper Z Endscripts e Subscript k Baseline circled times phi Subscript j Baseline left parenthesis k right parenthesis ModifyingAbove f With caret left parenthesis k right parenthesis EndMetric Subscript p Superscript q Baseline right parenthesis Superscript 1 divided by q Baseline less than infinity right brace EndLayout

with the usual modification if $q=\infty $ . The space $B^{s}_{p,q}({\mathbb T}; X)$ is independent from the choice of $\phi $ and different choices of $\phi $ lead to equivalent norms $\Vert \cdot \Vert _{B^{s}_{p,q}}$ on $B^{s}_{p,q}({\mathbb T}; X)$ . $B^{s}_{p,q}({\mathbb T}; X)$ equipped with the norm $\Vert \cdot \Vert _{B^{s}_{p,q}}$ is a Banach space. When $1\leq p < \infty $ and $s>0$ , we have $B^{s}_{p,q}({\mathbb T}; X) \subset L^p({\mathbb T}; X)$ and the embedding is continuous. See [Reference Arendt and Bu6, Section 2] for more information about the space $B^{s}_{p,q}({\mathbb T}; X)$ .

Let $1\leq p, q\leq \infty ,$ and $s> 0$ . We say that $(P_5)$ is $B_{p,q}^s$ -well-posed, if for each $f\in B_{p,q}^s({\mathbb T}; X)$ , there exists a unique $u\in B_{p,q}^s({\mathbb T}; D(A))$ such that $Mu\in B_{p,q}^{s+2}({\mathbb T}; X)$ and $(P_5)$ is satisfied a.e. on ${\mathbb T}$ . We recall that when $u\in B_{p,q}^{s+2}({\mathbb T}; X)$ for some $s>0$ , then u is twice differentiable a.e. on ${\mathbb T}$ and $u' \in B_{p,q}^{s+1}({\mathbb T}; X), \ u" \in B_{p,q}^s({\mathbb T}; X)$ [Reference Arendt and Bu6]. We remark that when $(P_5)$ is $B_{p,q}^s$ -well-posed, then each term appeared in $(P_5)$ is in $B_{p,q}^s({\mathbb T}; X)$ , this follows from Young’s inequality.

The solution space of the $B_{p,q}^s$ -well-posedness of $(P_5)$ is defined by

$$ \begin{align*} S_{p,q,s}(A,M):=\big\{u\in B_{p,q}^s(\mathbb{T}; D(A)):\ Mu\in B_{p,q}^{s+2}(\mathbb{T}; X)\big\},\nonumber \end{align*} $$

here we consider $D(A)$ and $D(M)$ as Banach spaces equipped with their graph norms. $S_{p,q,s}(A,M)$ is a Banach space equipped with the norm

$$ \begin{align*} \|u\|_{S_{p,q,s}(A,M)}:=\|u\|_{B_{p,q}^s}+\|Au\|_{B_{p,q}^s}+\|(Mu)'\|_{B_{p,q}^s}+\|(Mu)"\|_{B_{p,q}^s}.\nonumber \end{align*} $$

If $u\in S_{p,q,s}(A,M)$ , then $Mu$ and $(Mu)'$ are X-valued continuous functions on $\mathbb {T}$ , and $Mu(0)=Mu(2\pi )$ , $(Mu)'(0)=(Mu)'(2\pi )$ by [Reference Arendt and Bu5, Lemma 2.1].

If $(P_5)$ is $B_{p,q}^s$ -well-posed and $u\in S_{p,q,s}(A,M)$ is the unique strong $B_{p,q}^s$ -solution of $(P_5)$ , then there exists a constant $C>0$ such that for each $f\in B_{p,q}^s(\mathbb {T}; X)$ , we have

(4.1) $$ \begin{align} \|u\|_{S_{p,q,s}(A,M)}\leq C\|f\|_{B_{p,q}^s}. \end{align} $$

This can be obtained easily from the Closed Graph Theorem.

The main tool in the study of $B_{p,q}^s$ -well-posedness of $(P_5)$ is the operator-valued Fourier multiplier result established in [Reference Arendt and Bu6].

Definition 4.1 Let X be a Banach space, $1\leq p,q\leq \infty $ , $s\in {\mathbb R,}$ and let $(M_k)_{k\in {\mathbb Z}}\subset \mathcal {L}(X)$ . We say that $(M_{k})_{k\in {\mathbb Z}}$ is a $B^{s}_{p,q}$ -multiplier, if for each $f\in B^{s}_{p,q}({\mathbb T}; X)$ , there exists a (unique) $u\in B^{s}_{p,q}({\mathbb T}; X)$ , such that $\hat {u}(k)=M_{k}\hat {f}(k)$ for all $k\in {\mathbb Z}$ .

The following result has been obtained in [Reference Arendt and Bu6].

Theorem 4.2 Let X be a Banach space, $1\leq p,q\leq \infty $ , $s\in {\mathbb R,}$ and let $(M_k)_{k\in {\mathbb Z}}\subset \mathcal {L}(X)$ .

(i) If $(M_k)_{k\in {\mathbb Z}}$ is M-bounded of order $2$ , then $(M_k)_{k\in {\mathbb Z}}$ is a $B^{s}_{p,q}$ -multiplier.
(ii) If X is B-convex and $(M_k)_{k\in {\mathbb Z}}$ is M-bounded of order $1$ , then $(M_k)_{k\in {\mathbb Z}}$ is a $B^{s}_{p,q}$ -multiplier.

Recall that a Banach space X is B-convex if it does not contain $l_1^n$ uniformly. This is equivalent to say that X has Fourier type $1 < p \leq 2$ , that is, the Fourier transform is a bounded linear operator from $L^p({\mathbb R}; X)$ to $\ell ^q({\mathbb Z}; X)$ , where $1/p + 1/q = 1$ . It is well known that when $1 < p < \infty $ , then $L^p(\mu )$ has Fourier type $\min \{p, \frac {p}{p-1}\}$ .

In order to prove our main result, we need the following facts.

Proposition 4.3 Let A and M be closed linear operators defined on a Banach space X satisfying $D(A)\subset D(M)$ , where $1\leq p,q\leq \infty ,$ and $s>0$ . Assume that $\rho _{M}(A;\mu ,\nu ,\eta ,\tau )=\mathbb {Z}$ and the sets $\{k^2MN_k:\ k\in \mathbb {Z}\}$ , $\{AN_k:\ k\in \mathbb {Z}\},$ and $\{N_k:\ k\in \mathbb {Z}\}$ are norm bounded. If $\mu , \nu , \eta ,$ and $\tau $ are bounded scalar measures on ${\mathbb R}$ such that $(\alpha _k)_{k\in {\mathbb Z}}$ and $(\beta _k)_{k\in {\mathbb Z}}$ are $2$ -regular, where $N_k,\ \alpha _k$ , and $\beta _k$ are defined by (2.1). Then $(k^2MN_k)_{k\in\mathbb{Z}}$ , $(N_k)_{k\in \mathbb {Z}}$ , $(MN_k)_{k\in \mathbb {Z}}$ , and $(AN_k)_{k\in \mathbb {Z}}$ are $B_{p,q}^s$ -Fourier multipliers.

Proof Let $L_k =(N_k^{-1} - N_{k+1}^{-1})N_k$ when $k\in \mathbb {Z}$ . Then the set $\{kL_k: k\in \mathbb {Z}\}$ is norm bounded by (3.4). To show that $(k^2MN_k)_{k\in \mathbb {Z}}$ , $(N_k)_{k\in \mathbb {Z}}$ , $(MN_k)_{k\in \mathbb {Z}}$ , and $(AN_k)_{k\in \mathbb {Z}}$ are $B_{p,q}^s$ -Fourier multipliers, we only need to prove that the set $\{k^2\Delta L_k: k\in \mathbb {Z}\}$ is norm bounded by Theorems 3.3 and 4.2. We observe

(4.2) $$ \begin{align} \Delta L_k&=L_{k+1} - L_k\nonumber\\ &=-\Delta \alpha_{k+1}MN_{k+1} + \Delta \beta_{k+1}AN_{k+1} + \Delta \alpha_{k}MN_{k} - \Delta \beta_{k}AN_{k}\nonumber\\ &=-\Delta^2 \alpha_{k}MN_{k+1} - \Delta \alpha_kM(N_{k+1}-N_k) + \Delta^2\beta_kAN_{k+1}\nonumber\\ &\quad+\Delta \beta_kA(N_{k+1}-N_k)\nonumber\\ &=-\frac{\Delta^2\alpha_k}{\alpha_k}\frac{\alpha_k}{\alpha_{k+1}}(\alpha_{k+1}MN_{k+1}) - \frac{\Delta \alpha_k}{\alpha_k}\frac{\alpha_k}{\alpha_{k+1}}(\alpha_{k+1}MN_{k+1})L_k\nonumber\\ &\quad+\frac{\Delta^2\beta_k}{\beta_k}\frac{\beta_k}{\beta_{k+1}}(\beta_{k+1}AN_{k+1}) + \frac{\Delta \beta_k}{\beta_k}\frac{\beta_k}{\beta_{k+1}}(\beta_{k+1}AN_{k+1})L_k \end{align} $$

