Floquet theory

《Orderary differential equations》, Jake K. Hale

Floquet theory

考虑系统

\[\begin{equation} \dot{x}=A(t) x \end{equation} \]

其中\(A(t)\)是 \(T-\)周期的.

考虑三个函数类: \(\mathscr{D}=\mathscr{B}(-\infty,\infty),\mathscr{AP}\) 和 \(\mathscr{P}_T\)分别表示有界函数，概周期函数和\(T-\)周期函数的集合.

系统(1)称为关于某个函数类是非临界的如果系统(1)在此类中只有零解. 

Remark: 假设系统(1)的基本解矩阵为\(X(t)\), 那么存在非奇异矩阵\(C\)使得\(X(t+T)=X(t)C\). 存在复矩阵\(B\)使得\(e^{BT}=C\)[参见矩阵对数理论]. \(C\)的特征值称为系统(1)的特征乘子, \(B\)的特征值称为系统(1)的特征指数. 显然对于\(A\)是常矩阵的情况, 系统(1)的特征指数就是\(A\)的特征值加上\(2k\pi i\).

结论1: \(X(t)\)可以表示为

\[X(t)=P(t)e^{Bt},~~P(t+T)=P(t). \]

事实上, 令\(P(t)=X(t)e^{-Bt}\), 我们论证\(P(t+T)=P(t)\).

\[P(T+t)=X(t+T)e^{-BT}e^{-Bt} =X(t) C e^{-BT}e^{-Bt}=X(t)e^{-Bt}=P(t). \]

结论2: 系统(1)可以经过一个周期变换化为常系数的线性微分方程.

事实上, 令\(x=P(t)y\), 那么在此变换下系统(1)被变为

\[\dot{y}=By. \]

计算如下：

\[\dot{x}=A(t)x=A(t)P(t)y=\dot{P}(t)y+P(t)\dot{y} \]

下面计算\(\dot{P}(t)\):

\[\dot{X}(t)=A(t)X(t)\rightarrow \dot{P}(t)e^{Bt}+P(t)e^{Bt} B=A(t)P(t)e^{Bt}\\ \Rightarrow\dot{P}(t)=A(t)P(t)-P(t) B. \]

\[\Rightarrow A(t)P(t)y=A(t)P(t)y-P(t) By+P(t)\dot{y}\\ \Rightarrow \dot{y}=By. \]

结论3: \(\lambda\)是系统(1)的一个特征指数当且仅当\(e^{\lambda t}p(t),p(t+T)=p(t)\)是(1)的一个解.

\(\Rightarrow\) 如果\(\lambda\)是(1)的一个特征指数, \(e^{\lambda T}\)是\(C\)的一个特征值(\(X(T+t)=X(t)C\)), 设\(x_\lambda\)是对于的特征向量, 那么\(X(t)x_{\lambda}\)是一个解. 为了证明\(X(t)x_{\lambda}\)具有形式\(p(t)e^{\lambda t}\), 令\(p(t)=X(t)x_{\lambda} e^{-\lambda t}\)有周期\(T\). 验证如下:

\[p(t+T)=X(t+T)x_{\lambda} e^{-\lambda T}e^{-\lambda t} =X(t)Cx_{\lambda} e^{-\lambda T}e^{-\lambda t} =X(t)e^{\lambda T}x_{\lambda} e^{-\lambda T}e^{-\lambda t}\\ =X(t)x_{\lambda}e^{-\lambda t}=p(t). \]

\(\Leftarrow\) 假设\(e^{\lambda t}p(t)\)是(1)的一个解, 我们证明\(\lambda\)是(1)的一个特征指数. 假设\(e^{\lambda t}p(t)=X(t)x_0\), 那么

\[e^{\lambda t} e^{\lambda T} p(t+T)=X(t)Cx_0\\ \Rightarrow e^{\lambda T} X(t)x_0= e^{\lambda t} e^{\lambda T} p(t+T)=X(t)Cx_0\\ \Rightarrow X(t) [C-e^{\lambda T}]x_0=0\\ \Rightarrow [C-e^{\lambda T}]x_0=0 \]

\(x_0\ne 0\) 蕴含\(\lambda\)是系统(1)的一个特征指数.

