引言

近年来，具有跳跃项的半线性Duffing方程

$ x'' + a{x^ + } - b{x^ - } = f\left( {x,t} \right) $

(1)

已成为非线性振动理论研究的热点(见文献[1-10])，其中a和b是正常数，且a≠b；x⁺=max{x, 0}，x^－=max{－x, 0}，当方程中f(x, t)只与t有关时，方程(1)变为

$ x'' + a{x^ + } - b{x^ - } = f\left( t \right)。$

(2)

Dancer^[1]和Fucik^[2]研究了方程(2)的边值问题。Ortega^[3]研究了方程

$ x'' + a{x^ + } - b{x^ - } = 1 + \varepsilon h\left( t \right)。$

(3)

他证明了当ε充分小，h(t)∈C⁴(R /2π Z)时，一切解是有界的，即对于任意t∈ R，解x(t)

$ \mathop {\sup }\limits_{x \in {\bf{R}}} \left( {|x(t)| + \left| {{x^\prime }(t)} \right|} \right) < + \infty 。$

另一方面，当$\frac{1}{{\sqrt a }} + \frac{1}{{\sqrt b }} \in \bf{Q}$时，Alonso和Ortega^[4]在大初始条件下构造了一个2π无界周期解，这就意味着无界解和周期解同时存在。

Liu^[5]和Fabry^[6]研究了方程(2)，在一些合理假设下，当$\frac{1}{{\sqrt a }} + \frac{1}{{\sqrt b }} \in \bf{Q}$时，证明了方程(2)解的有界性。

Liu^[7]将方程(2)中f(t)是周期情形的结果推广到了拟周期情形。他首先建立了拟周期反转映射的不变曲线的存在性定理，并利用它证明了当f(t)是实解析拟周期函数时，方程(2)拟周期解的存在性和任意解的有界性。

文献[8]建立了光滑拟周期映射的不变曲线的存在性定理，并证明了当f(t)是光滑拟周期函数时，方程(2)的拟周期解的存在性和任意解的有界性。

Wang^[9]研究了方程(1)中函数f(x, t)=p(t)－φ(x)的情况，其中p(t)为光滑的2π周期函数，扰动项φ(x)为有界函数，利用Ortega建立的扭转定理，证明了周期解的有界性。

文献[10]研究了方程

$ {x^{\prime \prime }} + a{x^ + } - b{x^ - } = {G_x}(x,t) + p(t)。$

(4)

式中：p(t)∈C²³(R /2π Z)；G(x, t)∈C²¹(R×R /2π Z)，利用Moser小扭转定理证明了方程任意解的有界性。

受文献[8-9]启发，本文研究方程

$ {x^{\prime \prime }} + a{x^ + } - b{x^ - } + \varphi (x) = p(t)。$

(5)

式中：扰动项φ(x)为有界函数，且φ(0)=0；p(t)是光滑拟周期函数，其频率ω=(ω₁，ω₂，…，ω_n)满足Diophantine条件

$ |\langle k,\omega \rangle | = \left| {\sum\limits_{j = 1}^n {{k_j}} {\omega _j}} \right| \ge \frac{{{\sigma _0}}}{{|k{|^\mu }}},0 \ne k \in {{\bf{Z}}^n}, $

(6)

式中：|k| = |k₁| +| k₂| +…+ |k_n |；常数σ₀；μ>0。利用文献[8]中的扭转定理，证明了方程(5)的拟周期解的存在性和任意解的有界性。

先引入辅助函数^[4]

$ C\left( t \right) = \left\{ {\begin{array}{*{20}{c}} {\cos \sqrt a t,}&{0 \le \left| t \right| \le \frac{{\rm{ \mathsf{ π} }}}{{2\sqrt a }}}\\ { - \sqrt {\frac{a}{b}} \sin \sqrt b \left( {\left| t \right| - \frac{{\rm{ \mathsf{ π} }}}{{2\sqrt a }}} \right),}&{\frac{{\rm{ \mathsf{ π} }}}{{2\sqrt a }} \le \left| t \right| \le \frac{{\rm{ \mathsf{ π} }}}{2}\left( {\frac{1}{{\sqrt a }} + \frac{1}{{\sqrt b }}} \right)} \end{array}} \right., $

C(t)是初值问题

$ \left\{ {\begin{array}{*{20}{l}} {{x^{\prime \prime }} + a{x^ + } - b{x^ - } = 0}\\ {x\left( 0 \right) = 1,{x^\prime }\left( 0 \right) = 0} \end{array}} \right. $

的解。记S(t)=－C′(t)，则

(Ⅰ) C(－t)=C(t), S(－t)=－S(t)。

(Ⅱ)C(t), S(t)都是关于变量t的2πω₀周期函数，其中${\omega _0} = \left({\frac{1}{{\sqrt a }} + \frac{1}{{\sqrt b }}} \right)$。

(Ⅲ)S²(t)+a(C⁺(t))²+b(C^－(t))²≡a。

本文的主要结果如下：

定理1  设对任意的k∈ Z ⁿ\{0}有〈k, ωω₀〉$\notin $ Z，若(H₁)p(t)∈C^q+1(q>2n+1)，且

$ \int_0^{2{\rm{ \mathsf{ π} }}} p \left( {{t_0} + {\omega _0}\theta } \right)C\left( {{\omega _0}\theta } \right){\rm{d}}\theta \ne 0,{t_0} \in {\bf{R}}; $

$ \left( {{{\rm{H}}_2}} \right)\varphi \left( x \right) \in {C^q}\left( {\bf{R}} \right),\mathop {\lim }\limits_{t \to + \infty } {x^q}{\varphi ^{\left( q \right)}}\left( x \right) = 0, $

$\varphi (+ \infty) = \mathop {\lim }\limits_{x \to + \infty } \varphi (x), \varphi (- \infty) = \mathop {\lim }\limits_{x \to \infty } \varphi (x)$都是有限的；

$ \left( {{{\rm{H}}_3}} \right)b\varphi \left( { + \infty } \right) - a\varphi \left( { - \infty } \right) \ne {p_0}\left( {b - a} \right)。$

