Periodic solutions of four-order degenerate differential equations with finite delay in vector-valued function spaces

Abstract

In this paper, we mainly investigate the well-posedness of the four-order degenerate differential equation ($P_4$): $(Mu)''''(t) + \alpha (Lu)'''(t) + (Lu)''(t)$ $=\beta Au(t) + \gamma Bu'(t) + Gu'_t + Fu_t + f(t),\,( t\in [0,\,2\pi ])$ in periodic Lebesgue–Bochner spaces $L^p(\mathbb {T}; X)$ and periodic Besov spaces $B_{p,q}^s\;(\mathbb {T}; X)$, where $A$, $B$, $L$ and $M$ are closed linear operators on a Banach space $X$ such that $D(A)\cap D(B)\subset D(M)\cap D(L)$ and $\alpha,\,\beta,\,\gamma \in \mathbb {C}$, $G$ and $F$ are bounded linear operators from $L^p([-2\pi,\,0];X)$ (respectively $B_{p,q}^s([-2\pi,\,0];X)$) into $X$, $u_t(\cdot ) = u(t+\cdot )$ and $u'_t(\cdot ) = u'(t+\cdot )$ are defined on $[-2\pi,\,0]$ for $t\in [0,\, 2\pi ]$. We completely characterize the well-posedness of ($P_4$) in the above two function spaces by using known operator-valued Fourier multiplier theorems.

1. Introduction

The characterizations of the well-posedness for abstract degenerate differential equations with periodic initial conditions have been studied extensively in the last years. See e.g. [5–11], [14–20] and the references therein. For examples, Lizama and Ponce [16] considered the first-order degenerate equation:

(1.1)\begin{equation} (Mu)'(t)=Au(t)+f(t),\quad (t\in \mathbb{T}:=[0,2\pi]), \end{equation}

they gave necessary and sufficient conditions to guarantee the well-posedness of (1.1) in Lebesgue–Bochner spaces $L^p(\mathbb {T}; X)$, periodic Besov spaces $B_{p,q}^s(\mathbb {T}; X)$ and periodic Triebel–Lizorkin spaces $F_{p,q}^s(\mathbb {T}; X)$ under some appropriate assumptions on the modified resolvent operator determined by (1.1). Moreover, they also investigated the first-order degenerate equation with infinite delay [17]:

(1.2)\begin{equation} (Mu)'(t)=\alpha Au(t)+\int_{-\infty}^t a(t-s)Au(s){\rm d}s+f(t),\quad (t\in \mathbb{T}), \end{equation}

where $A$ and $M$ are closed linear operators defined on a Banach space $X$ with $D(A)\subseteq D(M)$, $a\in L^1(\mathbb {R}_+)$ is a scalar-valued kernel, $\alpha \in \mathbb {R}\backslash \left \{0\right \}$ and $f$ an $X$-valued function defined on $\mathbb {T}$.

Bu [9] considered a new second-order degenerate equation and gave necessary or sufficient conditions for this equation to be $L^p$-well-posed (respectively $B_{p,q}^s$-well-posed and $F_{p,q}^s$-well-posed), which recover some known results presented in [5, 6, 10] in the simpler case $M=I_X$. We notice that third-order differential equations also describe some kinds of models arising from natural phenomena, such as flexible space structures with internal damping, the well-posedness of third-order differential equations has been investigated extensively by many authors. See [1–3, 7, 8, 13, 14, 19] for more information and references therein. For example, Poblete and Pozo [19] studied the well-posedness for the abstract third-order equation:

(1.3)\begin{equation} \alpha u'''(t)+u''(t)=\beta Au(t)+\gamma Bu'(t)+f(t),\ (t\in \mathbb{T}), \end{equation}

where $A$ and $B$ are closed linear operators defined on a Banach space $X$ with $D(A)\cap D(B)\neq \emptyset$, the constants $\alpha,\,\beta,\,\gamma \in \mathbb {R}^+$ and $f$ belong to either the Lebesgue–Bochner spaces, or periodic Besov spaces, or periodic Triebel–Lizorkin spaces. They give necessary and sufficient conditions for (1.3) to be $L^p$-well-posed (respectively $B_{p,q}^s$-well-posed and $F_{p,q}^s$-well-posed) by using vector-valued Fourier theorems in the vector-valued function spaces.

In this paper, we study the following four-order degenerate differential equation:

(P ₄)\begin{align*} & (Mu)''''(t) + \alpha(Lu)'''(t) + (Lu)''(t)\nonumber\\ & \quad =\beta Au(t) + \gamma Bu'(t) + Gu'_t + Fu_t + f(t),\quad (t\in \mathbb{T}), \end{align*}

where $A$, $B$, $L$ and $M$ are closed linear operators on a Banach space $X$ such that $D(A)\cap D(B)\subset D(M)\cap D(L)$ and $\alpha,\,\beta,\,\gamma \in \mathbb {C}$, $G$ and $F$ are bounded linear operators from $L^p([-2\pi,\,0];X)$ (respectively $B_{p,q}^s([-2\pi,\,0];X)$) into $X$, $u_t(\cdot ) = u(t+\cdot )$ and $u'_t(\cdot ) = u'(\cdot +t)$ are defined on $[-2\pi,\,0]$ for $t\in [0,\, 2\pi ]$.

Let $f\in L^p(\mathbb {T}; X)$ be given, a function $u\in W_{\text {per}}^{1,p}(\mathbb {T}; X)\cap L^p(\mathbb {T}; D(A))$ is called a strong $L^p$-solution of ($P_4$), if $Mu\in W_{\text {per}}^{4,p}(\mathbb {T}; X),\,Lu\in W_{\text {per}}^{3,p}(\mathbb {T}; X),\,\ u'\in L^p(\mathbb {T}; D(B))$ and ($P_4$) is satisfied a.e. on $\mathbb {T}$, here we consider $D(A)$ and $D(B)$ as Banach spaces equipped with the graph norms. We say that ($P_4$) is $L^p$-well-posed, if for each $f\in L^p(\mathbb {T}; X)$, there exists a unique strong $L^p$-solution of ($P_4$). We introduce similarly the $B_{p,q}^s$-well-posedness of ($P_4$).

The main purpose of this paper is to give some characterizations of the well-posedness of ($P_4$) in Lebesgue–Bochner spaces $L^p(\mathbb {T}; X)$ and periodic Besov spaces $B_{p,q}^s(\mathbb {T}; X)$. The characterizations of the well-posedness of ($P_4$) involve the Rademacher boundedness (or norm boundedness) of the $M$-resolvent of $A$, $B$ and $L$ defined by ($P_4$). More precisely, we show that when $X$ is a UMD Banach space and $1 < p < \infty$, if $\{k(G_{k+1}-G_{k}):\ k\in \mathbb {Z}\}$ is Rademacher-bounded, then ($P_4$) is $L^p$-well-posed if and only if $\rho _{M}(A,\,B,\,L) = \mathbb {Z}$ (the $M$-resolvent set of $A$, $B$ and $L$ defined by ($P_4$)) and the sets

\[ \{k^4MN_k:\ k\in \mathbb{Z}\},\quad \{k^3LN_k:\ k\in \mathbb{Z}\}, \quad \{kBN_k:\ k\in \mathbb{Z}\}, \quad \{kN_k:\ k\in \mathbb{Z}\} \]

are Rademacher-bounded, where

\[ N_k=[(k^4M -(i\alpha k^3+k^2)L - \beta A - i\gamma kB - ikG_k - F_k]^{{-}1}, \]

$G_k,\, F_k,\,H_k \in \mathcal {L} (X)$ are defined by $G_k x = G(e_kx)$, $F_k x = F(e_kx)$, $x\in X$. Since this characterization of the $L^p$-well-posedness of ($P_4$) does not depend on the space parameter $1 < p < \infty$, we deduce that when $X$ is a UMD Banach space and the set $\{k(G_{k+1}-G_{k}):\ k\in \mathbb {Z}\}$ is Rademacher-bounded, then ($P_4$) is $L^p$-well-posed for some $1 < p < \infty$ if and only if it is $L^p$-well-posed for all $1 < p < \infty$.

We also give a similar characterization for the $B_{p,q}^s$-well-posedness of ($P_4$): let $X$ be a Banach space, $1\leq p,\,q\leq \infty,\,\ s>0$, assume that the sets $\{k(F_{k+2}-2F_{k+1}+F_k):\ k\in \mathbb {Z}\}$, $\{k(G_{k+1}-G_{k}):\ k\in \mathbb {Z}\}$ and $\{k^2(G_{k+2}-2G_{k+1}+G_k):\ k\in \mathbb {Z}\}$ are norm-bounded, then the problem ($P_4$) is $B_{p,q}^s$-well-posed if and only if $\subset \rho _{M}(A,\,B,\,L) = \mathbb {Z}$ and the sets

\[ \{k^4MN_k:\ k\in \mathbb{Z}\},\quad \{k^3LN_k:\ k\in \mathbb{Z}\},\quad \{kBN_k:\ k\in \mathbb{Z}\}, \quad \{kN_k:\ k\in \mathbb{Z}\} \]

are norm-bounded, where $N_k,\, \ F_k,\,\ G_k$ and $H_k$ are defined as in the $L^p$-well-posedness case. Since this characterization of the $B_{p,q}^s$-well-posedness of ($P_4$) does not depend on the parameters $1\leq p,\,q\leq \infty,\,\ s>0$, we deduce that when the sets $\{k(F_{k+2}-2F_{k+1}+F_k):\ k\in \mathbb {Z}\}$, $\{k(G_{k+1}-G_{k}): k\in \mathbb {Z}\}$ and $\{k^2(G_{k+2}-2G_{k+1}+G_k):\ k\in \mathbb {Z}\}$ are norm-bounded, then ($P_4$) is $B_{p,q}^s$-well-posed for some $1\leq p,\,q\leq \infty,\,\ s>0$ if and only if it is $B_{p,q}^s$-well-posed for all $1\leq p,\,q\leq \infty,\,\ s>0$.

