比例时滞复值BAM神经网络的稳定性研究

Research on Stability of Proportionally Delayed Complex-Valued BAM Neural Networks

摘要: BAM (Bidirectional Associative Memory)神经网络以其双向异联想性、较强的学习和自适应能力以及噪声容忍性好等特点，在模式分类和识别等方面具有广泛的应用前景。与实值神经网络相比，复值神经网络是一种基于复数运算的神经网络模型，可以有效地刻画如图像、声音等具有多个维度的信号，减少对信号的近似，从而提高模型的精度。因此，本文主要研究了一类比例时滞复值BAM神经网络的全局指数稳定性，利用Banach不动点定理，给出了这类神经网络全局指数稳定的充分条件。最后，举出具体的数值算例验证了结果的有效性。

Abstract: BAM (Bidirectional Associative Memory) neural networks have significant potential for applications in pattern classification and recognition due to their bidirectional associations, robust learning and adaptive capabilities, and excellent noise tolerance. Compared with real-valued neural networks, complex-valued neural networks, which are based on complex operations, can more effectively represent multi-dimensional signals such as images and sounds. They reduce signal approximation errors and enhance model accuracy. Consequently, this paper primarily focuses on the global exponential stability of a class of proportional delay complex-valued BAM neural networks. By applying the Banach fixed point theorem, the sufficient conditions for the global exponential stability of these neural networks are given. Finally, a numerical example is provided to demonstrate the effectiveness of the results.

1. 引言

BAM神经网络是由Kosko于1988年提出来的[1]，具有双向异联想的特性，而且具有较强的学习和自适应能力，被广泛用于模式识别、信号处理、自动控制工程、复杂优化等领域[2]-[5]，倍受学者们的关注。在处理光电子学、成像、遥感、量子神经器件等问题领域中，需要处理的问题是复值的，因此复值神经网络的应用显得十分重要[6]-[8]。研究表明，复值神经网络非常适合解释由二维傅里叶变换后的图像，而且这种网络能够执行存储和召回灰度图像的任务[9]。于是，人们对复值神经网络的动力学行为的研究日益增多[10]-[12]。赵莉莉利用实数集的指数二分性研究了一类具有时变时滞的复值BAM神经网络，得到了其概周期解存在、唯一，以及一致稳定的充分条件[11]。而闫欢等通过将复值神经网络分解为实部和虚部，基于Lyapunov函数方法和矩阵不等式技巧，得到了时间标度上具有时滞和脉冲影响的复值神经网络平衡点全局指数稳定性的充分条件[12]。

在许多实际系统中，如生物、人工神经网络、自动控制系统等，往往不可避免地会遇到时间延迟的问题，在实际当中，如LAD (Little’s Average Delay)算法中的时滞通常是比例时滞[13] [14]，从而科学家们又开始研究比例时滞神经网络[15]-[17]。王雅菁于2017年利用Lyapunov稳定性方法，考虑当输入值为常时函数的时候，研究了具有比例时滞的复值神经网络的全局指数稳定性和周期性[16]。而张磊等在激活函数不分解为实部函数和虚部函数的情形下，研究了带有比例时滞的复值神经网络全局指数稳定性问题。借助向量Lyapunov函数思想和同胚映射原理，并使用M-矩阵理论和不等式技巧，其建立了网络平衡点存在性、唯一性和全局指数稳定性的判定条件[17]，但是目前对比例时滞复值BAM神经网络稳定性的研究结果比较少。

求解神经网络稳定性的传统方法有：Lyapunov函数法、M-矩阵理论、指数二分法、线性矩阵不等式、Halanay不等式、同胚映射等方法，但这些方法通常都是建立在得到神经网络平衡点存在唯一性的基础上。本世纪初，Burton等首先利用了压缩映射原理研究了微分方程解的存在唯一性和稳定性[18]-[21]。最近，Rao等开始采用不动点原理来研究比较复杂的神经网络，同时得到了神经网络解的存在唯一性和神经网络的稳定性[22]-[25]。于是，也有很多学者利用不动点原理研究了脉冲神经网络。文献[26]则借助不动点理论和一些分析技巧研究了一类具比例时滞脉冲递归神经网络的稳定性问题。Chen等则利用不动点理论，给出了一类脉冲随机时滞回归神经网络p阶矩指数稳定的新的充分条件。所得到的结果既不需要激活函数的有界性、单调性和可微性，也不需要时变时滞的可微性[27]。