则方程(5)有无穷多个拟周期解，且所有解是有界的。

定理2  设存在k∈ Z ⁿ满足〈k, ωω₀〉∈ Z，记K:=$\left\{ {k \in {{\bf{Z}}^n}|\left\langle {k, \omega {\omega _0}} \right\rangle \in {\bf{Z}}} \right\}, {p_k}(t) = \sum\limits_{k \in K} {{p_k}} {{\rm{e}}^{{\rm{i}}\left\langle {k, \omega } \right\rangle t}}$，若

(H₄)p(t)∈C^q+1(q>2n+3)，且

$ \int_0^{2{\rm{ \mathsf{ π} }}} p \left( {{t_0} + {\omega _0}\theta } \right)C\left( {{\omega _0}\theta } \right){\rm{d}}\theta \ne 0,{t_0} \in {\bf{R}}; $

$ \left( {{{\rm{H}}_2}} \right)\varphi \left( x \right) \in {C^q}\left( {\bf{R}} \right),\mathop {\lim }\limits_{\left| x \right| \to \infty } {x^q}{\varphi ^{\left( q \right)}}\left( x \right) = 0, $

$\varphi (+ \infty) = \mathop {\lim }\limits_{x \to + \infty } \varphi (x), \varphi (- \infty) = \mathop {\lim }\limits_{x \to - \infty } \varphi (x)$存在有界；

$ \begin{array}{l} \left( {{{\rm{H}}_5}} \right)\sigma \omega _0^{\frac{3}{2}}\int_0^{2{\rm{ \mathsf{ π} }}} {\left[ {{p_K}\left( {{\tau _0} + {\omega _0}\theta } \right)C\left( {{\omega _0}\theta } \right)} \right]{\rm{d}}\theta } \ne \\ \;\;\;\;\;\;2\left( {\frac{1}{a}\varphi \left( { + \infty } \right) - \frac{1}{b}\varphi \left( { - \infty } \right)} \right),\;\;\forall {\tau _0} \in {\bf{R}}, \end{array} $

则方程(5)有无穷多个拟周期解，且所有解是有界的。

在下文中，规定c < 1和C>1是两个通用的正常数。

1 准备工作

方程(5)等价于下面的非自治Hamilton系统

$ \left\{ {\begin{array}{*{20}{l}} {{x^\prime } = - y = - \frac{{\partial H\left( {x,y,t} \right)}}{{\partial y}}}\\ {{y^\prime } = a{x^ + } - b{x^ - } + \varphi \left( x \right) - p\left( t \right) = \frac{{\partial H\left( {x,y,t} \right)}}{{\partial x}}} \end{array}} \right., $

(7)

其中

$ \begin{array}{l} H\left( {x,y,t} \right) = \frac{1}{2}{y^2} + \frac{1}{2}a{\left( {{x^ + }} \right)^2} + \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\frac{1}{2}b{\left( {{x^ - }} \right)^2} + \mathit{\Phi }(x) - xp(t),\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\mathit{\Phi }(x) = \int_0^x \varphi (s){\rm{d}}s。\end{array} $

容易证明下面的引理：

引理1 对任意x₀, y₀ ∈ R ², t₀∈ R，哈密顿系统(7)在整个t轴上存在满足z(t₀)=(x₀, y₀)的解为z(t)=(x(t; t₀, x₀, y₀), y(t; t₀, x₀, y₀))。

利用变换

$ (r,\theta ) \to (x,y):x = \sigma {r^{\frac{1}{2}}}C\left( {{\omega _0}\theta } \right),y = \sigma {r^{\frac{1}{2}}}S\left( {{\omega _0}\theta } \right), $

其中r>0, θmod(2π), $\sigma = \sqrt {\omega _0^{ - 1}{a^{ - 1}}} $，将系统(7)变换为关于(r, θ)的Hamilton系统

$ {r^\prime } = - \frac{{\partial h}}{{\partial \theta }}\left( {r,\theta ,t} \right),{\theta ^\prime } = \frac{{\partial h}}{{\partial r}}\left( {r,\theta ,t} \right), $

(8)

其中

$ h\left( {r,\theta ,t} \right) = \omega _0^{ - 1}r + {I_1}(r,\theta ) + {I_2}\left( {r,\theta ,t} \right), $

(9)

$ \begin{array}{*{20}{c}} {{I_1}(r,\theta ) = 2\mathit{\Phi }\left( {\sigma {r^{\frac{1}{2}}}C\left( {{\omega _0}\theta } \right)} \right),}\\ {{I_2}(r,\theta ,t) = - 2\sigma {r^{\frac{1}{2}}}p(t)C\left( {{\omega _0}\theta } \right)。} \end{array} $

(10)

由φ(x)∈C^q(R), p(t)∈C^q+1(R /2π Z)知，I₁, I₂关于r, θ分别为C^q+1, C²。记

$ J\left( r \right): = \frac{1}{{2{\rm{ \mathsf{ π} }}}}\int_0^{2{\rm{ \mathsf{ π} }}} {{I_1}} \left( {r,\theta } \right){\rm{d}}\theta 。$

(11)

有如下结论(见文献[4])：

$ \left( {\rm{I}} \right)\left| {{I_1}(r,\theta )} \right| \le C{r^{\frac{1}{2}}},\;|J(r)| \le C{r^{\frac{1}{2}}}, $

(12)

(Ⅱ)

$ \begin{array}{*{20}{c}} {\left| {\frac{{{\partial ^k}}}{{\partial {r^k}}}{I_1}(r,\theta )} \right| \le C{r^{ - k + \frac{1}{2}}},}\\ {\left| {\frac{{{d^k}}}{{d{r^k}}}J(r)} \right| \le C{r^{ - k + \frac{1}{2}}},1 \le k \le q + 1。} \end{array} $

(13)

(Ⅲ)若函数

$ \begin{array}{*{20}{c}} {R(h,t,\theta ) = {\omega _0}{I_1}\left( {{\omega _0}h - R,\theta } \right) - }\\ {2\sigma {\omega _0}{{\left( {{\omega _0}h - R} \right)}^{\frac{1}{2}}}p(t)C\left( {{\omega _0}\theta } \right)} \end{array} $