Our main tools in the investigation of the well-posedness of ($P_4$) are the operator-valued Fourier multiplier theorems obtained by Arendt and Bu [5, 6] on $L^p(\mathbb {T}; X)$ and $B_{p,q}^s(\mathbb {T}; X)$. In fact, our main idea is to transform the well-posedness of ($P_4$) to an operator-valued Fourier multiplier problem in the corresponding vector-valued function space.

This work is organized as follows: in § $2$, we study the well-posedness of ($P_4$) in vector-valued Lebesgue–Bochner spaces $L^p(\mathbb {T}; X)$. In § $3$, we consider the well-posedness of ($P_4$) in periodic Besov spaces $B_{p,q}^s(\mathbb {T}; X)$. In the last section, we give some examples of degenerate differential equations with finite delay to which our abstract results may be applied.

2. Well-posedness of ($P_4$) in Lebesgue–Bochner spaces

Let $X$ and $Y$ be complex Banach spaces and let $\mathbb {T}:=[0,\,2\pi ]$. We denote by $\mathcal {L}(X,\,Y)$ the space of all bounded linear operators from $X$ to $Y$. If $X=Y$, we will simply denote it by $\mathcal {L}(X)$. For $1\leq p<\infty$, we denote by $L^p(\mathbb {T}; X)$ the space of all equivalent class of $X$-valued measurable functions $f$ defined on $\mathbb {T}$ satisfying

\begin{align*} \left\Vert f\right\Vert_{L^p}:=\Bigg(\frac{1}{2\pi}\int_0^{2\pi}\left\Vert f(t)\right\Vert^p \,{\rm d}t\Big)^{1/p}<\infty. \end{align*}

For $f\in L^1(\mathbb {T}; X)$, the $k$-th Fourier coefficient of $f$ is defined by

\begin{align*} \hat{f}(k):=\frac{1}{2\pi}\int_{0}^{2\pi}e_{{-}k}(t)f(t)\,{\rm d}t, \end{align*}

where $k\in \mathbb {Z}$ and $e_k(t)=e^{ikt}$ when $t\in \mathbb {T}$.

Definition 2.1 Let $X$ and $Y$ be complex Banach spaces and $1\leq p<\infty$, we say that $(M_k)_{k\in \mathbb {Z}}\subset \mathcal {L}(X,\,Y)$ is an $L^p$-Fourier multiplier, if for each $f\in L^p(\mathbb {T}; X)$, there exists a unique $u\in L^p(\mathbb {T}; Y)$ such that $\hat {u}(k)=M_k\hat {f}(k)$ when $k\in \mathbb {Z}$.

From the closed graph theorem, if $(M_k)_{k\in \mathbb {Z}}\subset \mathcal {L}(X,\,Y)$ is an $L^p$-Fourier multiplier, then there exists a unique bounded linear operator $T\in \mathcal {L}(L^p(\mathbb {T}; X),\, L^p(\mathbb {T}; Y))$ satisfying $(Tf)^\wedge (k) = M_k\hat f(k)$ when $f\in L^p(\mathbb {T}; X)$ and $k\in \mathbb {Z}$. The operator-valued Fourier multiplier theorem on $L^p(\mathbb {T}; X)$ obtained in [5] involves the Rademacher boundedness for sets of bounded linear operators. Let $\gamma _j$ be the $j$-th Rademacher function on $[0,\,1]$ defined by $\gamma _j(t)=\rm {sgn}(\sin (2^{j-1}t))$ when $j\geq 1$. For $x\in X$, we denote by $\gamma _j\otimes x$ the vector-valued function $t\rightarrow r_j(t)x$ on $[0,\,1]$.

Definition 2.2 Let $X$ and $Y$ be Banach spaces. A set ${\bf {T}}\subset \mathcal {L}(X,\,Y)$ is said to be Rademacher-bounded (R-bounded, in short), if there exists $C>0$ such that

\[ \left\Vert \sum_{j=1}^n\gamma_j\otimes T_jx_j\right\Vert_{L^1([0,1];Y)}\leq C \left\Vert \sum_{j=1}^n\gamma_j\otimes x_j\right\Vert_{L^1([0,1];X)} \]

for all $T_1,\,\ldots,\,T_n\in {\bf {T}},\,x_1,\,\ldots,\,x_n\in X$ and $n\in \mathbb {N}$.

Remark 2.3

1. (i) Let ${\bf {S}},\,{\bf {T}}\subset \mathcal {L}(X)$ be $R$-bounded sets. Then it can be shown easily from the definition that ${\bf {ST}}:=\left \{ST:S\in {\bf {S}},\, T\in {\bf {T}}\right \}$ and ${\bf {S}}+{\bf {T}}:=\left \{S+T:S\in {\bf {S}},\, T\in {\bf {T}}\right \}$ are still $R$-bounded.

2. (ii) Let $X$ be a $\rm {UMD}$ Banach space and let $M_k=m_kI_X$ with $m_k\in \mathbb {C}$, where $I_X$ is the identity operator on $X$, if $\sup _{k\in \mathbb {Z}}\left \vert m_k\right \vert < \infty$ and $\sup _{k\in \mathbb {Z}}\left \vert k(m_{k+1}-m_k)\right \vert <\infty$, then $(M_k)_{k\in \mathbb {Z}}$ is an $L^p$-Fourier multiplier whenever $1 < p < \infty$ [5].

The main tool in our study of $L^p$-well-posedness of ($P_4$) is the $L^p$-Fourier multiplier theorem established in [5]. The following results will be very important in the proof of our main result of this section. For the concept of UMD Banach spaces, we refer the readers to [5] and references therein.

Theorem 2.4 [5, Theorem 1.3]

Let $X,\,Y$ be UMD Banach spaces and $(M_k)_{k\in \mathbb {Z}}\subset \mathcal {L}(X,\,Y)$. If the sets $\{M_k:\ k\in \mathbb {Z}\}$ and $\{k(M_{k+1}-M_k): \ k\in \mathbb {Z}\}$ are $R$-bounded, then $(M_k)_{k\in \mathbb {Z}}$ defines an $L^p$-Fourier multiplier whenever $1< p<\infty$.

Proposition 2.5 [5, 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.

Now we consider the following four-order degenerate differential equations with finite delays:

(P ₄)\begin{align*} & (Mu)''''(t) + \alpha(Lu)'''(t) + (Lu)''(t)\\ & \quad = \beta Au(t) + \gamma Bu'(t) + Gu'_t + Fu_t + f(t), \quad (t\in\mathbb{T}) \end{align*}

where $A,\, B,\, M$ and $L$ are closed linear operators on a Banach space $X$ satisfying $D(A)\cap D(B)\subset D(M)\cap D(L)$, $\alpha,\,\beta,\,\gamma \in \mathbb {C}$ are given and $F,\,G:L^p([-2\pi,\,0];X)\rightarrow X$ are bounded linear operators ($F$ and $G$ are known as the delay operators). Moreover, for fixed $t\in \mathbb {T}$, the functions $u_t$ and $u'_t$ are elements in $L^p([-2\pi,\,0];X)$ defined by $u_t(s)=u(t+s),\, \ u'_t(s) = u'(t+s)$ for $-2\pi \leq s\leq 0$, here we identify a function $u$ on $\mathbb {T}$ with its natural $2\pi$-periodic extension on $\mathbb {R}$.

Let $F,\,G\in \mathcal {L}((L^p[-2\pi,\,0];X),\,X)$ and $k\in \mathbb {Z}$. We define the linear operators $F_k,\,G_k\in \mathcal {L}(X)$ by

(2.1)\begin{equation} F_kx := F(e_kx), \quad G_kx := G(e_k x), \end{equation}

for $x\in X$, where $e_k(t) = e^{ikt}$ when $t\in \mathbb {T}$. It is clear that $\left \Vert F_k\right \Vert \leq \left \Vert F\right \Vert$ and $\left \Vert G_k\right \Vert \leq \left \Vert G\right \Vert$ as $\left \Vert e_k\right \Vert _p = 1$. It is easy to see that when $u\in L^p(\mathbb {T}; X)$, then

(2.2)\begin{equation} \widehat{Fu_.}(k) = F_k\hat u(k), \quad \widehat{Gu_.}(k) = G_k\hat u(k) \end{equation}

for $k\in \mathbb {Z}$. This implies that $(F_k)_{k\in \mathbb {Z}}$ and $(G_k)_{k\in \mathbb {Z}}$ are $L^p$-Fourier multipliers as

\[ \Vert Fu_t\Vert \leq \Vert F \Vert \Vert u_.\Vert_{L^p([{-}2\pi, 0]; X)} = \Vert F \Vert \Vert u\Vert_{L^p}, \]

and

\[ \Vert Gu_t\Vert \leq \Vert G \Vert \Vert u_.\Vert_{L^p([{-}2\pi, 0]; X)} = \Vert G \Vert \Vert u\Vert_{L^p}, \]

for $t\in \mathbb {T}$ so that $Fu_\cdot,\,\ Gu_\cdot,\,\ Hu_\cdot \in L^p(\mathbb {T}; X)$.

