一类奇数阶泛函微分方程周期解的存在性

Existence of Periodic Solutions for a Class of Odd Order Functional Differential Equations

摘要: 本文运用Mawhin延拓定理，研究了一类奇数阶泛函微分方程周期解的存在性问题，得到了新的判定准则。

Abstract: Using Mawhin’s continuation theorem we study the existence of periodic solutions for a class of odd order functional differential equations, and establish a new criterion.

1. 引言

由于泛函微分方程在许多数学模型中有着深刻的应用，因此吸引了国内外大量的学者进行研究 [1] [2] 。其中，周期解问题一直得到学者们的关注，但大部分都只是利用不动点理论、重合度理论或者临界点理论等等讨论了低阶的泛函微分方程 [3] - [15] ，涉及到高阶的却不多 [16] [17] [18] 。

本文将利用Mawhin延拓定理讨论一类奇数阶泛函微分方程

$x^{\left(2n+1\right)}\left(t\right)+{\displaystyle {\sum }_{i=0}^{2n-1}{a}_i\left(t\right)x^{\left(i\right)}\left(t\right)}+g\left(x\left(t-\tau \left(t\right)\right)\right)=p\left(t\right)$ (1)

周期解的存在性，其中 $\tau \left(t\right)$ 、 $a_i\left(t\right)\left(i=0,1,2,\cdots ,n\right)$ 和 $p\left(t\right)$ 都是定义在R上具有正周期T的实连续函数。并且 $a_0\left(t\right)>0$ ， $a_{2k-1}\left(t\right)>0\left(k=1,2,\cdots ,n\right)$ ， $g\left(x\right)$ 是定义在R上的实连续函数。

在第二节中，我们先给出一些预备知识，以及针对周期解存在性的条件做假设。在第三节中，我们将利用Mawhin延拓定理 [19] ，证明方程(1)的周期解存在性。

2. 预备知识

为了证明主要结果，我们需要介绍Mawhin延拓定理 [19] 。

令X和Y是两个Banach空间，且 $L:DomL\subset X\to Y$ 是一个线性映射， $N:X\to Y$ 是一个连续映射。若映射L满足：1) $dimKerL=codimImL<+\infty$ ；2) ImL在Y上是闭的。则称映射 是指标为零的Fredholm算子。若映射L是一个指标为零的Fredholm算子，则存在连续的投影算子 $P:X\to X$ 和 $Q:Y\to Y$ ，使得 $ImP=KerL$ 和 $ImL=KerQ=Im\left(I-Q\right)$ 成立。令 $K_p$ 表示 ${L|}_{DomL{{\displaystyle \cap }}^{\text{}}KerP}:\left(I-P\right)X\to ImL$ 的逆，若 $QN\left(\bar{\Omega }\right)$ 是有界的且 $\overline{K_p\left(I-Q\right)N\left(\stackrel{¯}{\Omega }\right)}$ 是紧的，则称映射N是 $\bar{\Omega }$ 上L-紧的，其中 $\Omega$ 是X上的有界开子集。

引理2.1 ( 延拓定理) [19] 令L是一个具有零指标的Fredholm算子，N是一个在 $\bar{\Omega }$ 上L-紧的非线性算子。如果

1) 对每个 $\lambda \in \left(0,1\right)$ 及 $x\in \partial \Omega ,Lx\ne \lambda Nx$ ；

2) 对每个 $x\in \partial \Omega \cap KerL,QNx\ne 0$ 及 $\mathrm{deg}\left(QN,\Omega \cap KerL,0\right)\ne 0$ ；

则方程 $Lx=Nx$ 在 $\bar{\Omega }\cap DomL$ 上至少有一个解。

在呈现我们主要结果之前，先作出如下假设：

(H₁)： $M_{2k-1}=\underset{t\in \left[0,T\right]}{\max }{a}_{2k-1}\left(t\right)\ge a_{2k-1}\left(t\right)\ge m_{2k-1}=\underset{t\in \left[0,T\right]}{\min }{a}_{2k-1}\left(t\right)>0\left(k=1,2,\cdots ,n\right)$ ；

(H₂)： $M_{2k-2}=\underset{t\in \left[0,T\right]}{\max }\left|a_{2k-2}\left(t\right)\right|\left(k=2,\cdots ,n-1\right),M_0=\underset{t\in \left[0,T\right]}{\max }{a}_0\left(t\right)\ge a_0\left(t\right)\ge m_0=\underset{t\in \left[0,T\right]}{\min }{a}_0\left(t\right)>0$ ；

(H₃)： $M_{2n-1}<{\left(\frac{\pi }{T}\right)}^2,\frac{M_{2n-2i+1}}{M_{2n-2i-1}}<{\left(\frac{\pi }{T}\right)}^2\left(i=2,3,\cdots ,n\right)$ ；