(14)

光滑且$|R(h, t, \theta)| \le \frac{1}{2}{\omega _0}h, h > > 1$，则

$ \left| {\frac{{{\partial ^{k + l}}}}{{\partial {h^k}\partial {t^l}}}R(h,t,\theta )} \right| \le C{h^{ - k + \frac{1}{2}}},k + l \le q + 1。$

(15)

(Ⅳ)若假设

$ \begin{array}{*{20}{c}} {R(h,t,\theta ) = {\omega _0}{I_1}\left( {{\omega _0}h - R,\theta } \right) - }\\ {2\sigma \omega _0^{\frac{3}{2}}{h^{\frac{1}{2}}}p(t)C\left( {{\omega _0}\theta } \right) - {R_1}(h,t,\theta ),} \end{array} $

(16)

其中

$ \begin{array}{*{20}{c}} {{R_1}(h,t,\theta ) = - {\omega _0}\int_0^1 {\frac{{\partial {I_1}}}{{\partial r}}} \left( {{\omega _0}h - sR,\theta } \right){\rm{d}}s + }\\ {\sigma {\omega _0}p(t)C\left( {{\omega _0}\theta } \right)\int_0^1 {{{\left( {{\omega _0}h - sR} \right)}^{ - \frac{1}{2}}}} {\rm{d}}s。} \end{array} $

(17)

则

$ \left| {\frac{{{\partial ^{k + l}}}}{{\partial {h^k}\partial {t^l}}}{R_1}(h,t,\theta )} \right| \le C{h^{ - k}},k + l \le q。$

(18)

$ \left( {\rm{V}} \right)\mathop {\lim }\limits_{r \to + \infty } \sqrt r {J^\prime }(r) = \frac{1}{{\rm{ \mathsf{ π} }}}\omega _0^{ - \frac{3}{2}}\left\{ {\frac{1}{a}\varphi ( + \infty ) - \frac{1}{b}\varphi ( - \infty )} \right\}。$

(19)

$ \left( {{\rm{VI}}} \right)\mathop {\lim }\limits_{r \to + \infty } \frac{{J(r)}}{{\sqrt r }} = \frac{2}{{\rm{ \mathsf{ π} }}}\omega _0^{ - \frac{3}{2}}\left\{ {\frac{1}{a}\varphi ( + \infty ) - \frac{1}{b}\varphi ( - \infty )} \right\}。$

(20)

(Ⅶ)令

$ \mathit{\Psi }(r) = \frac{{J(r)}}{{\sqrt r }} - \frac{2}{{\rm{ \mathsf{ π} }}}\omega _0^{ - \frac{3}{2}}\left\{ {\frac{1}{a}\varphi ( + \infty ) - \frac{1}{b}\varphi ( - \infty )} \right\}, $

(21)

则

$ \mathop {\lim }\limits_{r \to + \infty } \left| {{r^k}\frac{{{{\rm{d}}^k}}}{{{\rm{d}}{r^k}}}\mathit{\Psi }(r)} \right| = 0,\;0 \le k \le q + 1。$

(22)

下面对哈密顿系统(8)做正则变换。系统(8)的Hamilton函数h(r, θ, t)由(9)式给出。由于

$ r{\rm{d}}\theta - h{\rm{d}}t = - \left( {h{\rm{d}}t - r{\rm{d}}\theta } \right), $

这意味着若从(9)式能解出r=r(h, t, θ)作为h, t和θ的函数，于是

$ \frac{{{\rm{d}}h}}{{{\rm{d}}\theta }} = - \frac{{\partial r}}{{\partial t}}(h,t,\theta ),\;\;\frac{{{\rm{d}}t}}{{{\rm{d}}\theta }} = \frac{{\partial r}}{{\partial h}}(h,t,\theta )。$

(23)

方程(23)是一个Hamilton系统，它以r=r(h, t, θ)为其Hamilton函数，以h, t和θ分别作为作用变量、角变量和时间变量。由(9)知

$ \mathop {\lim }\limits_{r \to + \infty } \frac{h}{r} = \omega _0^{ - 1} > 0, $

$ \begin{array}{*{20}{c}} {\frac{{\partial h}}{{\partial r}} = \omega _0^{ - 1} + \frac{\partial }{{\partial r}}{I_1}\left( {r,\theta ,t} \right) + \frac{\partial }{{\partial r}}{I_2}\left( {r,\theta ,t} \right) > }\\ {\omega _0^{ - 1} - C{r^{ - \frac{1}{2}}} - C{r^{ - \frac{1}{2}}} > 0。} \end{array} $

(24)

由隐函数定理知，存在函数R=R(h, t, θ)，使得

$ r(h,t,\theta ) = {\omega _0}h - R(h,t,\theta ), $

且$|R(h, t, \theta)| \le \frac{1}{2}{\omega _0}h, h > > 1$。R(h, t, θ)关于h, t是C^q+1的。再由式(9)和(24)知，R(h, t, θ)满足式(14)。由式(16)，可将函数r写成如下形式

$ \begin{array}{*{20}{c}} {r(h,t,\theta ) = {\omega _0}h - {\omega _0}{I_1}\left( {{\omega _0}h,\theta } \right) + }\\ {2\sigma \omega _0^{\frac{3}{2}}{h^{\frac{1}{2}}}p(t)C\left( {{\omega _0}\theta } \right) - {R_1}(h,t,\theta ),} \end{array} $

(25)

其中R₁(h, t, θ)满足(18)。因此系统(8)转化为

$ \frac{{{\rm{d}}h}}{{{\rm{d}}\theta }} = - \frac{{\partial r}}{{\partial t}}(h,t,\theta ),\;\;\;\;\frac{{{\rm{d}}t}}{{{\rm{d}}\theta }} = \frac{{\partial r}}{{\partial h}}(h,t,\theta )。$

(26)