StartLayout 1st Row 1st Column normal upper Delta upper L Subscript k 2nd Column equals upper L Subscript k plus 1 Baseline minus upper L Subscript k Baseline 2nd Row 1st Column Blank 2nd Column equals minus normal upper Delta alpha Subscript k plus 1 Baseline upper M upper N Subscript k plus 1 Baseline plus normal upper Delta beta Subscript k plus 1 Baseline upper A upper N Subscript k plus 1 Baseline plus normal upper Delta alpha Subscript k Baseline upper M upper N Subscript k Baseline minus normal upper Delta beta Subscript k Baseline upper A upper N Subscript k Baseline 3rd Row 1st Column Blank 2nd Column equals minus normal upper Delta squared alpha Subscript k Baseline upper M upper N Subscript k plus 1 Baseline minus normal upper Delta alpha Subscript k Baseline upper M left parenthesis upper N Subscript k plus 1 Baseline minus upper N Subscript k Baseline right parenthesis plus normal upper Delta squared beta Subscript k Baseline upper A upper N Subscript k plus 1 Baseline 4th Row 1st Column Blank 2nd Column plus normal upper Delta beta Subscript k Baseline upper A left parenthesis upper N Subscript k plus 1 Baseline minus upper N Subscript k Baseline right parenthesis 5th Row 1st Column Blank 2nd Column equals minus StartFraction normal upper Delta squared alpha Subscript k Baseline Over alpha Subscript k Baseline EndFraction StartFraction alpha Subscript k Baseline Over alpha Subscript k plus 1 Baseline EndFraction left parenthesis alpha Subscript k plus 1 Baseline upper M upper N Subscript k plus 1 Baseline right parenthesis minus StartFraction normal upper Delta alpha Subscript k Baseline Over alpha Subscript k Baseline EndFraction StartFraction alpha Subscript k Baseline Over alpha Subscript k plus 1 Baseline EndFraction left parenthesis alpha Subscript k plus 1 Baseline upper M upper N Subscript k plus 1 Baseline right parenthesis upper L Subscript k Baseline 6th Row 1st Column Blank 2nd Column plus StartFraction normal upper Delta squared beta Subscript k Baseline Over beta Subscript k Baseline EndFraction StartFraction beta Subscript k Baseline Over beta Subscript k plus 1 Baseline EndFraction left parenthesis beta Subscript k plus 1 Baseline upper A upper N Subscript k plus 1 Baseline right parenthesis plus StartFraction normal upper Delta beta Subscript k Baseline Over beta Subscript k Baseline EndFraction StartFraction beta Subscript k Baseline Over beta Subscript k plus 1 Baseline EndFraction left parenthesis beta Subscript k plus 1 Baseline upper A upper N Subscript k plus 1 Baseline right parenthesis upper L Subscript k EndLayout

when $|k|\geq k_0$ . Since $\{k^2MN_k:k\in {\mathbb Z}\}$ is norm bounded, we have that $\{\alpha _kMN_k:k\in {\mathbb Z}\}$ , $\{kMN_k:k\in {\mathbb Z}\},$ and $\{MN_k:k\in {\mathbb Z}\}$ are also norm bounded. Here, we have used the fact that $\mu , \nu , \eta ,$ and $\tau $ are bounded scalar measures on ${\mathbb R}$ . Moreover, we deduce by (3.4) that $\{\beta _kAN_k:k\in {\mathbb Z}\}$ is also norm bounded. It follows from (3.4) that $\{kL_k: k\in \mathbb {Z}\}$ is norm bounded, here we have used the assumption that $(\alpha _k)_{k\in {\mathbb Z}}$ and $(\beta _k)_{k\in {\mathbb Z}}$ are $2$ -regular. Therefore, we obtain by (4.2) that the set $\{k^2\Delta L_k: k\in \mathbb {Z}\}$ is norm bounded. The proof is completed.

Now we are ready to state our main result of this section.

Theorem 4.4 Let X be a Banach space and let $1\leq p, q\leq \infty ,$ and $s> 0$ . Assume that A and M are closed operators in X such that $D(A)\subset D(M)$ , $\mu , \nu , \eta ,$ and $\tau $ are bounded scalar measures on ${\mathbb R}$ such that $(\alpha _k)_{k\in {\mathbb Z}}$ and $(\beta _k)_{k\in {\mathbb Z}}$ are $2$ -regular. Then the following assertions are equivalent:

(i) $(P_5)$ is $B_{p,q}^s$ -well-posed;
(ii) $\rho _{M}(A;\mu ,\nu ,\eta ,\tau )=\mathbb {Z}$ and $\{k^2MN_k:k\in {\mathbb Z}\}$ , $\{AN_k:\ k\in \mathbb {Z}\}$ , $\{N_k:k\in {\mathbb Z}\}$ are norm bounded, where $N_k$ is defined by (2.1).

Proof Now we assume that (ii) is true and we are going to show that $(P_5)$ is $B_{p,q}^s$ -well-posed. We first show the uniqueness. Let $u\in S_{p,q,s}(A,M)$ such that

$$ \begin{align*}\mu * (Mu)"(t)+ \nu * (Mu)'(t) + \eta * (Mu)(t) - \tau * Au(t) = 0 \end{align*} $$

a.e. on ${\mathbb T}$ and $ Mu(0) = Mu(2\pi ), \ (Mu)'(0) = (Mu)'(2\pi )$ . Taking Fourier coefficient on both sides, we find that $\big [(-k^2\tilde \mu (k) + ik\tilde \nu (k) + \tilde \eta (k))M - \tilde \tau (k)A\big ]\hat u(k) = 0$ for $k\in {\mathbb Z}$ . Here, we have used (2.2) and the fact that when $h\in B_{p,q}^s({\mathbb T}; D(A))$ , then $\hat h(k) \in D(A)$ for all $k\in {\mathbb Z}$ and $A\hat h(k) = \widehat {Ah}(k)$ [Reference Arendt and Bu5, Lemma 3.1]. It follows that $\hat u(k) = 0$ for $k\in {\mathbb Z}$ as $\rho _{M}(A;\mu ,\nu ,\eta ,\tau )={\mathbb Z}$ by assumption. Thus, $u = 0$ by the Uniqueness Theorem [Reference Arendt and Bu5, p. 314]. We have shown the uniqueness.

Now we show the existence. By Proposition 4.3, we obtain that $(MN_k)_{k\in {\mathbb Z}} $ , $(k^2MN_k)_{k\in {\mathbb Z}} $ , $(N_k)_{k\in {\mathbb Z}} $ , and $(AN_k)_{k\in {\mathbb Z}} $ are $B_{p,q}^s$ -multipliers. Then there exists $u_1, u_2, u_3, u\in B_{p,q}^s({\mathbb T}; X)$ such that

(4.3) $$ \begin{align} \hat u(k) = N_k\hat f(k),\ \widehat {u_1}(k) = AN_k\hat f(k) \end{align} $$

and

(4.4) $$ \begin{align} \widehat {u_2}(k) = MN_k\hat f(k),\ \widehat {u_3}(k) = -k^2MN_k\hat f(k) \end{align} $$

for $k\in {\mathbb Z}$ . We deduce from (4.3) that $\hat u(k) \in D(A)$ and $\widehat {u_1}(k) = AN_k\hat f(k) = A\hat u(k) $ for $k\in {\mathbb Z}$ . Hence, $u\in B_{p,q}^s({\mathbb T}; D(A))$ by [Reference Arendt and Bu5, Lemma 3.1]. From (4.4) that $\widehat {u_2}(k) = MN_k\hat f(k) = M\hat u(k) $ for $k\in {\mathbb Z}$ . Hence, $Mu\in B_{p,q}^s({\mathbb T}; X)$ by [Reference Arendt and Bu5, Lemma 3.1]. Again by (4.4), we have $\widehat {u_3}(k) = -k^2M\hat u(k)$ for $k\in {\mathbb Z}$ . This implies that $Mu\in B_{p,q}^{s+2}({\mathbb T}; X)$ and $Mu(0) = Mu(2\pi ), \ (Mu)'(0) = (Mu)'(2\pi ), \ (Mu)" = u_3$ .

From the equality $\hat u(k) = N_k\hat f(k)$ , we deduce that

$$ \begin{align*}\big[(-k^2\tilde \mu (k) + ik\tilde \nu(k) + \tilde\eta(k))M - \tilde\tau (k)A\big]\hat u(k) = \hat f(k) \end{align*} $$

for all $k\in {\mathbb Z}$ . This is equivalent to say that

$$ \begin{align*}\mu*(Mu)"(t)+ \nu *(Mu)'(t) + \eta * (Mu)(t) - \tau * Au(t) = f(t) \end{align*} $$

Now we give the proof for the implication (i) $\Rightarrow $ (ii). Assume that $(P_5)$ is $B_{p,q}^s$ -well-posed. Let $k\in \mathbb {Z}$ and $y\in X$ be fixed, we consider the function f defined by $f(t)=e^{ikt}y$ when $t\in \mathbb {T}$ . Then it is clear that $f\in B_{p,q}^s(\mathbb {T};X), \hat {f}(k)=y$ , and $\hat {f}(n)=0$ when $n\neq k$ . Since $(P_5)$ is $B_{p,q}^s$ -well-posed, there exists a unique $u\in S_{p,q,s}(A,M)$ such that

(4.5) $$ \begin{align} \mu*(Mu)"(t)+ \nu *(Mu)'(t) + \eta * (Mu)(t) - \tau * Au(t) = f(t) \end{align} $$

a.e. on $\mathbb {T}$ . We have $\hat {u}(n)\in D(A)$ when $n\in \mathbb {Z}$ by [Reference Arendt and Bu5, Lemma 3.1] as $u\in B_{p,q}^s(\mathbb {T}; D(A))$ . Taking Fourier transforms on both sides of (3.6), we obtain

(4.6) $$ \begin{align} \big[(-k^2\tilde \mu (k) + ik\tilde \nu(k) + \tilde\eta(k))M - \tilde\tau (k)A\big]\hat{u}(k)=y \end{align} $$

$$ \begin{align*}\big[(-k^2\tilde \mu (k) + ik\tilde \nu(k) + \tilde\eta(k))M - \tilde\tau(k)A\big]x=0. \end{align*} $$

Let u be the function given by $u(t)=e^{ikt}x$ when $t\in \mathbb {T}$ , then it is clear that $u\in S_{p,q,s}(A,M)$ and $(P_5)$ is satisfied a.e. on $\mathbb {T}$ when $f=0$ . Thus, u is a strong $B_{p,q}^s$ -solution of $(P_5)$ when taking $f=0$ . We get $x=0$ by the uniqueness assumption. We have shown that the operator $\big [(-k^2\tilde \mu (k) + ik\tilde \nu (k) + \tilde \eta (k))M - \tilde \tau (k)A\big ]$ from $D(A)$ into X is injective. Therefore, $\big [(-k^2\tilde \mu (k) + ik\tilde \nu (k) + \tilde \eta (k))M - \tilde \tau (k)A\big ]$ is bijective from $D(A)$ onto X.