Now we define the resolvent set of ($P_4$) by

\begin{align*} & \rho_{M}(A,B,L):= \big\{k\in\mathbb{Z}: k^4M - (\alpha ik^3+k^2)L\\ & \qquad - \beta A - i\gamma kB - ikG_k - F_k\text{ is invertible from }\\ & D(A)\cap D(B) \text{ onto }X \quad\text{and}\quad [k^4M - (\alpha ik^3+k^2)L - \beta A\\& \qquad - i\gamma kB - ikG_k - F_k]^{{-}1} \in \mathcal{L}(X) \big\}. \end{align*}

For the sake of simplicity, when $k\in \rho _{M}(A,\,B,\,L)$, we will use the following notation:

(2.3)\begin{equation} N_k=[a_kM - b_kL - \beta A - c_kB - ikG_k - F_k]^{{-}1}, \quad (k\in\mathbb{Z}), \end{equation}

where

(2.4)\begin{equation} a_k=k^4, \quad b_k=\alpha ik^3+k^2, \quad c_k=i\gamma k, \quad (k\in\mathbb{Z}). \end{equation}

If $k\in \rho _{M}(A,\,B,\,L)$, then $MN_k,\,\ LN_k,\,\ AN_k$ and $BN_k$ make sense as $D(A)\cap D(B)\subset D(M)\cap D(L)$ by assumption, and they belong to $\mathcal {L}(X)$ by the closed graph theorem and the closedness of $A,\,\ B,\,\ M$ and $L$.

Let $(L_k)_{k\in \mathbb {Z}}\subset \mathcal {L}(X,\,Y)$ be a given sequence of operators. We define

\[ (\triangle^0L)_k=L_k, \quad (\triangle L)_k=L_{k+1}-L_k, \quad (k\in\mathbb{Z}) \]

and for $n=2,\,3,\,\ldots,$ set

\[ (\triangle^nL)_k=\triangle (\triangle^{n-1}L)_k, \quad (k\in\mathbb{Z}). \]

Definition 2.6 A sequence $(d_k)_{k\in \mathbb {Z}}\subseteq \mathbb {C}\backslash \left \{0\right \}$ is called $1$-regular if the sequence $(k\frac {\triangle ^1d_k}{d_k})_{k\in \mathbb {Z}}$ is bounded; it is called $2$-regular if it is $1$-regular and the sequence $(k^2\frac {\triangle ^2d_k}{d_k})_{k\in \mathbb {Z}}$ is bounded; it is called $3$-regular if it is $2$-regular and the sequence $(k^3\frac {\triangle ^3d_k}{d_k})_{k\in \mathbb {Z}}$ is bounded.

Remark 2.7 It is easy to see that $(a_k)_{k\in \mathbb {N}}$, $(b_k)_{k\in \mathbb {N}}$ and $(c_k)_{k\in \mathbb {N}}$ are $3$-regular.

Definition 2.8 Let $1\leq p<\infty$, $n\geq 1$ be an integer and let $X$ be a Banach space, we define the the following vector-valued function spaces:

\begin{align*} W_{\text{per}}^{n,p}(\mathbb{T}; X)& :=\big\{u\in L^p(\mathbb{T}; X):\text{ there exists }v\in L^p(\mathbb{T}; X),\text{such that }\hat{v}(k)\\& =(ik)^n\hat{u}(k) \text{ for all } k\in \mathbb{Z}\big\}. \end{align*}

$W_{\text {per}}^{n,p}(\mathbb {T}; X)$ is the $n$-th $X$-valued periodic Sobolev space.

Remark 2.9 We have the following two useful properties concerning these spaces:

1. (i) Let $m,\,n\in \mathbb {N}$. If $n\leq m$, then $W_{\text {per}}^{m,p}(\mathbb {T}; X)\subseteq W_{\text {per}}^{n,p}(\mathbb {T}; X)$.

2. (ii) If $u\in W_{\text {per}}^{n,p}(\mathbb {T}; X)$, then for any $0\leq k\leq n-1$, we have $u^{(k)}(0)=u^{(k)}(2\pi )$.

Let $1\leq p<\infty$, we define the solution space of the $L^p$-well-posedness of ($P_4$) by

\begin{align*} & S_p(A,B,M,L):=\big\{u\in W_{\text{per}}^{1,p}(\mathbb{T}; X)\cap L^p(\mathbb{T}; D(A)): Mu\in W_{\text{per}}^{4,p}(\mathbb{T}; X), \\ & Lu\in W_{\text{per}}^{3,p}(\mathbb{T}; X), \ u'\in L^p(\mathbb{T}; D(B))\big\}, \end{align*}

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

\begin{align*} \left\Vert u\right\Vert_{S_p(A,B,M,L)}& :=\left\Vert u\right\Vert_{L^p}+\left\Vert Au\right\Vert_{L^p}+\left\Vert (Mu)'\right\Vert_{L^p}+\left\Vert (Mu)''\right\Vert_{L^p}+\left\Vert (Mu)'''\right\Vert_{L^p}\\ & \quad+\left\Vert (Mu)''''\right\Vert_{L^p}+\left\Vert (Lu)'\right\Vert_{L^p}+\left\Vert (Lu)''\right\Vert_{L^p}+\left\Vert (Lu)'''\right\Vert_{L^p}+\left\Vert Bu'\right\Vert_{L^p}. \end{align*}

If $u\in S_p(A,\,B,\,M,\,L)$, then $Mu$, $(Mu)'$, $(Mu)''$ and $(Mu)'''$ are $X$-valued continuous functions on $\mathbb {T}$, and $Mu(0)=Mu(2\pi )$, $(Mu)'(0)=(Mu)'(2\pi )$, $(Mu)''(0)=(Mu)''(2\pi )$, $(Mu)'''(0)=(Mu)'''(2\pi )$ by [5, Lemma 2.1].

Definition 2.10 Let $1\leq p<\infty$ and $f\in L^p(\mathbb {T}; X)$, $u\in S_p(A,\,B,\,M,\,L)$ is called a strong $L^p$-solution of ($P_4$), if ($P_4$) is satisfied a.e. on $\mathbb {T}$. We say that ($P_4$) is $L^p$-well-posed, if for each $f\in L^p(\mathbb {T}; X)$, there exists a unique strong $L^p$-solution of ($P_4$).

If ($P_4$) is $L^p$-well-posed, then there exists a constant $C>0$, such that for each $f\in L^p(\mathbb {T}; X)$, if $u\in S_p(A,\,B,\,M,\,L)$ is the unique strong $L^p$-solution of ($P_4$), we have

(2.5)\begin{equation} \left\Vert u\right\Vert_{S_p(A,B,M,L)}\leq C\left\Vert f\right\Vert_{L^p}. \end{equation}

This follows easily from the closed graph theorem.

In order to prove our main result of this section, we need the following preparations.

Proposition 2.11 Let $A$, $B$, $M$ and $L$ be closed linear operators defined on a UMD Banach space $X$ such that $D(A)\cap D(B)\subset D(M)\cap D(L)$, $1 < p < \infty$ and $\alpha,\,\ \beta,\, \gamma \in \mathbb {C}$. Let $F,\,G\in \mathcal {L}(L^p([-2\pi,\,0];X),\,X)$. Assume that $\rho _{M}(A,\,B,\,L) = \mathbb {Z}$ and the sets $\left \{a_kMN_k:k\in \mathbb {Z}\right \}$, $\left \{b_kLN_k:k\in \mathbb {Z}\right \}$, $\left \{c_kBN_k:k\in \mathbb {Z}\right \}$, $\{k\triangle G_k: k\in \mathbb {Z}\}$ and $\left \{kN_k:k\in \mathbb {Z}\right \}$ are $R$-bounded, then $\left (a_kMN_k\right )_{k\in \mathbb {Z}}$, $\left (b_kLN_k\right )_{k\in \mathbb {Z}}$, $\left (c_kBN_k\right )_{k\in \mathbb {Z}}$ and $\left (kN_k\right )_{k\in \mathbb {Z}}$ are $L^p$-Fourier multipliers.

Proof. We only need to show that the set $\{k(N_k^{-1} - N_{k+1}^{-1})N_k: k\in \mathbb {Z}\}$ is $R$-bounded by [11, Theorem 1.1] and theorem 2.4, here we have used the facts that $(a_k)_{k\in \mathbb {N}}$, $(b_k)_{k\in \mathbb {N}}$ and $(c_k)_{k\in \mathbb {N}}$ are $1$-regular sequences. It follows from the definition of $N_k$ that

(2.6)\begin{align} & (N_k^{{-}1}-N_{k+1}^{{-}1})N_k\nonumber\\ & \quad =[a_kM-b_kL-\beta A-c_kB-ikG_k-F_k-a_{k+1}M+b_{k+1}L+\beta A+c_{k+1}B\nonumber\\ & \qquad+i(k+1)G_{k+1}+F_{k+1}]N_k\nonumber\\ & \quad =[-\triangle a_kM+\triangle b_kL+\triangle c_kB+ik\triangle G_k+iG_{k+1}+\triangle F_k]N_k, \end{align}

which implies

(2.7)\begin{align} & k(N_k^{{-}1}-N_{k+1}^{{-}1})N_k\nonumber\\ & ={-}\frac{k\triangle a_k}{a_k}(a_kMN_k)+\frac{k\triangle b_k}{b_k}(b_kLN_k)+\frac{k\triangle c_k}{c_k}(c_kBN_k)\nonumber\\ & \quad+i(k\triangle G_k)(kN_k)+iG_{k+1}(kN_k)+\triangle F_k(kN_k), \end{align}

when $k\neq 0$. It follows from remark 2.3 that the products and sums of $R$-bounded sets are still $R$-bounded. Thus, the set $\{k(N_k^{-1} - N_{k+1}^{-1})N_k: k\in \mathbb {Z}\}$ is $R$-bounded. This completes the proof.

The following statement is the main result of this section which gives a necessary and sufficient condition for the $L^p$-well-posedness of ($P_4$).