引理2^[9]  存在正则变换Φ₁：h=ρ, t=τ+T(ρ, θ)，其中T(ρ, θ+2π)=T(ρ, θ)。在此变换下，Hamilton函数(25)变换为

$ \begin{array}{*{20}{c}} {\tilde r(\rho ,\tau ,\theta ) = {\omega _0}\rho - {\omega _0}J\left( {{\omega _0}\rho } \right) + }\\ {2\sigma \omega _0^{\frac{3}{2}}{\rho ^{\frac{1}{2}}}p(\tau )C\left( {{\omega _0}\theta } \right) - {{\tilde R}_1}(\rho ,\tau ,\theta ),} \end{array} $

(27)

且$\left| {\frac{{{\partial ^{k + l}}}}{{\partial {\rho ^k}\partial {\tau ^l}}}{\rm{ }}{{\tilde R}_1}(\rho, \tau, \theta)} \right| \le C{\rho ^{ - k}}, \quad k + l \le q$。

2 定理的证明

本节利用文献[8]中的小扭转定理来证明2个定理。考虑正则变换后的Hamilton系统

$ \frac{{{\rm{d}}\rho }}{{{\rm{d}}\theta }} = - \frac{{\partial \tilde r}}{{\partial \tau }}(\rho ,\tau ,\theta ),\;\;\;\;\frac{{{\rm{d}}\tau }}{{{\rm{d}}\theta }} = \frac{{\partial \tilde r}}{{\partial \rho }}(\rho ,\tau ,\theta ), $

(28)

其中Hamilton函数(ρ, τ, θ)由式(27)给出。

先给出系统(28)的Poincare映射的表达式。引入新变量$\nu \in \left[ {\frac{1}{2},1} \right]$和小参数δ>0，满足

$ \rho = {\delta ^{ - 2}}v,\;\;\;\;v \in \left[ {\frac{1}{2},1} \right], $

(29)

显然，系统(28)变换为

$ \frac{{{\rm{d}}v}}{{{\rm{d}}\theta }} = - \frac{{\partial H}}{{\partial \tau }}(v,\tau ,\theta ,\delta ),\;\;\;\;\frac{{{\rm{d}}\tau }}{{{\rm{d}}\theta }} = \frac{{\partial H}}{{\partial v}}(v,\tau ,\theta ,\delta ), $

(30)

其中

$ \begin{array}{*{20}{c}} {H(v,\tau ,\theta ,\delta ) = {\omega _0}v - {\delta ^2}{\omega _0}J\left( {{\omega _0}{\delta ^{ - 2}}v} \right) + }\\ {2\sigma \omega _0^{\frac{3}{2}}\delta {v^{\frac{1}{2}}}p(\tau )C\left( {{\omega _0}\theta } \right) - {\delta ^2}{{\tilde R}_1}\left( {{\delta ^{ - 2}}v,\tau ,\theta } \right)。} \end{array} $

由(21)知

$ \begin{array}{l} J\left( {{\omega _0}{\delta ^{ - 2}}v} \right) = \frac{2}{{\rm{ \mathsf{ π} }}}\omega _0^{ - 1}{\delta ^{ - 1}}{v^{\frac{1}{2}}}\left\{ {\frac{1}{a}\varphi ( + \infty ) - \frac{1}{b}\varphi ( - \infty )} \right\} + \\ \;\;\;\;\;\;\omega _0^{\frac{1}{2}}{\delta ^{ - 1}}{v^{\frac{1}{2}}}\mathit{\Psi }\left( {{\omega _0}{\delta ^{ - 2}}v} \right), \end{array} $

(31)

将J(ω₀δ^－2v)代入Hamilton函数H(v, τ, θ, δ)，得

$ \begin{array}{l} H(v,\tau ,\theta ,\delta ) = {\omega _0}v - 2\delta {v^{\frac{1}{2}}}\frac{1}{{\rm{ \mathsf{ π} }}}\left[ {\frac{1}{a}\varphi ( + \infty ) - \frac{1}{b}\varphi ( - \infty )} \right] + \\ \;\;\;\;\;\;2\delta {v^{\frac{1}{2}}}\sigma {\omega ^{\frac{3}{2}}}p(\tau )C\left( {{\omega _0}\theta } \right) + G(v,\tau ,\theta ,\delta ), \end{array} $

其中

$ G(v,\tau ,\theta ,\delta ) = - \omega _0^{\frac{3}{2}}\delta {v^{\frac{1}{2}}}\Psi \left( {{\omega _0}{\delta ^{ - 2}}v} \right) - {\delta ^2}{{\tilde R}_1}\left( {{\delta ^{ - 2}}v,\tau ,\theta } \right)。$

由注释(Ⅶ)和引理2得到

$ {\delta ^{ - 1}}\left| {\frac{{{\partial ^{k + l}}}}{{\partial {v^k}\partial {\tau ^l}}}G(v,\tau ,\theta ,\delta )} \right| \to 0,k + l \le q - 1,\delta \to {0^ + }, $

(32)

新的Hamilton函数H(v, τ, θ, δ)代入系统(29)得到

$ \left\{ \begin{array}{l} \frac{{{\rm{d}}v}}{{{\rm{d}}\theta }} = - 2\delta \sigma \omega _0^{\frac{3}{2}}{v^{\frac{1}{2}}}{p^\prime }(\tau )C\left( {{\omega _0}\theta } \right) - \frac{{\partial G(v,\tau ,\theta ,\delta )}}{{\partial \tau }},\\ \frac{{{\rm{d}}\tau }}{{{\rm{d}}\theta }} = {\omega _0} - \delta {v^{ - \frac{1}{2}}}\frac{1}{{\rm{ \mathsf{ π} }}}\left[ {\frac{1}{a}\varphi ( + \infty ) - \frac{1}{b}\varphi ( - \infty )} \right] + \\ \delta {v^{ - \frac{1}{2}}}\sigma \omega _0^{\frac{3}{2}}p(\tau )C\left( {{\omega _0}\theta } \right) + \frac{{\partial G}}{{\partial v}}(v,\tau ,\theta ,\delta ), \end{array} \right. $

(33)