引理 1:
系统(1)关于\(\mathscr{B}(-\infty,\infty)(\mathscr{{\rm or} \, AP})\) 是非临界的当且仅当(1)的特征乘子具有非零实部；系统(1)关于\(\mathscr{P}_T\)是非临界的当且仅当\(X(T)-I\)是非奇异的.

证明 :[有界和概周期的情况] \(\Rightarrow\) 如果(1)有特征乘子\(\lambda\)实部非零. \(Re(\lambda)>0\)蕴含\(e^{\lambda t}p(t)\)正向无界; \(Re(\lambda)<0\)蕴含\(e^{\lambda t}p(t)\)负向无界，均矛盾.

\(\Leftarrow\) 如果所有特征乘子具有零实部, 任一解具有形式\(x(t)=e^{iwt}p(t),p(t+T)=p(t)\),它是概周期的(概周期函数包含于有界函数).

[周期的情况] \(x(t)=X(t)x(0)\)是系统(1)的一个周期解\(\Leftrightarrow x(T)=x(0)\Leftrightarrow[X(T)-I]x(0)=0\) 对于某个\(x(0)\ne 0\)\(\Leftrightarrow X(T)-I\) 非奇异.

引理 2: [Fredholm's alternative]
考虑系统

\[\begin{equation} \dot{x}=A(t)x+f(t), \end{equation} \]

其中\(f\in\mathscr{D}\).
对于任何\(f\in\mathscr{P}_T\), 系统(2)在\(\mathscr{P}_T\)有解当且仅当

\[ \begin{equation} \int_{0}^{T}y(t)f(t)dt=0, \end{equation} \]

对伴随方程

\[ \begin{equation} \dot{y}=-yA(t) \end{equation} \]

的所有\(T-\)周期解成立. 如果(3)成立, 那么系统(2)有一个带\(r\)个参数的\(T-\)周期解族, \(r\)是系统(1)的线性无关的\(T-\)周期解的个数.

证明 :
利用常数变易公式得到(2)的解

\[x(t)=X(t,0)x(0)+\int_0^t X(t,s)f(s)ds, ~~X(0,0)=I. \]

\(x(T)=x(0)\)蕴含

\[X(T,0)x(0)+\int_0^T X(T,s)f(s)ds=x(0)\\ \Rightarrow x(0)+\int_0^T X^{-1}(s,0)f(s)ds=X^{-1}(T,0)x(0)\\ \Rightarrow [X^{-1}(T,0)-I]x(0)=\int_0^T X^{-1}(s,0)f(s)ds \Rightarrow Bx(0)=b,\\~B= [X^{-1}(T,0)-I],~b=\int_0^T X^{-1}(s,0)f(s)ds \]

满足\(Bx(0)=b\)的\(x(0)\)存在当且仅当\(\text{rank}(B)=\text{rank}(B,b)\)当且仅当\(b\)是\(B\)的列向量的线性组合, 这蕴含任何\(a: aB=0\)蕴含\(ab=0\). 令\(a=y(0)\), 那么\(aB=0\)蕴含\(y(T)=y(0)\), 即\(a\)是伴随方程的所有\(T-\)周期解的初值.

\[y(0)b=\int_0^T y(0)X^{-1}(s,0)f(s)ds=\int_0^Ty(t)f(s)ds,\]

从而对伴随方程的所有\(T-\)周期解\(y\), 有(2)成立.
剩下的部分是显然的.

定理1:
对于任何\(f\in \mathscr{D}\), 系统(2)在\(\mathscr{D}\)中有一个解当且仅当系统(1)关于\(\mathscr{D}\)是非临界的.
如果系统(1)关于\(\mathscr{D}\)是非临界的, 那么系统(2)在\(\mathscr{D}\)中有唯一解\(\mathscr{K}f\),它关于\(f\)是连续线性的, 存在\(K>0\)使得

\[|\mathscr{K}f|\le K|f|. \]

最后, 如果 \(\mathscr{D}=\mathscr{AP}\), 那么 \(m[\mathscr{K}f]\subset m[A,f].\)

证明:

• Case 1. \(\mathscr{D}=\mathscr{P}_T\).
  由 Fredholm's alterative, 任何\(f\in \mathscr{P}_T\), 系统(2)在\(\mathscr{P}_T\)中有一个解, 当且仅当(3)成立. 由于(2)中\(y(t)\)是伴随系统(4)的任何\(T-\)周期解, \(f\)是任何\(T\)周期函数，\(f\)的任意性表明伴随系统(4)在\(\mathscr{P}_T\)中只有零解，即(1)关于\(\mathscr{P}_T\)是非临界的. 对于非齐次系统的任何两个非零解T-周期解，它们的差是齐次系统的非零解T-周期解，从而由齐次系统只有零解是T-周期解得到非齐次系统有唯一的T-周期解[存在性有Fredholm's alterative保证了].