Theorem 2.12 Let $X$ be a UMD Banach space, $1< p<\infty$ and let $A,\, B,\, L$ and $M$ be closed linear operators on $X$ satisfying $D(A)\cap D(B)\subset D(M)\cap D(L)$ and $\alpha,\,\ \beta,\, \gamma \in \mathbb {C}$. Let $F,\,G\in \mathcal {L}(L^p([-2\pi,\,0];X),\,X)$ be such that the set $\{k\Delta G_k:\ k\in \mathbb {Z}\}$ is $R$-bounded. Then the following assertions are equivalent:

1. (i) ($P_4$) is $L^p$-well-posed;

2. (ii) $\rho _{M}(A,\,B,\,L)=\mathbb {Z}$, the sets $\{k^4MN_k:k\in \mathbb {Z}\},\, \{k^3LN_k:k\in \mathbb {Z}\}$, $\{kBN_k:k\in \mathbb {Z}\}$ and $\{kN_k:k\in \mathbb {Z}\}$ are $R$-bounded, where $N_k$ is defined by (2.3), the operators $F_k$ and $G_k$ are defined by (2.1).

Proof. First we show that the implication $(i)\Rightarrow (ii)$ holds true. We assume that ($P_4$) is $L^p$-well-posed and 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 ($P_4$) is $L^p$-well-posed, there exists a unique $u\in S_p(A,\,B,\,L,\,M)$ satisfying

(2.8)\begin{align} (Mu)''''(t) + \alpha(Lu)'''(t) + (Lu)''(t) = \beta Au(t) + \gamma Bu'(t) + Gu'_t + Fu_t + f(t) \end{align}

a.e. on $\mathbb {T}$. We have $\hat {u}(n)\in D(A)\cap D(B)$ when $n\in \mathbb {Z}$ by [5, Lemma 3.1] as $u\in L^p (\mathbb {T}; D(A))\cap L^p(\mathbb {T}; D(B))$. Taking Fourier transforms on both sides of (2.8), we obtain

(2.9)\begin{equation} [k^4M - (\alpha ik^3+k^2)L - \beta A - i\gamma kB - ikG_k - F_k]\hat{u}(k)=y \end{equation}

and $[n^4M - (\alpha in^3+n^2)L - \beta A - i\gamma nB - inG_n - F_n]\hat {u}(n)=0$ when $n\neq k$. This implies that the operator $k^4M - (\alpha ik^3+k^2)L - \beta A - i\gamma kB - ikG_k - F_k$ defined on $D(A)\cap D(B)$ with values in $X$ is surjective. To show that it is also injective, we let $x\in D(A)\cap D(B)$ be such that

\[ [k^4M - (\alpha ik^3+k^2)L - \beta A - i\gamma kB - ikG_k - F_k]x=0. \]

Let $u$ be the function given by $u(t)=e^{ikt}x$ when $t\in \mathbb {T}$, then it is clear that $u\in S_p(A,\,B,\,M,\,L)$ and ($P_4$) is satisfied a.e. on $\mathbb {T}$ when $f=0$. Thus, $u$ is a strong $L^p$-solution of ($P_4$) when taking $f=0$. We obtain $x=0$ by the uniqueness assumption. We have shown that the operator $k^4M - (\alpha ik^3+k^2)L - \beta A - i\gamma kB - ikG_k - F_k$ from $D(A)\cap D(B)$ into $X$ is injective. Therefore, $k^4M - (\alpha ik^3+k^2)L - \beta A - i\gamma kB - ikG_k - F_k$ is bijective from $D(A)\cap D(B)$ onto $X$.

Next we show that $[k^4M - (\alpha ik^3+k^2)L - \beta A - i\gamma kB - ikG_k - F_k]^{-1}\in \mathcal {L}(X)$. For $f(t)=e^{ikt}y$, we let $u\in S_p(A,\,B,\,M,\,L)$ be the unique strong $L^p$-solution of ($P_4$). Then

\[ \hat{u}(n)= \begin{cases} 0 & n\neq k,\\ [k^4M - (\alpha ik^3+k^2)L - \beta A - i\gamma kB - ikG_k - F_k]^{{-}1}y & n=k, \end{cases} \]

by (2.9). This implies that $u$ is given by

\[ u(t) = e^{ikt}[k^4M - (\alpha ik^3+k^2)L - \beta A - i\gamma kB - ikG_k - F_k]^{{-}1}y \]

when $t\in \mathbb {T}$. By (2.5), there exists a constant $C>0$ independent from $y$ and $k$, such that $\left \Vert u\right \Vert _{L^p}\leq C\left \Vert f\right \Vert _{L^p}$. This implies that

\begin{align*} \big\Vert [k^4M - (\alpha ik^3+k^2)L - \beta A - i\gamma kB - ikG_k - F_k]^{{-}1}y\big\Vert\leq C\left\Vert y\right\Vert \end{align*}

when $y\in X$, or equivalently

\begin{align*} \left\Vert [k^4M - (\alpha ik^3+k^2)L - \beta A - i\gamma kB - ikG_k - F_k]^{{-}1}\right\Vert\leq C. \end{align*}

We have shown that $k\in \rho _{M}(A,\,B,\,L)$ for all $k\in \mathbb {Z}$. Thus, $\rho _{M}(A,\,B,\,L)=\mathbb {Z}$.

Finally, we show that $(k^4MN_k)_{k\in \mathbb {Z}},\,\ (k^3LN_k)_{k\in \mathbb {Z}}$, $(kN_k)_{k\in \mathbb {Z}}$ and $(kBN_k)_{k\in \mathbb {Z}}$ define $L^p$-Fourier multipliers. Let $f\in L^p(\mathbb {T};X)$, then there exists $u\in S_p(A,\,B,\,M,\,L)$, a strong $L^p$-solution of ($P_4$) by assumption. Taking Fourier transforms on both sides of ($P_4$), we get that $\hat {u}(k)\in D(A)\cap D(B)$ by [5, Lemma 3.1] and

\[ [k^4M - (\alpha ik^3+k^2)L - \beta A - i\gamma kB - ikG_k - F_k]\hat{u}(k) = \hat{f}(k) \]

for $k\in \mathbb {Z}$. Since $k^4M - (\alpha ik^3+k^2)L - \beta A - i\gamma kB - ikG_k - F_k$ is invertible, we have

\[ \hat{u}(k) = [k^4M - (\alpha ik^3+k^2)L - \beta A - i\gamma kB - ikG_k - F_k]^{{-}1}\hat{f}(k) = N_k\hat f(k) \]

when $k\in \mathbb {Z}$. It follows from $u\in S_p(A,\,B,\,M,\,L)$ that $u\in L^{p}(\mathbb {T}; D(A))\cap W_{\text {per}}^{1,p}(\mathbb {T};X)$, $Mu\in W_{\text {per}}^{4,p}(\mathbb {T};X)$, $Lu \in W_{\text {per}}^{3,p}(\mathbb {T};X)$ and $u'\in L^p(\mathbb {T}; D(B))$. We have

\begin{align*} \widehat{(Mu)''''}(k)& =k^4M\hat{u}(k),\quad \widehat{(Lu)'''}(k)={-}ik^3L\hat{u}(k),\quad \widehat{Bu'}(k)\\& =ikB\hat{u}(k), \quad \widehat{u'}(k) = ik\hat u(k) \end{align*}

when $k\in \mathbb {Z}$. We conclude that $(k^4MN_k)_{k\in \mathbb {Z}},\, (k^3LN_k)_{k\in \mathbb {Z}}$, $(kBN_k)_{k\in \mathbb {Z}}$ and $(kN_k)_{k\in \mathbb {Z}}$ define $L^p$-Fourier multipliers as $(Mu)'''',\,\ (Lu)''',\,\ Bu',\,\ u'\in L^p(\mathbb {T};X)$. It follows from proposition 2.5 that the sets $\{k^4MN_k:k\in \mathbb {Z}\},\, \{k^3LN_k:k\in \mathbb {Z}\}$, $\{kBN_k:k\in \mathbb {Z}\}$ and $\{kN_k:k\in \mathbb {Z}\}$ are $R$-bounded. We have shown that the implication $(i)\Rightarrow (ii)$ is true.

Next we show that the implication $(ii)\Rightarrow (i)$ is valid. Assume that $\rho _{M}(A,\,B,\,L)=\mathbb {Z}$ and the sets $\{k^4MN_k:k\in \mathbb {Z}\},\, \{k^3LN_k:k\in \mathbb {Z}\}$, $\{kN_k:k\in \mathbb {Z}\}$ and $\{kBN_k:k\in \mathbb {Z}\}$ are $R$-bounded. It follows from proposition 2.11 that $(k^4MN_k)_{k\in \mathbb {Z}}$, $(k^3LN_k)_{k\in \mathbb {Z}}$, $(kBN_k)_{k\in \mathbb {Z}}$ and $(kN_k)_{k\in \mathbb {Z}}$ are $L^p$-Fourier multipliers. This implies that the sequences $(N_k)_{k\in \mathbb {Z}}$, $(BN_k)_{k\in \mathbb {Z}}$, $(k^2LN_k)_{k\in \mathbb {Z}}$, $(MN_k)_{k\in \mathbb {Z}}$, $(LN_k)_{k\in \mathbb {Z}}$ are $L^p$-Fourier multiplier. Here we have used the easy fact that $(d_k)_{k\in \mathbb {Z}}$ is an $L^p$-Fourier multiplier and the fact that the product of two $L^p$-Fourier multipliers is still an $L^p$-Fourier multiplier, where $d_k$ is defined by $d_k = 1/k$ when $k\not = 0$ and $d_0 =0$. In particular, considering $N_k\in \mathcal {L}(X,\, D(B))$, the sequence $(N_k)_{k\in \mathbb {Z}}$ is an $L^p$-Fourier multiplier. Then for all $f\in L^p(\mathbb {T};X)$, there exist $u_i\in L^p(\mathbb {T};X)$ ($1\leq i\leq 7$) and $u\in L^p(\mathbb {T}; D(B))$ satisfying