目前的研究主要是比例时滞实值BAM神经网络，比例时滞复值BAM神经网络稳定性研究比较少见，而比例时滞复值BAM神经网络在光电子学、成像和遥感中的应用十分广泛，因此本文主要研究比例时滞复值BAM神经网络。通过将实部和虚部分离的方法把复值神经网络转换为实值神经网络，接着构造合适的完备度量空间以及此空间上的一个映射，利用Banach不动点定理证明所构造的映射是压缩映射，给出了这类神经网络全局指数稳定的充分条件。对这种比例时滞复值BAM神经网络稳定性的研究，有利于图像处理、信号识别等问题的解决。并且，利用Banach不动点定理讨论比例时滞复值BAM神经网络的稳定性是对已有神经网络研究的补充。

2. 预备知识

2.1. 符号说明

$ℂ$ 和${ℂ}^{2n+2m}$ 别表示复数集和$2n+2m$ 维复数向量集。$C\left(\left[\rho t_0-t_0,0\right],{ℂ}^{2n+2m}\right)$ 表示从$\left[\rho t_0-t_0,0\right]$ 到$2n+2m$ 维复数空间${ℂ}^{2n+2m}$ 的连续函数的集合。$z=a+bi$ 表示一个复数，其中$i$ 是虚数单位，即$i=\sqrt{-1}$ 。$z^R$ 和$z^I$ 分别表示$z$ 的实部和虚部。$A$ 是一个矩阵，$A^{\text{T}}$ 代表$A$ 的转置。$\upsilon \left(t\right)\in ℂ$ 是一个复函数，$\Vert \upsilon \left(t\right)\Vert$ 表示$\upsilon \left(t\right)$ 的范数，${\upsilon }^R\left(t\right)$ 和${\upsilon }^I\left(t\right)$ 分别表示$\upsilon \left(t\right)$ 的实部和虚部，$\Vert \upsilon \left(t\right)\Vert =\max \left\{\sup \Vert {\upsilon }^R\left(t\right)\Vert ,\sup \Vert {\upsilon }^I\left(t\right)\Vert \right\}$ 。

2.2. 相关定义及引理

引理1 [28] 若$\left(\chi ,\rho \right)$ 是一个完备的度量空间，$\Gamma$ 是$\left(\chi ,\rho \right)\to \left(\chi ,\rho \right)$ 的一个压缩映射，则$\Gamma$ 在$\chi$ 上存在唯一的不动点。

引理2 [29] 若$f\left(z\right):ℂ\to ℂ$ ，则$f\left(z\right)$ 满足Lipschitz条件的充要条件是$f\left(z\right)$ 的实部和虚部都满足Lipschitz条件。

定义1 [25] 系统(3.5)和(3.6)是全局指数稳定的，如果对任意的${\left({\xi }^R\left(s\right),{\xi }^I\left(s\right),{\eta }^R\left(s\right),{\eta }^I\left(s\right)\right)}^{\text{T}}$ ，$s\in C\left(\left[\rho t_0-t_0,0\right],{ℝ}^{2n+2m}\right)$ ，都存在正常数$\alpha$ 和$\beta$ ，使得对任意的$t>0$ ，都有${\left({\upsilon }^R\left(t;s,\xi ,\eta \right),{\upsilon }^I\left(t;s,\xi ,\eta \right),{\omega }^R\left(t;s,\xi ,\eta \right),{\omega }^I\left(t;s,\xi ,\eta \right)\right)}^{\text{T}}\le \beta {\text{e}}^{-\alpha t}$ 。其中$\begin{array}{l}({\upsilon }^R\left(t;s,\xi ,\eta \right),{\upsilon }^I\left(t;s,\xi ,\eta \right),{\omega }^R\left(t;s,\xi ,\eta \right),{{\omega }^I\left(t;s,\xi ,\eta \right))}^{\text{T}}\\ =\underset{p\in N,q\in N}{\max }\left\{\sup \Vert {\upsilon }_p^R\left(t\right)\Vert ,\sup \Vert {\upsilon }_p^I\left(t\right)\Vert ,\sup \Vert {\omega }_q^R\left(t\right)\Vert ,\sup \Vert {\omega }_q^I\left(t\right)\Vert \right\}\end{array}$ ，${\xi }^R=\left({\xi }_1^R,{\xi }_2^R,\cdots ,{\xi }_p^R\right)$ ，${\xi }^I=\left({\xi }_1^I,{\xi }_2^I,\cdots ,{\xi }_p^I\right)$ ，${\eta }^R=\left({\eta }_1^R,{\eta }_2^R,\cdots ,{\eta }_q^R\right)$ ，${\eta }^I=\left({\eta }_1^I,{\eta }_2^I,\cdots ,{\eta }_q^I\right)$ ，${\upsilon }^R=\left({\upsilon }_1^R,{\upsilon }_2^R,\cdots ,{\upsilon }_p^R\right)$ ，${\upsilon }^I=\left({\upsilon }_1^I,{\upsilon }_2^I,\cdots ,{\upsilon }_p^I\right)$ ，${\omega }^R=\left({\omega }_1^R,{\omega }_2^R,\cdots ,{\omega }_q^R\right)$ ，${\omega }^I=\left({\omega }_1^I,{\omega }_2^I,\cdots ,{\omega }_p^I\right)$ 。

