最近, 关于具有周期边值条件的退化微分方程的适定性受到了学者们的极大关注, 参见文献[1-8].例如, Lizama和Ponce^[1]研究了一阶退化微分方程

$ \begin{eqnarray} (Mu)'(t) = Au(t) + f(t)\ \ (t\in {\Bbb T} := [0, 2\pi]), \end{eqnarray} $

(1.1)

且具有周期边值条件$(Mu)(0) = (Mu)(2\pi)$, 这里$A$, $M$为复Banach空间$X$上闭线性算子.利用向量值函数空间上的算子值傅里叶乘子定理, 他们在Lebesgue-Bochner空间$L^p({\Bbb T}, X)$, 周期Besov空间$B_{p, q}^s({\Bbb T}, X)$和周期Triebel-Lizorkin空间$F_{p, q}^s({\Bbb T}, X)$上给出了方程$(1.1)$具有适定性的充要条件.进一步, Bu^[3]考虑了二阶退化微分方程

$ \begin{eqnarray} (Mu')'(t) = Au(t) + f(t)\ \ (t\in {\Bbb T}), \end{eqnarray} $

(1.2)

且具有周期边值条件$u(0) = u(2\pi), $ $ (Mu')(0) = (Mu')(2\pi)$, 这里$A$, $M$为复Banach空间$X$上闭线性算子.他也给出了方程$(1.2)$在空间$L^p({\Bbb T}, X)$, $B_{p, q}^s({\Bbb T}, X)$和$F_{p, q}^s({\Bbb T}, X)$上具有相应适定特征的充分条件或者必要条件.同时Bu^[4]也考虑了具有周期边值条件$Mu(0) = Mu(2\pi)$的一阶有限时滞退化微分方程

$ \begin{eqnarray} (Mu)'(t) = Au(t) + Fu_t + f(t)\ \ (t\in {\Bbb T}). \end{eqnarray} $

(1.3)

关于退化微分方程的详细研究参见文献[9-10].

我们研究下面二阶有限时滞退化微分方程

$ (Mu')'(t) = Au(t) + Bu'(t) + Fu_t + f(t)\ \ (t\in{\Bbb T}), $

($P_2$)

且具有周期边界条件$u(0) = u(2\pi), (Mu')(0) = (Mu')(2\pi)$, 其中$A, B, M$为Banach空间$X$上闭线性算子满足$D(A)\cap D(B)\subset D(M)$, $F$为$L^p([-2\pi, 0], X)$ (或$B_{p, q}^s([-2\pi, 0], X)$)到$X$上有界线性算子, $u_t$定义为: $u_t(s) = u(t+s)\ (s\in [-2\pi, 0])$, $f\in L^p({\Bbb T}, X)$ (或$B_{p, q}^s({\Bbb T}, X)). $

当$M = I_X, \ A = aG, \ B = \alpha G, \ F = 0$时, 其中$I_X$为恒等算子, $(P_2)$变为下述二阶微分方程

$ u''(t) - aGu(t) - \alpha Gu'(t) = f(t)\ \ (t\in{\Bbb T}). $

(1.4)

Keyantuo和Lizama^[11]研究了方程(1.4)的适定性.确切地, 他们证明了:当$X$为UMD空间且$1 < p <\infty$, 则方程$(1.4)$是$L^p$ -适定的当且仅当

$ \Big\{\frac{- k^2}{a + \alpha ik}:k\in{\Bbb Z}\Big\}\subseteq \rho(G)\ \mbox{且集合 }\ \Big\{\frac{- k^2}{a + \alpha ik}(\frac{- k^2}{a + \alpha ik}I - G)^{-1}: k\in{\Bbb Z}\Big\} $

是Rademacher有界的(参见文献[11, 定理2.11]).他们也证明了:方程$(1.4)$是$B_{p, q}^s$ (或$C_{2\pi}^s$) -适定的当且仅当

$ \Big\{\frac{- k^2}{a + \alpha ik}:k\in{\Bbb Z}\Big\}\subseteq \rho(G) \ \mbox{且}\ \Big\{\frac{- k^2}{a + \alpha ik}(\frac{- k^2}{a + \alpha ik}I - G)^{-1}: k\in{\Bbb Z}\Big\} $

是范数有界的(参见文献[11, 定理3.6, 3.8]).

当$A = aG, \ B = \alpha G, \ F = 0$时, Cai^[12]研究了$(P_2)$的$L^p$ (或$B_{p, q}^s)$ -适定性特征.具体地, 他证明了:若$X$为UMD空间且$1 < p < \infty$, 假设$i{\Bbb Z}\subset\rho_M(G)$且集合$\{k^2MN_k: k\in{\Bbb Z}\}$和$\{kN_k: k\in{\Bbb Z}\}$是Rademacher有界的, 则$(P_2)$是$L^p$ -适定的(参见文献[12, 定理2.8]); 设$X$为Banach空间, 若$(P_2)$是$L^p$ -适定的, 则集合{$k^2MN_k: k\in{\Bbb Z}$}是Rademacher有界的(参见文献[12, 定理2.9]); 设$X$为Banach空间, $1\leq p, q\leq\infty, s>0$, 假设$i{\Bbb Z}\subset\rho_M(G)$且集合$\{k^2MN_k: k\in{\Bbb Z}\}$和$\{kN_k: k\in{\Bbb Z}\}$是范数有界的, 则$(P_2)$是$B_{p, q}^s$ -适定的(参见文献[12, 定理3.7]); 设$X$为Banach空间, $1\leq p, q\leq\infty, s>0$, 若$(P_2)$是$B_{p, q}^s$ -适定的, 则$i{\Bbb Z}\subset\rho_M(G)$且集合$\{k^2MN_k: k\in{\Bbb Z}\}$是范数有界的, 其中$N_k=[k^2M + (a + \alpha ik)G]^{-1}\ (k\in{\Bbb Z})$ (参见文献[12, 定理3.8]).

让$1 \leq p < \infty$.称$(P_2)$是$L^p$ -适定的, 若任给$f\in L^p({\Bbb T}, X)$, 存在唯一$u\in L^p({\Bbb T}, D(A))\cap W_{{\rm per}}^{1, p}({\Bbb T}, X), \ u'\in L^p({\Bbb T}, D(B))\cap L^p({\Bbb T}, D(M))$, $Mu'\in W_{{\rm per}}^{1, p}({\Bbb T}, X)$且$(P_2)$关于${\Bbb T}$几乎处处成立, 这里$W_{{\rm per}}^{1, p}({\Bbb T}, X)$为$X$ -值一阶周期Sobolev空间.类似地, 我们可以引入$(P_2)$的$B_{p, q}^s$ -适定性的定义.