(2.10)\begin{align} \hat{u}_1(k) & = k^4MN_k\hat{f}(k),\quad \hat u_2(k)=ikN_k\hat f(k),\nonumber\\ \hat u_3(k) & = MN_k\hat f(k),\quad \hat u_4(k) = LN_k\hat f (k) \end{align}

(2.11)\begin{align} \hat u_5(k) & =ikBN_k\hat f(k),\quad \hat u_6(k) ={-}ik^3LN_k\hat f(k),\nonumber\\ \hat u_7(k) & ={-}k^2LN_k\hat f(k), \hat{u}(k) = N_k\hat{f}(k) \end{align}

for $k\in \mathbb {Z}$. Hence, $\hat u_2(k) = ik\hat u(k)$ for $k\in \mathbb {Z}$ by (2.10). This implies that $u\in W_{\text {per}}^{1,p}(\mathbb {T};X)$. It follows from (2.11) that $\widehat {u'}(k) = ik\hat u(k) = ikN_k\hat f(k)$ when $k\in \mathbb {Z}$. This together with $\hat u_5(k) =ikBN_k\hat f(k)$ when $k\in \mathbb {Z}$ implies that $u'\in L^p(\mathbb {T}; D(B))$ [5, Lemma 3.1]. By (2.10) and (2.11), we have $\hat u_3(k)= M\hat u(k)$ when $k\in \mathbb {Z}$. Hence, $u\in L^p(\mathbb {T}; D(M))$ and $Mu = u_3$. Similarly, by using (2.10) and (2.11), we have $\hat u_4(k) = L\hat u(k)$ when $k\in \mathbb {Z}$. Thus, $u\in L^p(\mathbb {T}; D(L))$ and $Lu = u_4$ [5, Lemma 3.1]. By (2.10) and the fact that $Mu = u_3$, we deduce $\hat u_1(k) = (ik)^4\widehat {Mu}(k) = (ik)^4\hat u_3(k)$ when $k\in \mathbb {Z}$. Thus, $Mu \in W_{\text {per}}^{4,p}(\mathbb {T}; X)$. Similarly, using (2.11) and the fact hat $Lu = u_4$, we deduce that $Lu \in W_{\text {per}}^{3,p}(\mathbb {T}; X)$.

We note that $(G_k)_{k\in \mathbb {Z}}$ and $(F_k)_{k\in \mathbb {Z}}$ are $L^p$-Fourier multipliers by (2.2), where $G_k,\, \ F_k$ and $H_k$ are defined by (2.1). Thus, $(ikG_kN_k)_{k\in \mathbb {Z}}$ and $(F_kD_k)_{k\in \mathbb {Z}}$ are $L^p$-Fourier multipliers as the product of two $L^p$-Fourier multipliers is still an $L^p$-Fourier multiplier. We have

\[ \beta AN_k = k^4MN_k - (\alpha ik^3+k^2)LN_k - i\gamma kBN_k - ikG_kN_k - F_kN_k - I_X \]

for $k\in \mathbb {Z}$. It follows that $\left (AN_k\right )_{k\in \mathbb {Z}}$ is also an $L^p$-Fourier multiplier as the sum of $L^p$-Fourier multipliers is an $L^p$-Fourier multiplier. We deduce from (2.11) and [5, Lemma 3.1] that $u\in L^p(\mathbb {T};D(A))$. We have shown that $u\in S_p(A,\,B,\,M,\,L)$. This shows the existence of strong $L^p$-solution.

To show uniqueness of strong $L^p$-solution, we let $u\in S_p(A,\,B,\,M,\,L)$ be such that

\[ (Mu)'''(t) + \alpha(Lu)'''(t) + (Nu)''(t) = \beta Au(t) + \gamma Bu'(t)+ Gu'_t + Fu_t \]

a.e. on $\mathbb {T}$. Taking the Fourier transforms on both sides, we deduce that

\[ [k^4M - (\alpha ik^3+k^2)L - \beta A - i\gamma kB - ikG_k - F_k]\hat{u}(k)=0 \]

when $k\in \mathbb {Z}$. Since $\rho _{M}(A,\,B,\,L)=\mathbb {Z}$, this implies that $\hat {u}(k)=0$ when $k\in \mathbb {Z}$ and thus $u=0$. This shows that the solution is unique. This completes the proof.

We notice that the assumption that the underlying Banach space $X$ is a UMD space in theorem 2.12 was only used in the implication $(ii)\Rightarrow (i)$. Since the second statement of theorem 2.12 does not depend on the space parameter $1 < p < \infty$, theorem 2.12 has the following immediate consequence.

Corollary 2.13 Let $X$ be a $\rm {UMD}$ Banach space, let $A,\,B,\,L$ and $M$ be closed linear operators on $X$ satisfying $D(A)\cap D(B)\subset D(M)\cap D(L)$, and $\alpha,\,\ \beta,\,\ \gamma \in \mathbb {C}$. Then if ($P_4$) is $L^p$-well-posed for some $1 < p < \infty$, then it is $L^p$-well-posed for all $1 < p < \infty$.

3. Well-posedness of ($P_4$) in Besov spaces

In this section, we consider the well-posedness of ($P_4$) in periodic Besov spaces $B_{p,q}^s(\mathbb {T}; X)$. Firstly, we briefly recall the definition of periodic Besov spaces in the vector-valued case introduced in [6]. 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 $\left \Vert f\right \Vert _{\alpha }=\sup _{x\in \mathbb {T}}\left \vert f^{(\alpha )}(x)\right \vert$ for $\alpha \in \mathbb {N}_0:=\mathbb {N}\cup \left \{0\right \}$. Let $\mathcal {D}'(\mathbb {T}; X):=\mathcal {L}(\mathcal {D}(\mathbb {T}),\,X)$ be the space of all continuous linear operator from $\mathcal {D}(\mathbb {T})$ to $X$. We consider the dyadic-like subsets of $\mathbb {R}$:

\[ I_0=\left\{t\in\mathbb{R}:\left\vert t\right\vert\leq2\right\},I_k=\left\{t\in \mathbb{R}:2^{k-1}<\left\vert t\right\vert\leq 2^{k+1}\right\} \]

for $k\in \mathbb {N}$. Let $\phi (\mathbb {R})$ be the set of all systems $\phi =(\phi _k)_{k\in \mathbb {N}_0}\subset \mathcal {S}(\mathbb {R})$ satisfying $\text {supp}(\phi _k)\subset \bar {I}_k$ for each $k\in \mathbb {N}_0$, $\sum _{k\in \mathbb {N}_0}\phi _k(x)=1$ for $x\in \mathbb {R}$, and for each $\alpha \in \mathbb {N}_0$, $\sup _{ x\in \mathbb {R},\, k\in \mathbb {N}_0 }2^{k\alpha }\vert \phi _k^{(\alpha )}(x)\vert <\infty$. Let $\phi =(\phi _k)_{k\in \mathbb {N}_0}\subset \phi (\mathbb {R})$ 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_{p,q}^s(\mathbb{T}; X)& =\Biggl\{f\in\mathcal {D}'(\mathbb{T}; X): \left\Vert f\right\Vert_{B_{p,q}^s}\\& :=\Biggl(\sum_{j\geq0}2^{sjq}\Big\Vert\sum_{k\in\mathbb{Z}}e_k\otimes \phi_j(k)\hat{f}(k)\Big\Vert_p^q\Biggr)^{1/q}<\infty\Biggr\} \end{align*}

with the usual modification if $q=\infty$. The space $B_{p,q}^s(\mathbb {T}; X)$ is independent from the choice of $\phi$ and different choices of $\phi$ lead to equivalent norms on $B_{p,q}^s(\mathbb {T}; X)$. $B_{p,q}^s(\mathbb {T}; X)$ equipped with the norm $\left \Vert \cdot \right \Vert _{B_{p,q}^s}$ is a Banach space. See [6, Section 2] for more information about the space $B_{p,q}^s(\mathbb {T}; X)$. It is well known that if $s_1\leq s_2$, then $B_{p,q}^{s_1}(\mathbb {T}; X)\subset B_{p,q}^{s_2}(\mathbb {T}; X)$ and the embedding is continuous [6, Theorem 2.3]. When $s>0$, it is shown in [6, Theorem 2.3] that $B_{p,q}^s(\mathbb {T}; X)\subset L^p(\mathbb {T}; X)$, $f\in B_{p,q}^{s+1}(\mathbb {T}; X)$ if and only if $f$ is differentiable a.e. on $\mathbb {T}$ and $f'\in B_{p,q}^s(\mathbb {T}; X)$. This implies that if $u\in B_{p,q}^s(\mathbb {T}; X)$ is such that there exists $v\in B_{p,q}^s(\mathbb {T}; X)$ satisfying $\hat {v}(k)=ik\hat {u}(k)$ when $k\in \mathbb {Z}$, then $u\in B_{p,q}^{s+1}(\mathbb {T}; X)$ and $u'=v$.

Let $1\leq p,\,q\leq \infty,\, s>0$ be fixed. We consider the following four-order degenerate differential equations with finite delay:

(P ₄)\begin{align*} & (Mu)''''(t) + \alpha (Lu)'''(t) + (Lu)''(t)\\ & \quad = \beta Au(t) + \gamma Bu'(t) + Gu'_t + Fu_t + f(t),\quad (t\in\mathbb{T}) \end{align*}

where $A,\, B,\, M$ and $L$ are closed linear operators on a Banach space $X$ satisfying $D(A)\cap D(B)\subset D(M)\cap D(L)$ and $\alpha,\,\ \beta,\,\ \gamma \in \mathbb {C}$, $f\in B_{p,q}^s(\mathbb {T};X)$ is given, and $F,\,G:B_{p,q}^s([-2\pi,\,0];X)\rightarrow X$ are bounded linear operators. Moreover, for fixed $t\in \mathbb {T}$, $u_t\in B_{p,q}^s([-2\pi,\,0];X)$ is defined by $u_t(s)=u(t+s)$ for $-2\pi \leq s\leq 0$, here we identify a function $u$ on $\mathbb {T}$ with its natural $2\pi$-periodic extension on $\mathbb {R}$.