Next, we show that $\big [(-k^2\tilde \mu (k) + ik\tilde \nu (k) + \tilde \eta (k))M - \tilde \tau (k)A\big ]^{-1}\in \mathcal {L}(X)$ . For $f(t)=e^{ikt}y$ , we let $u\in S_{p,q,s}(A,M)$ such that u is the unique strong $B_{p,q}^s$ -solution of $(P_5)$ . Then

(4.7) $$ \begin{align} \hat{u}(n)= \begin{cases}0&n\neq k,\\ \big[(-k^2\tilde \mu (k) + ik\tilde \nu(k) + \tilde\eta(k))M - \tilde\tau(k)A\big]^{-1}y&n=k, \end{cases} \end{align} $$

by (3.7). This implies that u is given by

$$ \begin{align*}u(t) = e^{ikt}\big[(-k^2\tilde \mu (k) + ik\tilde \nu(k) + \tilde\eta(k))M - \tilde\tau(k)A\big]^{-1}y \end{align*} $$

when $t\in \mathbb {T}$ . By (3.5), there exists a constant $C>0$ independent from y and k, such that $\|u\|_{S_{p,q,s}(A,M)}\leq C\|f\|_{B_{p,q}^s}$ . This implies that

$$ \begin{align*}\big\Vert \big[N(k)y\big\Vert\leq C\|y\|,\big\Vert-k^2MN(k)y\big\Vert\leq C\|y\| \end{align*} $$

when $y\in X$ , or equivalently,

$$ \begin{align*}\big\|\big[N(k)\big\|\leq C,\big\|k^2MN(k)\big\|\leq C. \end{align*} $$

Therefore, we obtain that $k\in \rho _{M}(A;\mu ,\nu ,\eta ,\tau )$ for all $k\in \mathbb {Z}$ . We have shown that the implication (i) $\Rightarrow $ (ii) is true. This finishes the proof.

Since the second assertion of Theorem 4.4 does not depend on the choice of $1\leq p, q\leq \infty ,$ and $s> 0$ , we deduce that $(P_5)$ is $B^{s}_{p,q}$ -well-posed for some $1\leq p, q\leq \infty ,$ and $s> 0$ if and only if it is $B^{s}_{p,q}$ -well-posed for all $1\leq p, q\leq \infty ,$ and $s> 0$ . Furthermore, by Theorem 4.2, if the underlying Banach space X is B-convex, then a first-order Marcinkiewicz-type condition is already sufficient for $(M_k)_{k\in {\mathbb Z}}$ to be a $B^{s}_{p,q}$ -multiplier. This observation and the proof of Theorem 3.4 show that when X is B-convex, then Theorem 4.4 remains valid under a weaker condition on the measures $\mu , \nu , \eta ,$ and $\tau $ : we only need to assume that the scalar sequences involved are $1$ -regular. We recall that, in the proof of Theorem 3.4, we only use Theorem 3.1 in which only a Marcinkiewicz-type condition of order $1$ is involved.

Corollary 4.5 Let X be a B-convex Banach space and let $1\leq p, q\leq \infty ,$ and $s> 0$ . Assume that A and M are closed operators in X such that $D(A)\subset D(M)$ , $\mu , \nu , \eta ,$ and $\tau $ are bounded scalar measures on ${\mathbb R}$ such that $(\alpha _k)_{k\in {\mathbb Z}}$ and $(\beta _k)_{k\in {\mathbb Z}}$ are $1$ -regular. Then the following assertions are equivalent:

(i) $(P_5)$ is $B_{p,q}^s$ -well-posed;
(ii) $\rho _{M}(A;\mu ,\nu ,\eta ,\tau )=\mathbb {Z}$ and $\{k^2MN_k:k\in {\mathbb Z}\}$ , $\{AN_k:\ k\in \mathbb {Z}\}$ , $\{N_k:k\in {\mathbb Z}\}$ are norm bounded, where $N_k$ is defined by (2.1).

Periodic Hölder continuous function space on ${\mathbb T}$ is a particular case of periodic Besov space $B^{s}_{p,q}({\mathbb T}; X)$ . It follows from [Reference Arendt and Bu6, Theorem 3.1] that $B^{\alpha }_{\infty ,\infty }({\mathbb T}; X)=C^\alpha _{\mathrm{per}}({\mathbb T}; X)$ whenever $0 <\alpha < 1$ , where $C^\alpha _{\mathrm{per}}({\mathbb T}; X)$ is the space of all X-valued functions f defined on ${\mathbb T}$ satisfying $f(0) = f(2\pi )$ and $\sup _{x\not =y}\frac {\Vert f(x) - f(y)\Vert }{\vert x - y\vert ^\alpha } <\infty $ . Moreover, $\Vert f\Vert _{C^\alpha _{\mathrm{per}}} =\max _{t\in {\mathbb T}}\Vert f(t)\Vert + \sup _{x\not = y}\frac {\Vert f(x) -f(y)\Vert }{\vert x - y\vert ^\alpha }$ on $C^\alpha _{\mathrm{per}}({\mathbb T}; X)$ defines an equivalent norm of $B^{\alpha }_{\infty ,\infty }({\mathbb T}; X)$ . If $0 < \alpha < 1$ , we say that $(P_5)$ is $C^\alpha _{\mathrm{per}}$ -well-posed, if for every $f\in C^\alpha _{\mathrm{per}}({\mathbb T}; X)$ , there exists a unique $u(t)\in D(A)$ such that $Mu\in C^{\alpha +2}_{\mathrm{per}}({\mathbb T}; X)$ and equation $(P_5)$ holds true for all $t\in {\mathbb T}$ , and $u\in C^\alpha _{\mathrm{per}}({\mathbb T}; D(A))$ , where $ C^{\alpha +2}_{\mathrm{per}}({\mathbb T}; X)$ is the space of all functions $u\in C^2({\mathbb T}; X)$ such that $u',\ u"\in C^{\alpha }_{\mathrm{per}}({\mathbb T}; X)$ , $u(0) = u(2\pi )$ , and $u'(0) = u'(2\pi )$ . Theorem 4.4 and Corollary 4.5 have the following useful corollary.

Corollary 4.6 Let X be a Banach space and let $0 < \alpha < 1$ . Assume that A and M are closed operators in X such that $D(A)\subset D(M)$ , $\mu , \nu , \eta ,$ and $\tau $ are bounded scalar measures on ${\mathbb R}$ such that $(\alpha _k)_{k\in {\mathbb Z}}$ and $(\beta _k)_{k\in {\mathbb Z}}$ are $2$ -regular. Then the following assertions are equivalent:

(i) $(P_5)$ is $C^\alpha _{\mathrm{per}}$ -well-posed;
(ii) $\rho _{M}(A;\mu ,\nu ,\eta ,\tau )=\mathbb {Z}$ and $\{k^2MN_k:k\in {\mathbb Z}\}$ , $\{AN_k:\ k\in \mathbb {Z}\}$ , $\{N_k:k\in {\mathbb Z}\}$ are norm bounded, where $N_k$ is defined by (2.1).

If furthermore X is B-convex, then the two assertions above are still equivalent if we replace the assumptions that $(\alpha _k)_{k\in {\mathbb Z}}$ and $(\beta _k)_{k\in {\mathbb Z}}$ are $2$ -regular by the weaker assumptions that they are $1$ -regular.

In the rest of this section, we study ${F}^{s}_{p,q}$ -well-posedness of $(P_5)$ . We briefly recall the definition of periodic Triebel–Lizorkin spaces in the vector-valued case introduced in [Reference Bu and Kim10]. Let $\phi =(\phi _{k})_{k\in {\mathbb N}_{0}}\in \phi ({\mathbb R})$ be fixed. For $1\leq p <\infty $ , $1\leq q \leq \infty $ , and $s\in {\mathbb R}$ , X-valued periodic Triebel–Lizorkin space is defined by

$$ \begin{align*} {F}^{s}_{p,q}({\mathbb T}; X) = &\left\{\vphantom{\left(\sum_{j\geq0}2^{sjq}\left\Vert\sum_{k\in{\mathbb Z}}e_{k}\otimes\phi_{j}(k)\hat{f}(k)\right\Vert_{X}^{q}\right)^{1/q}}f\in{\mathcal D}'({\mathbb T}; X):\ \left\|f\right\|_{{F}^{s}_{p,q}}\right. \nonumber\\&\quad \left.= \left\Vert \left(\sum_{j\geq0}2^{sjq}\left\Vert\sum_{k\in{\mathbb Z}}e_{k}\otimes\phi_{j}(k)\hat{f}(k)\right\Vert_{X}^{q}\right)^{1/q}\right\|_{L_{p}}<\infty\right\} \end{align*} $$

StartLayout 1st Row 1st Column upper F Subscript p comma q Superscript s Baseline left parenthesis double struck upper T semicolon upper X right parenthesis equals 2nd Column left brace f element of script upper D prime left parenthesis double struck upper T semicolon upper X right parenthesis colon StartMetric f EndMetric Subscript upper F Sub Subscript p comma q Sub Superscript s Subscript Baseline 2nd Row 1st Column Blank 2nd Column equals StartMetric left parenthesis sigma summation Underscript j greater than or equals 0 Endscripts 2 Superscript s j q Baseline StartMetric sigma summation Underscript k element of double struck upper Z Endscripts e Subscript k Baseline circled times phi Subscript j Baseline left parenthesis k right parenthesis ModifyingAbove f With caret left parenthesis k right parenthesis EndMetric Subscript upper X Superscript q Baseline right parenthesis Superscript 1 divided by q Baseline EndMetric Subscript upper L Sub Subscript p Subscript Baseline less than infinity right brace EndLayout

with the usual modification if $q=\infty $ . The space ${F}^{s}_{p,q}({\mathbb T}; X)$ is independent from the choice of $\phi $ and different choices of $\phi $ lead to equivalent norms on ${F}^{s}_{p,q}({\mathbb T}; X)$ . ${F}^{s}_{p,q}({\mathbb T}; X)$ equipped with the norm $\Vert \cdot \Vert _{{F}^{s}_{p,q}}$ is a Banach space. See [Reference Bu and Kim10, Section 2] for more information about the spaces ${F}^{s}_{p,q}({\mathbb T}; X)$ .

Let $1\leq p <\infty $ , $1\leq q \leq \infty $ , $s\in {\mathbb R}$ , and let $(M_k)_{k\in \mathbb {Z}}\subset \mathcal {L}(X)$ . We will say that $(M_{k})_{k\in {\mathbb Z}}$ is an ${F}^{s}_{p,q}$ -multiplier, if for each $f\in {F}^{s}_{p,q}({\mathbb T}; X)$ , there exists $g\in {F}^{s}_{p,q}({\mathbb T}; X)$ , such that $\hat {g}(k)=M_{k}\hat {f}(k)$ for all $k\in {\mathbb Z}$ . In this case, it follows from the Closed Graph Theorem that there exists a constant $C>0$ such that for $f\in {F}^{s}_{p,q}({\mathbb T}; X)$ , we have $\|\sum _{k\in {\mathbb Z}}e_{k}\otimes {M}_{k}\hat {f}(k)\|_{{F} ^{s}_{p,q}}\leq {C}\|f\|_{{F} ^{s}_{p,q}}$ .