引理2 [30] 设函数$f\left(z\right)=m\left(x,y\right)+in\left(x,y\right)$ 在点集$E$ 上有定义，$z_0=m_0+i{n}_0$ 为$E$ 的聚点，则$\underset{\begin{array}{c}z\to z_0\\ z\in E\end{array}}{\lim }=a+ib$ 的充要条件是$\underset{\begin{array}{c}\left(x,y\right)\to \left(m_0,n_0\right)\\ \left(x,y\right)\in E\end{array}}{\lim }m\left(x,y\right)=a$ ，$\underset{\begin{array}{c}\left(x,y\right)\to \left(m_0,n_0\right)\\ \left(x,y\right)\in E\end{array}}{\lim }n\left(x,y\right)=b$ 。

3. 模型建立

文献[15]中，研究了以下具有比例时滞的BAM神经网络：

$\{\begin{array}{l}{\dot{x}}_i\left(t\right)=-a_i{x}_i\left(t\right)+{\displaystyle \sum _{j=1}^m{c}_{ij}}{f}_j\left(y_j\left(t\right)\right)+{\displaystyle \sum _{j=1}^m{c}_{ij}^{\tau }}{f}_j\left(y_j\left(q_{1}t\right)\right),i\in N\\ {\dot{y}}_j\left(t\right)=-b_j{y}_j\left(t\right)+{\displaystyle \sum _{i=1}^n{d}_{ji}}{g}_i\left(x_i\left(t\right)\right)+{\displaystyle \sum _{i=1}^n{d}_{ji}^{\tau }}{g}_i\left(x_i\left(q_{2}t\right)\right),j\in M\end{array}$

并利用Brouwer不动点定理得到了系统平衡点全局指数稳定的充分条件。而在通信系统、量子计算、信号处理等领域，我们考虑的信号通常是复数值的。复数的二维性质可以使神经网络编码更多的信息，从而在相同网络规模下提升模型的表达能力。同时，比例时滞在复数域中可能引发幅值与相位的耦合动态行为，复值模型能够更精准地刻画这类现象。所以，我们考虑如下具有比例时滞的复值BAM神经网络模型：

$\{\begin{array}{l}{\dot{\upsilon }}_p\left(t\right)=-a_p{\upsilon }_p\left(t\right)+{\displaystyle \sum _{q=1}^m{c}_{1pq}}{f}_q\left({\omega }_q\left(t\right)\right)+{\displaystyle \sum _{q=1}^m{c}_{2pq}}{f}_q\left({\omega }_q\left({\theta }_{q}t\right)\right),p\in N\\ {\dot{\omega }}_q\left(t\right)=-b_q{\omega }_q\left(t\right)+{\displaystyle \sum _{p=1}^n{d}_{1qp}}{g}_p\left({\upsilon }_p\left(t\right)\right)+{\displaystyle \sum _{p=1}^n{d}_{2qp}}{g}_p\left({\upsilon }_p\left({\sigma }_{p}t\right)\right),q\in M\end{array}$ (3.1)

其中，$t\ge t_0$ ，$N=1,2,\cdots ,n$ ，$M=1,2,\cdots ,m$ 。${\upsilon }_p\left(t\right)$ 和${\omega }_q\left(t\right)\in ℂ$ 分别表示在$t$ 时刻第$p$ 个和第$q$ 个神经元的状态变量。$a_p>0$ 和$b_q>0$ 表示自反馈连接权重。$c_{1pq}$ 、$c_{2pq}$ 、$d_{1qp}$ 、$d_{2qp}$ 表示复的突触连接权重和复的时滞突触连接权重，表示信号的延迟和无延迟传播同时出现。${\theta }_q$ 、${\sigma }_p$ 称为比例时滞因子，满足$0\le {\theta }_q,{\sigma }_p\le 1$ ，${\theta }_q\left(t\right)=t-\left(1-{\theta }_q\right)t$ ，${\sigma }_p\left(t\right)=t-\left(1-{\sigma }_p\right)t$ 。其中$\left(1-{\theta }_q\right)t$ ，$\left(1-{\sigma }_p\right)t$ 是轴突信号传输比例时滞函数，它们与时间成正比，故称比例时滞，且当$t\to 0$ 时$\left(1-{\sigma }_p\right)t\to +\infty$ ，$\left(1-{\theta }_q\right)t\to +\infty$ ；$g_p(\cdot ):ℂ\to ℂ$ 和$f_q(\cdot ):ℂ\to ℂ$ 是非线性输出函数。