本文在Lebesgue-Bochner空间$L^p({\Bbb T}, X)$和周期Besov空间$B_{p, q}^s({\Bbb T}, X)$上, 研究$(P_2)$的$L^p$ -适定性(或$B_{p, q}^s$ -适定性)特征.具体地, 我们将证明:若$X$为UMD空间且$1 < p < \infty$, 则$(P_2)$是$L^p$ -适定的当且仅当$\rho_M(A, B) = {\Bbb Z}$且集合$\{k^2MN_k: k\in{\Bbb Z}\}$, $\{kBN_k: k\in{\Bbb Z}\}$和$\{kN_k: k\in{\Bbb Z}\}$是Rademacher有界的; 若$X$为Banach空间, $1 \leq p, q \leq \infty, s>0$且集合$\{k(F_{k+2}-2F_{k+1} + F_k): k\in {\Bbb Z}\}$为范数有界的, 则$(P_2)$是$B_{p, q}^s$ -适定的当且仅当$\rho_M(A, B) = {\Bbb Z}$且集合$\{k^2MN_k: k\in {\Bbb Z}\}$, $\{kBN_k: k\in {\Bbb Z}\}$和$\{kN_k: k\in {\Bbb Z}\}$是范数有界的, 这里$N_k = [k^2M + A + ikB + F_k]^{-1}$ $(k\in {\Bbb Z})$, $F_k$为$X$上有界线性算子且其定义为$F_k x := F(e_kx)\ (k\in{\Bbb Z}, \ x\in X)$, $e_k(t) = e^{ikt}\ (t\in{\Bbb T})$.我们的主要结果包含文献[11-12]中相应结果作为特殊情形.与问题$(1.3)$相比较, 我们研究二阶混合退化微分方程, 由于算子$B$的出现和阶数的增加, 方程变得更加复杂, 相应的解空间和预解集也变得更加复杂, 这需要更细心的计算和推导, 故本文的结果在一定程度上推广了文献[4]中结果.

本文结构如下:第2节在Lebesgue-Bochner空间$L^p({\Bbb T}, X)$上研究$(P_2)$的$L^p$ -适定性.第3节在Besov空间$B_{p, q}^s({\Bbb T}, X)$上研究$(P_2)$的$B_{p, q}^s$ -适定性.第4节将主要结果应用到具体的例子.

设$X$, $Y$为复Banach空间且${\Bbb T} := [0, 2\pi]$.记${\cal L}(X, Y)$为$X$到$Y$上所有有界线性算子构成的集合.若$X=Y$, 简记为${\cal L}(X)$.给定$1 \leq p < \infty$, 记$L^p({\Bbb T}, X)$为${\Bbb T}$上所有$X$ -值可测函数$f$构成的集合且范数定义为

$ \left\| f \right\|_{L^p} := \bigg(\int_0^{2\pi}\left\| {f(t)} \right\|^p\frac{{\rm d}t}{2\pi}\bigg)^{1/p} < \infty. $

任给$f\in L^1({\Bbb T}, X)$, 记

$ \hat{f}(k) := \frac{1}{2\pi}\int_{0}^{2\pi}e_{-k}(t)f(t){\rm d}t\ \ (k\in {\Bbb Z}), $

为$f$的第$k$个傅里叶系数, 这里$e_k(t) = e^{ikt}\ (t\in{\Bbb T})$.

Banach空间$X$称为UMD空间, 若对某个(等价地, 对任意) $1 < p < \infty$, Riesz投影$Rf := \sum\limits_{k\geq0} \hat{f}(k)e_k$是$L^p({\Bbb T}, X)$上有界算子^[13].例如$L^p$ -空间是UMD空间且每个UMD空间是超自反的.

定义$\gamma_j(t) = \mbox{sgn}(\sin(2^{j}\pi t))$为$[0, 1]$上的第$j$个Rademacher函数, 这里$j\geq1$.令$(\gamma_j\otimes x)(t) = \gamma_j(t)x$, $(x\in X, t\in[0, 1])$.

定义2.1  设$X$, $Y$为Banach空间.称${\bf{T}}\subset {\cal L}(X, Y)$是Rademacher有界的(简写为${\rm R}$ -有界的), 若存在常数$C>0$, 使得对任意$T_1, \cdots, T_n\in {\bf{T}}, x_1, \cdots, x_n\in X$, $n\in {\Bbb N}$有

$ \begin{eqnarray} \bigg\|\sum\limits_{j=1}^n\gamma_j\otimes T_jx_j\bigg\|_{L^1([0, 1], Y)}\leq C \bigg\|\sum\limits_{j=1}^n\gamma_j\otimes x_j\bigg\|_{L^1([0, 1], X)}. \end{eqnarray} $

(2.1)

注2.2  若${\bf{S}}, {\bf{T}}\subset {\cal L}(X)$为${\rm R}$ -有界的.则

$ {\bf{ST}} := \left\{ {ST:S\in {\bf{S}}, T\in {\bf{T}}} \right\} \mbox{ 与 }\ \ {\bf{S}}+{\bf{T}} := \left\{ {S+T:S\in {\bf{S}}, T\in {\bf{T}}} \right\} $

也为${\rm R}$ -有界的.易知, 若$\Omega\subset{\Bbb C}$有界, 则集合{$\lambda I_X:\lambda\in \Omega$}为${\rm R}$ -有界的(参见文献[13, Theorem 4.4]).

定义2.3  设$X, Y$为Banach空间, $1\leq p < \infty$.称$(M_k)_{k\in {\Bbb Z}}\subset{\cal L}(X, Y)$是$L^p$ -傅里叶乘子, 若任给$f\in L^p({\Bbb T}, X)$, 存在$u\in L^p({\Bbb T}, Y)$满足$\hat{u}(k) = M_k\hat{f}(k)$ $(k\in {\Bbb Z})$.

命题2.4^{[6, Proposition 1.11]}  设$X, Y$为Banach空间, $1 \leq p < \infty$.若$(M_k)_{k\in {\Bbb Z}}\subset{\cal L}(X, Y)$是$L^p$ -傅里叶乘子, 则集合$\{M_k: k\in{\Bbb Z}\}$是${\rm R}$ -有界的.

定理2.5^{[6, Theorem 1.3]}  设$X, Y$为$\rm UMD$空间, $1 < p < \infty$, $(M_k)_{k\in {\Bbb Z}}\subset{\cal L}(X, Y)$.若集合$\{M_k: k\in{\Bbb Z}\}$与$\{k(M_{k+1} -M_k): k\in {\Bbb Z}\}$均为${\rm R}$ -有界的, 则$(M_k)_{k\in {\Bbb Z}}$为$L^p$ -傅里叶乘子.

设$1 \leq p < \infty$, 一阶周期"Sobolev"空间定义为^[6]

$ W_{\rm per}^{1, p}({\Bbb T}, X) :=\big\{u\in L^p({\Bbb T}, X): \mbox{ 存在$v\in L^p({\Bbb T}, X)$ 使得 $\hat{v}(k) = ik\hat{u}(k), \forall\ k\in {\Bbb Z}\big\}.$} $

设$u\in L^p({\Bbb T}, X)$, 则$u\in W_{{\rm per}}^{1, p}({\Bbb T}, X)$当且仅当$u$在${\Bbb T}$上几乎处处可导且有$u'\in L^p({\Bbb T}, X)$, 从而$u$连续且满足$u(0) = u(2\pi)$ (参见文献[6, Lemma 2.1]).