在初始条件(v(v₀, τ₀, 0), τ(v₀, τ₀, 0))=(v₀, τ₀)下，系统(33)存在解(v(v₀, τ₀, θ), τ(v₀, τ₀, θ))，可设它有如下表达式

$ \left\{ \begin{array}{l} v\left( {{v_0},{\tau _0},\theta } \right) = {v_0} + \delta {F_2}\left( {{v_0},{\tau _0},\theta } \right)\\ \tau \left( {{v_0},{\tau _0},\theta } \right) = {\tau _0} + {\omega _0}\theta + \delta {F_1}\left( {{v_0},{\tau _0},\theta } \right) \end{array} \right.。$

(34)

因此系统(33)的Poincare映射P₁为

$ \begin{array}{*{20}{c}} {{P_1}\left( {{v_0},{\tau _0}} \right) = \left( {{v_0} + \delta {F_2}\left( {{v_0},{\tau _0},\theta } \right),{\tau _0} + {\omega _0}\theta + } \right.}\\ {\left. {\delta {F_1}\left( {{v_0},{\tau _0},\theta } \right)} \right)。} \end{array} $

对(34)两边求导得

$ \left\{ \begin{array}{l} \frac{{{\rm{d}}{F_1}}}{{{\rm{d}}\theta }} = - \left( {{v_0} + \delta {F_2}} \right) - \frac{1}{2}\frac{1}{\pi }\left[ {\frac{1}{a}\varphi ( + \infty ) - \frac{1}{b}\varphi ( - \infty )} \right] \cdot \\ {\left( {{v_0} + \delta {F_2}} \right)^{ - \frac{1}{2}}}\sigma \omega _0^{\frac{3}{2}}p(\tau )C\left( {{\omega _0}\theta } \right) + {\delta ^{ - 1}}\frac{{\partial G}}{{\partial v}}(v,\tau ,\theta ,\delta ),\\ \frac{{{\rm{d}}{F_2}}}{{{\rm{d}}\theta }} = - 2\sigma \omega _0^{\frac{3}{2}}{\left( {{v_0} + \delta {F_2}} \right)^{\frac{1}{2}}}{p^\prime }(\tau )C\left( {{\omega _0}\theta } \right) - \\ {\delta ^{ - 1}}\frac{{\partial G(v,\tau ,\theta ,\delta )}}{{\partial \tau }}。\end{array} \right. $

(35)

由式(32)和(35)可得，当δ→0⁺, k+l≤q－2, 时，

$ \left| {\frac{{{\partial ^{k + l}}}}{{\partial v_0^k\partial \tau _0^l}}{F_1}\left( {{v_0},{\tau _0},\theta } \right)} \right| \le {C_0}, $

$ \left| {\frac{{{\partial ^{k + l}}}}{{\partial v_0^k\partial \tau _0^l}}{F_2}\left( {{v_0},{\tau _0},\theta } \right)} \right| \le {C_0}, $

其中C₀是与δ无关的常数。此时记作

$ {F_1}\left( {{v_0},{\tau _0},\theta } \right) = {O_{q - 2}}(1),{F_2}\left( {{v_0},{\tau _0},\theta } \right) = {O_{q - 2}}(1)。$

若当δ→0⁺，k+l≤q－2时

$ \left| {\frac{{{\partial ^{k + l}}}}{{\partial v_0^k\partial \tau _0^l}}{F_1}\left( {{v_0},{\tau _0},\theta } \right)} \right| \to 0, $

$ \left| {\frac{{{\partial ^{k + l}}}}{{\partial v_0^k\partial \tau _0^l}}{F_2}\left( {{v_0},{\tau _0},\theta } \right)} \right| \to 0, $

记作

$ {F_1}\left( {{v_0},{\tau _0},\theta } \right) = {o_{q - 2}}(1),{F_2}\left( {{v_0},{\tau _0},\theta } \right) = {o_{q - 2}}(1)。$

因此

$ \begin{array}{*{20}{c}} {v\left( {{v_0},{\tau _0},\theta } \right) = {v_0} + \delta {O_{q - 2}}(1),}\\ {\tau \left( {{v_0},{\tau _0},\theta } \right) = {\tau _0} + {\omega _0}\theta + \delta {O_{q - 2}}(1)。} \end{array} $

(36)

由式(35)直接计算知

$ \begin{array}{l} {F_1}\left( {{v_0},{\tau _0},2{\rm{ \mathsf{ π} }}} \right) = \\ - \int_0^{2{\rm{ \mathsf{ π} }}} {\left( {{v_0} + \delta {F_2}} \right)} - \frac{1}{2}\frac{1}{{\rm{ \mathsf{ π} }}}\left[ {\frac{1}{a}\varphi ( + \infty ) - \frac{1}{b}\varphi ( - \infty )} \right]{\rm{d}}\theta + \\ \int_0^{2{\rm{ \mathsf{ π} }}} {{{\left( {{v_0} + \delta {F_2}} \right)}^{ - \frac{1}{2}}}} \sigma \omega _0^{\frac{3}{2}}p(\tau )C\left( {{\omega _0}\theta } \right){\rm{d}}\theta + {o_{q - 2}}(1) = \\ - \int_0^{2{\rm{ \mathsf{ π} }}} {v_0^{ - \frac{1}{2}}} \frac{1}{{\rm{ \mathsf{ π} }}}\left[ {\frac{1}{a}\varphi ( + \infty ) - \frac{1}{b}\varphi ( - \infty )} \right]{\rm{d}}\theta + \\ \int_0^{2{\rm{ \mathsf{ π} }}} {v_0^{ - \frac{1}{2}}} \sigma {\omega ^{\frac{3}{2}}}p(\tau )C\left( {{\omega _0}\theta } \right){\rm{d}}\theta + {o_{q - 2}}(1) = \\ - v_0^{ - \frac{1}{2}}2\left( {\frac{1}{a}\varphi ( + \infty ) - \frac{1}{b}\varphi ( - \infty )} \right) + \\ v_0^{ - \frac{1}{2}}\sigma {\omega ^{\frac{3}{2}}}\int_0^{2{\rm{ \mathsf{ π} }}} {\left[ {p\left( {{\tau _0} + {\omega _0}\theta } \right)C\left( {{\omega _0}\theta } \right)} \right]} {\rm{d}}\theta + {o_{q - 2}}(1), \end{array} $