系统(3.1)的初始条件如下：

$\{\begin{array}{l}{\upsilon }_p\left(s\right)={\xi }_p\left(s\right),p\in N\\ {\omega }_q\left(s\right)={\eta }_q\left(s\right),q\in M\end{array}\forall s\in \left[\rho t_0,t_0\right].$ (3.2)

其中，$\rho =\underset{\begin{array}{c}p\in N\\ q\in M\end{array}}{\min }\left\{{\sigma }_p,{\theta }_q\right\}$ ，${\xi }_p(\cdot )$ 和${\eta }_q(\cdot )$ 都是$\left[\rho t_0-t_0,0\right]$ 上的有界函数。

做平移变换，系统(3.1)变成如下系统：

$\{\begin{array}{l}{\dot{\upsilon }}_p\left(t\right)=-a_p{\upsilon }_p\left(t\right)+{\displaystyle \sum _{q=1}^m{c}_{1pq}}{f}_q\left({\omega }_q\left(t\right)\right)+{\displaystyle \sum _{q=1}^m{c}_{2pq}}{f}_q\left({\omega }_q\left({\theta }_{q}t\right)\right),p\in N\\ {\dot{\omega }}_q\left(t\right)=-b_q{\omega }_q\left(t\right)+{\displaystyle \sum _{p=1}^n{d}_{1qp}}{g}_p\left({\upsilon }_p\left(t\right)\right)+{\displaystyle \sum _{p=1}^n{d}_{2qp}}{g}_p\left({\upsilon }_p\left({\sigma }_{p}t\right)\right),q\in M\end{array}$ (3.3)

其中，$t\in \left[0,+\infty \right)$ ，并且带有以下初值条件：

$\{\begin{array}{ll}{\upsilon }_p\left(s\right)={\xi }_p\left(s\right),\hfill & p\in N\hfill \\ {\omega }_q\left(s\right)={\eta }_q\left(s\right),\hfill & q\in M\hfill \end{array}\forall s\in \left[\rho t_0-t_0,0\right].$ (3.4)

易知系统(3.1)和(3.2)的稳定性和系统(3.3)和(3.4)的稳定性是等价的。接下来，我们主要讨论系统(3.3)和(3.4)的稳定性。

对于复值神经网络(3.3)，我们作如下假设：

假设1 ${\upsilon }_p\left(t\right)$ 、${\omega }_q\left(t\right)$ 、$c_{1qp}\left(t\right)$ 、$c_{2qp}\left(t\right)$ 、$d_{1qp}\left(t\right)$ 和$d_{2qp}\left(t\right)$ 可分别表示为：

${\upsilon }_p\left(t\right)={\upsilon }_p^R\left(t\right)+i{\upsilon }_p^I\left(t\right),{\omega }_q\left(t\right)={\omega }_q^R\left(t\right)+i{\omega }_q^I\left(t\right),$

$c_{1pq}\left(t\right)=c_{1pq}^R\left(t\right)+i{c}_{1pq}^I\left(t\right),c_{2pq}\left(t\right)=c_{2pq}^R\left(t\right)+i{c}_{2pq}^I\left(t\right),$

$d_{1qp}\left(t\right)=d_{1qp}^R\left(t\right)+i{d}_{1qp}^I\left(t\right),d_{2qp}\left(t\right)=d_{2qp}^R\left(t\right)+i{d}_{2qp}^I\left(t\right),$

其中，${\upsilon }_p^R\left(t\right)$ 、${\upsilon }_p^I\left(t\right)$ 、${\omega }_q^R\left(t\right)$ 、${\omega }_q^I\left(t\right)$ 、$c_{1pq}^R\left(t\right)$ 、$c_{1pq}^I\left(t\right)$ 、$c_{2pq}^R\left(t\right)$ 、$c_{2pq}^I\left(t\right)$ 、$d_{1qp}^R\left(t\right)$ 、$d_{1qp}^I\left(t\right)$ 、$d_{2qp}^R\left(t\right)$ 、$d_{2qp}^I\left(t\right)$ 都是实值函数。

假设2 $f_q\left({\omega }_q\left(t\right)\right)$ 和$g_p\left({\upsilon }_p\left(t\right)\right)$ 可分别表示为：$f_q\left({\omega }_q\left(t\right)\right)=f_q^R\left({\omega }_q\left(t\right)\right)+i{f}_q^I\left({\omega }_q\left(t\right)\right)$ 和$g_p\left({\upsilon }_p\left(t\right)\right)=g_p^R\left({\upsilon }_p\left(t\right)\right)+i{g}_p^I\left({\upsilon }_i\left(t\right)\right)$ 。