(H₄)：存在正常数r使得 $\frac{m_0-r}{2}>\frac{T\delta M_0^2}{T\delta M_0+M_1}$ ，且 $A-DB>0$ 和 $1-A^{*}>0$ 。其中 $\delta ={\text{e}}^{\frac{M_0}{M_1}T}/{\text{e}}^{\frac{M_0}{M_1}T}-1$ ， $A=1-A^{*}$ ，

$A^{*}=M_{2n-1}{\left(\frac{T}{2}\right)}^2+M_{2n-2}{\left(\frac{T}{2}\right)}^3+\cdots +M_1{\left(\frac{T}{2}\right)}^{2n},$

$B=\left(M_{2n-1}-m_{2n-1}\right)\left(\frac{T}{2}\right)+M_{2n-2}{\left(\frac{T}{2}\right)}^2+\cdots +M_2{\left(\frac{T}{2}\right)}^{2n-2}+\left(M_1-m_1\right){\left(\frac{T}{2}\right)}^{2n-1},$

$D=\frac{T}{m_0-r}\left(\frac{M_0{M}_1}{T\delta M_0+M_1}+\frac{m_0+r}{2}\right).$

3. 主要结果的证明

定理3.1 若假设(H₁)~(H₄)成立，且满足

${\lim }_{\left|x\right|\to \infty }\sup \left|\frac{g\left(x\right)}{x}\right|\le r,$ (2)

${\lim }_{\left|x\right|\to \infty }\mathrm{sgn}\left(x\right)g\left(x\right)=+\infty ,$ (3)

则方程(1)至少存在一个T周期解。

为了证明定理3.1，我们要做如下准备工作。令

$X:=\left\{x|x\in C^{2n}\left(R,R\right),x\left(t+T\right)=x\left(t\right),\forall t\in R\right\},$

和 $x^{\left(0\right)}\left(t\right)=x\left(t\right)$ ，并在空间X上定义如下范数：

$\Vert x\Vert =\underset{0\le j\le 2n}{\max }\underset{t\in \left[0,T\right]}{\max }\left|x^{\left(2n\right)}\left(t\right)\right|.$

类似地，令 $Y:=\left\{y|y\in C\left(R,R\right),y\left(t+T\right)=y\left(t\right),\forall t\in R\right\}$ ，在空间Y上定义如下范数：

${\Vert y\Vert }_0=\underset{t\in \left[0,T\right]}{\max }\left|y\left(t\right)\right|.$

显然， $\left(X,\Vert \cdot \Vert \right)$ 和 $\left(Y,{\Vert \cdot \Vert }_0\right)$ 都是Banach空间。

分别定义算子 $L:X\to Y$ 和 $N:X\to Y$ ，如下

$Lx\left(t\right)=x^{\left(2n+1\right)}\left(t\right),t\in R;Nx\left(t\right)=p\left(t\right)-{\displaystyle {\sum }_{i=0}^{2n-1}{a}_i\left(t\right)x^{\left(i\right)}\left(t\right)}-g\left(x\left(t-\tau \left(t\right)\right)\right),t\in R.$ (4)

易知算子L是一个指标为零的Fredholm算子，再定义投影算子P、Q分别为

$Px\left(t\right)=x\left(0\right)=x\left(T\right),\forall t\in R,x\in X;Qy\left(t\right)=\frac{1}{T}{\displaystyle {\int }_0^{T}y\left(s\right)\text{d}s},\forall t\in R,y\in Y.$ (5)

容易验证，算子N是一个在 $\bar{\Omega }$ 上是L-紧的非线性算子。

考虑如下的辅助方程

$x^{\left(2n+1\right)}\left(t\right)+\lambda {\displaystyle {\sum }_{i=0}^{2n-1}{a}_i\left(t\right)x^{\left(i\right)}\left(t\right)}+\lambda g\left(x\left(t-\tau \left(t\right)\right)\right)=\lambda p\left(t\right),$ (6)

其中 $0<\lambda <1$ 。

引理3.2 [20] 令 $x\left(t\right)\in C^n\left(R,R\right)\cap C_T$ ，则

$x^{\left(i\right)}\le \frac{1}{2}{\displaystyle {\int }_0^T\left|x^{\left(i+1\right)}\left(s\right)\right|\text{d}s},i=1,2,\cdots ,n-1.$

其中 $n\ge 2$ ，且 $C_T:=\left\{x|x\in C\left(R,R\right),x\left(t+T\right)=x\left(t\right),\forall t\in R\right\}$ 。

引理3.3 [21] 设M，λ是两个正数，且满足 $0<M<{\left(\frac{\pi }{T}\right)}^2$ 和 $0<\lambda <1$ ，则对任意的函数 $\phi \left(t\right)\left(t\in \left[0,T\right]\right)$ ，方程