Let $F,\,G\in \mathcal {L}(B_{p,q}^s[-2\pi,\,0];X),\,X)$ and $k\in \mathbb {Z}$. We define the linear operators $F_k,\,\ G_k$ by

(3.1)\begin{equation} F_kx := F(e_k*\otimes x), \quad G_kx := G(e_k\otimes x) \end{equation}

when $x\in X$. It is clear that there exists a constant $C>0$ such that $\left \Vert e_k\otimes x\right \Vert _{B_{p,q}^s(\mathbb {T}; X)}\leq C\left \Vert x\right \Vert$ when $k\in \mathbb {Z}$. Thus,

(3.2)\begin{equation} \left\Vert F_k\right\Vert\leq C\left\Vert F\right\Vert,\quad\left\Vert G_k\right\Vert\leq C\left\Vert G\right\Vert \end{equation}

whenever $k\in \mathbb {Z}$. It can be seen easily that when $u\in B_{p,q}^s(\mathbb {T}; X)$, then

\[ \widehat{Fu_.}(k) = F_k\hat u(k), \quad \widehat{Gu_.}(k) = G_k\hat u(k) \]

for $k\in \mathbb {Z}$. The resolvent set of ($P_4$) in the $B_{p,q}^s$-well-posedness setting is defined by

\begin{align*} & \rho_{M}(A,B,L):= \big\{k\in\mathbb{Z}: k^4M - (\alpha ik^3+k^2)L - \beta A\\ & \quad - i\gamma kB - ikG_k - F_k\text{ is invertible from }\\ & D(A)\cap D(B) \text{ onto }X \ \text{and}\ [k^4M - (\alpha ik^3+k^2)L\\ & \quad - \beta A - i\gamma kB - ikG_k - F_k]^{{-}1} \in \mathcal{L}(X) \big\}. \end{align*}

For the sake of simplicity, when $k\in \rho _{M}(A,\,B,\,L)$, we will use the following notation:

(3.3)\begin{equation} N_k=[k^4M - (\alpha ik^3+k^2)L - \beta A - i\gamma kB- ikG_k - F_k]^{{-}1}. \end{equation}

If $k\in \rho _{M}(A,\,B,\,L)$, then $MN_k,\,\ LN_k,\,\ AN_k$ and $BN_k$ make sense as $D(A)\cap D(B)\subset D(M)\cap D(L)$ by assumption, and they belong to $\mathcal {L}(X)$ by the closed graph theorem and the closedness of $A,\,\ B,\,\ M$ and $L$.

Let $1\leq p,\,q\leq \infty,\,s>0$. It is noted that that the functions $Gu_.$ and $Fu'_.$ are uniformly bounded on $\mathbb {T}$, but they are not necessarily in $B_{p,q}^s(\mathbb {T};X)$. We define the solution space of $B_{p,q}^s$-well-posedness for ($P_4$) by

\begin{align*} S_{p,q,s}(A, B, M, L)& :=\big\{u\in B_{p,q}^s(\mathbb{T};D(A))\cap B_{p,q}^{1+s}(\mathbb{T};X)\\& \qquad:\ Mu\in B_{p,q}^{4+s}(\mathbb{T};X), Lu\in B_{p,q}^{2+s}(\mathbb{T};X),\\ & \qquad u'\in B_{p,q}^s(\mathbb{T}; D(B))\ \mbox{and}\ Fu_., Gu'_. \in B_{p,q}^{s}(\mathbb{T};X)\big\}. \end{align*}

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

\begin{align*} \left\Vert u\right\Vert_{S_{p,q,s}(A, B, M, L)}& :=\left\Vert u\right\Vert_{B_{p,q}^{1+s}(\mathbb{T}; X)}+\left\Vert u\right\Vert_{B_{p,q}^s(\mathbb{T}; D(A))}\\ & \quad+\left\Vert Mu\right\Vert_{B_{p,q}^{4+s}(\mathbb{T}; X)}+\left\Vert Lu\right\Vert_{B_{p,q}^{3+s}(\mathbb{T}; X)}\\ & \quad + \left\Vert u'\right\Vert_{B_{p,q}^{s}(\mathbb{T}; D(B))} + \left\Vert Fu_.\right\Vert_{B_{p,q}^s(\mathbb{T}; X)} + \left\Vert Gu'_.\right\Vert_{B_{p,q}^s(\mathbb{T}; X)}. \end{align*}

If $u\in S_{p,q,s}(A,\, B,\, M,\, L)$, then $Mu$, $(Mu)'$, $(Mu)''$ and $(Mu)'''$ are $X$-valued continuous function on $\mathbb {T}$, and $Mu(0)=Mu(2\pi )$,$(Mu)'(0)=(Mu)'(2\pi )$, $(Mu)''(0)=(Mu)''(2\pi )$ and $(Mu)'''(0)=(Mu)'''(2\pi )$ by [5, Lemma 2.1].

Now we give the definition of the $B_{p,q}^s$-well-posedness of ($P_4$).

Definition 3.1 Let $1\leq p,\,q\leq \infty,\,s>0$ and $f\in B_{p,q}^s(\mathbb {T}; X)$, $u\in S_{p,q,s}(A,\,B, M,\,L)$ is called a strong $B_{p,q}^s$-solution of ($P_4$), if ($P_4$) is satisfied a.e. on $\mathbb {T}$. We say that ($P_4$) is $B_{p,q}^s$-well-posed, if for each $f\in B_{p,q}^s(\mathbb {T}; X)$, there exists a unique strong $B_{p,q}^s$-solution of ($P_4$).

If ($P_4$) is $B_{p,q}^s$-well-posed and $u\in S_{p,q,s}(A,\,B,\,M,\,L)$ is the unique strong $B_{p,q}^s$-solution of ($P_4$), there exists a constant $C>0$ such that for each $f\in B_{p,q}^s(\mathbb {T}; X)$, we have

(3.4)\begin{equation} \left\Vert u\right\Vert_{S_{p,q,s}(A,B,M,L)}\leq C\left\Vert f\right\Vert_{B_{p,q}^s}. \end{equation}

This is an easy result that can be obtained by the closedness of the operators $A$, $B$, $M$ and $L$ and the closed graph theorem.

Next we give the definition of operator-valued Fourier multipliers in the context of periodic Besov spaces, which is important in the proof of our main result of this section.

Definition 3.2 Let $X,\,Y$ be Banach spaces, $1\leq p,\,q\leq \infty,\,s\in \mathbb {R}$ and let $\left (M_k\right )_{k\in \mathbb {Z}}\subset \mathcal {L}(X,\,Y)$. We say that $\left (M_k\right )_{k\in \mathbb {Z}}$ is a $B_{p,q}^s$-Fourier multiplier, if for each $f\in B_{p,q}^s(\mathbb {T}; X)$, there exists $u\in B_{p,q}^s(\mathbb {T}; Y)$, such that $\hat {u}(k)=M_k\hat {f}(k)$ for all $k\in \mathbb {Z}$.

The following result has been obtained in [6, Theorem 4.5] which gives a sufficient condition for an operator-valued sequence to be a $B_{p,q}^s$-Fourier multiplier. For the notions of $B$-convex Banach spaces, we refer the readers to [6] and references therein.

Theorem 3.3 Let $X,\,Y$ be Banach spaces and let $\left (M_k\right )_{k\in \mathbb {Z}}\subset \mathcal {L}(X,\,Y)$. We assume that

(3.5)\begin{align} \sup_{k\in\mathbb{Z}}\big(\left\Vert M_k\right\Vert+\left\Vert k\bigtriangleup M_k\right\Vert\big)& =\sup_{k\in\mathbb{Z}}\big(\left\Vert M_k\right\Vert+\left\Vert k(M_{k+1}-M_k)\right\Vert\big)<\infty, \end{align}

(3.6)\begin{align} \sup_{k\in\mathbb{Z}}\left\Vert k^2\bigtriangleup^2 M_k\right\Vert& =\sup_{k\in\mathbb{Z}}\left\Vert k^2\big(M_{k+2}-2M_{k+1}+M_{k}\big)\right\Vert<\infty. \end{align}

Then for $1\leq p,\,q\leq \infty,\,s\in \mathbb {R}$, $\left (M_k\right )_{k\in \mathbb {Z}}$ is an $B_{p,q}^s$-multiplier. If $X$ is $B$-convex, then the first-order condition (3.5) is already sufficient for $\left (M_k\right )_{k\in \mathbb {Z}}$ to be a $B_{p,q}^s$-multiplier.

Remark 3.4

1. (i) If $\left (M_k\right )_{k\in \mathbb {Z}}$ is a $B_{p,q}^s$-Fourier multiplier, then there exists a bounded linear operator $T$ from $B_{p,q}^s(\mathbb {T}; X)$ to $B_{p,q}^s(\mathbb {T}; Y)$ satisfying $\widehat {Tf}(k) = M_k\hat f(k)$ when $k\in \mathbb {Z}$. This implies in particular that $\left (M_k\right )_{k\in \mathbb {Z}}$ must be bounded.