由假设1和假设2，复值系统(3.3)可转化为如下实值系统：

$\{\begin{array}{l}{\dot{\upsilon }}_p^R\left(t\right)=-a_p{\upsilon }_p^R\left(t\right)+{\displaystyle \sum _{q=1}^m{c}_{1pq}^R}{f}_q^R\left({\omega }_q\left(t\right)\right)-{\displaystyle \sum _{q=1}^m{c}_{1pq}^I}{f}_q^I\left({\omega }_q\left(t\right)\right)+{\displaystyle \sum _{q=1}^m{c}_{2pq}^R}{f}_q^R\left({\omega }_q\left({\theta }_{q}t\right)\right)-{\displaystyle \sum _{q=1}^m{c}_{2pq}^I}{f}_q^I\left({\omega }_q\left({\theta }_{q}t\right)\right),\\ {\dot{\upsilon }}_p^I\left(t\right)=-a_p{\upsilon }_p^I\left(t\right)+{\displaystyle \sum _{q=1}^m{c}_{1pq}^R}{f}_q^I\left({\omega }_q\left(t\right)\right)+{\displaystyle \sum _{q=1}^m{c}_{1pq}^I}{f}_q^R\left({\omega }_q\left(t\right)\right)+{\displaystyle \sum _{q=1}^m{c}_{2pq}^R}{f}_q^I({\omega }_q({\theta }_{q}t))+{\displaystyle \sum _{q=1}^m{c}_{2pq}^I}{f}_q^R\left({\omega }_q\left({\theta }_{q}t\right)\right),\\ {\dot{\omega }}_q^R\left(t\right)=-b_q{\omega }_q^R\left(t\right)+{\displaystyle \sum _{p=1}^n{d}_{1qp}^R}{g}_p^R\left({\upsilon }_p\left(t\right)\right)-{\displaystyle \sum _{p=1}^n{d}_{1qp}^I}{g}_p^I\left({\upsilon }_p\left(t\right)\right)+{\displaystyle \sum _{p=1}^n{d}_{2qp}^R}{g}_p^R\left({\upsilon }_p\left({\sigma }_{p}t\right)\right)-{\displaystyle \sum _{p=1}^n{d}_{2qp}^I}{g}_p^I\left({\upsilon }_p\left({\sigma }_{p}t\right)\right),\\ {\dot{\omega }}_q^I\left(t\right)=-b_q{\omega }_q^I\left(t\right)+{\displaystyle \sum _{p=1}^n{d}_{1qp}^R}{g}_p^I\left({\upsilon }_p\left(t\right)\right)+{\displaystyle \sum _{p=1}^n{d}_{1qp}^I}{g}_p^R\left({\upsilon }_p\left(t\right)\right)+{\displaystyle \sum _{p=1}^n{d}_{2qp}^R}{g}_p^I\left({\upsilon }_p\left({\sigma }_{p}t\right)\right)+{\displaystyle \sum _{p=1}^n{d}_{2qp}^I}{g}_p^R\left({\upsilon }_p\left({\sigma }_{p}t\right)\right),\end{array}$ (3.5)

其中，$p\in N,q\in M,t\in \left[0,+\infty \right)$ 。

因此，对系统(3.3)和(3.4)稳定性的研究可以转化为对系统(3.5)带有以下初始条件的稳定性研究：

$\{\begin{array}{l}{\upsilon }_p^R\left(s\right)={\xi }_p^R\left(s\right)\\ {\upsilon }_p^I\left(s\right)={\xi }_p^I\left(s\right)\\ {\omega }_q^R\left(s\right)={\eta }_q^R\left(s\right)\\ {\omega }_q^I\left(s\right)={\eta }_q^I\left(s\right)\end{array},\forall s\in \left[\rho t_0-t_0,0\right]$ (3.6)

对实值系统(3.5)和(3.6)作如下假设：

假设3 $f_q\left(0\right)=0$ ，$g_p\left(0\right)=0$ ；且存在正常数$F_q$ 和$G_p$ ，使得对任意的$\alpha ,\beta \in ℂ$ ，都有：

$\Vert f_q\left(\alpha \right)-f_q\left(\beta \right)\Vert \le F_q\Vert \alpha -\beta \Vert$ ; $\Vert g_p\left(\alpha \right)-g_p\left(\beta \right)\Vert \le G_p\Vert \alpha -\beta \Vert$ , $p\in N$ , $q\in M$ .