设$F\in (L^p([-2\pi, 0], X), X)$.定义$F_k$为$F_kx = F(e_kx)\ (x\in X, \ k\in{\Bbb Z})$.易知$\|F_k\|\leq\|F\|$且$\widehat{Fu_\cdot}(k) = F_k\hat{u}(k)$.进一步, 因为

$ \|Fu_t\|\leq\|F\|\|u_\cdot\|_p = \|F\|\|u\|_p\ \ (t\in{\Bbb Z}), $

则$(F_k)_{k\in{\Bbb Z}}$为$L^p$ -傅里叶乘子.

考虑下述二阶有限时滞退化微分方程

$ (Mu')'(t)= Au(t) + Bu'(t) + Fu_t + f(t)\ \ (t\in{\Bbb T}), $

($P_2$)

且具有周期边值条件$u(0) = u(2\pi), (Mu')(0) = (Mu')(2\pi)$, 这里$A, B, M$为Banach空间$X$上闭线性算子满足$D(A)\cap D(B)\subset D(M)$, $F$为$L^p([-2\pi, 0], X)$到$X$上有界线性算子, $u_t$定义为: $u_t(s) = u(t+s)$ $ (s\in [-2\pi, 0])$, $f\in L^p({\Bbb T}, X)$.

定义$A, \ B$的$M$ -预解集为

$ \begin{eqnarray*} \rho_M(A, B):=&\big\{&k\in {\Bbb Z}:(-k^2M - A - ik B - F_k):D(A)\cap D(B)\rightarrow X \mbox{ 可逆且 }\\ &&(-k^2M - A - ik B - F_k)^{-1}\in {\cal L}(X) \big\}. \end{eqnarray*} $

设$1 \leq p < \infty$. $(P_2)$在$L^p$ -适定性意义下的解空间定义为

$ \begin{eqnarray*} S_p(A, B, M):=&\big\{&u\in W_{{\rm per}}^{1, p}({\Bbb T}, X)\cap L^p({\Bbb T}, D(A)):\\ &&u'\in L^p({\Bbb T}, D(M))\cap L^p({\Bbb T}, D(B)), Mu'\in W_{{\rm per}}^{1, p}({\Bbb T}, X)\big\}, \end{eqnarray*} $

其中$D(A)$, $D(B)$与$D(M)$赋予图像范数成为Banach空间. $S_p(A, B, M)$的范数定义为

$ \|u\|_{S_p(A, B, M)}:=\|u\|_{L^p}+\|u'\|_{L^p}+\|Au\|_{L^p}+\|Bu'\|_{L^p}+\|Mu'\|_{L^p}+\|(Mu')'\|_{L^p}. $

易知$S_p(A, B, M)$在赋予此范数下为Banach空间.根据文献[6, Lemma 2.1]知, 若$u\in S_p(A, B, M)$, 则$u$与$Mu'$在${\Bbb T}$上是$X$ -值连续的且满足$u(0) = u(2\pi)$, $(Mu')(0) = (Mu')(2\pi)$.

定义2.6  设$1 \leq p < \infty$, $f\in L^p({\Bbb T}, X)$.称$u\in S_p(A, B, M)$为$(P_2)$的$L^p$ -强解, 若$(P_2)$在${\Bbb T}$上几乎处处成立.称$(P_2)$是$L^p$ -适定的, 若任给$f\in L^p({\Bbb T}, X)$, $(P_2)$存在唯一$L^p$ -强解.

若$(P_2)$是$L^p$ -适定的且$u\in S_p(A, B, M)$为$(P_2)$的$L^p$ -强解, 则存在常数$C>0$使得任给$f\in L^p({\Bbb T}, X)$, 有

$ \begin{equation} \|u\|_{L^p} + \|u'\|_{L^p} + \|Au\|_{L^p} + \|Bu'\|_{L^p} + \|Mu'\|_{L^p} + \|(Mu')'\|_{L^p}\leq C\|f\|_{L^p}. \end{equation} $

(2.2)

由闭图像定理和$A, B, M$的闭性可得到此结果.

命题2.7  设$A, B, M$为UMD空间$X$上闭线性算子满足$D(A)\cap D(B)\subseteq D(M)$.设$1 < p < \infty$, $N_k = (k^2M + A + ikB + F_k)^{-1}\ (k\in{\Bbb Z})$, $\rho_M(A, B) = {\Bbb Z}$.若集合$\{k^2MN_k: k\in{\Bbb Z}\}$, $\{kN_k: k\in{\Bbb Z}\}$和$\{kBN_k: k\in{\Bbb Z}\}$是${\rm R}$ -有界的, 则$(k^2MN_k)_{k\in{\Bbb Z}}$, $(N_k)_{k\in{\Bbb Z}}$, $(kN_k)_{k\in{\Bbb Z}}$和$(kBN_k)_{k\in{\Bbb Z}}$是$L^p$ -傅里叶乘子.

证  令$M_k = k^2MN_k, \ S_k = kBN_k, \ P_k = kN_k$ $(k\in {\Bbb Z})$.假设集合$\{k^2MN_k: k\in{\Bbb Z}\}$, $\{kN_k: k\in{\Bbb Z}\}$和$\{kBN_k: k\in{\Bbb Z}\}$是${\rm R}$ -有界的.由定理2.5, 只需证明集合$\{k(M_{k+1} -M_k): k\in{\Bbb Z}\}$, $\{k(N_{k+1} -N_k):k \in{\Bbb Z}\}$, $\{k(S_{k+1} -S_k):k\in{\Bbb Z}\}$与$\{k(P_{k+1} -P_k):k\in{\Bbb Z}\}$是${\rm R}$ -有界的.事实上, 观察

$ \begin{eqnarray} N_{k+1} - N_k& =& N_{k+1}(N_{k}^{-1} - N_{k+1}^{-1})N_k\\ & =& N_{k+1}[k^2M + A +ikB + F_k-(k+1)^2M-A-i(k+1)B -F_{k+1}]N_k\\ & = &- (2k+1)N_{k+1}MN_k - iN_{k+1}BN_k - N_{k+1}(F_{k+1} - F_k)N_k. \end{eqnarray} $

(2.3)

根据假设和$(2.3)$式知集合

$ \begin{eqnarray} \{k^2(N_{k+1} - N_k): k\in{\Bbb Z}\}, \{k^2B(N_{k+1} - N_k): k\in{\Bbb Z}\}, \{ k^3M(N_{k+1} - N_k): k\in{\Bbb Z}\} \end{eqnarray} $

(2.4)

为${\rm R}$ -有界的.另外一方面, 观察

$ \begin{eqnarray} k(M_{k+1} - M_k) &= &k[(k+1)^2MN_{k+1}-k^2MN_k]\\ & =& k[k^2MN_{k+1}-k^2MN_k + (2k+1)MN_{k+1}]\\ & = &k^3M(N_{k+1} - N_k) + k(2k+1)MN_{k+1}, \end{eqnarray} $

(2.5)

$ \begin{equation} k(S_{k+1} - S_k) = k[(k+1)BN_{k+1}-kBN_k] = k^2B(N_{k+1} - N_k) + kBN_{k+1}, \end{equation} $

(2.6)

$ \begin{equation} k(P_{k+1} -P_k)= k[(k+1)N_{k+1}-kN_k] = k^2(N_{k+1} - N_k) + kN_{k+1}. \end{equation} $

(2.7)

根据(2.4)-(2.7)式和假设知集合$\{k(M_{k+1} -M_k):k\in{\Bbb Z}\}$, $\{k(N_{k+1} -N_k):k\in{\Bbb Z}\}$, $\{k(S_{k+1} -S_k):k\in{\Bbb Z}\}$与$\{k(P_{k+1} -P_k):k\in{\Bbb Z}\}$是${\rm R}$ -有界的.