$\{\begin{array}{l}{x}^{″}\left(t\right)+\lambda Mx\left(t\right)=\lambda \phi \left(t\right)\hfill \\ x\left(0\right)=x\left(T\right),x^{\prime }\left(0\right)=x^{\prime }(\; T\; )\hfill \end{array}$

有唯一解，其表达形式如下：

$x\left(t\right)={\displaystyle {\int }_0^{T}G\left(t,s\right)\lambda \phi \left(s\right)\text{d}s},$

其中， $\alpha =\sqrt{\lambda M}$ ，

$G\left(t,s\right)=\{\begin{array}{l}\omega \left(t-s\right),\left(k-1\right)T\le s\le t\le kT,\hfill \\ \omega \left(T+t-s\right),\left(k-1\right)T\le t\le s\le kT\left(k\in N\right).\hfill \end{array}$

和

$\omega \left(t\right)=\frac{\cos \alpha \left(t-\frac{T}{2}\right)}{2\alpha \sin \frac{\alpha T}{2}}.$

引理3.4 若定理3.1中的条件被满足，且 $x\left(t\right)$ 是方程(6)的一个T周期解，则存在独立于λ的常数 $D_i\left(i=0,1,\cdots ,2n\right)$ ，使得

$x^{\left(i\right)}\le D_i,t\in \left[0,T\right],i=0,1,\cdots ,2n.$ (7)

证明：设 $x\left(t\right)$ 是方程(6)的一个T周期解。令

$\epsilon =\frac{m_0-r}{2}-\frac{T\delta M_0^2}{T\delta M_0+M_1}.$

由(2)知，存在一个正常数 ${\bar{M}}_1>0$ ，使得

$\left|g\left(x\left(t\right)\right)\right|\le r\left|x\left(t\right)\right|,\left|x\left(t\right)\right|>{\bar{M}}_1,t\in R.$ (8)

令

$E_1=\left\{t|t\in \left[0,T\right],\left|x\left(t\right)\right|\rangle {\bar{M}}_1\right\},E_2=\left[0,T\right]\backslash E_1.$ (9)

和

$\rho ={\max }_{E_2}g\left(x\right).$ (10)

由(6)、(8)、(9)和(10)式和引理3.2，有

$\begin{array}{c}{\Vert x^{\left(2n\right)}\Vert }_0\le \frac{1}{2}{\displaystyle {\int }_0^T\left|x^{\left(2n+1\right)}\left(t\right)\right|\text{d}}t\\ \le \frac{\lambda }{2}{\displaystyle {\int }_0^T\left[\left|{\displaystyle {\sum }_{i=0}^{2n-1}{a}_i\left(t\right)x^{\left(i\right)}\left(t\right)}\right|+\left|g\left(x\left(t-\tau \left(t\right)\right)\right)\right|+\left|p\left(t\right)\right|\right]\text{d}t}\\ \le \frac{\lambda T}{2}\left[M_{2n-1}{\Vert x^{\left(2n-1\right)}\Vert }_0+M_{2n-2}{\Vert x^{\left(2n-2\right)}\Vert }_0+\cdots +M_0{\Vert x^{\left(0\right)}\Vert }_0\right]+\frac{\lambda }{2}{\displaystyle {\int }_0^T\left|g\left(x\left(t-\tau \left(t\right)\right)\right)\right|\text{d}}t+\frac{\lambda T}{2}{\Vert p\Vert }_0\end{array}$

$\begin{array}{c}\le \frac{T}{2}\left[M_{2n-1}\frac{T}{2}+M_{2n-2}{\left(\frac{T}{2}\right)}^2+\cdots +M_1{\left(\frac{T}{2}\right)}^{2n-1}\right]{\Vert x^{\left(2n\right)}\Vert }_0+\frac{T}{2}{M}_0{\Vert x\Vert }_0\\ +\frac{1}{2}\left[{\displaystyle {\int }_{E_1}\left|g\left(x\left(t-\tau \left(t\right)\right)\right)\right|\text{d}t}+{\displaystyle {\int }_{E_2}\left|g\left(x\left(t-\tau \left(t\right)\right)\right)\right|\text{d}t}\right]+\frac{T}{2}{\Vert p\Vert }_0\\ \le A^{*}{\Vert x^{\left(2n\right)}\Vert }_0+\frac{T}{2}\left(M_0+r+\epsilon \right){\Vert x\Vert }_0+\frac{T}{2}\left(\rho +{\Vert p\Vert }_0\right)\\ =A^{*}{\Vert x^{\left(2n\right)}\Vert }_0+\frac{m_0-r}{2}D{\Vert x\Vert }_0+\frac{T}{2}C.\end{array}$ (11)

其中， $C=\rho +{\Vert p\Vert }_0,D=\frac{T}{m_0-r}\left(\frac{M_0{M}_1}{T\delta M_0+M_1}+\frac{m_0+r}{2}\right)$ 。