$ \begin{array}{l} {F_2}\left( {{v_0},{\tau _0},2{\rm{ \mathsf{ π} }}} \right) = \\ - 2\sigma \omega _0^{\frac{3}{2}}\int_0^{2{\rm{ \mathsf{ π} }}} {{{\left( {{v_0} + \delta {F_2}} \right)}^{\frac{1}{2}}}} {p^\prime }(\tau )C\left( {{\omega _0}\theta } \right)d\theta + {o_{q - 2}}(1) = \\ - 2\sigma \omega _0^{\frac{3}{2}}v_0^{\frac{1}{2}}\int_0^{2{\rm{ \mathsf{ π} }}} {{p^\prime }} (\tau )C\left( {{\omega _0}\theta } \right){\rm{d}}\theta + {o_{q - 2}}(1) = \\ - 2\sigma \omega _0^{\frac{3}{2}}v_0^{\frac{1}{2}}\int_0^{2{\rm{ \mathsf{ π} }}} {{p^\prime }} \left( {{\tau _0} + {\omega _0}\theta } \right)C\left( {{\omega _0}\theta } \right){\rm{d}}\theta + {o_{q - 2}}(1)。\end{array} $

故Poincare映射P₁的表达式为：

$ \left\{ {\begin{array}{*{20}{l}} {{\tau _1} = {\tau _0} + 2{\rm{ \mathsf{ π} }}{\omega _0} - \delta {l_1}\left( {{v_0},{\tau _0}} \right) + \delta {o_{q - 2}}(1),}\\ {{v_1} = {v_0} - \delta {l_2}\left( {{v_0},{\tau _0}} \right) + \delta {o_{q - 2}}(1),} \end{array}} \right. $

(37)

其中

$ \left\{ \begin{array}{l} {l_1}\left( {{v_0},{\tau _0}} \right) = v_0^{ - \frac{1}{2}}2\left( {\frac{1}{a}\varphi ( + \infty ) - \frac{1}{b}\varphi ( - \infty )} \right) - \\ v_0^{ - \frac{1}{2}}\sigma \omega _0^{\frac{3}{2}}\int_0^{2{\rm{ \mathsf{ π} }}} {\left[ {p\left( {{\tau _0} + {\omega _0}\theta } \right)C\left( {{\omega _0}\theta } \right)} \right]{\rm{d}}\theta } ,\\ {l_2}\left( {{v_0},{\tau _0}} \right) = 2\sigma \omega _0^{\frac{3}{2}}v_0^{\frac{1}{2}}\int_0^{2{\rm{ \mathsf{ π} }}} {{p^\prime }} \left( {{\tau _0} + {\omega _0}\theta } \right)C\left( {{\omega _0}\theta } \right)d\theta 。\end{array} \right. $

(38)

引入变换$\tau = \tau, \; u = \frac{1}{v}$，则映射P₁变换为：

$ {P_2}\left\{ {\begin{array}{*{20}{l}} {{\tau _1} = {\tau _0} + 2{\rm{ \mathsf{ π} }}{\omega _0} + \delta {l_1}\left( {{u_0},{\tau _0}} \right) + \delta {o_{q - 2}}(1),}\\ {{u_1} = {u_0} + \delta {l_2}\left( {{u_0},{\tau _0}} \right) + \delta {o_{q - 2}}(1),} \end{array}} \right. $

(39)

$ \begin{array}{*{20}{c}} {其中\;{l_1}\left( {{u_0},{\tau _0}} \right) = - 2u_0^{\frac{1}{2}}\left( {\frac{1}{a}\varphi ( + \infty ) - \frac{1}{b}\varphi ( - \infty )} \right) + }\\ {u_0^{\frac{1}{2}}\sigma \omega _0^{\frac{3}{2}}\int_0^{2{\rm{ \mathsf{ π} }}} {\left[ {p\left( {{\tau _0} + {\omega _0}\theta } \right)C\left( {{\omega _0}\theta } \right)} \right]{\rm{d}}\theta } ,}\\ {{l_2}\left( {{u_0},{\tau _0}} \right) = 2\sigma \omega _0^{\frac{3}{2}}u_0^{\frac{3}{2}}\int_0^{2{\rm{ \mathsf{ π} }}} {{p^\prime }} \left( {{\tau _0} + {\omega _0}\theta } \right)C\left( {{\omega _0}\theta } \right){\rm{d}}\theta 。} \end{array} $

(40)

假设函数p(t)具有如下Fourier展开式

$ p\left( t \right) = \sum\limits_k {{p_k}} {{\rm{e}}^{{\rm{i}}\left\langle {k,\omega } \right\rangle t}}。$

定理1的证明：若对任意的k∈ Z ⁿ\{0}有〈k, ωω₀〉$ \notin $ Z，由文献[8]中的定理3.1知，若δ充分小且

$\mathop {\lim }\limits_{T \to + \infty } \frac{1}{T}\int_0^T {\frac{{\partial {l_1}\left({{u_0}, {\tau _0}} \right)}}{{{u_0}}}} {\rm{d}}{\tau _0} \ne 0, \mathop {\lim }\limits_{T \to + \infty } \frac{1}{T}\int_0^T {{l_2}} \left({{u_0}, {\tau _0}} \right){\rm{d}}{\tau _0} = 0$时，P₁有不变曲线。由式(40)知