给定\(f\in \mathscr{P}_T\), 非齐次系统在\(\mathscr{P}_T\)有唯一解这个事实蕴含着\(\mathscr{K}f\)关于\(f\)是线性的. 下面证明存在\(K\)使得

\[|\mathscr{K}f|\le K|f|. \]

由

\[\mathscr{K}f(t)=X(t,0)x_0+\int_{0}^{t}X(t,s)f(s)ds,~~\mathscr{K}f(0)=x_0 \]

和\(\mathscr{K}f(t)=\mathscr{K}f(t+T)\)知:

\[X(t,0)x_0+\int_{0}^{t}X(t,s)f(s)ds=X(t+T,0)x_0+\int_{0}^{t+T}X(t+T,s)f(s)ds\\ \Rightarrow [X(t,0)-X(t+T,0)]x_0=\int_{0}^{t}[X(t+T,s)-X(t,s)]f(s)ds+\int_{t}^{t+T}X(t+T,s)f(s)ds \]

上式左边\(=[I-X(t+T,t)]X(t,0)x_0\),
且\([X(t+T,s)-X(t,s)]=[X(t+T,t)-I]X(t,s)\),由此得到

\[X(t,0)x_0+\int_0^tX(t,s)f(s)ds=\int_t^{t+T}[I-X(t+T,t)]^{-1}X(t+T,s)f(s)ds\\ =\int_t^{t+T}[X^{-1}(t+T,t)-I]^{-1}X(t,s)f(s)ds\\ =\int_0^{T}[X^{-1}(t+T,t)-I]^{-1}X(t,t+s)f(t+s)ds \]

即

\[\mathscr{K}f(t)=\int_0^{T}[X^{-1}(t+T,t)-I]^{-1}X(t,t+s)f(t+s)ds. \]

那么

\[|\mathscr{K}f|\le K|f| \]

其中

\[K=T\sup_{0\le s,t\le T}| [X^{-1}(t+T,t)-I]^{-1}X(t,t+s)| \]

[[[

• 推导 \(K<\infty\)：
  \(|X(t,t+s)|\)关于第一，第二变元连续，在紧区间上有界.下面论证 \(|[X^{-1}(t+T,t)-I]^{-1}|\)有界. 系统(1)关于\(\mathscr{P}_T\)非临界蕴含

\[x(s)=X(s,t)x(t)\ne x(s+T)=X(s+T,t)x(t),~~\forall x(t)\ne 0,\forall s, \\ \Rightarrow [X(s+T,t)-X(s,t)]x(t)\ne 0, \forall x(t)\ne 0,\forall s, \\ \Rightarrow [X(s+T,t)-X(s,t)]^{-1}存在, \forall s,t \]

\[\Rightarrow [X(t+T,t)-X(t,t)]^{-1}= [X(t+T,t)-I]^{-1}存在\\ \Rightarrow [X(t+T,t)-I]^{-1}=-X(t+T,t)^{-1}(X(t+T,t)^{-1}-I)^{-1}存在 \\ \]

\[\Rightarrow (X(t+T,t)^{-1}-I)^{-1}存在关于t连续 \\ \Rightarrow \sup_{0\le t\le T}| (X(t+T,t)^{-1}-I)^{-1}|<\infty. \]

]]]

• Case 2. \(\mathscr{D}=\mathscr{B}(-\infty,\infty)\).
  设\(X(t)\)是系统(1)的基本解矩阵, Floquet表示蕴含\(X(t)=P(t)e^{Bt}\), \(P(t+T)=P(t)\). 令\(x=P(t)y\), 那么系统(1)化为:

\[ \begin{equation} \dot{y}=By+P^{-1}(t)f(t)=:By+g(t). \end{equation} \]

上述系统写为:

\[\begin{align} &\dot{u}=B_+u+g_+\\ &\dot{v}=B_0v+g_0\\ &\dot{w}=B_-w+g_- \end{align} \]