由假设(H₄)和(11)式，有

${\Vert x^{\left(2n\right)}\Vert }_0\le {\left(1-A^{*}\right)}^{-1}\left(\frac{m_0-r}{2}D{\Vert x\Vert }_0+\frac{T}{2}C\right).$ (12)

由(6)式和引理3.3，得

$\begin{array}{c}{x}^{\left(2n-1\right)}\left(t\right)=\lambda {\displaystyle {\int }_0^T{G}_1\left(t,t_1\right)\left[\left(M_{2n-1}-a_{2n-1}\left(t_1\right)\right)x^{\left(2n-1\right)}\left(t_1\right)+p\left(t_1\right)-g\left(x\left(t_1-\tau \left(t_1\right)\right)\right)\right]\text{d}{t}_1}\\ -\lambda {\displaystyle {\int }_0^T{G}_1\left(t,t_1\right){\displaystyle {\sum }_{i=0}^{2n-2}{a}_i\left(t_1\right)x^{\left(i\right)}\left(t_1\right)\text{d}{t}_1}},\end{array}$ (13)

其中， ${\alpha }_1=\sqrt{\lambda M_{2n-1}}$ ，

$G_1\left(t,t_1\right)=\{\begin{array}{l}{\omega }_1\left(t-t_1\right),\left(k-1\right)T\le t_1\le t\le kT,\hfill \\ {\omega }_1\left(T+t-t_1\right),\left(k-1\right)T\le t\le t_1\le kT\left(k\in N\right).\hfill \end{array}$

和

${\omega }_1\left(t\right)=\frac{\cos {\alpha }_1\left(t-\frac{T}{2}\right)}{2{\alpha }_1\sin \frac{{\alpha }_{1}T}{2}},{\displaystyle {\int }_0^T{G}_1\left(t,t_1\right)\text{d}{t}_1}=\frac{1}{\lambda M_{2n-1}}.$

由(13)式和引理3.3，得

$\begin{array}{c}{x}^{\left(2n-3\right)}\left(t\right)=\lambda {\displaystyle {\int }_0^T{G}_2\left(t,t_1\right){\displaystyle {\int }_0^T{G}_1\left(t_1,t_2\right)\left[p\left(t_2\right)-g\left(x\left(t_2-\tau \left(t_2\right)\right)\right)\right]\text{d}{t}_2}}\text{d}{t}_1\\ +\lambda {\displaystyle {\int }_0^T{G}_2\left(t,t_1\right){\displaystyle {\int }_0^T{G}_1\left(t_1,t_2\right)\left(M_{2n-1}-a_{2n-1}\left(t_2\right)\right)x^{\left(2n-1\right)}\left(t_2\right)\text{d}{t}_2\text{d}{t}_1}}\\ +\lambda {\displaystyle {\int }_0^T{G}_2\left(t,t_1\right)\left[\frac{M_{2n-3}}{M_{2n-1}}{x}^{\left(2n-3\right)}\left(t_1\right)-\lambda {\displaystyle {\int }_0^T{G}_1\left(t_1,t_2\right)a_{2n-3}\left(t_2\right)x^{\left(2n-3\right)}\left(t_2\right)\text{d}{t}_2}\right]\text{d}}{t}_1\\ -\lambda {\displaystyle {\int }_0^T{G}_2\left(t,t_1\right){\displaystyle {\int }_0^T{G}_1\left(t_1,t_2\right)\left[{\displaystyle {\sum }_{i=0}^{2n-4}{a}_i\left(t_2\right)x^{\left(i\right)}\left(t_2\right)+a_{2n-2}\left(t_2\right)x^{\left(2n-2\right)}}\left(t_2\right)\right]\text{d}{t}_2}\text{d}{t}_1},\end{array}$ (14)

其中， ${\alpha }_2=\sqrt{\frac{M_{2n-1}}{M_{2n-3}}}$ ，

$G_2\left(t,t_2\right)=\{\begin{array}{l}{\omega }_2\left(t-t_2\right),\left(k-1\right)T\le t_2\le t\le kT,\hfill \\ {\omega }_2\left(T+t-t_2\right),\left(k-1\right)T\le t\le t_2\le kT\left(k\in N\right).\hfill \end{array}$

和

${\omega }_2\left(t\right)=\frac{\cos {\alpha }_2\left(t-\frac{T}{2}\right)}{2{\alpha }_2\sin \frac{{\alpha }_{2}T}{2}},{\displaystyle {\int }_0^T{G}_2\left(t,t_2\right)\text{d}{t}_2}=\frac{M_{2n-1}}{M_{2n-3}}.$