Let $1\leq p < \infty , \ 1\leq q \leq \infty ,$ and $s> 0$ . We say that $(P_5)$ is $F_{p,q}^s$ -well-posed, if for each $f\in F_{p,q}^s({\mathbb T}; X)$ , there exists a unique $u\in F_{p,q}^s({\mathbb T}; D(A))$ such that $Mu\in F_{p,q}^{s+2}({\mathbb T}; X)$ and $(P_5)$ is satisfied a.e. on ${\mathbb T}$ . We recall that when $u\in F_{p,q}^{s+2}({\mathbb T}; X)$ for some $s>0$ , then u is twice differentiable a.e. on ${\mathbb T}$ and $u' \in F_{p,q}^{s+1}({\mathbb T}; X), \ u" \in F_{p,q}^s({\mathbb T}; X)$ [Reference Bu and Kim10]. We remark that when $(P_5)$ is $F_{p,q}^s$ -well-posed, then each term appeared in $(P_5)$ is in $F_{p,q}^s({\mathbb T}; X)$ , this follows from Young’s inequality.

The following ${F} ^{s}_{p,q}$ -multiplier theorem is due to Bu and Kim [Reference Bu and Kim10, Theorem 3.2].

Theorem 4.7 Let X be a Banach space and let $(M_{k})_{k\in {\mathbb Z}}\subset \mathcal {L}(X)$ . Assume that $(M_{k})_{k\in {\mathbb Z}}$ is M-bounded of order $3$ , then it is an ${F} ^{s}_{p,q}$ -multiplier whenever $1\leq p <\infty $ , $1\leq q \leq \infty $ , and $s\in {\mathbb R}$ . If $(M_{k})_{k\in {\mathbb Z}}$ is M-bounded of order $2$ , then it is an ${F} ^{s}_{p,q}$ -multiplier whenever $1 < p <\infty $ , $1 < q \leq \infty ,$ and $s\in {\mathbb R}$ .

In order to prove $F_{p,q}^s$ -well-posedness of $(P_5)$ , we need the following preparations.

Lemma 4.8 Let $(c_k)_{\in {\mathbb Z}}\subset {\mathbb C}$ be a $3$ -regular sequence and denote $d_k=\frac {\Delta c_k}{c_k}$ , then the sequences $(k^2\Delta d_k)_{\in {\mathbb Z}}$ and $(k^3\Delta ^2 d_k)_{\in {\mathbb Z}}$ are bounded.

Proof By the definition of $d_n$ , we have

(4.8) $$ \begin{align} \Delta d_k&=\frac{\Delta c_{k+1}}{c_{k+1}}-\frac{\Delta c_k}{c_k}\nonumber\\ &=\frac{\Delta c_{k+1}-\Delta c_k}{c_{k+1}}+(\frac{1}{c_{k+1}}-\frac{1}{c_k})\Delta c_k \nonumber\\ &=\Delta(\frac{1}{c_k})\Delta c_k+\frac{\Delta^2c_k}{c_{k+1}}, \end{align} $$

StartLayout 1st Row 1st Column normal upper Delta d Subscript k 2nd Column equals StartFraction normal upper Delta c Subscript k plus 1 Baseline Over c Subscript k plus 1 Baseline EndFraction minus StartFraction normal upper Delta c Subscript k Baseline Over c Subscript k Baseline EndFraction 2nd Row 1st Column Blank 2nd Column equals StartFraction normal upper Delta c Subscript k plus 1 Baseline minus normal upper Delta c Subscript k Baseline Over c Subscript k plus 1 Baseline EndFraction plus left parenthesis StartFraction 1 Over c Subscript k plus 1 Baseline EndFraction minus StartFraction 1 Over c Subscript k Baseline EndFraction right parenthesis normal upper Delta c Subscript k Baseline 3rd Row 1st Column Blank 2nd Column equals normal upper Delta left parenthesis StartFraction 1 Over c Subscript k Baseline EndFraction right parenthesis normal upper Delta c Subscript k Baseline plus StartFraction normal upper Delta squared c Subscript k Baseline Over c Subscript k plus 1 Baseline EndFraction comma EndLayout

which implies

(4.9) $$ \begin{align} \Delta^2d_k&=\Delta(\frac{1}{c_{k+1}})\Delta c_{k+1}+\frac{\Delta^2c_{k+1}}{c_{k+2}}-\Delta(\frac{1}{c_k})\Delta c_k-\frac{\Delta^2c_k}{c_{k+1}}\nonumber\\ &=\Delta^2(\frac{1}{c_k})(\Delta c_{k+1})+\Delta(\frac{1}{c_k})(\Delta^2 c_k)+\Delta(\frac{1}{c_{k+1}})\Delta^2c_{k+1}+\frac{1}{c_{k+1}}(\Delta^3c_k) \end{align} $$

StartLayout 1st Row 1st Column normal upper Delta squared d Subscript k 2nd Column equals normal upper Delta left parenthesis StartFraction 1 Over c Subscript k plus 1 Baseline EndFraction right parenthesis normal upper Delta c Subscript k plus 1 Baseline plus StartFraction normal upper Delta squared c Subscript k plus 1 Baseline Over c Subscript k plus 2 Baseline EndFraction minus normal upper Delta left parenthesis StartFraction 1 Over c Subscript k Baseline EndFraction right parenthesis normal upper Delta c Subscript k Baseline minus StartFraction normal upper Delta squared c Subscript k Baseline Over c Subscript k plus 1 Baseline EndFraction 2nd Row 1st Column Blank 2nd Column equals normal upper Delta squared left parenthesis StartFraction 1 Over c Subscript k Baseline EndFraction right parenthesis left parenthesis normal upper Delta c Subscript k plus 1 Baseline right parenthesis plus normal upper Delta left parenthesis StartFraction 1 Over c Subscript k Baseline EndFraction right parenthesis left parenthesis normal upper Delta squared c Subscript k Baseline right parenthesis plus normal upper Delta left parenthesis StartFraction 1 Over c Subscript k plus 1 Baseline EndFraction right parenthesis normal upper Delta squared c Subscript k plus 1 Baseline plus StartFraction 1 Over c Subscript k plus 1 Baseline EndFraction left parenthesis normal upper Delta cubed c Subscript k Baseline right parenthesis EndLayout

when $|k|\geq k_0$ . Therefore, we obtain that the sequences $(k^2\Delta d_k)_{\in {\mathbb Z}}$ and $(k^3\Delta ^2 d_k)_{\in {\mathbb Z}}$ are bounded as $(c_k)_{\in {\mathbb Z}}$ is $3$ -regular. This completes the proof.

Theorem 4.9 Let $T:D(T)\subset X\rightarrow X$ be a closed linear operator defined in Banach space X. For each $k\in {\mathbb Z}$ , let $H_k:X\rightarrow D(T)$ be a sequence of bounded and linear operators such that $0\in \rho (H_k)$ for all $k\in {\mathbb Z}$ . Let $M_k:=c_kTH_k$ and $L_k:=(H_k^{-1}-H_{k+1}^{-1})H_k$ . If $(c_k)_{k\in {\mathbb Z}}$ is 3-regular and the sets $\{M_k:k\in {\mathbb Z}\}$ , $\{k^3\Delta ^2 L_k:k\in {\mathbb Z}\}$ , $\{k^2\Delta L_k:k\in {\mathbb Z}\}$ , and $\{kL_k:k\in {\mathbb Z}\}$ are norm bounded, then $\{M_k:k\in {\mathbb Z}\}$ is M-bounded of order 3.

Proof By virtue of the definition of $M_k$ , we observe

(4.10) $$ \begin{align} \Delta M_k&=c_{k+1}TH_{k+1}-c_kTH_k\nonumber\\ &=c_{k+1}T\Delta H_k+(\Delta c_k)TH_k\nonumber\\ &=c_{k+1}TH_{k+1}(H_k^{-1}-H_{k+1}^{-1})H_k+\frac{\Delta c_k}{c_k}c_kTH_k.\nonumber\\ &=d_kM_k+M_{k+1}L_k. \end{align} $$

StartLayout 1st Row 1st Column normal upper Delta upper M Subscript k 2nd Column equals c Subscript k plus 1 Baseline upper T upper H Subscript k plus 1 Baseline minus c Subscript k Baseline upper T upper H Subscript k Baseline 2nd Row 1st Column Blank 2nd Column equals c Subscript k plus 1 Baseline upper T normal upper Delta upper H Subscript k Baseline plus left parenthesis normal upper Delta c Subscript k Baseline right parenthesis upper T upper H Subscript k Baseline 3rd Row 1st Column Blank 2nd Column equals c Subscript k plus 1 Baseline upper T upper H Subscript k plus 1 Baseline left parenthesis upper H Subscript k Superscript negative 1 Baseline minus upper H Subscript k plus 1 Superscript negative 1 Baseline right parenthesis upper H Subscript k Baseline plus StartFraction normal upper Delta c Subscript k Baseline Over c Subscript k Baseline EndFraction c Subscript k Baseline upper T upper H Subscript k Baseline period 4th Row 1st Column Blank 2nd Column equals d Subscript k Baseline upper M Subscript k Baseline plus upper M Subscript k plus 1 Baseline upper L Subscript k Baseline period EndLayout

It follows that

(4.11) $$ \begin{align} \Delta^2 M_k&=d_{k+1}M_{k+1}+M_{k+2}L_{k+1}-d_kM_k-M_{k+1}L_k\nonumber\\ &=d_{k+1}\Delta M_k+(\Delta d_k)M_k+M_{k+2}(\Delta L_k)+(\Delta M_{k+1})L_k. \end{align} $$

Moreover, we obtain

(4.12) $$ \begin{align} \Delta^3 M_k&=d_{k+2}\Delta M_{k+1}+(\Delta d_{k+1})M_{k+1}+M_{k+3}(\Delta L_{k+1})+(\Delta M_{k+2})L_{k+1}\nonumber\\ &\quad-d_{k+1}\Delta M_k-(\Delta d_k)M_k-M_{k+2}(\Delta L_k)-(\Delta M_{k+1})L_k\nonumber\\ &=d_{k+2}(\Delta^2M_k)+(\Delta d_{k+1})(\Delta M_k)+(\Delta d_{k+1})(\Delta M_k)+(\Delta^2d_k)M_k\nonumber\\ &\quad +M_{k+3}(\Delta^2L_k)+(\Delta M_{k+2})(\Delta L_k)+(\Delta M_{k+2})(\Delta L_k)+(\Delta^2M_{k+1})L_k. \end{align} $$