下面结果给出$(P_2)$是$L^p$ -适定的一个充要条件.

定理2.8  设$X$为UMD空间, $1 < p < \infty$且$A, B, M$为$X$上闭线性算子满足$D(A)\cap D(B)\subset D(M)$.则下述论断等价.

(ⅰ) $\rho_M(A, B) = {\Bbb Z}$且集合$\{k^2MN_k: k\in {\Bbb Z}\}$, $\{kN_k: k\in {\Bbb Z}\}$和$\{kBN_k: k\in {\Bbb Z}\}$是${\rm R}$ -有界的, 其中$N_k=[k^2M + A + ik B + F_k]^{-1}\ (k\in{\Bbb Z})$.

(ⅱ) $(P_2)$是$L^p$ -适定的.

证   ${\rm (ⅰ)}\Rightarrow {\rm (ⅱ)}$.假设$\rho_M(A, B) = {\Bbb Z}$且集合$\{k^2MN_k: k\in {\Bbb Z}\}$, $\{kN_k: k\in {\Bbb Z}\}$和$\{kBN_k: k\in {\Bbb Z}\}$是${\rm R}$ -有界的.令$M_k = k^2MN_k$, $S_k = kBN_k$, $P_k = kN_k\ (k\in{\Bbb Z})$.根据命题2.7知$(k^2MN_k)_{k\in {\Bbb Z}}$, $(N_k)_{k\in{\Bbb Z}}$, $(kN_k)_{k\in{\Bbb Z}}$和$(kBN_k)_{k\in{\Bbb Z}}$是$L^p$ -傅里叶乘子.则任给$f\in L^p({\Bbb T}, X)$, 存在$u, \ v, \ w, \ g\in L^p({\Bbb T}, X)$使得

$ \begin{eqnarray} \hat{u}(k) = M_k\hat{f}(k), \ \hat{v}(k) = - iS_k\hat{f}(k), \ \hat{w}(k) = - N_k\hat{f}(k), \ \hat{g}(k) = - iP_k\hat{f}(k)\ (k\in{\Bbb Z}). \end{eqnarray} $

(2.8)

因此$\hat{g}(k) = ik\hat{w}(k)\ (k\in{\Bbb Z})$.则根据文献[6, Lemma 3.1]知$w\in W_{{\rm per}}^{1, p}({\Bbb T}, X)$且$w'=g$.根据$(2.8)$式知$\hat{v}(k) = ikB\hat{w}(k)\ (k\in{\Bbb Z})$.再根据文献[6, Lemma 3.1]知$w'(t)\in D(B)$且$w'\in L^p({\Bbb T}, D(B))$.另外易知$(F_kN_k)_{k\in{\Bbb Z}}$也为$L^p$ -傅里叶乘子, 根据等式

$ k^2MN_k + AN_k + ikBN_k + F_kN_k = I_X $

知$(AN_k)_{k\in{\Bbb Z}}$也为$L^p$ -傅里叶乘子.从而存在$h\in L^p({\Bbb T}, X)$使得

$ \hat{h}(k) = - AN_k\hat{f}(k) = A\hat{w}(k). $

由文献[6, Lemma 3.1]知$w(t)\in D(A)$且$w\in L^p({\Bbb T}, D(A))$.注意到$(M_k)_{k\in{\Bbb Z}}$与$(\frac{-i}{k}I_X)_{k\in{\Bbb Z}}$是$L^p$ -傅里叶乘子, 则$(-ikMN_k)_{k\in{\Bbb Z}}$是$L^p$ -傅里叶乘子.因此存在$b\in L^p({\Bbb T}, X)$满足

$ \hat{b}(k) = - ikMN_k\hat{f}(k) = ikM\hat{w}(k) = \widehat{Mw'}(k). $

于是$w'\in L^p({\Bbb T}, D(M))$.由(2.8)式知

$ \hat{u}(k) = M_k\hat{f}(k) = k^2MN_k\hat{f}(k) = -k^2M\hat{w}(k) = \widehat{(Mw')'}(k). $

于是$Mu'\in W_{{\rm per}}^{1, p}({\Bbb T}, X)$.我们已经证明了$w\in S_p(A, B, M)$.再次根据(2.8)式知

$ -k^2M\hat{w}(k)= A\hat{w}(k) + ikB\hat{w}(k) + F_k\hat{w}(k) + \hat{f}(k)\ \ (k\in{\Bbb Z}). $

因此$(Mw')'(t) = Aw(t) + Bw'(t) + Fw_t + f(t)$关于${\Bbb T}$几乎处处成立^[6].故$w$为$(P_2)$的$L^p$ -强解.这就证明了存在性.

下面证明唯一性.设$u\in S_p(A, B, M)$使得

$ (Mu')'(t) - Au(t) - Bu'(t) - Fu_t=0 $

在${\Bbb T}$上几乎处处成立.两边取傅里叶变换得

$ [-k^2M-A-ikB-F_k]\hat{u}(k)=0\ \ (k\in{\Bbb Z}). $

因为$k\in \rho_M(A, B)\ (k\in{\Bbb Z})$, 故$u=0$.因此$(P_2)$是$L^p$ -适定的.

${\rm (ⅱ)}\Rightarrow {\rm (ⅰ)}$.假设$(P_2)$是$L^p$ -适定的.设$k\in{\Bbb Z}$且$y\in X$, 定义$f(t)=e^{ikt}y\ (t\in {\Bbb T})$.则$f\in L^p({\Bbb T}, X), \ \hat{f}(k)=y$且有$\hat{f}(n)=0\ (n\neq k)$.因为$(P_2)$是$L^p$ -适定的, 故存在唯一$u\in S_p(A, B, M)$使得

$ (Mu')'(t) = Au(t) + Bu'(t) + Fu_t + f(t) $

在${\Bbb T}$上几乎处处成立.于是根据文献[6, Lemma 3.1]知$\hat{u}(n)\in D(A)\cap D(B)\ (n\in{\Bbb Z})$.两边取傅里叶变换得

$ \begin{eqnarray} [-k^2M-A-ikB- F_k]\hat{u}(k)=y \end{eqnarray} $

(2.9)

且$[-n^2M-A-in B-F_n]\hat{u}(n)=0\ (n\neq k)$.因此$ -k^2M -A -ikB -F_k$是满射.下证$ -k^2M -A -ikB -F_k$是单射.取$x\in D(A)\cap D(B)$使得

$ [-k^2M-A-ikB- F_k]x=0. $

令$u(t) = e^{ikt}x\ (t\in{\Bbb T})$, 则$u\in S_p(A, B, M)$且当$f = 0$时, $(P_2)$几乎处处成立.于是当$f = 0$时, $u$为$(P_2)$的$L^p$ -强解.由唯一性假设知$x = 0$.于是$ -k^2M -A -ikB -F_k$是单射.因此$ -k^2M -A -ikB -F_k$是$D(A)\cap D(B)$到$X$上的满射.