利用数学归纳法，可以得到

$\begin{array}{c}{x}^{\prime }\left(t\right)\\ =\lambda {\displaystyle {\int }_0^T{G}_n\left(t,t_1\right)}\cdots {\displaystyle {\int }_0^T{G}_1\left(t_{n-1},t_n\right)\left[p\left(t_n\right)-g\left(x\left(t_n-\tau \left(t_n\right)\right)\right)\right]\text{d}{t}_n}\cdots \text{d}{t}_1\\ +\lambda {\displaystyle {\int }_0^T{G}_n\left(t,t_1\right)}\cdots {\displaystyle {\int }_0^T{G}_1\left(t_{n-1},t_n\right)\left(M_{2n-1}-a_{2n-1}\left(t_n\right)\right)x^{\left(2n-1\right)}\left(t_n\right)\text{d}{t}_n}\cdots \text{d}{t}_1\\ +{\displaystyle {\int }_0^T{G}_n\left(t,t_1\right)}\cdots {\displaystyle {\int }_0^T{G}_2\left(t_{n-2},t_{n-1}\right)}\left[\frac{M_{2n-3}}{M_{2n-1}}{x}^{\left(2n-3\right)}\left(t_{n-1}\right)-\lambda {\displaystyle {\int }_0^T{G}_1\left(t_{n-1},t_n\right)a_{2n-3}\left(t_n\right)x^{\left(2n-3\right)}\left(t_n\right)\text{d}{t}_n}\right]\text{d}{t}_{n-1}\cdots \text{d}{t}_1\end{array}$

$\begin{array}{c}+{\displaystyle {\int }_0^T{G}_n\left(t,t_1\right)}\cdots {\displaystyle {\int }_0^T{G}_3\left(t_{n-3},t_{n-2}\right)}\left[\frac{M_{2n-5}}{M_{2n-3}}{x}^{\left(2n-5\right)}\left(t_{n-2}\right)-\lambda {\displaystyle {\int }_0^T{G}_2\left(t_{n-2},t_{n-1}\right){\displaystyle {\int }_0^T{G}_1\left(t_{n-1},t_n\right)a_{2n-5}\left(t_n\right)x^{\left(2n-5\right)}\left(t_n\right)\text{d}{t}_n\text{d}{t}_{n-1}}}\right]\text{d}{t}_{n-2}\cdots \text{d}{t}_1\\ +\cdots +\cdots \\ +{\displaystyle {\int }_0^T{G}_n\left(t,t_1\right)}\left[\frac{M_1}{M_3}{x}^{\left(1\right)}\left(t_1\right)-\lambda {\displaystyle {\int }_0^T{G}_{n-1}\left(t_1,t_2\right)}{\displaystyle {\int }_0^T{G}_{n-2}\left(t_2,t_3\right)}\cdots {\displaystyle {\int }_0^T{G}_1\left(t_{n-1},t_n\right)a_1\left(t_n\right)x^{\left(1\right)}\left(t_n\right)\text{d}{t}_n}\cdots \text{d}{t}_2\right]\text{d}{t}_1\\ -\lambda {\displaystyle {\int }_0^T{G}_n\left(t,t_1\right)}\cdots {\displaystyle {\int }_0^T{G}_1\left(t_{n-1},t_n\right)}\left[{\displaystyle \underset{k=1}{\overset{n-1}{\sum }}{a}_{2k}\left(t_n\right)x^{\left(2k\right)}\left(t_n\right)}\right]\text{d}{t}_n\cdots \text{d}{t}_1-\lambda {\displaystyle {\int }_0^T{G}_n\left(t,t_1\right)}\cdots {\displaystyle {\int }_0^T{G}_1\left(t_{n-1},t_n\right)}\left[a_0\left(t_n\right)x^{\left(0\right)}\left(t_n\right)\right]\text{d}{t}_n\cdots \text{d}{t}_1.\end{array}$ (15)

其中， ${\alpha }_i=\sqrt{\frac{M_{2n-2i+1}}{M_{2n-2i+3}}},2\le i\le n$ ，

$G_i\left(t,t_1\right)=\{\begin{array}{l}{\omega }_i\left(t-t_i\right),\left(k-1\right)T\le t_i\le t\le kT,\hfill \\ {\omega }_i\left(T+t-t_i\right),\left(k-1\right)T\le t\le t_i\le kT\left(k\in N\right).\hfill \end{array}$

和

${\omega }_i\left(t\right)=\frac{\cos {\alpha }_i\left(t-\frac{T}{2}\right)}{2{\alpha }_i\sin \frac{{\alpha }_{i}T}{2}},{\displaystyle {\int }_0^T{G}_i\left(t,t_i\right)\text{d}{t}_i}=\frac{M_{2n-2i+3}}{M_{2n-2i+1}},2\le i\le n.$