[[[

(5)的推导: 假设\(\dot{x}=A(t)x\)有基本解矩阵为\(X(t)=P(t)e^{bt}\), 那么 在上面 结论2 的证明中我们已经得到

\[\dot{P}(t)=A(t)P(t)-P(t) B. \]

变换\(x=P(t)y\)把系统(2)变为

\[\dot{x}=A(t)x+f(t)\\ \rightarrow A(t)P(t)y+f(t)=\dot{P}(t)y+P(t)\dot{y}\\ A(t)P(t)y+f(t)=A(t)P(t)y-P(t) By+P(t)\dot{y}\\ \dot{y}=By+P^{-1}(t)f(t)=By+g(t). \]

]]]

变换\(x=P(t)y,P(t+T)=P(t)\)蕴含\(x\in \mathscr{B}(-\infty,\infty)\Leftrightarrow y\in \mathscr{B}(-\infty,\infty)\). \(g(t)=P^{-1}(t)f(t)\) 表明 \(f\in \mathscr{B}(-\infty,\infty)\Leftrightarrow g\in\mathscr{B}(-\infty,\infty).\)

下面我们考虑系统(5).

• 一方面我们需要证明：当(7)缺席时, 任何\(g\in \mathscr{B}(-\infty,\infty)\), 系统(5)在\(\mathscr{B}(-\infty,\infty)\) 中有一解. 因为(6)和(8)是独立的, 因此只要证明它们分别存在有界解即可.

(6)的解为:

\[u(t)=e^{B_+t}u(\sigma)+\int_{\sigma}^te^{B_+(t-s)}g_+(s)ds \]

注意到\(B_+\)的特征值的实部为正, 故\(u(t)\)有界必须有\(e^{B_+t}u(\sigma)=0\)[否则\(t\to +\infty\)时这一项无界], 此外, \(\int_{\sigma}^te^{B_+(t-s)}g_+(s)ds\)要有界，我们需要条件\(t-s\le 0\), 因为\(s\)介于\(t\)和\(\sigma\)之间, 所以\(\sigma=+\infty\), 于是, (6)的有界解表达式为

\[(\mathscr{K}u) (t)=\int_{+\infty}^te^{B_+(t-s)}g_+(s)ds=\int_{+\infty}^0 e^{-B_+s}g_+(t+s)ds. \]

(8)的解为:

\[w(t)=e^{B_-t}w(\sigma)+\int_{\sigma}^te^{B_-(t-s)}g_-(s)ds \]

注意到\(B_-\)的特征值的实部为负, 故\(u(t)\)有界必须有\(e^{B_-t}u(\sigma)=0\)[否则\(t\to -\infty\)时这一项无界], 此外, \(\int_{\sigma}^te^{B_-(t-s)}g_-(s)ds\)要有界，我们需要条件\(t-s\ge 0\), 因为\(s\)介于\(t\)和\(\sigma\)之间, 所以\(\sigma=-\infty\), 于是, (6)的有界解表达式为

\[(\mathscr{K}w) (t)=\int_{-\infty}^te^{B_-(t-s)}g_-(s)ds=\int_{-\infty}^0 e^{-B_-s}g_-(t+s)ds. \]

因此, 给定任何\(g\in \mathscr{B}(-\infty,\infty)\), (5)在\(\mathscr{B}(-\infty,\infty)\)中有解

\[(\mathscr{K}g)(t)=(\int_{+\infty}^0 e^{-B_+s}g_+(t+s)ds,\int_{-\infty}^0 e^{-B_-s}g_-(t+s)ds)^T. \]

• 另一方面我们需要证明： 如果任何\(g\in \mathscr{B}(-\infty,\infty)\), 系统(5)在\(\mathscr{B}(-\infty,\infty)\) 中有一解，那么(7)缺席.这等价于证明当(7)不缺席时, 存在\(g\in \mathscr{B}(-\infty,\infty)\), 使得系统(5)的任何解无界.