2. (ii) If $\left (M_k\right )_{k\in \mathbb {Z}}$ and $\left (N_k\right )_{k\in \mathbb {Z}}$ are $B_{p,q}^s$-Fourier multipliers, it can be seen easily that the product sequence $\left (M_kN_k\right )_{k\in \mathbb {Z}}$ and the sum sequence $\left (M_k+N_k\right )_{k\in \mathbb {Z}}$ are still $B_{p,q}^s$-Fourier multipliers.

3. (iii) Let $c_k=\frac {1}{k}$ when $k\neq 0$ and $c_0=1$, then it is easy to see that the sequence $\left (c_kI_X\right )_{k\in \mathbb {Z}}$ satisfies the conditions (3.2) and (3.3). Thus, the sequence $\left (c_kI_X\right )_{k\in \mathbb {Z}}$ is a $B_{p,q}^s$-Fourier multiplier by theorem 3.3.

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

Proposition 3.5 Let $A,\, B,\, M$ and $L$ be closed linear operators defined on a Banach space $X$ satisfying $D(A)\cap D(B)\subset D(M)\cap D(L),\,\ \alpha,\,\ \beta,\,\ \gamma \in \mathbb {C}$ and let $F,\,G\in \mathcal {L}(B_{p,q}^s([-2\pi,\,0];X),\,X)$, where $1\leq p,\,q\leq \infty$ and $s>0$. Assume that $\rho _M(A,\,B,\,L)=\mathbb {Z}$ and the sets $\{k\Delta ^2F_k:\ k\in \mathbb {Z}\}$, $\{k\Delta G_{k}:\ k\in \mathbb {Z}\},\,\{k^2\Delta ^2G_k:\ k\in \mathbb {Z}\}, \left \{k^4MN_k:\ k\in \mathbb {Z}\right \},\,\ \{k^3LN_k:k\in \mathbb {Z}\},\,\ \{kBN_k:k\in \mathbb {Z}\}$ and $\left \{kN_k:\ k\in \mathbb {Z}\right \}$ are norm-bounded, where $N_k$ is defined by (3.3), the operators $F_k,\,\ G_k,\,\ H_k$ are defined by (3.1). Then $(k^4MN_k)_{k\in \mathbb {Z}}$, $(k^3LN_k)_{k\in \mathbb {Z}}$, $(kBN_k)_{k\in \mathbb {Z}}$, $(N_k)_{k\in \mathbb {Z}}$, $(kN_k)_{k\in \mathbb {Z}}$, $(F_kN_k)_{k\in \mathbb {Z}}$ and $(kG_kN_k)_{k\in \mathbb {Z}}$ are $B_{p,q}^s$-Fourier multipliers.

Proof. It follows immediately from the norm boundedness of the set $\{kN_k:k\in \mathbb {Z}\}$ that the set $\{N_k:k\in \mathbb {Z}\}$ is norm-bounded. 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 the proof of proposition 2.11. Since remark 2.7 and the sequence $(k^j)_{k\in \mathbb {Z}}$ is $2$-regular when $0\leq j\leq 3$, to show that $(k^4MN_k)_{k\in \mathbb {Z}}$, $(k^3LN_k)_{k\in \mathbb {Z}}$, $(kBN_k)_{k\in \mathbb {Z}}$, $(N_k)_{k\in \mathbb {Z}}$ and $(kN_k)_{k\in \mathbb {Z}}$ are $B_{p,q}^s$-Fourier multipliers, we only need to show that the set $\{k^2\Delta L_k: k\in \mathbb {Z}\}$ is norm-bounded by [11, Theorem 1.1] and theorem 3.3. We have

\begin{align*} L_k = L_k^{(1)} + L_k^{(2)}, \end{align*}

where

\begin{align*} & L_k^{(1)} :={-}\Delta a_kMN_k + \Delta b_kLN_k + \Delta c_kBN_k,\\ & L_k^{(2)}:=ik\Delta G_kN_k + iG_{k+1}N_k + \Delta F_kN_k, \end{align*}

when $k\in \mathbb {Z}$ by (2.6). We observe that

(3.7)\begin{align} \Delta L_k^{(1)}& ={-}\Delta a_{k+1}MN_{k+1} + \Delta b_{k+1}LN_{k+1}\nonumber\\ & \quad + \Delta c_{k+1}BN_{k+1} + \Delta a_kMN_k - \Delta b_kLN_k - \Delta c_kBN_k\nonumber\\ & ={-}\Delta^2a_kMN_{k+1} - \Delta a_kM\Delta N_k + \Delta^2b_kLN_{k+1}\nonumber\\ & \quad + \Delta b_kL\Delta N_k + \Delta^2 c_kBN_{k+1} + \Delta c_k B\Delta N_k\nonumber\\ & ={-}\Delta^2a_kMN_{k+1} - \Delta a_kMN_{k+1}L_k + \Delta^2b_kLN_{k+1}\nonumber\\ & \quad + \Delta b_kLN_{k+1}L_k + \Delta^2 c_kBN_{k+1} + \Delta c_k BN_{k+1}L_k, \end{align}

and

(3.8)\begin{align} \Delta L_k^{(2)}& =i(k+1)\Delta G_{k+1}N_{k+1} + iG_{k+2}N_{k+1}\nonumber\\ & \quad + \Delta F_{k+1}N_{k+1}-ik\Delta G_kN_k- iG_{k+1}N_k - \Delta F_kN_k\nonumber\\ & =ik\Delta^2G_kN_{k+1} + ik\Delta G_k\Delta N_k + i\Delta G_{k+1}N_{k+1} + i\Delta G_{k+1}N_{k+1}\nonumber\\ & \quad + iG_{k+1}\Delta N_k + \Delta^2 F_kN_{k+1} +\Delta F_k\Delta N_k\nonumber\\ & =ik\Delta^2G_kN_{k+1} + ik\Delta G_k\Delta N_k + 2i\Delta G_{k+1}N_{k+1} + iG_{k+1}\Delta N_k\nonumber\\ & \quad + \Delta^2 F_kN_{k+1} +\Delta F_k\Delta N_k\nonumber\\ & =ik\Delta^2G_kN_{k+1} + ik\Delta G_k N_{k+1}L_k + 2i\Delta G_{k+1}N_{k+1}\nonumber\\ & \quad + iG_{k+1}N_{k+1}L_k + \Delta^2 F_kN_{k+1} +\Delta F_k\Delta N_k, \end{align}

when $k\in \mathbb {Z}$. It follows from (3.7) and (3.8) that the sets$\{k^2\Delta L_k^{(1)}: k\in \mathbb {Z}\}$ and $\{k^2\Delta L_k^{(2)}: k\in \mathbb {Z}\}$ are norm-bounded by the norm boundedness of the sets $\{kL_k: k\in \mathbb {Z}\}$ and the assumptions that the sets $\{k\Delta ^2F_k:\ k\in \mathbb {Z}\}$, $\{k\Delta G_{k}: k\in \mathbb {Z}\},\,\{k^2\Delta ^2G_k:\ k\in \mathbb {Z}\},\, \left \{k^4MN_k:\ k\in \mathbb {Z}\right \},\,\{k^3LN_k:k\in \mathbb {Z}\},\,\{kBN_k:k\in \mathbb {Z}\}$ and $\left \{kN_k:\ k\in \mathbb {Z}\right \}$ are norm-bounded.

It remains to show that the sequences $(F_kN_k)_{ k\in \mathbb {Z}}$ and $(kG_kN_k)_{ k\in \mathbb {Z}}$ satisfy (3.5) and (3.6). This follows easily from the norm boundedness of the sets $\{k\Delta ^2F_k:\ k\in \mathbb {Z}\}$, $\{k\Delta G_{k}: k\in \mathbb {Z}\}$ and $\{k^2\Delta ^2G_k:\ k\in \mathbb {Z}\}$. We omit the details. The proof is completed.

Next we give a necessary and sufficient condition for $B_{p,q}^s$-well-posedness of ($P_4$). Its proof is just an easy adaptation of the proof of theorem 2.12 by using proposition 3.5. We omit the detail.

Theorem 3.6 Let $X$ be a Banach space, $1\leq p,\,q\leq \infty,\, s>0$, let $A,\, B,\, M$ and $L$ be closed linear operators on $X$ satisfying $D(A)\cap D(B)\subset D(M)\cap D(L)$ and $\alpha,\, \beta,\,\ \gamma \in \mathbb {C}$. Let $F,\,G\in \mathcal {L}(B_{p,q}^s([-2\pi,\,0];X),\,X)$. We assume that the sets $\{k\Delta ^2F_k:\ k\in \mathbb {Z}\}$, $\{k\Delta G_{k}: k\in \mathbb {Z}\}$ and $\{k^2\Delta ^2G_k:\ k\in \mathbb {Z}\}$ are norm-bounded. Then the following assertions are equivalent:

1. (i) ($P_4$) is $B_{p,q}^s$-well-posed.

2. (ii) $\rho _{M}(A,\,B,\,L)=\mathbb {Z}$ and the sets $\left \{k^4MN_k: k\in \mathbb {Z}\right \},\,\ \{k^3LN_k:k\in \mathbb {Z}\},\,\ \{kBN_k:k\in \mathbb {Z}\}$ and $\left \{kN_k:\ k\in \mathbb {Z}\right \}$ are norm-bounded, where $N_k$ is defined by (3.3).

When the underlying Banach space $X$ is $B$-convex, the first-order Marcinkiewicz-type condition (3.5) is already sufficient for an operator-valued sequence to be a $B_{p,q}^s$-Fourier multiplier. This remark together with the proof of theorem 2.12 gives immediately the following result which gives an characterization of the $B_{p,q}^s$-well-posedness of ($P_4$) under a weaker condition on the sequence $(G_k)_{k\in \mathbb {Z}}$ when the underlying Banach space is $B$-convex.