4. 主要结果

对于系统(3.5)和(3.6)，若存在常数$0<\varpi <1$ ，且有$\frac{1}{a_p}{\displaystyle \sum _{q=1}^m\left(\left|c_{1pq}^R\right|+\left|c_{1pq}^I\right|+\left|c_{2pq}^R\right|+\left|c_{2pq}^I\right|\right)}\cdot F_q<\varpi$ ，$\frac{1}{b_q}{\displaystyle \sum _{p=1}^n\left(\left|d_{1qp}^R\right|+\left|d_{1qp}^I\right|+\left|d_{2qp}^R\right|+\left|d_{2qp}^I\right|\right)}\cdot G_p<\varpi$ $\left(p\in N,q\in M\right)$ ，则系统(3.5)和(3.6)是全局指数稳定的。

证明：首先定义乘积空间$\Xi ={\Xi }_1\times {\Xi }_1\times {\Xi }_1\times {\Xi }_1$ ，其中${\Xi }_1$ 是由$z_1\left(t\right):\left[0,+\infty \right)\to {ℝ}^n$ 构成的，${\Xi }_2$ 是由$z_2\left(t\right):\left[0,+\infty \right)\to {ℝ}^n$ 构成的，${\Xi }_3$ 是由$z_3\left(t\right):\left[0,+\infty \right)\to {ℝ}^m$ 构成的，${\Xi }_4$ 是由$z_4\left(t\right):\left[0,+\infty \right)\to {ℝ}^m$ 构成的，且满足以下条件：

1) $z_r$ 在$t\in \left[0,+\infty \right)$ 上连续，$r=1,2,3,4$ ；

2) 当$t\in \left[\rho t_0-t_0,0\right]$ 时，$z_1\left(t\right)={\xi }^R\left(t\right)$ ，$z_2\left(t\right)={\xi }^I\left(t\right)$ ，$z_3\left(t\right)={\eta }^R\left(t\right)$ ，$z_4\left(t\right)={\eta }^I\left(t\right)$ ；

3) 当$t\to +\infty$ 时，${\text{e}}^{\gamma t}{z}_1\left(t\right)\to 0\in {ℝ}^n$ ，${\text{e}}^{\gamma t}{z}_2\left(t\right)\to 0\in {ℝ}^n$ ，${\text{e}}^{\gamma t}{z}_3\left(t\right)\to 0\in {ℝ}^m$ ，${\text{e}}^{\gamma t}{z}_4\left(t\right)\to 0\in {ℝ}^m$ ，其中$\gamma$ 是正常数，满足$\gamma <\min \left\{a_p,b_q\right\}$ 。

再定义空间$\Xi$ 上的距离如下：

$dist\left(\widehat{z},\overline{z}\right)=\underset{1\le r\le 2n+2m}{\max }\left\{\sup \Vert {\widehat{z}}^r\left(t\right)-{\overline{z}}^r\left(t\right)\Vert \right\},$

其中

$\widehat{z}=\widehat{z}\left(t\right)={\left({\widehat{z}}_1\left(t\right),{\widehat{z}}_2\left(t\right),{\widehat{z}}_3\left(t\right),{\widehat{z}}_4\left(t\right)\right)}^{\text{T}}={\left({\widehat{z}}^{\left(1\right)}\left(t\right),{\widehat{z}}^{\left(2\right)}\left(t\right),\cdots ,{\widehat{z}}^{\left(2n+2m\right)}\left(t\right)\right)}^{\text{T}}\in \Xi ;$

$\overline{z}=\overline{z}\left(t\right)={\left({\overline{z}}_1\left(t\right),{\overline{z}}_2\left(t\right),{\overline{z}}_3\left(t\right),{\overline{z}}_4\left(t\right)\right)}^{\text{T}}={\left({\overline{z}}^{\left(1\right)}\left(t\right),{\overline{z}}^{\left(2\right)}\left(t\right),\cdots ,{\overline{z}}^{\left(2n+2m\right)}\left(t\right)\right)}^{\text{T}}\in \Xi ;$

${\widehat{z}}_1\left(t\right)={\left({\widehat{z}}^{\left(1\right)}\left(t\right),{\widehat{z}}^{\left(2\right)}\left(t\right),\cdots ,{\widehat{z}}^{\left(n\right)}\left(t\right)\right)}^{\text{T}}\in {\Xi }_1;$

${\widehat{z}}_2\left(t\right)={\left({\widehat{z}}^{\left(n+1\right)}\left(t\right),{\widehat{z}}^{\left(n+2\right)}\left(t\right),\cdots ,{\widehat{z}}^{\left(2n\right)}\left(t\right)\right)}^{\text{T}}\in {\Xi }_2;$

${\widehat{z}}_3\left(t\right)={\left({\widehat{z}}^{\left(2n+1\right)}\left(t\right),{\widehat{z}}^{\left(2n+2\right)}\left(t\right),\cdots ,{\widehat{z}}^{\left(2n+m\right)}\left(t\right)\right)}^{\text{T}}\in {\Xi }_3;$

