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 $t\in \mathbb {T}$.

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 [Reference Arendt and Bu5] 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 $n\in \mathbb {N}$.

The main tool in our study of $L^p$-well-posedness of ($P_4$) is the $L^p$-Fourier multiplier theorem established in [Reference Arendt and Bu5]. 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 [Reference Arendt and Bu5] and references therein.

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 2.8 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.

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 [Reference Arendt and Bu5, Lemma 2.1].

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.

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 [Reference Conejero, Lizama, Murillo-Arcila and Seoane-Sepulveda11, 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 ($P_4$).

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 [Reference Arendt and Bu5, 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 [Reference Arendt and Bu5, 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 $u'\in L^p(\mathbb {T}; D(B))$ [Reference Arendt and Bu5, 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 $Lu = u_4$ [Reference Arendt and Bu5, 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 [Reference Arendt and Bu5, 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.

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 [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 $\left \Vert f\right \Vert _{\alpha }=\sup _{x\in \mathbb {T}}\left \vert f^{(\alpha )}(x)\right \vert$ 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 [Reference Arendt and Bu6, 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 [Reference Arendt and Bu6, Theorem 2.3]. When $s>0$, it is shown in [Reference Arendt and Bu6, 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 [Reference Arendt and Bu5, Lemma 2.1].

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

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

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

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 [Reference Conejero, Lizama, Murillo-Arcila and Seoane-Sepulveda11, 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.

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.

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.

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 [Reference Favini and Yagi12, 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 [Reference Arendt and Bu5].

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 [Reference Favini and Yagi12, 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 $H$ [Reference Arendt, Batty, Hieber and Neubrander4, 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 [Reference Arendt and Bu5]. 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.