再根据(15)式和常数变易法，

$\begin{array}{c}x\left(t\right)=\left({\text{e}}^{\frac{M_0}{M_1}T}-1\right){\displaystyle {\int }_0^t\Phi \left(t,s\right)}\{\lambda {\displaystyle {\int }_0^T{G}_n\left(s,t_1\right)}\cdots {\displaystyle {\int }_0^T{G}_1\left(t_{n-1},t_n\right)}\left[p\left(t_n\right)-g\left(x\left(t_n-\tau \left(t_n\right)\right)\right)\right]\text{d}{t}_n\cdots \text{d}{t}_1\\ +\lambda {\displaystyle {\int }_0^T{G}_n\left(s,t_1\right)}\cdots {\displaystyle {\int }_0^T{G}_1\left(t_{n-1},t_n\right)}\left(M_{2n-1}-a_{2n-1}\left(t_n\right)\right)x^{\left(2n-1\right)}\left(t_n\right)\text{d}{t}_n\cdots \text{d}{t}_1\\ +{\displaystyle {\int }_0^T{G}_n\left(s,t_1\right)}\cdots {\displaystyle {\int }_0^T{G}_2\left(t_{n-2},t_{n-1}\right)}\left[\frac{M_{2n-3}}{M_{2n-1}}{x}^{\left(2n-3\right)}\left(t_{n-1}\right)-\lambda {\displaystyle {\int }_0^T{G}_1\left(t_{n-1},t_n\right)a_{2n-3}\left(t_n\right)x^{\left(2n-3\right)}\left(t_n\right)\text{d}{t}_n}\right]\text{d}{t}_{n-1}\cdots \text{d}{t}_1\\ +\cdots +\cdots \end{array}$

$\begin{array}{c}+{\displaystyle {\int }_0^T{G}_n\left(s,t_1\right)}\left[\frac{M_1}{M_3}{x}^{\left(1\right)}\left(t_1\right)-\lambda {\displaystyle {\int }_0^T{G}_{n-1}\left(t_1,t_2\right)}{\displaystyle {\int }_0^T{G}_{n-2}\left(t_2,t_3\right)}\cdots {\displaystyle {\int }_0^T{G}_1\left(t_{n-1},t_n\right)a_1\left(t_n\right)x^{\left(1\right)}\left(t_n\right)\text{d}{t}_n}\cdots \text{d}{t}_2\right]\text{d}{t}_1\\ -\lambda {\displaystyle {\int }_0^T{G}_n\left(s,t_1\right)}\cdots {\displaystyle {\int }_0^T{G}_1\left(t_{n-1},t_n\right)}\left[{\displaystyle \underset{k=1}{\overset{n-1}{\sum }}{a}_{2k}\left(t_n\right)x^{\left(2k\right)}\left(t_n\right)}\right]\text{d}{t}_n\cdots \text{d}{t}_1\}\text{d}s\\ +\left({\text{e}}^{\frac{M_0}{M_1}T}-1\right){\displaystyle {\int }_o^t\Phi \left(t,s\right)}\left\{\frac{M_0}{M_1}x\left(s\right)-\lambda {\displaystyle {\int }_0^T{G}_n\left(s,t_1\right)}\cdots {\displaystyle {\int }_0^T{G}_1\left(t_{n-1},t_n\right)}\left[a_0\left(t_n\right)x^{\left(0\right)}\left(t_n\right)\right]\text{d}{t}_n\cdots \text{d}{t}_1\right\}\text{d}s\\ +{\displaystyle {\int }_0^T\Phi \left(t,s\right)}\{\lambda {\displaystyle {\int }_0^T{G}_n\left(s,t_1\right)}\cdots {\displaystyle {\int }_0^T{G}_1\left(t_{n-1},t_n\right)}\left[p\left(t_n\right)-g\left(x\left(t_n-\tau \left(t_n\right)\right)\right)\right]\text{d}{t}_n\cdots \text{d}{t}_1\\ +\lambda {\displaystyle {\int }_0^T{G}_n\left(s,t_1\right)}\cdots {\displaystyle {\int }_0^T{G}_1\left(t_{n-1},t_n\right)}\left(M_{2n-1}-a_{2n-1}\left(t_n\right)\right)x^{\left(2n-1\right)}\left(t_n\right)\text{d}{t}_n\cdots \text{d}{t}_1\end{array}$