Theorem 3.7 Let $X$ be a $B$-convex Banach space, $1\leq p,\,q\leq \infty,\, s>0$, let $A,\, B,\, M$ and $L$ be closed linear operators on $X$ satisfying $D(A)\cap D(B)\subset D(M)\cap D(L)$ and $\alpha,\,\ \beta,\,\ \gamma \in \mathbb {C}$. Let $F,\,G\in \mathcal {L}(B_{p,q}^s([-2\pi,\,0];X),\,X)$. We assume that $\left \{k\Delta G_{k}:k\in \mathbb {Z}\right \}$ is norm-bounded. Then the following assertions are equivalent:

1. (i) ($P_4$) is $B_{p,q}^s$-well-posed.

2. (ii) $\rho _M(A,\,B,\,L)=\mathbb {Z}$ and the sets $\left \{k^4MN_k:\ k\in \mathbb {Z}\right \},\, \{k^3LN_k:k\in \mathbb {Z}\},\,\ \{kBN_k:k\in \mathbb {Z}\}$ and $\left \{kN_k: k\in \mathbb {Z}\right \}$ are norm-bounded, where $N_k$ is defined by (3.3).

Since the second statement of theorem 3.6 does not depend on the parameters $1\leq p,\,q\leq \infty,\, s>0$, theorem 3.6 has the following immediate consequence.

Corollary 3.8 Let $X$ be a Banach space, $1\leq p,\,q\leq \infty,\, s>0$, let $A,\, B,\, M$ and $L$ be closed linear operators on $X$ satisfying $D(A)\cap D(B)\subset D(M)\cap D(L)$ and $\alpha,\, \beta,\,\ \gamma \in \mathbb {C}$. Let $F,\,G\in \mathcal {L}(B_{p,q}^s([-2\pi,\,0];X),\,X)$. We assume that the sets $\{k\Delta ^2F_k:\ k\in \mathbb {Z}\}$, $\{k\Delta G_{k}: k\in \mathbb {Z}\}$ and $\{k^2\Delta ^2G_k:\ k\in \mathbb {Z}\}$ are norm-bounded. Then if ($P_4$) is $B_{p,q}^s$-well-posed for some $1\leq p,\,q\leq \infty,\, s>0$, then it is $B_{p,q}^s$-well-posed for all $1\leq p,\,q\leq \infty,\, s>0$.

4. Applications

Example 4.1 Let $\Omega$ be a bounded domain in $\mathbb {R}^k$ with smooth boundary, $m$ be a given non-negative-bounded measurable function on $\Omega$ and let $\alpha,\, \gamma \in \mathbb {C},\,\ \beta >0$ be given. We let $X$ be the Hilbert space $H^{-1}(\Omega )$, and let $F,\,G\in \mathcal {L}(L^p([-2\pi,\,0];X),\,X)$ for some $1 < p< \infty$. We consider the problem

\[ \left\{\begin{array}{@{}ll} \dfrac{\partial^4}{\partial t^4} (m(x)u(t, x)) + \alpha\dfrac{\partial^3}{\partial t^3} (m(x){u(t, x)}) + \dfrac{\partial^2}{\partial t^2} (m(x){u(t, x)})\\ = \beta \Delta u(t,x)+ \gamma\Delta\dfrac{\partial}{\partial t}u(t, x) + Gu'_t({\cdot}, x) + Fu_t({\cdot}, x) + f(t, x),\ (t, x) \in \mathbb{T}\times \Omega,\\ u(t, x)= 0, \ (t, x) \in \mathbb{T}\times \partial \Omega. \end{array} \right. \]

where $f$ is defined on $\mathbb {T}\times \Omega$ and the Laplacian $\Delta$ only acts on the space variable $x\in \Omega$, $u_t'$ and $u_t$ are defined by $u_t' (s,\, x) = u'(t+s,\, x)$ and $u_t' (s,\, x) = u(t+s,\, x)$ when $t\in \mathbb {T},\, \ s\in [-2\pi,\, 0]$ and $x\in \Omega$.

Let $M$ be the multiplication operator on $X$ by $m$, then there exist constants $C >0,\, \beta > 0$, such that

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

whenever $Re (z)\leq \beta ( 1 + \vert Im (z)\vert )$ by [12, 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

(P ₁)\begin{align*} & (Mu)''''(t) + \alpha (Mu)'''(t) + (Mu)''(t)\\ & \quad = \beta Au(t) + \gamma A u'(t) + Gu'_t + Fu_t + f(t),\quad (t\in \mathbb{T}) \end{align*}

a differential equation on $\mathbb {T}$ with values in $X$, where $f\in L^p(\mathbb {T}; X)$ and the solution $u\in W_{\text {per}}^{1,p}(\mathbb {T}; D(A))$ satisfies $\ Mu\in W_{\text {per}}^{4,p}(\mathbb {T}; X)$.

We assume that $\rho _{M}(A,\,A,\,M)=\mathbb {Z}$ and the set $\left \{k\Delta G_{k}:k\in \mathbb {Z}\right \}$ is norm-bounded. Furthermore, we 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

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

whenever $Re z\leq \beta ( 1 + \vert Im z\vert )$ by (4.1). We claim that ($P_1$) is $L^p$-well-posed. Indeed, the operator $(k^4 - \alpha ik^3 - k^2)M - (\beta +ik) A - ikG_k - F_k: D(A) \to X$ is bijective and $[(k^4 - \alpha ik^3 - k^2)M - (\beta +ik) A - ikG_k - F_k]^{-1} \in \mathcal {L} (X)$ whenever $k\in \mathbb {Z}$ by the assumption $\rho _{M}(A,\,A,\,M)=\mathbb {Z}$. It follows that the sets

\[ \{k^2MN_k: k\in \mathbb{Z}\},\ \{\Delta N_k: k\in \mathbb{Z}\},\quad \{kN_k: k\in \mathbb{Z}\} \]

are norm-bounded by (4.1) and (4.2), where $N_k = [(k^4 - \alpha ik^3 - k^2)M - (\beta +ik) A - ikG_k - F_k]^{-1}$. Here we have used the uniform boundedness of the sequences $(F_k)_{k\in \mathbb {Z}}$ and $(G_k)_{k\in \mathbb {Z}}$. Thus, the problem ($P_1$) is $L^p$-well-posed by theorem 2.12. 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 [5].

Under the same assumptions, we obtain the $B_{p,q}^s$-well-posedness of ($P_1$) when $1\leq p,\, q\leq \infty$ by corollary 3.8.

Example 4.2 Let $H$ be a Hilbert space, $P$ 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 exists $C > 0$ and $\beta >0$ such that

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

whenever $Re z\geq -\beta ( 1 + \vert Im z\vert )$ by [12, page 73]. If $M$ is regular enough so that $0\in \rho (M)$, then

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

whenever $Re z\geq -\beta ( 1 + \vert Im z\vert )$ by (4.3).

Let $\Omega = (0,\, 1)$ and let $H = L^2(\Omega )$. It is clear that the operator $\frac {{\rm d}^2}{{\rm d}x^2}$ with domain $H^2(\Omega )\cap H_0^1(\Omega )$ generates a contraction semigroup on $H$ and $P = - \frac {{\rm d}^2}{{\rm d}x^2}$ is positive and self-adjoint in $H$ [4, Example 3.4.7]. Hence, $1\in \rho (\frac {{\rm d}^2}{{\rm d}x^2})$, or equivalently $M = I_X+ P$ has a bounded inverse. Let $\alpha,\, \gamma \in \mathbb {C}$ and $\beta < 0$ be fixed and let $F,\,G\in \mathcal {L}(L^p([-2\pi,\,0];X),\,X)$ for some $1 < p< \infty$, we consider the following equations:

\[ \left\{\begin{array}{@{}l} \dfrac{\partial^4}{\partial t^4} (1- \dfrac{\partial^2}{\partial x^2})u(t, x) +\alpha \dfrac{\partial^3}{\partial t^3} (1- \dfrac{\partial^2}{\partial x^2})u(t, x) + \dfrac{\partial^2}{\partial t^2} (1- \dfrac{\partial^2}{\partial x^2})u(t, x)\\ = \beta \dfrac{\partial^4}{\partial x^4} u(t, x) + \gamma\dfrac{\partial^4}{\partial x^4} \dfrac{\partial}{\partial t}u(t, x)\\ \quad + Gu'_t({\cdot}, x) + Fu_t ({\cdot}, x) + f(t, x), \quad (t, x) \in \mathbb{T}\times \Omega,\\ u(t, 0) = u(t, 1)= \dfrac{\partial^2}{\partial x^2}u(t, 0) = \dfrac{\partial^2}{\partial x^2}u(t, 1) = 0, \ t \in \mathbb{T}. \end{array} \right. \]

This equation can be rewritten in the compact form:

(P ₂)\begin{align*} & (Mu)''''(t) + \alpha (Mu)'''(t) + (Mu)''(t)\\ & \quad = \beta Au(t) + \gamma A u'(t) + Gu'_t + Fu_t + f(t), \quad (t\in \mathbb{T}) \end{align*}

a differential equation on $\mathbb {T}$ with values in $H$, where $f\in L^p(\mathbb {T}; H)$ and the solution $u$ is in $u\in W_{\text {per}}^{1,p}(\mathbb {T}; D(A))$, satisfies $Mu\in W_{\text {per}}^{4,p}(\mathbb {T}; H)$, 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 $\rho _{M}(A,\,A,\,M)=\mathbb {Z}$, one can obtain the $L^p$-well-posedness of ($P_2$) by using (4.3), (4.4) and theorem 2.12 under suitable assumption on the delay operator $G$. 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 [5]. One can also obtain the $B_{p,q}^s$-well-posedness pf ($P_2$) when $1\leq p,\, q\leq \infty$ by using theorem 3.6 or corollary 3.8.