StartLayout 1st Row 1st Column normal upper Delta cubed upper M Subscript k 2nd Column equals d Subscript k plus 2 Baseline normal upper Delta upper M Subscript k plus 1 Baseline plus left parenthesis normal upper Delta d Subscript k plus 1 Baseline right parenthesis upper M Subscript k plus 1 Baseline plus upper M Subscript k plus 3 Baseline left parenthesis normal upper Delta upper L Subscript k plus 1 Baseline right parenthesis plus left parenthesis normal upper Delta upper M Subscript k plus 2 Baseline right parenthesis upper L Subscript k plus 1 Baseline 2nd Row 1st Column Blank 2nd Column minus d Subscript k plus 1 Baseline normal upper Delta upper M Subscript k minus left parenthesis normal upper Delta d Subscript k Baseline right parenthesis upper M Subscript k minus upper M Subscript k plus 2 Baseline left parenthesis normal upper Delta upper L Subscript k Baseline right parenthesis minus left parenthesis normal upper Delta upper M Subscript k plus 1 Baseline right parenthesis upper L Subscript k 3rd Row 1st Column Blank 2nd Column equals d Subscript k plus 2 Baseline left parenthesis normal upper Delta squared upper M Subscript k Baseline right parenthesis plus left parenthesis normal upper Delta d Subscript k plus 1 Baseline right parenthesis left parenthesis normal upper Delta upper M Subscript k Baseline right parenthesis plus left parenthesis normal upper Delta d Subscript k plus 1 Baseline right parenthesis left parenthesis normal upper Delta upper M Subscript k Baseline right parenthesis plus left parenthesis normal upper Delta squared d Subscript k Baseline right parenthesis upper M Subscript k Baseline 4th Row 1st Column Blank 2nd Column plus upper M Subscript k plus 3 Baseline left parenthesis normal upper Delta squared upper L Subscript k Baseline right parenthesis plus left parenthesis normal upper Delta upper M Subscript k plus 2 Baseline right parenthesis left parenthesis normal upper Delta upper L Subscript k Baseline right parenthesis plus left parenthesis normal upper Delta upper M Subscript k plus 2 Baseline right parenthesis left parenthesis normal upper Delta upper L Subscript k Baseline right parenthesis plus left parenthesis normal upper Delta squared upper M Subscript k plus 1 Baseline right parenthesis upper L Subscript k Baseline period EndLayout

By Lemma 4.8, we obtain that sequences $(k^2\Delta d_k)_{\in {\mathbb Z}}$ and $(k^3\Delta ^2 d_k)_{\in {\mathbb Z}}$ are bounded. Thus, we have from (4.10) that $\{k\Delta M_k:k\in {\mathbb Z}\}$ is norm bounded as $(kd_k)_{\in {\mathbb Z}}$ and $\{kL_k:k\in {\mathbb Z}\}$ are bounded. Furthermore, we conclude from (4.11) that ${\{k^2\Delta ^2 M_k:k\in {\mathbb Z}\}}$ is norm bounded as $(kd_k)_{\in {\mathbb Z}}$ , $(k^2\Delta d_k)_{\in {\mathbb Z}}$ , $\{kL_k:k\in {\mathbb Z}\},$ and $\{k^2\Delta L_k:k\in {\mathbb Z}\}$ are bounded. Therefore, we get from (4.12) that $\{M_k:k\in {\mathbb Z}\}$ is M-bounded of order 3. The proof is completed.

Proposition 4.10 Let A and M be closed linear operators defined on a Banach space X satisfying $D(A)\subset D(M)$ , where $1\leq p<\infty $ , $1\leq q\leq \infty ,$ and $s>0$ . Assume that $\rho _M(A)=\mathbb {Z}$ and the sets $\{k^2MN_k:\ k\in \mathbb {Z}\}$ , $\{AN_k:\ k\in \mathbb {Z}\},$ and $\{N_k:\ k\in \mathbb {Z}\}$ are norm bounded. If $\mu , \nu , \eta ,$ and $\tau $ are bounded scalar measures on ${\mathbb R}$ such that $(\alpha _k)_{k\in {\mathbb Z}}$ and $(\beta _k)_{k\in {\mathbb Z}}$ are $3$ -regular, where $N_k$ , $\alpha _k$ , and $\beta _k$ are defined by (2.1). Then $(k^2MN_k)_{k\in \mathbb {Z}}$ , $(N_k)_{k\in \mathbb {Z}}$ , $(MN_k)_{k\in \mathbb {Z}}$ , and $(AN_k)_{k\in \mathbb {Z}}$ are $F_{p,q}^s$ -Fourier multipliers.

Proof Let $L_k =(N_k^{-1} - N_{k+1}^{-1})N_k$ when $k\in \mathbb {Z}$ . By Proposition 4.3, we obtain that the set $\{k^2\Delta L_k:k\in {\mathbb Z}\}$ is norm bounded. In order to finish the proof, we only show that the set $\{k^3\Delta ^2 L_k:k\in {\mathbb Z}\}$ is norm bounded by Theorem 4.9. In fact, it follows from (4.2) that

$$ \begin{align*}\Delta L_k=-(\Delta^2\alpha_k)MN_{k+1}-(\Delta \alpha_k)MN_{k+1}L_k +(\Delta^2\beta_k)AN_{k+1}+(\Delta \beta_k)AN_{k+1}L_k, \end{align*} $$

which implies

$$ \begin{align*} &\quad\Delta^2 L_k\nonumber\\ &=-(\Delta^2\alpha_{k+1})MN_{k+2}-(\Delta \alpha_{k+1})MN_{k+2}L_{k+1} +(\Delta^2\beta_{k+1})AN_{k+2}+(\Delta \beta_{k+1})AN_{k+2}L_{k+1}\nonumber\\ &\quad+(\Delta^2\alpha_k)MN_{k+1}+(\Delta \alpha_k)MN_{k+1}L_k -(\Delta^2\beta_k)AN_{k+1}-(\Delta \beta_k)AN_{k+1}L_k\nonumber\\ &=-(\Delta^3\alpha_k)MN_{k+2}-(\Delta^2\alpha_k)MN_{k+2}L_{k+1}-(\Delta^2\alpha_k)MN_{k+2}L_{k+1}\nonumber\\ &\quad-(\Delta \alpha_k)M(N_{k+2}L_{k+1}-N_{k+1}L_k)+(\Delta^3\beta_k)AN_{k+2}+(\Delta^2\beta_k)AN_{k+2}L_{k+1}\nonumber\\ &\quad+(\Delta^2\beta_k)AN_{k+2}L_{k+1}+(\Delta \beta_k)A(N_{k+2}L_{k+1}-N_{k+1}L_k)\nonumber\\ &=-(\Delta^3\alpha_k)MN_{k+2}-(\Delta^2\alpha_k)MN_{k+2}L_{k+1}-(\Delta^2\alpha_k)MN_{k+2}L_{k+1}\nonumber\\ &\quad-(\Delta \alpha_k)MN_{k+2}L_{k+1}^2-(\Delta \alpha_k)MN_{k+1}(\Delta L_k)+(\Delta^3\beta_k)AN_{k+2}+(\Delta^2\beta_k)AN_{k+2}L_{k+1}\nonumber\\ &\quad+(\Delta^2\beta_k)AN_{k+2}L_{k+1}+(\Delta \beta_k)AN_{k+2}L_{k+1}^2+(\Delta \beta_k)AN_{k+1}(\Delta L_k)\nonumber \end{align*} $$

StartLayout 1st Row 1st Column Blank 2nd Column normal upper Delta squared upper L Subscript k Baseline 2nd Row 1st Column Blank 2nd Column equals minus left parenthesis normal upper Delta squared alpha Subscript k plus 1 Baseline right parenthesis upper M upper N Subscript k plus 2 Baseline minus left parenthesis normal upper Delta alpha Subscript k plus 1 Baseline right parenthesis upper M upper N Subscript k plus 2 Baseline upper L Subscript k plus 1 Baseline plus left parenthesis normal upper Delta squared beta Subscript k plus 1 Baseline right parenthesis upper A upper N Subscript k plus 2 Baseline plus left parenthesis normal upper Delta beta Subscript k plus 1 Baseline right parenthesis upper A upper N Subscript k plus 2 Baseline upper L Subscript k plus 1 Baseline 3rd Row 1st Column Blank 2nd Column plus left parenthesis normal upper Delta squared alpha Subscript k Baseline right parenthesis upper M upper N Subscript k plus 1 plus left parenthesis normal upper Delta alpha Subscript k Baseline right parenthesis upper M upper N Subscript k plus 1 Baseline upper L Subscript k minus left parenthesis normal upper Delta squared beta Subscript k Baseline right parenthesis upper A upper N Subscript k plus 1 minus left parenthesis normal upper Delta beta Subscript k Baseline right parenthesis upper A upper N Subscript k plus 1 Baseline upper L Subscript k 4th Row 1st Column Blank 2nd Column equals minus left parenthesis normal upper Delta cubed alpha Subscript k Baseline right parenthesis upper M upper N Subscript k plus 2 Baseline minus left parenthesis normal upper Delta squared alpha Subscript k Baseline right parenthesis upper M upper N Subscript k plus 2 Baseline upper L Subscript k plus 1 Baseline minus left parenthesis normal upper Delta squared alpha Subscript k Baseline right parenthesis upper M upper N Subscript k plus 2 Baseline upper L Subscript k plus 1 Baseline 5th Row 1st Column Blank 2nd Column minus left parenthesis normal upper Delta alpha Subscript k Baseline right parenthesis upper M left parenthesis upper N Subscript k plus 2 Baseline upper L Subscript k plus 1 Baseline minus upper N Subscript k plus 1 Baseline upper L Subscript k Baseline right parenthesis plus left parenthesis normal upper Delta cubed beta Subscript k Baseline right parenthesis upper A upper N Subscript k plus 2 plus left parenthesis normal upper Delta squared beta Subscript k Baseline right parenthesis upper A upper N Subscript k plus 2 Baseline upper L Subscript k plus 1 6th Row 1st Column Blank 2nd Column plus left parenthesis normal upper Delta squared beta Subscript k Baseline right parenthesis upper A upper N Subscript k plus 2 Baseline upper L Subscript k plus 1 plus left parenthesis normal upper Delta beta Subscript k Baseline right parenthesis upper A left parenthesis upper N Subscript k plus 2 Baseline upper L Subscript k plus 1 Baseline minus upper N Subscript k plus 1 Baseline upper L Subscript k Baseline right parenthesis 7th Row 1st Column Blank 2nd Column equals minus left parenthesis normal upper Delta cubed alpha Subscript k Baseline right parenthesis upper M upper N Subscript k plus 2 Baseline minus left parenthesis normal upper Delta squared alpha Subscript k Baseline right parenthesis upper M upper N Subscript k plus 2 Baseline upper L Subscript k plus 1 Baseline minus left parenthesis normal upper Delta squared alpha Subscript k Baseline right parenthesis upper M upper N Subscript k plus 2 Baseline upper L Subscript k plus 1 Baseline 8th Row 1st Column Blank 2nd Column minus left parenthesis normal upper Delta alpha Subscript k Baseline right parenthesis upper M upper N Subscript k plus 2 Baseline upper L Subscript k plus 1 Superscript 2 minus left parenthesis normal upper Delta alpha Subscript k Baseline right parenthesis upper M upper N Subscript k plus 1 Baseline left parenthesis normal upper Delta upper L Subscript k Baseline right parenthesis plus left parenthesis normal upper Delta cubed beta Subscript k Baseline right parenthesis upper A upper N Subscript k plus 2 plus left parenthesis normal upper Delta squared beta Subscript k Baseline right parenthesis upper A upper N Subscript k plus 2 Baseline upper L Subscript k plus 1 9th Row 1st Column Blank 2nd Column plus left parenthesis normal upper Delta squared beta Subscript k Baseline right parenthesis upper A upper N Subscript k plus 2 Baseline upper L Subscript k plus 1 plus left parenthesis normal upper Delta beta Subscript k Baseline right parenthesis upper A upper N Subscript k plus 2 Baseline upper L Subscript k plus 1 Superscript 2 plus left parenthesis normal upper Delta beta Subscript k Baseline right parenthesis upper A upper N Subscript k plus 1 Baseline left parenthesis normal upper Delta upper L Subscript k Baseline right parenthesis EndLayout