下面证明$[-k^2M-A-ikB-F_k]^{-1}\in{\cal L}(X)$.设$f(t) = e^{ikt}y$, $u\in S_p(A, B, M)$为$(P_2)$的$L^p$ -强解.则

$ \begin{eqnarray*} \hat{u}(n)= \left\{\begin{array}{ll}0& (n\neq k), \\ {}[-k^2M-A-ikB- F_k]^{-1}~~& (n = k). \end{array}\right. \end{eqnarray*} $

于是$u(t) = e^{ikt}[-k^2M-A-ikB-F_k]^{-1}y$.根据$(2.2)$式, 存在独立于$y$和$k$的常数$C>0$使得

$ \|u\|_{L^p} + \|u'\|_{L^p} +\|Au\|_{L^p} +\|Bu'\|_{L^p} + \|Mu'\|_{L^p} + \|(Mu')'\|_{L^p}\leq C\|f\|_{L^p}. $

于是任给$y\in X$, 有$\|[-k^2M-A-ikB-F_k]^{-1}y\|\leq C\|y\|$.因此

$ \|-k^2M -A -ikB - F_k]^{-1}\|\leq C. $

我们已经得到了$k\in \rho_M(A, B)\ (k\in{\Bbb Z})$.则$\rho_M(A, B) = {\Bbb Z}$.

最后证明, 若$N_k = [k^2M + A + ikB + F_k]^{-1}\ (k\in{\Bbb Z})$, 则集合$\{k^2MN_k: k\in{\Bbb Z}\}$, $\{kBN_k: k\in{\Bbb Z}\}$和$\{kN_k: k\in{\Bbb Z}\}$是${\rm R}$ -有界的.事实上, 设$f\in L^p({\Bbb T}, X)$, 则存在$u\in S_p(A, B, M)$为$(P_2)$的$L^p$ -强解. $(P_2)$两边取傅里叶变换有$\hat{u}(k)\in D(A)\cap D(B)$且

$ [-k^2M-A-ikB- F_k]\hat{u}(k) = \hat{f}(k)\ (k\in{\Bbb Z}). $

因为$ -k^2M -A -ikB -F_k$可逆, 则

$ \hat{u}(k) = [-k^2M-A-ikB- F_k]^{-1}\hat{f}(k)\ (k\in{\Bbb Z}). $

由$u\in S_p(A, B, M)$知

$ \widehat{(Mu')'}(k) = - k^2M\hat{u}(k), \ \widehat{Bu'}(k) = ikB\hat{u}(k), \ \widehat{u'}(k) = ik\hat{u}(k)\ (k\in{\Bbb Z}). $

于是

$ \widehat{(Mu')'}(k) = k^2MN_k\hat{f}(k), \ \widehat{Bu'}(k) = - ikBN_k\hat{f}(k), \ \widehat{u'}(k) = - ikN_k\hat{f}(k)\ (k\in{\Bbb Z}). $

又因为$(Mu')', \ Bu', \ u'\in L^p({\Bbb T}, X)$, 从而$(k^2MN_k)_{k\in{\Bbb Z}}$, $(kBN_k)_{k\in{\Bbb Z}}$和$(kN_k)_{k\in{\Bbb Z}}$是$L^p$ -傅里叶乘子.根据命题2.4知集合$\{k^2MN_k: k\in{\Bbb Z}\}$, $\{kBN_k: k\in{\Bbb Z}\}$和$\{kN_k: k\in{\Bbb Z}\}$是${\rm R}$ -有界的.

因为定理2.8与参数$p$无关, 于是我们有下面推论.

推论2.9  设$X$为UMD空间且$A, B, M$为$X$上闭线性算子满足$D(A)\cap D(B)\subset D(M)$.若关于某个$1 < p < \infty$, $(P_2)$是$L^p$ -适定的, 则任给$1 < p < \infty$, $(P_2)$也是$L^p$ -适定的.

Arendt和Bu^[7]引进了向量值周期Besov空间$B_{p, q}^s({\Bbb T}, X)$并研究了其上的算子值傅里叶乘子.设${\cal S}({\Bbb R})$为Schwartz空间, $ {\cal D}({\Bbb T})$为定义在${\Bbb T}$上的$C^\infty$函数全体构成的集合.半范族$\|f\|_{\alpha} = \sup\limits_{x\in{\Bbb T}}|{f^{(\alpha)}(x)}|$给出了${\cal D}({\Bbb T})$上的局部凸拓扑, 其中$\alpha\in{\Bbb N}_0 := {\Bbb N}\cup ${0}.令$ {\cal D}'({\Bbb T}; X) := {\cal L}({\cal D}({\Bbb T}), X)$为从${\cal D}({\Bbb T})$到$X$上的所有连续线性算子构成的空间.考虑${\Bbb R}$上的二进制覆盖$(I_k)_{k\in{\Bbb N}_0}$

$ \begin{eqnarray*} I_0 = \Big\{t\in{\Bbb R}:\left| t \right|\leq2\Big\}, \ \ I_k = \Big\{t\in {\Bbb R}:2^{k-1} < \left| t \right|\leq 2^{k+1}\Big\}\ (k\in{\Bbb N}). \end{eqnarray*} $

设$\phi({\Bbb R})$为所有满足下列条件的函数列$\phi = (\phi_k)_{k\in{\Bbb N}_0} \subset {\cal S}({\Bbb R})$构成的集合:若$\mbox{supp}(\phi_k)\subset\bar{I}_k\ (k\in{\Bbb N}_0)$, 则有$\sum\limits_{k\in{\Bbb N}_0}\phi_k(x)=1\ (x\in {\Bbb R})$和$\sup\limits_{ x\in{\Bbb R}, k\in{\Bbb N}_0 }2^{k\alpha}\vert\phi_k^{(\alpha)}(x)\vert < \infty\ (\alpha\in{\Bbb N}_0)$.

设$1 \leq p, q \leq \infty, s\in{\Bbb R}$, $\phi = (\phi_k)_{k\in{\Bbb N}_0}\in\phi({\Bbb R})$. $X$ -值周期Besov空间定义为

$ \begin{eqnarray*} B_{p, q}^s({\Bbb T}, X)& := &\Big\{f\in{\cal D}'({\Bbb T};X): \|f\|_{B_{p, q}^s} := \\ && \Big(\sum\limits_{j\geq0}2^{sjq}\big\|\sum\limits_{k\in{\Bbb Z}}e_k\otimes \phi_j(k)\hat{f}(k)\big\|_{L^p}^q\Big)^{1/q}<\infty\Big\}\ (1\leq q<\infty), \end{eqnarray*} $

且

$ \begin{eqnarray*} B_{p, \infty}^s({\Bbb T}, X) := \Big\{f\in{\cal D}'({\Bbb T};X): \|f\|_{B_{p, \infty}^s} := \sup\limits_{j\geq0}2^{sjq}\big\|\sum\limits_{k\in{\Bbb Z}}e_k\otimes \phi_j(k)\hat{f}(k)\big\|_{L^p}<\infty\Big\}. \end{eqnarray*} $