${\widehat{z}}_4\left(t\right)={\left({\widehat{z}}^{\left(2n+m+1\right)}\left(t\right),{\widehat{z}}^{\left(2n+m+2\right)}\left(t\right),\cdots ,{\widehat{z}}^{\left(2n+2m\right)}\left(t\right)\right)}^{\text{T}}\in {\Xi }_4;$

${\overline{z}}_1\left(t\right)={\left({\overline{z}}^{\left(1\right)}\left(t\right),{\overline{z}}^{\left(2\right)}\left(t\right),\cdots ,{\overline{z}}^{\left(n\right)}\left(t\right)\right)}^{\text{T}}\in {\Xi }_1;$

${\overline{z}}_2\left(t\right)={\left({\overline{z}}^{\left(n+1\right)}\left(t\right),{\overline{z}}^{\left(n+2\right)}\left(t\right),\cdots ,{\overline{z}}^{\left(2n\right)}\left(t\right)\right)}^{\text{T}}\in {\Xi }_2;$

${\overline{z}}_3\left(t\right)={\left({\overline{z}}^{\left(2n+1\right)}\left(t\right),{\overline{z}}^{\left(2n+2\right)}\left(t\right),\cdots ,{\overline{z}}^{\left(2n+m\right)}\left(t\right)\right)}^{\text{T}}\in {\Xi }_3;$

${\overline{z}}_4\left(t\right)={\left({\overline{z}}^{\left(2n+m+1\right)}\left(t\right),{\overline{z}}^{\left(2n+m+2\right)}\left(t\right),\cdots ,{\overline{z}}^{\left(2n+2m\right)}\left(t\right)\right)}^{\text{T}}\in {\Xi }_4.$

不难证明空间$\Xi$ 是一个完备度量空间。

1) 构造空间$\Xi$ 上的映射。

设${\left({\upsilon }_1^R\left(t\right),\cdots ,{\upsilon }_n^R\left(t\right),{\upsilon }_1^I\left(t\right),\cdots ,{\upsilon }_n^I\left(t\right),{\omega }_1^R\left(t\right),\cdots ,{\omega }_m^R\left(t\right),{\omega }_1^I\left(t\right),\cdots ,{\omega }_m^I\left(t\right)\right)}^{\text{T}}$ 是系统(3.5)的解，那么当$t\in \left[0,+\infty \right]$ 时，有

$\begin{array}{c}\frac{\text{d}}{\text{d}t}\left({\text{e}}^{a_{p}t}{\upsilon }_p^R\left(t\right)\right)=a_p{e}^{a_{p}t}{\upsilon }_p^R\left(t\right)+{\text{e}}^{a_{p}t}[{\displaystyle \sum _{q=1}^m{c}_{1pq}^R}{f}_q^R\left({\omega }_q\left(t\right)\right)-{\displaystyle \sum _{q=1}^m{c}_{1pq}^I}{f}_q^I\left({\omega }_q\left(t\right)\right)\\ +{\displaystyle \sum _{q=1}^m{c}_{2pq}^R}{f}_q^R\left({\omega }_q\left({\theta }_{q}t\right)\right)-{\displaystyle \sum _{q=1}^m{c}_{2pq}^I}{f}_q^I\left({\omega }_q\left({\theta }_{q}t\right)\right)],\end{array}$

等式两边同时积分，有

$\begin{array}{c}{\upsilon }_p^R\left(t\right)={\text{e}}^{-a_{p}t}{\xi }_p^R\left(0\right)+{\text{e}}^{-a_{p}t}{\displaystyle {\int }_0^t{e}^{a_{p}s}}[{\displaystyle \sum _{q=1}^m{c}_{1pq}^R}{f}_q^R\left({\omega }_q\left(s\right)\right)-{\displaystyle \sum _{q=1}^m{c}_{1pq}^I}{f}_q^I\left({\omega }_q\left(s\right)\right)\\ +{\displaystyle \sum _{q=1}^m{c}_{2pq}^R}{f}_q^R\left({\omega }_q\left({\theta }_{q}s\right)\right)-{\displaystyle \sum _{q=1}^m{c}_{2pq}^I}{f}_q^I\left({\omega }_q\left({\theta }_{q}s\right)\right)]\text{d}s,\end{array}$