如果(7)不缺席, 那么假设\(B_0=\text{diag}(B_{01},\cdots,B_{0m})\), \(B_{0j}=iw_jI+R_j\), \(R_j\)只有零特征值. 考虑其中一个系统:

\[\dot{w}=B_{0j}w+g_j=(iw_jI+R_j)w+g_j(t), \\ (\frac{d}{dt}(we^{-iw_jt}) )I=R_jwe^{-iw_jt}+g_j(t)e^{-iw_jt}. \]

令\(x=we^{-iw_jt}\), 上式等价于

\[\begin{equation} \dot{x}=Rx+G(t), ~~G(t)=g_j(t)e^{-iw_jt}. \end{equation} \]

\(R\)的秩小于\(R\)的阶数, 从而存在\(a\ne 0\)使得\(aR=0\). 对于这样的\(a\), 上式为

\[a\dot{x}=aG(t). \]

取\(G=a^*\)(\(a^*\)是\(a\)的共轭转置), 即 \(g_j:=a^* e^{iw_jt}\), 那么

\[a\dot{x}=|a|^2>0, \]

这蕴含对于这个\(G\), (9)的任何解满足

\[ax(t)\to\infty,~~t\to \infty, \]

故任何解无界.

下面证明\(\mathscr{K}\)的线性性和有界性：线性性的证明完全相同于Case 1. 下面证明有界性：

存在正数 \(K\) 和 \(\alpha\) 使得:

\[|e^{-B_+t}|\le Ke^{-\alpha t},~~t\ge 0,\\ |e^{-B_-t}|\le Ke^{\alpha t},~~t\le 0. \]

由此得到:

\[|\mathscr{K}u|=|\int_{+\infty}^0 e^{-B_+s}g_+(t+s)ds|\le \int^{+\infty}_0 Ke^{-\alpha t}ds|g_+|=(K/\alpha)|g_+|,\\ |\mathscr{K}w|\le (K/\alpha)|g_-| . \]

从而

\[|\mathscr{K}g|\le (K/\alpha)|g|. \]

• Case 3. \(\mathscr{D}=\mathscr{AP}\).

注意到\(x=P(t)y\), \(P(t)\)周期蕴含\(x\in \mathscr{AP}\Leftrightarrow y\in\mathscr{AP}\).

一方面我们要证明：当(7)缺席时, 任何\(g\in \mathscr{AP}\), 系统(5)在\(\mathscr{AP}\)中有一个解.

注意到当(7)缺席时, 任何\(g\in \mathscr{AP}\), 系统(5)在\(\mathscr{B}(-\infty,\infty)\)中有一个解

\[(\mathscr{K}g)(t)=(\int_{+\infty}^0 e^{-B_+s}g_+(t+s)ds,\int_{-\infty}^0 e^{-B_-s}g_-(t+s)ds)^T. \]

因为\(\mathscr{K}\)是有界线性的, 任何\(g\in \mathscr{AP}\), 对于实数集的任何子序列\(\alpha^{\prime}=\{\alpha_n^{\prime}\}\), 存在\(\alpha=\{\alpha_n\}\subset \alpha^\prime\)使得\(\{g(t+\alpha_n)=g_n(t)\}\)一致收敛在\((-\infty,\infty)\)[概周期函数的平移仍然是概周期的，这可以根据定义直接验证]. 假设\(\{g(t+\alpha_n)=g_n(t)\}\)一致收敛于\(u(t)\)[\(u(t)\in \mathscr{AP}\)见P-340Theorem 1,(5)]. 于是

\[|(\mathscr{K}g_n)(t)-(\mathscr{K}u)(t)|\le K|g_n(t)-u(t)|\to 0,~{\rm uniformly}, \]

这表明\((\mathscr{K}g)\in \mathscr{AP}.\)

另一方面我们要证明：如果任何 \(g\in \mathscr{AP}\), 系统(5)在\(\mathscr{AP}\)中有一个解, 那么(7)缺席. 反设(7)不缺席, 那么我们在Case 2中构造的\(g_j=a^* e^{iw_jt}\)是概周期的，使得(5)的任何解无界，当然也就不是概周期的, 与假设矛盾.
这样我们就完成了整个证明.

上面我们考虑了\(A(t)\)是周期的情形, 下面我们考虑\(A(t)\)有界的情形

定理2 假设\(A(t)\)的\(n\times n\)的矩阵, 关于\(t\)连续, 且\(|A(t)|\le M\), 那么对任何\(f\in \mathscr{B}[0,\infty)\), 系统(2)的每一个解都有界当且仅当系统(1)是一致渐近稳定的.