$ \begin{array}{l} \mathop {\lim }\limits_{T \to + \infty } \frac{1}{T}\int_0^{\rm{T}} {\frac{{\partial {l_1}\left( {{u_0},{\tau _0}} \right)}}{{{u_0}}}{\rm{d}}{\tau _0}} = \\ \mathop {\lim }\limits_{T \to + \infty } \frac{1}{T}\left[ {\int_0^{\rm{T}} {\left( { - \frac{1}{2}} \right)} u_0^{ - \frac{1}{2}}2\left( {\frac{1}{a}\varphi ( + \infty ) - \frac{1}{b}\varphi ( - \infty )} \right){\rm{d}}{\tau _0}} \right.\\ \left. {\int_0^{\rm{T}} {\frac{{\sigma \omega _0^{\frac{j}{2}}u_0^{ - \frac{1}{2}}}}{2}} \int_0^{2{\rm{ \mathsf{ π} }}} {\left[ {p\left( {{\tau _0} + {\omega _0}\theta } \right)C\left( {{\omega _0}\theta } \right)} \right]{\rm{d}}\theta {\rm{d}}{\tau _0}} } \right] = \\ u_0^ - \frac{1}{2}\left\{ {\left( { - \frac{1}{a}\varphi ( + \infty ) + \frac{1}{b}\varphi ( - \infty )} \right) + {p_0}\left( {\frac{1}{a} - \frac{1}{b}} \right)} \right\} \ne 0, \end{array} $

即bφ(+∞)－aφ(－∞)≠p₀(b－a)；

由式(8)及Fubini定理知

$ \begin{array}{l} \mathop {\lim }\limits_{T \to + \infty } \frac{1}{T}\int_0^{\rm{T}} {{l_2}} \left( {{u_0},{\tau _0}} \right){\rm{d}}{\tau _0} = \\ \mathop {\lim }\limits_{T \to + \infty } \frac{1}{T}\int_0^{\rm{T}} 2 \sigma \omega _0^{\frac{3}{2}}u_0^{\frac{3}{2}}\int_0^{2{\rm{ \mathsf{ π} }}} {{p^\prime }} \left( {{\tau _0} + {\omega _0}\theta } \right)C\left( {{\omega _0}\theta } \right){\rm{d}}\theta {\rm{d}}{\tau _0} = \\ \mathop {\lim }\limits_{T \to + \infty } \int_0^{2{\rm{ \mathsf{ π} }}} 2 \sigma \omega _0^{\frac{3}{2}}u_0^{\frac{3}{2}}\left[ {\frac{1}{T}\int_0^T {{p^\prime }} \left( {{\tau _0} + {\omega _0}\theta } \right){\rm{d}}{\tau _0}} \right]C\left( {{\omega _0}\theta } \right){\rm{d}}\theta = 0 \end{array} $

满足文献[8]中定理3.1的所有条件，因此系统(33)的Poincare映射有拟周期不变曲线，频率为(ω₁，ω₂，…，ω_n)。从而系统(7)有无穷多个拟周期解，且所有解都是有界的。定理证毕。

注1：从定理1的证明过程知，系统(7)有无穷多个频率为

$ \left( {{\omega _1},{\omega _2}, \cdots ,{\omega _n},\frac{{2{\rm{ \mathsf{ π} }}{\omega _0}}}{{2{\rm{ \mathsf{ π} }}{\omega _0} + \delta \alpha }}} \right) $

的拟周期解，其中α满足下面的条件

$ \left| {\langle k,\omega \rangle \frac{{2{\rm{ \mathsf{ π} }}{\omega _0} + \delta \alpha }}{{2{\rm{ \mathsf{ π} }}}} - j} \right| \ge \frac{{\delta \gamma }}{{|k{|^\tau }}},\;\;\;\;k \in {{\bf{Z}}^n}\backslash \left\{ 0 \right\},j \in {\bf{Z}}, $

$ \alpha \in \left[ {2\eta + {{12}^{ - 3}}\gamma ,2\sqrt 2 \eta - {{12}^{ - 3}}\gamma } \right], $

其中常数γ，δ充分小，

$ \eta = {p_0}\left( {\frac{1}{a} - \frac{1}{b}} \right) - \left( {\frac{1}{a}\varphi ( + \infty ) - \frac{1}{b}\varphi ( - \infty )} \right)。$

定理2的证明：设存在k∈ Z ⁿ满足〈k, ωω₀〉∈ Z，记K:=$\left\{ {k \in {{\bf{Z}}^n}|\left\langle {k, \omega {\omega _0}} \right\rangle \in {\bf{Z}}} \right\}, {p_K}(t) = \sum\limits_{k \in K} {{p_k}} {{\rm{e}}^{ik\left\langle {k, w} \right\rangle t}}$，根据文献[8]中定理3.4证明过程知，

$ l = {l_1}\left( {{u_0},{\tau _0}} \right),\quad m = {l_2}\left( {{u_0},{\tau _0}} \right), $

$ \left\{ {\begin{array}{*{20}{l}} {\bar l = {{\bar l}_1}\left( {{u_0},{\tau _0}} \right) = - u_0^{\frac{1}{2}}2\left( {\frac{1}{a}\varphi ( + \infty ) - \frac{1}{b}\varphi ( - \infty )} \right) + }\\ {\sigma \omega _0^{\frac{3}{2}}u_0^{\frac{1}{2}}\int_0^{2{\rm{ \mathsf{ π} }}} {\left[ {{p_K}\left( {{\tau _0} + {\omega _0}\theta } \right)C\left( {{\omega _0}\theta } \right)} \right]{\rm{d}}\theta } ,}\\ {\bar m = {{\bar l}_2}\left( {{u_0},{\tau _0}} \right) = 2\sigma \omega _0^{\frac{3}{2}}u_0^{\frac{3}{2}}\int_0^{2{\rm{ \mathsf{ π} }}} {p_K^\prime } \left( {{\tau _0} + {\omega _0}\theta } \right)C\left( {{\omega _0}\theta } \right){\rm{d}}\theta 。} \end{array}} \right. $

由于∀τ₀∈ R，

$ \begin{array}{l} u_0^{\frac{1}{2}}2\left( {\frac{1}{a}\varphi ( + \infty ) - \frac{1}{b}\varphi ( - \infty )} \right) \ne \\ \sigma \omega _0^{\frac{3}{2}}u_0^{\frac{1}{2}}\int_0^{2{\rm{ \mathsf{ π} }}} {\left[ {{p_K}\left( {{\tau _0} + {\omega _0}\theta } \right)C\left( {{\omega _0}\theta } \right)} \right]{\rm{d}}\theta } , \end{array} $