$\begin{array}{l}+{\displaystyle {\int }_0^T{G}_n\left(s,t_1\right)}\cdots {\displaystyle {\int }_0^T{G}_2\left(t_{n-2},t_{n-1}\right)}\left[\frac{M_{2n-3}}{M_{2n-1}}{x}^{\left(2n-3\right)}\left(t_{n-1}\right)-\lambda {\displaystyle {\int }_0^T{G}_1\left(t_{n-1},t_n\right)a_{2n-3}\left(t_n\right)x^{\left(2n-3\right)}\left(t_n\right)\text{d}{t}_n}\right]\text{d}{t}_{n-1}\cdots \text{d}{t}_1\\ +\cdots +\cdots \\ +{\displaystyle {\int }_0^T{G}_n\left(s,t_1\right)}\left[\frac{M_3}{M_1}{x}^{\left(1\right)}\left(t_1\right)-\lambda {\displaystyle {\int }_0^T{G}_{n-1}\left(t_1,t_2\right){\displaystyle {\int }_0^T{G}_{n-2}\left(t_2,t_3\right)}}\cdots {\displaystyle {\int }_0^T{G}_1\left(t_{n-1},t_n\right)a_1\left(t_n\right)x^{\left(1\right)}\left(t_n\right)\text{d}{t}_n}\cdots \text{d}{t}_2\right]\text{d}{t}_1\\ -\lambda {\displaystyle {\int }_0^T{G}_n\left(s,t_1\right)}\cdots {\displaystyle {\int }_0^T{G}_1\left(t_{n-1},t_n\right)}\left[{\displaystyle \underset{k=1}{\overset{n-1}{\sum }}{a}_{2k}\left(t_n\right)x^{\left(2k\right)}\left(t_n\right)}\right]\text{d}{t}_n\cdots \text{d}{t}_1\}\text{d}s\\ +{\displaystyle {\int }_0^T\Phi \left(t,s\right)}\left\{\frac{M_0}{M_1}x\left(s\right)-\lambda {\displaystyle {\int }_0^T{G}_n\left(s,t_1\right)}\cdots {\displaystyle {\int }_0^T{G}_1\left(t_{n-1},t_n\right)}\left[a_0\left(t_n\right)x^{\left(0\right)}\left(t_n\right)\right]\text{d}{t}_n\cdots \text{d}{t}_1\right\}\text{d}s.\end{array}$ (16)

其中

$\Phi \left(t,s\right)=\frac{{\text{e}}^{\frac{M_0}{M_1}\left(s-t\right)}}{{\text{e}}^{\frac{M_0}{M_1}T}-1},\Phi \left(t,s\right)\le \Phi \left(t,t+T\right)=\frac{{\text{e}}^{\frac{M_0}{M_1}T}}{{\text{e}}^{\frac{M_0}{M_1}T}-1}=\delta ,$

且

${\displaystyle {\int }_0^t\Phi \left(t,s\right)\text{d}s}=\frac{M_1\left(1-{\text{e}}^{-\frac{M_0}{M_1}T}\right)}{M_0\left({\text{e}}^{\frac{M_0}{M_1}T}-1\right)}\le \frac{M_1}{M_0\left({\text{e}}^{\frac{M_0}{M_1}T}-1\right)}.$

由(8)、(9)、(10)、(16)式和引理3.2，得

$\begin{array}{c}{\Vert x\Vert }_0\le \left(\frac{1}{M_0}+\frac{T\delta }{M_1}\right)\{{\Vert p\Vert }_0+\rho +[M_0-m_0+\left(r+\epsilon \right)\left]{\Vert x\Vert }_0+\right[\left(M_{2n-1}-m_{2n-1}\right)\left(\frac{T}{2}\right)+M_{2n-2}{\left(\frac{T}{2}\right)}^2\\ +\left(M_{2n-3}-m_{2n-3}\right){\left(\frac{T}{2}\right)}^3+M_{2n-4}{\left(\frac{T}{2}\right)}^4+\cdots +\cdots \left(M_3-m_3\right){\left(\frac{T}{2}\right)}^{2n-3}+M_2{\left(\frac{T}{2}\right)}^{2n-2}\\ +\left(M_1-m_1\right){\left(\frac{T}{2}\right)}^{2n-1}]{\Vert x^{\left(2n\right)}\Vert }_0\},\end{array}$ (17)

经化简后，有

$\begin{array}{l}\frac{M_0{M}_1}{T\delta M_0+M_1}{\Vert x\Vert }_0-\left(M_0-m_0+r+\epsilon \right){\Vert x\Vert }_0\le {\Vert p\Vert }_0+\rho +[\left(M_{2n-1}-m_{2n-1}\right)\left(\frac{T}{2}\right)+M_{2n-2}{\left(\frac{T}{2}\right)}^2\\ +\left(M_{2n-3}-m_{2n-3}\right){\left(\frac{T}{2}\right)}^3+\cdots +M_2{\left(\frac{T}{2}\right)}^{2n-2}+\left(M_1-m_1\right){\left(\frac{T}{2}\right)}^{2n-1}]{\Vert x^{\left(2n\right)}\Vert }_0,\end{array}$ (18)

因此，根据假设(H₄)、(18)式和ε的取值，可得

${\Vert x\Vert }_0\le \frac{2\left(B{\Vert x^{\left(2n\right)}\Vert }_0+C\right)}{m_0-r}.$ (19)

结合(12)式和(19)式，得

$\left(1-A^{*}\right){\Vert x^{\left(2n\right)}\Vert }_0\le D\left(B{\Vert x^{\left(2n\right)}\Vert }_0+C\right)+\frac{TC}{2},$ (20)