证明. (\(\Rightarrow\)) 系统(2)的满足\(x(0)=0\)的解为:

\[x(t)=\int_0^tX(t,s)f(s)ds. \]

考虑算子族\(T_t:\mathscr{B}[0,\infty)\to\mathscr{B}[0,\infty), t\ge 0\), 它们的定义如下

\[\begin{equation} (T_tf)(\alpha)=\left\{ \begin{aligned} &\int_0^\alpha X(t,s)f(s)ds,~~&0\le \alpha\le t,\\ &\int_0^t X(t,s)f(s)ds, &t\le \alpha <\infty. \end{aligned} \right. \end{equation} \]

给定任何\(f\in \mathscr{B}[0,\infty)\), 题设蕴含

\[|\int_0^t X(t,s)f(s)ds|\le M(f),~~\forall t\ge 0 \]

这蕴含

\[\sup_{t\ge 0}|T_tf|\le M(f). \]

一致有界原理表明\(\sup_{t\ge 0}|T_t|\le K,\) 进而,

\[|T_t f|\le K|f|,~~\Rightarrow |\int_0^t X(t,s) f(s)ds|\le K|f|,~~\forall f\in \mathscr{B}[0,\infty). \]

特别地, 对\(f\equiv 1\), 上式为 \(|\int_0^t X(t,s)ds|\le K\).

\(|\int_0^t X(t,s)ds|\le K\)蕴含 \(|X(t,s)|<c\): \(X(t,s)X(s,t)+I\Rightarrow X(t,s)\)作为\(s\)的函数是伴随方程\(\dot{y}=-yA(t)\)的基本解矩阵, 即

\[\frac{d}{ds}X(t,s)=-X(t,s)A(s)\\ \Rightarrow I-X(t,s)=-\int_{s}^{t}X(t,s)A(s)ds \\ \Rightarrow X(t,s)=I+ \int_{s}^{t}X(t,s)A(s)ds.\\ \Rightarrow |X(t,s)|\le 1+MK, 0\le s\le t<\infty. \]

从而系统(1)是一致稳定的.

下面证明吸引性：\(|X(t,\tau)|\to 0,~~t\to \infty.\)

\[(t-\tau)|X(t,\tau)|=|\int_\tau^t X(t,\tau)ds|=|\int_\tau^t X(t,s)X(s,\tau)ds|\le (1+MK)|\int_\tau^t X(t,s)ds|<\infty, \]

这蕴含\(|X(t,\tau) |\to 0,t\to \infty\). 从而系统(1)是一致渐近稳定的.

下面我们证明：如果系统(1)是一致渐近稳定的那么任何 \(f\in \mathscr{B}[0,\infty)\), 系统(2)的任何解都是有界的.

系统(1)是一致渐近稳定的蕴含

\[|X(t,s)|\le Ke^{-\alpha(t-s)}, K>0,\alpha>0,0\le s\le t<\infty. \]

由此,

\[|x(t)|=|X(t,0)(0)x(0)+\int_0^t X(t,s)f(s)ds|\\ =|X(t,0)(0)x(0)|+\int_0^t Ke^{-\alpha(t-s)}ds |f|<\infty. \]

至此, 我们完成了整个证明.

Weakly Nonlinear Equations---Noncritical case

Throughout this section, it will be assumed that \(A\) is a continuous \(n \times n\) matrix in \(\mathscr{P}_{\boldsymbol{T}}, \Omega(\rho, \sigma)=\left\{x\right.\) in \(C^n, \varepsilon\) in \(\left.C^r:|x| \leqq \rho,|\varepsilon| \leqq \sigma\right\}, \eta(\rho, \sigma), M(\sigma)\), \(\rho \geqq 0, \sigma \geqq 0\), are continuous functions which are nondecreasing in both variables, \(\eta(0,0)=0, \quad M(0)=0\), and \(\mathscr{Lih} (\eta, M)=\left\{q: \quad R \times \Omega\left(\rho_0, \varepsilon_0\right) \rightarrow\right.\) \(C^n: q\) continuous, \(|q(t, 0, \varepsilon)| \leqq M(|\varepsilon|),|q(t, x, \varepsilon)-q(t, y, \varepsilon)| \leqq \eta(\rho, \sigma)|x-y|\), for all \((t, x, \epsilon),(t, y, \epsilon)\) in \(R \times \Omega(\rho, \sigma), 0 \leqq \rho \leqq \rho_0, 0<\sigma<\varepsilon_0\) and \(q(t, x, \varepsilon)\) is continuous in \(x, \varepsilon\) uniformly for \(t\) in \(\mathbb{R}\)}.