不妨设

$ \begin{array}{l} u_0^{\frac{1}{2}}2\left( {\frac{1}{a}\varphi ( + \infty ) - \frac{1}{b}\varphi ( - \infty )} \right) > \\ \sigma \omega _0^{\frac{3}{2}}u_0^{\frac{1}{2}}\int_0^{2{\rm{ \mathsf{ π} }}} {\left[ {{p_K}\left( {{\tau _0} + {\omega _0}\theta } \right)C\left( {{\omega _0}\theta } \right)} \right]{\rm{d}}\theta } , \end{array} $

则l - ₁(u₀, τ₀) < 0,

$ \begin{array}{l} {{\bar l}_1}\left( {{u_0},{\tau _0}} \right) < 0,\\ \frac{{\partial {{\bar l}_1}}}{{\partial {u_0}}}\left( {{u_0},{\tau _0}} \right) = - u_0^{ - \frac{1}{2}}\left( {\frac{1}{a}\varphi ( + \infty ) - \frac{1}{b}\varphi ( - \infty )} \right) + \\ \frac{{\sigma \omega _0^{\frac{3}{2}}u_0^{\frac{1}{2}}}}{2}\int_0^{2{\rm{ \mathsf{ π} }}} {\left[ {{p_K}\left( {{\tau _0} + {\omega _0}\theta } \right)C\left( {{\omega _0}\theta } \right)} \right]{\rm{d}}\theta } < 0。\end{array} $

令

$ \begin{array}{l} I\left( {{u_0},{\tau _0}} \right) = u_0^{\frac{1}{2}}\left[ {2\left( {\frac{1}{a}\varphi ( + \infty ) - \frac{1}{b}\varphi ( - \infty )} \right) - } \right.\\ {\left. {\sigma \omega _0^{\frac{3}{2}}\int_0^{2{\rm{ \mathsf{ π} }}} {\left[ {{p_K}\left( {{\tau _0} + {\omega _0}\theta } \right)C\left( {{\omega _0}\theta } \right)} \right]{\rm{d}}\theta } } \right]^{ - 1}}, \end{array} $

则

$ \begin{array}{l} \frac{{\partial I}}{{\partial {\tau _0}}}\left( {{u_0},{\tau _0}} \right) = \\ u_0^{\frac{1}{2}}\sigma \omega _0^{\frac{3}{2}}\int_0^{2{\rm{ \mathsf{ π} }}} {\left[ {p_K^\prime \left( {{\tau _0} + {\omega _0}\theta } \right)C\left( {{\omega _0}\theta } \right)} \right]{\rm{d}}\theta } \cdot \\ \left[ {2\left( {\frac{1}{a}\varphi ( + \infty ) - \frac{1}{b}\varphi ( - \infty )} \right) - } \right.\\ {\left. {\sigma {\omega ^{\frac{3}{2}}}\int_0^{2{\rm{ \mathsf{ π} }}} {\left[ {{p_K}\left( {{\tau _0} + {\omega _0}\theta } \right)C\left( {{\omega _0}\theta } \right)} \right]{\rm{d}}\theta } } \right]^{ - 1}}, \end{array} $

$ \begin{array}{l} \frac{{\partial I}}{{\partial {u_0}}}\left( {{u_0},{\tau _0}} \right) = \frac{1}{2}u_0^{ - \frac{1}{2}}\left[ {2\left( {\frac{1}{a}\varphi ( + \infty ) - \frac{1}{b}\varphi ( - \infty )} \right) - } \right.\\ {\left. {\sigma \omega _0^{\frac{3}{2}}\int_0^{2{\rm{ \mathsf{ π} }}} {\left[ {{p_K}\left( {{\tau _0} + {\omega _0}\theta } \right)C\left( {{\omega _0}\theta } \right)} \right]{\rm{d}}\theta } } \right]^{ - 1}}, \end{array} $

$ {{\bar l}_1}\left( {{u_0},{\tau _0}} \right)\frac{{\partial I}}{{\partial {\tau _0}}}\left( {{u_0},{\tau _0}} \right) + {{\bar l}_2}\left( {{u_0},{\tau _0}} \right)\frac{{\partial I}}{{\partial {u_0}}}\left( {{u_0},{\tau _0}} \right) = 0, $

满足了文献[8]中定理3.4的所有条件，因此系统(33)的Poincare映射有拟周期不变曲线，频率为ω₁，ω₂，…，ω_n。因此系统(7)有无穷多个拟周期解，且所有解都是有界的。定理证毕。

注2：从定理2的证明过程知，系统(7)有无穷多个拟周期解，其频率为

$ \left( {{\omega _1},{\omega _2}, \cdots ,{\omega _n},\frac{{2{\rm{ \mathsf{ π} }}{\omega _0}}}{{2{\rm{ \mathsf{ π} }}{\omega _0} + \delta \alpha }}} \right), $

其中α满足下面的条件

$ \left| {\langle k,\omega \rangle \frac{{2{\rm{ \mathsf{ π} }}{\omega _0} + \delta \alpha }}{{2\pi }} - j} \right| \ge \frac{{\delta \gamma }}{{{{\left| k \right|}^\tau }}},\;\;\;\;k \in {{\bf{Z}}^n}\backslash \left\{ 0 \right\},j \in {\bf{Z}}, $

$ \alpha \in \left[ {\mathit{\Omega }\left( 1 \right) + {{12}^{ - 3}}\gamma ,\mathit{\Omega }\left( 2 \right) - {{12}^{ - 3}}\gamma } \right], $

其中$\Omega (h) = 2\pi {\omega _0}/\int_0^{2\pi {\omega _0}} {\frac{{{\rm{d}}x}}{{\bar l(x, Y(x, h))}}}, Y(x, h)$是I(x, Y)=h的解，常数γ，δ充分小。

致谢: 本文的研究和写作过程中，朴大雄教授给予了悉心地指导，并提出了很多宝贵意见，作者在此向他表示诚挚的谢意。

AMS Subject Classification: 34C11；34C27；37E40；37J40