空间$B_{p, q}^s({\Bbb T}, X)$与$\phi$的选择无关, 不同的$\phi$定义的范数等价. $B_{p, q}^s({\Bbb T}, X)$在赋予范数$\|\cdot\|_{B_{p, q}^s}$下为Banach空间.关于$B_{p, q}^s({\Bbb T}, X)$更详细的研究参见文献[7]中第二部分内容.若$s_1\leq s_2$, 则\ $B_{p, q}^{s_1}({\Bbb T}, X)\subset B_{p, q}^{s_2}({\Bbb T}, X)$且其嵌入是连续的(参见文献[7, 定理2.3]).当$s>0$, 有$B_{p, q}^s({\Bbb T}, X)\subset L^p({\Bbb T}, X)$且$f\in B_{p, q}^{s+1}({\Bbb T}, X)$当且仅当$f$几乎处处可导且$f'\in B_{p, q}^s({\Bbb T}, X)$ (参见文献[7, 定理2.3]).若$u\in B_{p, q}^s({\Bbb T}, X)$且存在$v\in B_{p, q}^s({\Bbb T}, X)$使得$\hat{v}(k) = ik\hat{u}(k)$ $(k\in{\Bbb Z})$, 则$u\in B_{p, q}^{s+1}({\Bbb T}, X)$且$u' = v$.

定义3.1  设$X, Y$为Banach空间, $1 \leq p, q \leq \infty, s\in{\Bbb R}$, $(M_k)_{k\in{\Bbb Z}}\subset {\cal L}(X, Y)$.称$(M_k)_{k\in{\Bbb Z}}$是$B_{p, q}^s$ -傅里叶乘子, 若任给$f\in B_{p, q}^s({\Bbb T}, X)$, 存在唯一的$u\in B_{p, q}^s({\Bbb T}, Y)$满足$\hat{u}(k) = M_k\hat{f}(k)$ $ (k\in {\Bbb Z})$.

Arendt和Bu^[7]证明了下述算子值$B_{p, q}^s$ -傅里叶乘子定理.

定理3.2  设$X, Y$为Banach空间, $1\leq p, q\leq\infty, s\in{\Bbb R}$, $\{M_k:k\in{\Bbb Z}\}\subset {\cal L}(X, Y)$.若

$ \begin{equation} \sup\limits_{k\in{\Bbb Z}}\|M_k\| < \infty, \, \, \sup\limits_{k\in{\Bbb Z}}\|k(M_{k+1} - M_k)\| < \infty, \end{equation} $

(3.1)

$ \begin{equation} \sup\limits_{k\in{\Bbb Z}}\|k^2(M_{k+2} - 2M_{k+1} + M_{k})\| < \infty. \end{equation} $

(3.2)

则$(M_k)_{k\in{\Bbb Z}}$是$B_{p, q}^s$ -傅里叶乘子.

注3.3   (ⅰ)~若$(M_k)_{k\in{\Bbb Z}}$和$(N_k)_{k\in{\Bbb Z}}$是$B_{p, q}^s$ -傅里叶乘子, 则$(M_kN_k)_{k\in{\Bbb Z}}$与$(M_k + N_k)_{k\in{\Bbb Z}}$也是$B_{p, q}^s$ -傅里叶乘子

(ⅱ)~易知$\{\frac{1}{k}I_X: k\in{\Bbb Z}\}$满足条件(3.1)和(3.2).根据定理3.2知$(\frac{1}{k}I_X)_{k\in{\Bbb Z}}$是$B_{p, q}^s$ -傅里叶乘子.

考虑下述二阶有限时滞退化微分方程

$ (Mu')'(t) = Au(t) + Bu'(t) + Fu_t + f(t), (t\in{\Bbb T}), $

($P_2$)

且具有周期边值条件$u(0) = u(2\pi), (Mu')(0) = (Mu')(2\pi)$, 这里$A, B, M$为Banach空间$X$上闭线性算子满足$D(A)\cap D(B)\subset D(M)$, $F:B_{p, q}^s([-2\pi, 0], X)\rightarrow X$为有界线性算子, $u_t$定义为: $u_t(s) = u(t+s)\ (s\in [-2\pi, 0])$, $f\in B_{p, q}^s({\Bbb T}, X)$.

设$F\in(B_{p, q}^s([-2\pi, 0], X), X)$.定义$F_k$为$F_kx = F(e_kx)\ (k\in{\Bbb Z}, \ x\in X)$.因为存在常数$C>0$使得$\|e_k\otimes x\|_{B_{p, q}^s}\leq C\|x\|\ (x\in X)$, 则$\|F_k\| \leq \|F\|$且$\widehat{Fu_\cdot}(k) = F_k\hat{u}(k)$, $u\in B_{p, q}^s({\Bbb T}, X)$.

设$1 \leq p, q \leq \infty, s>0$. $(P_2)$在$B_{p, q}^s$ -适定性意义下的解空间定义为

$ \begin{eqnarray*} S_{p, q, s}(A, B, M)& :=& \big\{u\in B_{p, q}^{s+1}({\Bbb T}, X)\cap B_{p, q}^s({\Bbb T}, D(A)):u'\in B_{p, q}^s({\Bbb T}, D(M))\cap B_{p, q}^s({\Bbb T}, D(B)), \\ &&\ \ Mu'\in B_{p, q}^{1+s}({\Bbb T}, X), Fu_\cdot\in B_{p, q}^{s}({\Bbb T}, X)\big\}, \end{eqnarray*} $

其中$D(A)$, $D(B)$与$D(M)$赋予图像范数成为Banach空间. $S_{p, q, s}(A, B, M)$的范数定义为

$ \begin{eqnarray*} \|u\|_{S_{p, q, s}(A, B, M)} &:= &\|u\|_{B_{p, q}^s} + \|u'\|_{B_{p, q}^s} + \|Au\|_{B_{p, q}^s} + \|Bu'\|_{B_{p, q}^s}\\ & &+ \|Mu'\|_{B_{p, q}^s} + \|(Mu')'\|_{B_{p, q}^s} + \|Fu_\cdot\|_{B_{p, q}^s}. \end{eqnarray*} $

易知$S_{p, q, s}(A, B, M)$在赋予此范数下为Banach空间.根据文献[6, Lemma 2.1]知, 若$u\in S_{p, q, s}(A, B, M)$, 则$u$与$Mu'$在${\Bbb T}$上是$X$ -值连续的且满足$u(0)=u(2\pi)$, $(Mu')(0)=(Mu')(2\pi)$.

定义3.4  设$1 \leq p, q \leq \infty, s>0$, $f\in B_{p, q}^s({\Bbb T}, X)$.称$u\in S_{p, q, s}(A, B, M)$为$(P_2)$的$B_{p, q}^s$ -强解, 若$(P_2)$在${\Bbb T}$上几乎处处成立.称$(P_2)$是$B_{p, q}^s$ -适定的, 若任给$f\in B_{p, q}^s({\Bbb T}, X)$, $(P_2)$存在唯一$B_{p, q}^s$ -强解.