If \(q\) is in \(\mathscr{Lih} (\eta, M)\), then automatically \(q(\cdot, x, \varepsilon)\) is in \(\mathscr{B}(-\infty, \infty)\). In fact, \(|q(t, x, \varepsilon)| \leqq \eta(\rho, \sigma)|x|+M(|\varepsilon|)\) for \((t, x, \varepsilon)\) in \(R \times \Omega(\rho, \sigma)\). A function \(q\) will be said to be in \(\mathscr{A} \mathscr{P} \cap \mathscr{Lih}(\eta, M)\) if \(q\) is in \(\mathscr{Lih}(\eta, M)\) and, for each fixed \(\varepsilon\), \(q(t, x, \varepsilon)\) is almost periodic in \(t\) uniformly with respect to \(x\) for \(x\) in compact sets. A function \(q\) will be said to be in \(\mathscr{P}_T \cap \mathscr{Lih}(\eta, M)\) if \(q\) is in \(\mathscr{Lih}(\eta, M)\) and \(q(t+T, x, \varepsilon)=q(t, x, \varepsilon)\) for all \((t, x, \varepsilon)\) in \(R \times \Omega\left(\rho_0, \varepsilon_0\right)\).

A function \(q\) will clearly be in \(\mathscr{Lih}(\eta, M)\) for some \(\eta, M\) if \(q(t, x, \varepsilon) \rightarrow 0\) and \(\partial q(t, x, \varepsilon) / \partial x \rightarrow 0\) as \(x \rightarrow 0, \varepsilon \rightarrow 0\) uniformly in \(t\).

In this section, results are given concerning the existence of bounded, almost periodic and periodic solutions of the nonlinear equation

\[\dot{x}=A(t)x+q(t,x,\varepsilon), \tag{2.1} \]

where \(q\) is in \(\mathscr{Lih}\)(\eta,M)$ and the homogeneous equation (1) is noncritical. More specifically, we prove

THEOREM 2.1 Suppose \(\mathscr{D}\) is one of the classes \(\mathscr{B}(-\infty,\infty),\mathscr{AP}\) or \(\mathscr{P}_T\). If \(q\) is in \(\mathscr{D}\cap \mathscr{Lih}(\eta,M)\) and system (1) is noncritical with respect to \(\mathscr{D}\) then there are constants \(\rho_1>0,\varepsilon_1>0\) and a function \(x^*(t,\varepsilon)\) continuous in \(t,\varepsilon\) for \(-\infty<t<\infty,0\le |\varepsilon|\le \varepsilon_1, x^*(t,0)=0, x^*(t,\varepsilon)\,{\rm in }\, \mathscr{D}, |x^*(t,\varepsilon)|\le \rho_1\), such that \(x^*\) is a solution of system (2.1) and is the only solution of (2.1)
in \(\mathscr{D}\) which has norm \(\le \rho_1\), \(0\le |\varepsilon|\le \varepsilon_1\).

Proof. For a given \(\rho_1\), \(0<\rho_1\le \rho_0\), let \(\mathscr{D}_{\rho_1}=\{x \, {\rm in}\, \mathscr{D}: |x|\le \rho_1\}\). Then \(\mathscr{D}_{\rho_1}\) is a bounded, closed subset of the Banach space \(\mathscr{D}\). For any \(x\in \mathscr{D}_{\rho_1}\), \(q(\cdot,x(\cdot),\varepsilon)\) is in \(\mathscr{D}\).
Any given \(x\) in \(\mathscr{D}_{\rho_1}\), the assumption that system (1) is noncritical with respect to \(\mathscr{D}\) implies that system

\[\dot{z}=A(t)z+q(t,x(t),\varepsilon) \]

has a unique solution \(z=\mathscr{K}q(t,x(t),\varepsilon)\) in \(\mathscr{D}\) and Theorem 1 implies that \(\mathscr{K}\) is continuous and linear. Thus \(z=\mathscr{K}q(t,x(t),\varepsilon)\) define a transformation from \(\mathscr{D}_{\rho_1}\) into \(\mathscr{D}\), \(x\in \mathscr{D}_{\rho_1}\), given by

\[w=(\mathscr{T}x)(t)=\mathscr{K}q(t,x(t),\varepsilon). \]

It is clear that

\[\frac{d}{dt}(\mathscr{T}x)(t)=A(t)(\mathscr{T}x)(t)+q(t,x(t),\varepsilon), \]

which implies that the fixed point of \(\mathscr{T}\) coincides with the solution of (2.1) in \(\mathscr{D}\).