因此有

$\left(A-DB\right){\Vert x^{\left(2n\right)}\Vert }_0\le DC+\frac{TC}{2},$ (21)

其中， $C=\rho +{\Vert p\Vert }_0,D=\frac{T}{m_0-r}\left(\frac{M_0{M}_1}{T\delta M_0+M_1}+\frac{m_0+r}{2}\right).$

再由假设(H₄)和(21)式，得到

${\Vert x^{\left(2n\right)}\Vert }_0\le \frac{DC+TC/2}{A-DB}=D_{2n}.$ (22)

联立(19)和(22)式，得

${\Vert x\Vert }_0\le \frac{2\left(B{D}_{2n}+C\right)}{m_0-r}=D_0.$ (23)

最后，由(22)、(23)式和引理3.2，得到

${\Vert x^{\left(i\right)}\Vert }_0\le D_i,\left(0\le i\le 2n\right).$

引理3.4得证。

定理3.1的证明：设 $x\left(t\right)$ 是方程(6)的一个T周期解。由引理3.4知，存在独立于λ的常数 $D_i\left(i=0,1,\cdots ,2n\right)$ ，使得(7)式成立。由(3)式知，存在正常数 $M_2>0$ ，使得

$\mathrm{sgn}\left(x\right)g\left(x\right)>0,\left|x\left(t\right)\right|>M_2,t\in R.$ (24)

取一正常数 $\bar{D}>{\max }_{0\le i\le 2n}\left\{D_i\right\}+M_2$ ，令

$\Omega :=\left\{x\in X|\Vert x\Vert <\bar{D}\right\}.$

此时 是指标为零的Fredholm算子，N是在 $\bar{\Omega }$ 上L-紧的非线性算子。对任意有界的周期解 $x\left(t\right)$ ，当 $x\in \partial \Omega \cap DomL$ ， $\lambda \in \left(0,1\right)$ 时，有 $Lx\ne \lambda Nx$ 。

当 $x\in \partial \Omega \cap KerL$ 时，有 $x=\bar{D}$ 或 $x=-\bar{D}$ ，再结合(3)和(5)式可得

$\begin{array}{l}{x}^{\left(i\right)}\left(t\right)=0,i=1,2,\cdots ,n-1,\\ -\frac{1}{T}{\displaystyle {\int }_0^T\left[{\displaystyle {\sum }_{i=0}^{2n-1}{a}_0\left(t\right)\bar{D}}+g\left(\bar{D}\right)-p\left(t\right)\right]\text{d}t}<0,\\ \frac{1}{T}{\displaystyle {\int }_0^T\left[{\displaystyle {\sum }_{i=0}^{2n-1}{a}_0\left(t\right)\bar{D}}-g\left(-\bar{D}\right)+p\left(t\right)\right]\text{d}}t>0.\end{array}$ (25)

然后由上式可知，当 $x\in \partial \Omega \cap KerL$ 时，有

$QNx=\frac{1}{T}{\displaystyle {\int }_0^T\left[-{\displaystyle {\sum }_{i=0}^{2n-1}{a}_i\left(t\right)x^{\left(i\right)}\left(t\right)}-g\left(x\left(t-\tau \left(t\right)\right)\right)+p\left(t\right)\right]\text{d}t}\ne 0.$ (26)

故对任意的 $x\in \partial \Omega \cap KerL$ 和 $\eta \in \left(0,1\right)$ ，有

$xH\left(x,\eta \right)=-\eta x^2+\frac{x}{T}\left(1-\eta \right){\displaystyle {\int }_0^T\left[-{\displaystyle {\sum }_{i=0}^{2n-1}{a}_i\left(t\right)x^{\left(i\right)}\left(t\right)}-g\left(x\left(t-\tau \left(t\right)\right)\right)+p\left(t\right)\right]\text{d}t}\ne 0.$ (27)

因此， $H\left(x,\eta \right)$ 是一个同伦映射。进而有

$\begin{array}{c}\mathrm{deg}\left(QN,\Omega \cap KerL,0\right)=\mathrm{deg}\left\{\frac{1}{T}{\displaystyle {\int }_0^T\left[-{\displaystyle \underset{i=0}{\overset{2n-1}{\sum }}{a}_i\left(t\right)x^{\left(i\right)}\left(t\right)}-g\left(x\left(t-\tau \left(t\right)\right)\right)+p\left(t\right)\right]\text{d}t},\Omega \cap KerL,0\right\}\\ =\mathrm{deg}\left(-x,\Omega \cap KerL,0\right)\ne 0.\end{array}$ (28)

根据引理2.1可知，方程 $Lx=Nx$ 在 $\bar{\Omega }\cap DomL$ 上至少存在一个解。因此，方程(1)至少存在一个 周期解。定理3.1得证。

基金项目

广东省自然科学基金资助项目(2018A030313871)。

NOTES

^*通讯作者。