若$(P_2)$是$B_{p, q}^s$ -适定的且$u\in S_{p, q, s}(A, B, M)$为$(P_2)$的$B_{p, q}^s$ -强解, 则存在常数$C>0$使得任给$f\in B_{p, q}^s({\Bbb T}, X)$, 有

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

(3.3)

这里用到闭图像定理和$A, B, M$的闭性.

命题3.5  设$1 \leq p, q \leq \infty, s>0$, $A, B, M$为Banach空间$X$上闭线性算子满足$D(A)\cap D(B)\subseteq D(M)$.假设$\rho_M(A, B) = {\Bbb Z}$且集合$\{k(F_{k+2} -2F_{k+1} + F_k): k\in{\Bbb Z}\}$, $\{k^2MN_k: k\in{\Bbb Z}\}$, $\{kN_k: k\in{\Bbb Z}\}$和$\{kBN_k: k\in{\Bbb Z}\}$是范数有界的, 则$(k^2MN_k)_{k\in {\Bbb Z}}$, $(N_k)_{k\in{\Bbb Z}}$, $(kN_k)_{k\in{\Bbb Z}}$, $(kBN_k)_{k\in{\Bbb Z}}$和$(F_kN_k)_{k\in{\Bbb Z}}$是$B_{p, q}^s$ -傅里叶乘子, 这里$N_k = (k^2M + A + ikB + F_k)^{-1}\ (k\in{\Bbb Z})$.

证     令$M_k = k^2MN_k, \ S_k = kBN_k, \ P_k=kN_k$, $(k\in {\Bbb Z})$.假设$\rho_M(A, B) = {\Bbb Z}$且集合$\{k(F_{k+2} -2F_{k+1} + F_k): k\in{\Bbb Z}\}$, $\{k^2MN_k: k\in{\Bbb Z}\}$, $\{kN_k: k\in{\Bbb Z}\}$和$\{kBN_k: k\in{\Bbb Z}\}$是范数有界的.根据(2.3)式知

$ \begin{equation} \sup\limits_{k\in{\Bbb Z}}\|k^2(N_{k+1} - N_k)\| < \infty, \ \sup\limits_{k\in{\Bbb Z}}\|k^2B(N_{k+1} - N_k)\| < \infty, \ \sup\limits_{k\in{\Bbb Z}}\|k^3M(N_{k+1} - N_k)\| < \infty. \end{equation} $

(3.4)

由(2.5)-(2.7), (3.4)式得

$ \begin{eqnarray} \sup\limits_{k\in{\Bbb Z}}\|k(M_{k+1} - M_k)\| < \infty, \ \sup\limits_{k\in{\Bbb Z}}\|k(S_{k+1} - S_k)\| < \infty, \ \sup\limits_{k\in{\Bbb Z}}\|k(P_{k+1} - P_k)\| < \infty. \end{eqnarray} $

(3.5)

观察

$ \begin{eqnarray} &&N_{k+2} - 2N_{k+1} + N_k\\ &= &N_{k+2}[N_k^{-1}-2N_{k+2}^{-1}N_{k+1}N_k^{-1} + N_{k+2}^{-1}]N_k\\ &=& N_{k+2}\Big\{N_k^{-1}-2[N_{k+1}^{-1} + (2k+3)M + iB + F_{k+2}- F_{k+1}]N_{k+1}N_k^{-1}\\ && + [N_k^{-1} + (4k + 4)M + 2iB + F_{k+2}-F_k]\Big\}N_k\\ &=& N_{k+2}\Big\{ - 4kM(N_{k+1} - N_k) -6MN_{k+1} + 4MN_k - 2iB(N_{k+1} - N_k)\\ && - 2(F_{k+2} - F_{k+1})N_{k+1} + (F_{k+2} - F_k)N_k\Big\}\\ &=& N_{k+2}\Big\{ - 4kM(N_{k+1} - N_k) -6MN_{k+1} + 4MN_k - 2iB(N_{k+1} - N_k)\\ &&- (F_{k+2} - 2F_{k+1} + F_k)N_{k+1} - (F_{k+2} - F_k)(N_{k+1} - N_k)\Big\}. \end{eqnarray} $

(3.6)

根据(3.4)-(3.6)式得

$ \begin{equation} \sup\limits_{k\in{\Bbb Z}}\|k^3(N_{k+2} -2N_{k+1} + N_k)\| < \infty, \ \sup\limits_{k\in{\Bbb Z}}\|k^3B(N_{k+2} -2N_{k+1} + N_k)\| < \infty, \end{equation} $

(3.7)

$ \begin{equation} \sup\limits_{k\in{\Bbb Z}}\|k^4M(N_{k+2} -2N_{k+1} + N_k)\| < \infty. \end{equation} $

(3.8)

另外一方面, 注意到

$ \begin{eqnarray} &&k^2(M_{k+2} - 2M_{k+1} + M_k)\\ &=& k^2\Big\{(k + 2)^2MN_{k+2} - 2(k + 1)^2MN_{k+1} + k^2MN_k\Big\}\\ &= &k^4M(N_{k+2} - 2N_{k+1} + N_k) + k^2\Big\{(4k + 4)MN_{k+2} - 2(2k + 1)MN_{k+1}\Big\}\\ &= &k^4M(N_{k+2} - 2N_{k+1} + N_k) + 4k^3M(N_{k+2} - N_{k+1}) + 4k^2MN_{k+2} - 2k^2MN_{k+1}, \end{eqnarray} $

(3.9)

$ \begin{eqnarray} k^2(S_{k+2} - 2S_{k+1} + S_k) &=& k^2\Big\{(k + 2)BN_{k+2} - 2(k + 1)BN_{k+1} + kBN_k\Big\}\\ &= &k^3B(N_{k+2} - 2N_{k+1} + N_k) + 2k^2B(N_{k+2} - N_{k+1}), \end{eqnarray} $

(3.10)

$ \begin{eqnarray} k^2(P_{k+2} - 2P_{k+1} + P_k) &= &k^2\Big\{(k + 2)N_{k+2} - 2(k + 1)N_{k+1} + kN_k\Big\}\\ &=& k^3(N_{k+2} - 2N_{k+1} + N_k) + 2k^2(N_{k+2} - N_{k+1}), \end{eqnarray} $

(3.11)

$ \begin{eqnarray} &&k^2(F_{k+2}N_{k+2} - 2F_{k+1}N_{k+1} + F_kN_k)\\ &=& k^2(F_{k+2} - 2F_{k+1} + F_k)N_{k+2} + 2k^2F_{k+1}(N_{k+2} - N_{k+1})\\ &&- k^2F_k[(N_{k+2}-N_{k+1})+(N_{k+1}-N_k)], \end{eqnarray} $

(3.12)

由(3.4), (3.7)-(3.12)式知

$ \begin{equation} \sup\limits_{k\in{\Bbb Z}}\|k^2(M_{k+2} -2M_{k+1} + M_k)\| < \infty, \ \sup\limits_{k\in{\Bbb Z}}\|k^2(S_{k+2} -2S_{k+1} + S_k)\| < \infty, \end{equation} $

(3.13)

$ \begin{equation} \sup\limits_{k\in{\Bbb Z}}\|k^2(P_{k+2} -2P_{k+1} + P_k)\| < \infty, \ \sup\limits_{k\in{\Bbb Z}}\|k^2(F_{k+2}N_{k+2} - 2F_{k+1}N_{k+1} + F_kN_k)\| < \infty. \end{equation} $

(3.14)

根据定理3.2知$(k^2MN_k)_{k\in{\Bbb Z}}$, $(N_k)_{k\in{\Bbb Z}}$, $(kN_k)_{k\in{\Bbb Z}}$, $(kBN_k)_{k\in{\Bbb Z}}$和$(F_kN_k)_{k\in{\Bbb Z}}$是$B_{p, q}^s$ -傅里叶乘子.