We use the contraction principle to show that the operator \(\mathscr{T}\) has a unique fixed point in \(\mathscr{D}_{\rho_1}\) for some \(\rho_1\) sufficiently small. Since

\[|\mathscr{T}(x)(t)|=|\mathscr{K}q(t,x,\varepsilon)|\le K[|q(t,x,\varepsilon)-q(t,0,\varepsilon|+|q(t,0,\varepsilon|]\\ \le K[\eta(\rho_1,\varepsilon_1)\rho_1+M(\varepsilon_1)], \]

which implies
\(|\mathscr{T}x|\le \rho_1\)
when \(\rho_1\) and \(\varepsilon_1\) are chosen so that

\[K[\eta(\rho_1,\varepsilon_1)\rho_1+M(\varepsilon_1)]<\rho_1. \]

To prove \(\mathscr{T}\) is a self-mapping on \(\mathscr{D}_{\rho_1}\), it suffies to show that \((\mathscr{T}x)(t,\varepsilon)\) is almost periodic for each \(0\le|\varepsilon|\le \varepsilon_1\).
The fact that \(q(t,x,\varepsilon)\) is almost periodic in \(t\) uniformly for \(x\) in a compact set implies that \(q(t,x(t),\varepsilon)\) is almost periodic for every \(x\in \mathscr{AP}\) for each \(\varepsilon\). This implies that for each sequence \(\alpha^\prime=\{\alpha_n^\prime\}\) of \(\mathbb{R}\), there is a subsequence \(\alpha=\{\alpha_n\}\) such that \(q(t+\alpha_n,x(t+\alpha),\varepsilon)\) converges uniformly to \(u(t,\varepsilon)\in \mathscr{AP}\) for each \(\varepsilon\). Thus,

\[|(\mathscr{T}x)(t+\alpha_n)-\mathscr{K}u(t+\varepsilon)|=|(\mathscr{K}q)(t+\alpha_n,x(t+\alpha_n),\varepsilon)-\mathscr{K}u(t+\varepsilon)|\\ =|\mathscr{K}[q(t+\alpha_n,x(t+\alpha_n),\varepsilon)-u(t+\varepsilon)]|\\ \le K|q(t+\alpha_n,x(t+\alpha_n),\varepsilon)-u(t+\varepsilon)| \to 0,~~{\rm uniformly,} \]

which implies \((\mathscr{T}x)\in \mathscr{AP}\).

Additioally,

\[|\mathscr{x}(t)-\mathscr{y}(t)|=|\mathscr{K}q(t,x,\varepsilon)-\mathscr{K}q(t,y,\varepsilon)|\le K|q(t,x,\varepsilon)-q(t,y,\varepsilon)|\\ \le K\eta(\rho_1,\varepsilon_1)|x-y|=\theta|x-y| \]

with \(\theta<1\) when \(\rho_1,\varepsilon_1\) are sufficiently small. The contraction principle is applied to obtain system (2.1) has a unique solution \(x^*(t,\varepsilon)\) in \(\mathscr{D}_{\rho_1}\).

• The fact that \(q(t, x, \varepsilon)\) is continuous in \(x, \varepsilon\) uniformly for \(t\) in \(\mathbb{R}\) implies that \(x^*(t,\varepsilon)\) is continuous in \(t,\varepsilon\)[cf.p-20,Th3.2].

• For \(\varepsilon=0\), \(x=0\) is obviously a solution of (2.1) in \(\mathscr{D}_{\rho_1}\). Uniqueness of the solution implies \(x^*(t,0)=0\).

Finally, \(m[x^*(\cdot,\varepsilon)]\subset m[q,A]\) follows from Theorem 8 of the Appendix. This proves the theorem.