when $|k|\geq k_0$ . Therefore, the set $\{k^3\Delta ^2 L_k: k\in \mathbb {Z}\}$ is norm bounded as the sets $\{kL_k: k\in \mathbb {Z}\}$ , $\{k^2\Delta L_k: k\in \mathbb {Z}\}$ and $\{\alpha _kMN_k:k\in \mathbb {Z}\}$ , $\{\beta _kAN_k:\ k\in \mathbb {Z}\}$ are norm bounded. This finishes the proof.

The solution space of the $F_{p,q}^s$ -well-posedness of $(P_5)$ is defined by

$$ \begin{align*} S_{p,q,s}'(A,M):=\big\{u\in F_{p,q}^s(\mathbb{T}; D(A)):\ Mu\in F_{p,q}^{s+2}(\mathbb{T}; X)\big\},\nonumber \end{align*} $$

here we consider $D(A)$ and $D(M)$ as Banach spaces equipped with their graph norms. $S_{p,q,s}'(A,M)$ is a Banach space equipped with the norm

$$ \begin{align*} \|u\|_{S_{p,q,s}'(A,M)}:=\|u\|_{F_{p,q}^s}+\|Au\|_{F_{p,q}^s}+\|(Mu)'\|_{F_{p,q}^s}+\|(Mu)"\|_{F_{p,q}^s}.\nonumber \end{align*} $$

If $u\in S_{p,q,s}'(A,M)$ , then $Mu$ and $(Mu)'$ are X-valued continuous functions on $\mathbb {T}$ , and $Mu(0)=Mu(2\pi )$ , $(Mu)'(0)=(Mu)'(2\pi )$ by [Reference Arendt and Bu5, Lemma 2.1].

If $(P_5)$ is $F_{p,q}^s$ -well-posed and $u\in S_{p,q,s}'(A,M)$ is the unique strong $F_{p,q}^s$ -solution of $(P_5)$ , then there exists a constant $C>0$ such that for each $f\in F_{p,q}^s(\mathbb {T}; X)$ , we have

(4.13) $$ \begin{align} \|u\|_{S_{p,q,s}'(A,M)}\leq C\|f\|_{F_{p,q}^s}. \end{align} $$

This can be obtained easily from the Closed Graph Theorem.

Theorem 4.11 Let X be a Banach space and let $1\leq p < \infty , \ 1\leq q \leq \infty ,$ and $s> 0$ . Assume that A and M are closed operators in X such that $D(A)\subset D(M)$ , $\mu , \nu , \eta ,$ and $\tau $ are bounded scalar measures on ${\mathbb R}$ such that $(\alpha _k)_{k\in {\mathbb Z}}$ and $(\beta _k)_{k\in {\mathbb Z}}$ are $3$ -regular, where $\alpha _k$ and $\beta _k$ are defined by (2.1). Then the following assertions are equivalent:

(i) $(P_5)$ is $F_{p,q}^s$ -well-posed.
(ii) $\rho _{M}(A;\mu ,\nu ,\eta ,\tau )=\mathbb {Z}$ and $\{k^2MN_k:k\in {\mathbb Z}\}$ , $\{AN_k:\ k\in \mathbb {Z}\}$ , $\{N_k:k\in {\mathbb Z}\}$ are norm bounded, where $N_k$ is defined by (2.1).

Proof Now we assume that (ii) is true and we are going to show that $(P_5)$ is $F_{p,q}^s$ -well-posed. We first show the uniqueness. Let $u\in S_{p,q,s}'(A,M)$ such that

$$ \begin{align*}\mu * (Mu)"(t)+ \nu * (Mu)'(t) + \eta * (Mu)(t) - \tau * Au(t) = 0 \end{align*} $$

a.e. on ${\mathbb T}$ and $ Mu(0) = Mu(2\pi ), \ (Mu)'(0) = (Mu)'(2\pi )$ . Taking Fourier coefficient on both sides, we find that $\big [(-k^2\tilde \mu (k) + ik\tilde \nu (k) + \tilde \eta (k))M - \tilde \tau (k)A\big ]\hat u(k) = 0$ for ${k\in {\mathbb Z}}$ . Here, we have used (2.2) and the fact that when $h\in F_{p,q}^s({\mathbb T}; D(A))$ , then $\hat h(k) \in D(A)$ for all $k\in {\mathbb Z}$ and $A\hat h(k) = \widehat {Ah}(k)$ [Reference Arendt and Bu5, Lemma 3.1]. It follows that $\hat u(k) = 0$ for $k\in {\mathbb Z}$ as $\rho _{M}(A;\mu ,\nu ,\eta ,\tau )={\mathbb Z}$ by assumption. Thus, $u = 0$ by the Uniqueness Theorem [Reference Arendt and Bu5, p. 314]. We have shown the uniqueness.

Now we show the existence. By Proposition 4.10, we obtain that $(MN_k)_{k\in {\mathbb Z}} $ , $(k^2MN_k)_{k\in {\mathbb Z}} $ , $(N_k)_{k\in {\mathbb Z}} $ , and $(AN_k)_{k\in {\mathbb Z}} $ are $F_{p,q}^s$ -multipliers. Then there exists $u_1, u_2, u_3, u\in F_{p,q}^s({\mathbb T}; X)$ such that

(4.14) $$ \begin{align} \hat u(k) = N_k\hat f(k),\ \widehat {u_1}(k) = AN_k\hat f(k) \end{align} $$

and

(4.15) $$ \begin{align} \widehat {u_2}(k) = MN_k\hat f(k),\ \widehat {u_3}(k) = -k^2MN_k\hat f(k) \end{align} $$

for $k\in {\mathbb Z}$ . We deduce from (4.14) that $\hat u(k) \in D(A)$ and $\widehat {u_1}(k) = AN_k\hat f(k) = A\hat u(k) $ for $k\in {\mathbb Z}$ . Hence, $u\in F_{p,q}^s({\mathbb T}; D(A))$ by [Reference Arendt and Bu5, Lemma 3.1]. From (4.15) that $\widehat {u_2}(k) = MN_k\hat f(k) = M\hat u(k) $ for $k\in {\mathbb Z}$ . Hence, $Mu\in F_{p,q}^s({\mathbb T}; X)$ by [Reference Arendt and Bu5, Lemma 3.1]. Again by (4.15), we have $\widehat {u_3}(k) = -k^2M\hat u(k)$ for $k\in {\mathbb Z}$ . This implies that $Mu\in F_{p,q}^{s+2}({\mathbb T}; X)$ and $Mu(0) = Mu(2\pi ), \ (Mu)'(0) = (Mu)'(2\pi ), \ (Mu)" = u_3$ .

From the equality $\hat u(k) = N_k\hat f(k)$ , we deduce that

$$ \begin{align*}\big[(-k^2\tilde \mu (k) + ik\tilde \nu(k) + \tilde\eta(k))M - \tilde\tau (k)A\big]\hat u(k) = \hat f(k) \end{align*} $$

for all $k\in {\mathbb Z}$ . This is equivalent to say that

$$ \begin{align*}\mu*(Mu)"(t)+ \nu *(Mu)'(t) + \eta * (Mu)(t) - \tau * Au(t) = f(t) \end{align*} $$

Now we give the proof for the implication (i) $\Rightarrow $ (ii). Assume that $(P_5)$ is $F_{p,q}^s$ -well-posed. Let $k\in \mathbb {Z}$ and $y\in X$ be fixed, we consider the function f defined by $f(t)=e^{ikt}y$ when $t\in \mathbb {T}$ . Then it is clear that $f\in F_{p,q}^s(\mathbb {T};X), \hat {f}(k)=y,$ and $\hat {f}(n)=0$ when $n\neq k$ . Since $(P_5)$ is $F_{p,q}^s$ -well-posed, there exists a unique $u\in S_{p,q,s}'(A,M)$ such that

(4.16) $$ \begin{align} \mu*(Mu)"(t)+ \nu *(Mu)'(t) + \eta * (Mu)(t) - \tau * Au(t) = f(t) \end{align} $$

a.e. on $\mathbb {T}$ . We have $\hat {u}(n)\in D(A)$ when $n\in \mathbb {Z}$ by [Reference Arendt and Bu5, Lemma 3.1] as $u\in F_{p,q}^s(\mathbb {T}; D(A))$ . Taking Fourier transforms on both sides of (4.16), we obtain

(4.17) $$ \begin{align} \big[(-k^2\tilde \mu (k) + ik\tilde \nu(k) + \tilde\eta(k))M - \tilde\tau (k)A\big]\hat{u}(k)=y \end{align} $$

$$ \begin{align*}\big[(-k^2\tilde \mu (k) + ik\tilde \nu(k) + \tilde\eta(k))M - \tilde\tau(k)A\big]x=0. \end{align*} $$

Let u be the function given by $u(t)=e^{ikt}x$ when $t\in \mathbb {T}$ , then it is clear that $u\in S_{p,q,s}'(A,M)$ and $(P_5)$ is satisfied a.e. on $\mathbb {T}$ when $f=0$ . Thus, u is a strong $F_{p,q}^s$ -solution of $(P_5)$ when taking $f=0$ . We get $x=0$ by the uniqueness assumption. We have shown that the operator $\big [(-k^2\tilde \mu (k) + ik\tilde \nu (k) + \tilde \eta (k))M - \tilde \tau (k)A\big ]$ from $D(A)$ into X is injective. Therefore, $\big [(-k^2\tilde \mu (k) + ik\tilde \nu (k) + \tilde \eta (k))M - \tilde \tau (k)A\big ]$ is bijective from $D(A)$ onto X.