下面结果给出$(P_2)$是$B_{p, q}^s$ -适定的一个充要条件.证明与定理$2.8$类似, 我们略去细节.

定理3.6  设$1 \leq p, q \leq \infty, s>0$, $A, B, M$为Banach空间$X$上闭线性算子满足$D(A)\cap D(B)\subset D(M)$.假设集合$\{k(F_{k+2} -2F_{k+1} + F_k):k\in{\Bbb Z}\}$范数有界.则下述论断等价.

(ⅰ) $\rho_M(A, B) = {\Bbb Z}$且集合$\{k^2MN_k: k\in{\Bbb Z}\}$, $\{kN_k: k\in {\Bbb Z}\}$和$\{kBN_k: k\in{\Bbb Z}\}$是范数有界的, 其中$N_k = [k^2M + A + ik B + F_k]^{-1}\ (k\in{\Bbb Z})$.

(ⅱ) $(P_2)$是$B_{p, q}^s$ -适定的.

因为定理3.6与参数$p, q, s$无关, 于是我们有下面推论.

推论3.7  设$X$为Banach空间且$A, B, M$为$X$上闭线性算子满足$D(A)\cap D(B)\subset D(M)$.若关于某个$1 \leq p, q \leq \infty, s>0$, $(P_2)$是$B_{p, q}^s$ -适定的, 则任给$1 \leq p, q \leq \infty, s>0$, $(P_2)$也是$B_{p, q}^s$ -适定的.

我们将主要结果定理2.8和定理3.6应用到下述例子.

例4.1  设$\Omega$为${\Bbb R}^n$上具有光滑边界的有界区域, $m$为$\Omega$上非负有界可测函数, $f$为$[0, 2\pi]\times\Omega$上函数, $X = H^{-1}(\Omega)$.考虑下述有限时滞微分方程:

$ \begin{equation} \left\{ \begin{array}{ll} \frac{\partial^2}{\partial t^2}(m(x)u(t, x)) = \Delta u + B\frac{\partial}{\partial t}(u(t, x)) + Fu_t + f(t, x), &(t, x)\in[0, 2\pi]\times\Omega, \\[2mm] u(t, x)=0,&(t, x)\in[0, 2\pi]\times\partial\Omega, \\ u(0, x)=u(2\pi, x),&x\in\Omega, \\[2mm] \frac{\partial u(t, x)}{\partial t}|_{t=0}=\frac{\partial u(t, x)}{\partial t}|_{t=2\pi},&x\in\Omega, \end{array} \right. \end{equation} $

(4.1)

这里$u_t(s, x) := u(t+s, x)\ (s\in[-2\pi, 0], \ x\in\Omega)$, $F:L^p([-2\pi, 0];X)\rightarrow X$为有界线性算子, $B$为$X$上有界线性算子, $1 < p < \infty$.

定义$M$为函数空间$H^{-1}(\Omega)$上乘积算子$m$且其定义域为$D(M)$.根据文献[9, Section 3.7]知, 若考虑$X$上Dirichlet拉普拉斯算子$\Delta$, 则存在常数$C>0$使得

$ \|M(zM - \Delta)^{-1}\|\leq\frac{C}{1+\left| z \right|}, $

其中${\rm Re}(z)\geq-\beta(1+\left| {\rm Im}(z) \right|)$, 这里的常数$\beta$仅仅与$m$有关.从而

$ \|M(k^2M + \Delta)^{-1}\|\leq\frac{C}{1+\left| k \right|^2}\ \ (k\in{\Bbb Z}). $

(4.2)

若假设$m^{-1}$足够正则且满足$m^{-1}$为$H^{-1}(\Omega)$上有界算子, 则存在常数$C_1>0$使得

$ \|(k^2M + \Delta)^{-1}\|\leq\frac{C_1}{1+\left| k \right|^2}\ \ (k\in{\Bbb Z}). $

(4.3)

进一步假设$D(\Delta)\cap D(B)\subset D(M)$且$\rho_M(A, B) = {\Bbb Z}$, 则$k^2M + \Delta + ikB + F_k$为$D(\Delta)\cap D(B)$到$X$上双射且$(k^2M + \Delta +ikB + F_k)^{-1} \in{\cal L}(X)$.观察

$ k^2M + \Delta + ikB + F_k = [I + (ikB + F_k)(k^2M + \Delta)^{-1}](k^2M + \Delta)\ \ (k\in{\Bbb Z}). $

根据(4.3)式知$\lim\limits_{k\rightarrow \infty} \Vert(ikB + F_k)(k^2M + \Delta)^{-1}\Vert = 0$, 这里用到了$(F_k)_{k\in{\Bbb Z}}$与算子$B$的范数有界性.于是当$\vert k\vert$足够大时, $I + (ikB + F_k)(k^2M+\Delta)^{-1}$可逆.对于这样的$k$, 我们有

$ [k^2M + \Delta + ikB + F_k]^{-1} = (k^2M + \Delta)^{-1}[I + (ikB + F_k)(k^2M + \Delta)^{-1}]^{-1}. $

由(4.2)和(4.3)式得

$ \sup\limits_{k\in{\Bbb Z}}\|k^2M(k^2M + \Delta + ikB + F_k)^{-1}\|<\infty, $

$ \sup\limits_{k\in{\Bbb Z}}\|kB(k^2M + \Delta + ikB + F_k)^{-1}\|<\infty, $

$ \sup\limits_{k\in{\Bbb Z}}\|k(k^2M + \Delta + ikB + F_k)^{-1}\|<\infty. $

因此集合$\{k^2M(k^2M + \Delta + ikB + F_k)^{-1}: k\in{\Bbb Z}\}$, $\{kB(k^2M + \Delta + ikB + F_k)^{-1}: k\in{\Bbb Z}\}$和$\{k^2(k^2M + \Delta + ikB + F_k)^{-1}: k\in{\Bbb Z}\}$是$R$ -有界的.这里用到了Hilbert空间中集合的${\rm R}$ -有界性与范数有界性等价(参见文献[6, Proposition 1.13]).根据定理2.8知方程$(4.1)$是$L^p$ -适定的.如果考虑$F \in {\cal L}(B_{p, q}^s([-2\pi, 0]; X), X)$且进一步假设集合$\{k(F_{k+2} -2F_{k+1} + F_k): k\in{\Bbb Z}\}$是范数有界的, 则根据定理3.6知方程$(4.1)$是$B_{p, q}^s$ -适定的.