Next, we show that $\big [(-k^2\tilde \mu (k) + ik\tilde \nu (k) + \tilde \eta (k))M - \tilde \tau (k)A\big ]^{-1}\in \mathcal {L}(X)$ . For $f(t)=e^{ikt}y$ , we let $u\in S_{p,q,s}'(A,M)$ such that u is the unique strong $F_{p,q}^s$ -solution of $(P_5)$ . Then

(4.18) $$ \begin{align} \hat{u}(n)= \begin{cases}0&n\neq k,\\ \big[(-k^2\tilde \mu (k) + ik\tilde \nu(k) + \tilde\eta(k))M - \tilde\tau(k)A\big]^{-1}y&n=k, \end{cases} \end{align} $$

by (3.7). This implies that u is given by

$$ \begin{align*}u(t) = e^{ikt}\big[(-k^2\tilde \mu (k) + ik\tilde \nu(k) + \tilde\eta(k))M - \tilde\tau(k)A\big]^{-1}y \end{align*} $$

when $t\in \mathbb {T}$ . By (4.13), there exists a constant $C>0$ independent from y and k, such that $\|u\|_{S_{p,q,s}'(A,M)}\leq C\|f\|_{F_{p,q}^s}$ . This implies that

$$ \begin{align*}\big\Vert \big[N(k)y\big\Vert\leq C\|y\|,\big\Vert-k^2MN(k)y\big\Vert\leq C\|y\| \end{align*} $$

when $y\in X$ , or equivalently,

$$ \begin{align*}\big\|\big[N(k)\big\|\leq C,\big\|k^2MN(k)\big\|\leq C. \end{align*} $$

Therefore, we obtain that $k\in \rho _{M}(A;\mu ,\nu ,\eta ,\tau )$ for all $k\in \mathbb {Z}$ . We have shown that the implication (i) $\Rightarrow $ (ii) is true. This finishes the proof.

Concluding remark: $(P_2)$ is a special case of $(P_5)$ when taking $M=I_X, \mu = \delta _0, \ \nu = \alpha \delta _0 + b, \ \eta = 0,$ and $\tau = \delta _0 - a$ for some $a, b\in L^1({\mathbb R}_+)$ . It is easy to verify that, when $\alpha \not = 0,$ our hypotheses on $\mu , \ \nu ,\ \eta ,$ and $\tau $ are weaker than the corresponding hypotheses imposed on $a, b\in L^1({\mathbb R}_+)$ in [Reference Bu and Fang9]. Thus, we may consider the results obtained in this article as generalizations of results established in [Reference Bu and Fang9].

5 Applications

Example 5.1 Let $\Omega $ be a bounded domain in $\mathbb {R}^k$ with smooth boundary, m be a given nonnegative bounded measurable function on $\Omega $ , the kernel $a, b\in L^1({\mathbb R}_+),$ and $1< p < \infty $ . We let X be the Hilbert space $H^{-1}(\Omega )$ and consider the second-order problem

$$ \begin{align*} \quad \left\{ \begin{array}{@{}ll} \frac{\partial^2}{\partial t^2} (m(x)u(t, x)) + \int_{-\infty}^0 a(t-s)\frac{\partial}{\partial s} (m(x){u(s, x)}) ds\\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ + \int_{-\infty}^0 b(t-s)m(x)u(s, x)) ds = Au(t, x) + f(t, x)\ \ \ \ (t, x) \in \mathbb{T}\times \Omega,\nonumber\\ \\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ u(t, x)= 0, \ \ \ \ \ \ \ \ \ \ (t, x) \in \mathbb{T}\times \partial \Omega,\nonumber \end{array} \right. \end{align*} $$

StartLayout 1st Row StartLayout Enlarged left brace 1st Row StartFraction partial differential squared Over partial differential t squared EndFraction left parenthesis m left parenthesis x right parenthesis u left parenthesis t comma x right parenthesis right parenthesis plus integral Subscript negative infinity Superscript 0 Baseline a left parenthesis t minus s right parenthesis StartFraction partial differential Over partial differential s EndFraction left parenthesis m left parenthesis x right parenthesis u left parenthesis s comma x right parenthesis right parenthesis d s 2nd Row plus integral Subscript negative infinity Superscript 0 Baseline b left parenthesis t minus s right parenthesis m left parenthesis x right parenthesis u left parenthesis s comma x right parenthesis right parenthesis d s equals upper A u left parenthesis t comma x right parenthesis plus f left parenthesis t comma x right parenthesis left parenthesis t comma x right parenthesis element of double struck upper T times normal upper Omega comma 3rd Row Blank 4th Row u left parenthesis t comma x right parenthesis equals 0 comma left parenthesis t comma x right parenthesis element of double struck upper T times partial differential normal upper Omega comma EndLayout EndLayout

where f is defined on $\mathbb {T}\times \Omega $ and the Laplacian $\Delta $ only acts on the space variable $x\in \Omega $ .

Let M be the multiplication operator on X by m with domain of definition $D(M) = \{u\in X: mu\in X\}$ . Then there exist constants $C>0,\ \rho > 0$ , such that

(5.1) $$ \begin{align}\big\Vert M(zM + \Delta)^{-1}\big\Vert \leq \frac{C}{1 + \vert z\vert} \end{align} $$

whenever $Re (z)\leq \rho ( 1 + \vert Im (z)\vert )$ by [Reference Favini and Yagi13, Section 3.7], where $\Delta $ is the Laplacian on $H^{-1}(\Omega )$ with Dirichlet boundary condition. Let $A = \Delta $ and we assume that $D(A)\subset D(M)$ . Then the above equation may be rewritten in the form

$$ \begin{align*}(Mu)"(t) + \int_{-\infty}^0 a(t-s)(Mu)'(s) ds + \int_{-\infty}^0 b(t-s)(Mu)(s) ds = Au(t) + f(t), \end{align*} $$

or equivalently,

(P) $$ \begin{align}\mu * (Mu)"(t) + \nu *{(Mu)}'(t) + \eta *(Mu)(t) = \tau * Au(t) + f(t) \end{align} $$

with $\mu = \tau =\delta _0$ and $\nu = a\, dt, \ \eta = b\, dt$ , which is an ordinary differential equation on $\mathbb {T}$ with values in X, where $f\in L^p(\mathbb {T}; X)$ is given, and the solution $u\in W_{\text {per}}^{1,p}(\mathbb {T}; D(A))$ satisfies $\ Mu\in W_{\text {per}}^{2,p}(\mathbb {T}; X)$ .

We notice that when $a\in L^1(\mathbb {R}_+)$ , then a can be extended to $\mathbb {R}$ by defining $a(t)=0$ when $t<0$ . With this convention, we have

$$\begin{align*}\hat{a}(t)=\int_0^\infty e^{-ist}a(s)ds=\int_{-\infty}^{\infty}e^{-ist}a(s)dt=\tilde{\mu}(t) \end{align*}$$

for all $t\in \mathbb {R}$ , where $\mu =a dt$ .

We assume that $\rho _{M}(A;\mu ,\nu ,\eta ,\tau )=\mathbb {Z}$ and let $N_k=[(k^2 -ik \hat a(k) - \hat b(k))M +A]^{-1}$ when $k\in {\mathbb Z}$ . Assume that $m> 0$ a.e. on $\Omega $ and m is regular enough so that the multiplication operator by $m^{-1}$ is bounded on $H^{-1}(\Omega )$ , then

(5.2) $$ \begin{align}\big\Vert (zM + \Delta)^{-1}\big\Vert \leq \frac{C}{1 + \vert z\vert} \end{align} $$

whenever $Re (z)\leq \rho ( 1 + \vert Im z\vert )$ by (5.1). We assume furthermore that the sequence $k^2 -ik \hat a(k) - \hat b(k)$ is $1$ -regular and $\{AN_k:\ k\in \mathbb {Z}\}$ is norm bounded. Then $(P)$ is $L^p$ -well-posed. Indeed, the operator $(k^2 -ik \hat a(k) - \hat b(k))M +A: D(A) \to X$ is bijective and $[(k^2 -ik \hat a(k) - \hat b(k))M +A]^{-1} \in \mathcal {L} (X)$ whenever $k\in \mathbb {Z}$ by the assumption $\rho _{M}(A;\mu ,\nu ,\eta ,\tau )=\mathbb {Z}$ . It follows that the sets

$$ \begin{align*}\{k^2MN_k: k\in \mathbb{Z}\}, \ \ \{N_k: k\in \mathbb{Z}\}\end{align*} $$

are norm bounded by (5.1) and (5.2). We conclude that the problem $(P)$ is $L^p$ -well-posed by Theorem 3.4. Here, we have used the fact that $H^{-1} (\Omega )$ is a Hilbert space and the fact that every norm bounded subset of $\mathcal {L}(X)$ is R-bounded when X is isomorphic to a Hilbert space [Reference Arendt and Bu5].

If we assume furthermore that the sequence $k^2 -ik \hat a(k) - \hat b(k)$ is $2$ -regular and $\{AN_k:\ k\in \mathbb {Z}\}$ is norm bounded, then we obtain the $B_{p,q}^s$ -well-posedness of $(P)$ when $s>0,\ 1\leq p, q\leq \infty $ by Theorem 4.4.

Example 5.2 Let H be a Hilbert space, and P be a densely defined positive self-adjoint operator on H with $P\geq \delta> 0$ . Let $M = P- \epsilon $ with $\epsilon < \delta $ , and let $ A = \sum _{i=0}^k a_iP^i$ with $a_i\geq 0,\ a_k> 0$ , where k is an integer $\geq 2$ . Then there exist $C> 0$ and $\rho>0$ such that

(5.3) $$ \begin{align}\big\Vert M(zM + A)^{-1}\big\Vert \leq \frac{C}{1 + \vert z\vert} \end{align} $$

whenever $Re (z)\geq -\rho ( 1 + \vert Im z\vert )$ by [Reference Favini and Yagi13, p. 73]. If M is regular enough so that $0\in \rho (M)$ , then

(5.4) $$ \begin{align}\big\Vert (zM + A)^{-1}\big\Vert \leq \frac{C}{1 + \vert z\vert} \end{align} $$

whenever $Re (z)\geq -\rho ( 1 + \vert Im z\vert )$ by (5.3).

Let $\Omega = (0, 1)$ and let $H = L^2(\Omega )$ . It is clear that the operator $\frac {d^2}{dx^2}$ with domain $H^2(\Omega )\cap H_0^1(\Omega )$ generates a contraction semigroup on H and $P = - \frac {d^2}{dx^2}$ is positive and self-adjoint in H [Reference Arendt, Batty, Hieber and Neubrander4, Example 3.4.7]. Hence, $M = 1+ P$ has a bounded inverse. Let $a, b\in L^1({\mathbb R}_+)$ and let $1 < p< \infty $ . We consider the following equation:

$$ \begin{align*} \quad \left\{ \begin{array}{@{}ll} \frac{\partial^2}{\partial t^2} (1- \frac{\partial^2}{\partial x^2})u(t, x) +\int_{-\infty}^0 a(t-s) \frac{\partial}{\partial t} (1- \frac{\partial^2}{\partial x^2})u(s, x)ds + \int_{-\infty}^0 b(t-s) (1- \frac{\partial^2}{\partial x^2})u(s, x)ds\\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ = \frac{\partial^4}{\partial x^4} u(t, x) + f(t, x), \ \ \ \ \ \ \ \ (t, x) \in \mathbb{T}\times \Omega,\nonumber\\ \\ u(t, 0) = u(t, 1)= \frac{\partial^2}{\partial x^2}u(t, 0) = \frac{\partial^2}{\partial x^2}u(t, 1) = 0, \ \ \ \ \ \ \ t \in \mathbb{T}.\nonumber \end{array} \right. \end{align*} $$

This equation can be rewritten in the more compact form:

(P^′) $$\begin{align}\begin{array}{@{}l} (Mu)"(t) + a*(Mu)'(t) + b*(Mu)(t) = (Au)(t) + f(t),\ \ (t\in \mathbb{T}),\\ \end{array} \end{align} $$

which is an ordinary differential equation on $\mathbb {T}$ with values in H, where $f\in L^p(\mathbb {T}; H)$ and the solution $u\in W_{\text {per}}^{1,p}(\mathbb {T}; D(A))$ satisfies $\ Mu\in W_{\text {per}}^{2,p}(\mathbb {T};\, H)$ , where $M = 1- \frac {\partial ^2}{\partial x^2}$ and $A= \Delta ^2$ , here we consider $\Delta $ as the Laplacian on $L^2(\Omega )$ with Dirichlet boundary condition. If $\rho _{M}(A;\mu ,\nu ,\eta ,\tau )=\mathbb {Z}$ , the sequence $k^2 -ik \hat a(k) - \hat b(k)$ is $1$ -regular and $\{AN_k:\ k\in \mathbb {Z}\}$ is norm bounded, one can obtain the $L^p$ -well-posedness of $(P')$ by using (5.3), (5.4), and Theorem 3.4. Here again, we have used the fact that $L^2 (\Omega )$ is a Hilbert space and the fact that every norm bounded subset of $\mathcal {L}(X)$ is R-bounded when X is isomorphic to a Hilbert space [Reference Arendt and Bu5]. One can also obtain the $B_{p,q}^s$ -well-posedness of $(P_2)$ when $s>0,\ 1\leq p, q\leq \infty $ and the sequence $k^2 -ik \hat a(k) - \hat b(k)$ is $2$ -regular and $\{AN_k:\ k\in \mathbb {Z}\}$ is norm bounded by using Theorem 4.4.

Acknowledgements

The authors are ranked in alphabetic order of their names and should be treated equally.

Footnotes

This work was supported by the Natural Science Foundation of China (Grant No. 12571141) and the Chongqing Talent Support Program (Grant No. cstc2024ycjh-bgzxm0121).

References

Alvarez, E. and Lizama, C., A characterization of ${L}^p$

-maximal regularity for time-fractional systems in UMD spaces and applications . J. Differential Equations 389(2024), 257–284.10.1016/j.jde.2024.01.021CrossRef Google Scholar

Alvarez, E., Lizama, C., and Murillo-Arcila, M., Maximal regularity of solutions for the tempered fractional Cauchy problem . J. Funct. Anal. 290(2026), 111196, 35 pp.10.1016/j.jfa.2025.111196CrossRef Google Scholar

Arendt, W., Batty, C., and Bu, S., Fourier multipliers for Hölder continuous functions and maximal regularity . Stud. Math. 160(2004), 23–51.10.4064/sm160-1-2CrossRef Google Scholar

Arendt, W., Batty, C., Hieber, M., and Neubrander, F., Vector-valued Laplace transforms and Cauchy problems, Birkhäuser, Basel, 2001.10.1007/978-3-0348-5075-9CrossRef Google Scholar

Arendt, W. and Bu, S., The operator-valued Marcinkiewicz multiplier theorem and maximal regularity . Math. Z. 240(2002), no. 2, 311–343.10.1007/s002090100384CrossRef Google Scholar

Arendt, W. and Bu, S., Operator-valued Fourier multipliers on periodic Besov spaces and applications . Proc. Edinb. Math. Soc. 47(2004), 15–33.10.1017/S0013091502000378CrossRef Google Scholar

Bourgain, J., Vector-valued singular integrals and the ${H}^1$

-BMO duality. In: Probability theory and harmonic analysis (Cleveland, OH, 1983), Monographs and Textbooks in Pure and Applied Mathematics, 98, Dekker, New York, NY, 1988, pp. 1–19.Google Scholar

Bu, S. and Fang, Y., Maximal regularity for integro-differential equations in periodic Triebel-Lizorkin spaces . Taiwanese J. Math. 12(2008), 281–292.10.11650/twjm/1500574153CrossRef Google Scholar

Bu, S. and Fang, Y., Periodic solutions for second order integro-differential equations with infinite delay in Banach spaces . Stud. Math. 184(2008), 103–119.10.4064/sm184-2-1CrossRef Google Scholar

Bu, S. and Kim, J., Operator-valued Fourier multipliers on periodic Triebel spaces . Acta Math. Sin. (Engl. Ser.) 21(2005), 1049–1056.10.1007/s10114-004-0453-9CrossRef Google Scholar

Chill, R. and Srivastava, S., ${L}^p$

-maximal regularity for second order Cauchy problems . Math. Z. 251(2005), 751–781.10.1007/s00209-005-0815-8CrossRef Google Scholar

Conejero, J. A., Lizama, C., Murillo-Arcila, M., and Seoane-Sepúlveda, J. B., Well-posedness degenerate third order equations with delay and applications to inverse problems . Israel J. Math. 229(2019), 219–254.10.1007/s11856-018-1796-8CrossRef Google Scholar

Favini, A. and Yagi, A., Degenerate differential equations in Banach spaces, Pure Mathematics and Applied Mathematics, 215, Dekker, New York, NY, 1999.Google Scholar

Keyantuo, V. and Lizama, C., Fourier multipliers and integro-differential equations in Banach spaces . J. Lond. Math. Soc. 69(2004), 737–750.10.1112/S0024610704005198CrossRef Google Scholar

Keyantuo, V., Lizama, C., and Poblete, V., Periodic solutions of integro-differential equations in vector-valued function spaces . J. Differential Equations 246(2009), 1007–1037.10.1016/j.jde.2008.09.007CrossRef Google Scholar

Pietsch, A. and Wenzel, J., Orthonormal systems and Banach space geometry, Encyclopedia of Mathematics and its Applications, 70, Cambridge University Press, 1998.10.1017/CBO9780511526145CrossRef Google Scholar

Ponce, R., Well-posedness of second order differential equations with memory . Math. Nachr. 295(2022), 2246–2264.10.1002/mana.202000113CrossRef Google Scholar

Ponce, R., Periodic solutions to second-order differential equations with fading memory . Math. Z. 307(2024), Article no. 5, 19 pp.10.1007/s00209-024-03474-zCrossRef Google Scholar

Ponce, R., Approximate controllability of abstract discrete fractional systems of order $1<\alpha <2$

via resolvent sequences . J. Optim. Theory Appl. 203(2024), 359–385.10.1007/s10957-024-02516-0CrossRef Google Scholar

Weis, L., Operator-valued Fourier multipliers and maximal ${L}_p$

-regularity . Math. Ann. 319(2001), 735–758.10.1007/PL00004457CrossRef Google Scholar

Weis, L., A new approach to maximal ${L}_p$

-regularity. In: Evolution equations and their applications in physical and life sciences (Bad Herrenalb, 1998), Lecture Notes in Pure and Applied Mathematics, 215, Dekker, New York, NY, 2001, pp. 195–214.Google Scholar

Article contents

Well-posedness of second-order integro-differential equations

Abstract

Keywords

MSC classification

Information

1 Introduction

2 Preliminaries

Definition 2.1 [Reference Arendt and Bu5]

Definition 2.4 [Reference Keyantuo, Lizama and Poblete15]

Theorem 2.5 [Reference Keyantuo, Lizama and Poblete15, Theorem 4.6]

3 $L^p$ -well-posedness of $(P_5)$

Theorem 3.1 [Reference Arendt and Bu5, Theorem 1.3]

Proposition 3.2 [Reference Arendt and Bu5, Proposition 1.11]

Theorem 3.3 [Reference Conejero, Lizama, Murillo-Arcila and Seoane-Sepúlveda12, Theorem 1.1]

4 $B_{p,q}^s$ -well-posedness and $F_{p,q}^s$ -well-posedness of $(P_5)$

5 Applications

Acknowledgements

Footnotes

References

Save article to Kindle

Save article to Dropbox

Save article to Google Drive

Reply to: Submit a response

Your details

You have entered the maximum number of contributors

Conflicting interests