根据同样的步骤，当$t\in \left[0,+\infty \right]$ 时，可以得到以下三个式子：

$\begin{array}{c}{\upsilon }_p^I\left(t\right)={\text{e}}^{-a_{p}t}{\xi }_p^I\left(0\right)+{\text{e}}^{-a_{p}t}{\displaystyle {\int }_0^t{\text{e}}^{a_{p}s}}[{\displaystyle \sum _{q=1}^m{c}_{1pq}^R}{f}_q^I\left({\omega }_q\left(s\right)\right)+{\displaystyle \sum _{q=1}^m{c}_{1pq}^I}{f}_q^R\left({\omega }_q\left(s\right)\right)\\ +{\displaystyle \sum _{q=1}^m{c}_{2pq}^R}{f}_q^I\left({\omega }_q\left({\theta }_{q}s\right)\right)+{\displaystyle \sum _{q=1}^m{c}_{2pq}^I}{f}_q^R\left({\omega }_q\left({\theta }_{q}s\right)\right)]\text{d}s,\end{array}$

$\begin{array}{c}{\omega }_q^R\left(t\right)={\text{e}}^{-b_{q}t}{\eta }_q^R\left(0\right)+{\text{e}}^{-b_{q}t}{\displaystyle {\int }_0^t{e}^{b_{q}s}}[{\displaystyle \sum _{p=1}^n{d}_{1qp}^R}{g}_p^R\left({\upsilon }_p\left(s\right)\right)-{\displaystyle \sum _{p=1}^n{d}_{1qp}^I}{g}_p^I\left({\upsilon }_p\left(s\right)\right)\\ +{\displaystyle \sum _{p=1}^n{d}_{2qp}^R}{g}_p^R\left({\upsilon }_p\left({\sigma }_{p}s\right)\right)-{\displaystyle \sum _{p=1}^n{d}_{2qp}^I}{g}_p^I\left({\upsilon }_p\left({\sigma }_{p}s\right)\right)]\text{d}s,\end{array}$

和

$\begin{array}{c}{\omega }_q^I\left(t\right)={\text{e}}^{-b_{q}t}{\eta }_q^I\left(0\right)+{\text{e}}^{-b_{q}t}{\displaystyle {\int }_0^t{\text{e}}^{b_{q}s}}[{\displaystyle \sum _{p=1}^n{d}_{1qp}^R}{g}_p^I\left({\upsilon }_p\left(s\right)\right)+{\displaystyle \sum _{p=1}^n{d}_{1qp}^I}{g}_p^R\left({\upsilon }_p\left(s\right)\right)\\ +{\displaystyle \sum _{p=1}^n{d}_{2qp}^R}{g}_p^I\left({\upsilon }_p\left({\sigma }_{p}s\right)\right)+{\displaystyle \sum _{p=1}^n{d}_{2qp}^I}{g}_p^R\left({\upsilon }_p\left({\sigma }_{p}s\right)\right)]\text{d}s.\end{array}$

因此，可以按如下定义构造空间$\Xi$ 上的映射$\Lambda$ ：

$\{\begin{array}{l}\Lambda \left({\upsilon }_p^R\left(t\right)\right)={\text{e}}^{-a_{p}t}{\xi }_p^R\left(0\right)+{\text{e}}^{-a_{p}t}{\displaystyle {\int }_0^t{\text{e}}^{a_{p}s}}[{\displaystyle \sum _{q=1}^m{c}_{1pq}^R}{f}_q^R\left({\omega }_q\left(s\right)\right)-{\displaystyle \sum _{q=1}^m{c}_{1pq}^I}{f}_q^I\left({\omega }_q\left(s\right)\right)\\ +{\displaystyle \sum _{q=1}^m{c}_{2pq}^R}{f}_q^R\left({\omega }_q\left({\theta }_{q}s\right)\right)-{\displaystyle \sum _{q=1}^m{c}_{2pq}^I}{f}_q^I\left({\omega }_q\left({\theta }_{q}s\right)\right)]\text{d}s,\\ \Lambda \left({\upsilon }_p^I\left(t\right)\right)={\text{e}}^{-a_{p}t}{\xi }_p^I\left(0\right)+{\text{e}}^{-a_{p}t}{\displaystyle {\int }_0^t{\text{e}}^{a_{p}s}}[{\displaystyle \sum _{q=1}^m{c}_{1pq}^R}{f}_q^I\left({\omega }_q\left(s\right)\right)+{\displaystyle \sum _{q=1}^m{c}_{1pq}^I}{f}_q^R\left({\omega }_q\left(s\right)\right)\\ +{\displaystyle \sum _{q=1}^m{c}_{2pq}^R}{f}_q^I\left({\omega }_q\left({\theta }_{q}s\right)\right)+{\displaystyle \sum _{q=1}^m{c}_{2pq}^I}{f}_q^R\left({\omega }_q\left({\theta }_{q}s\right)\right)]\text{d}s,\\ \Lambda \left({\omega }_q^R\left(t\right)\right)={\text{e}}^{-b_{q}t}{\eta }_q^R\left(0\right)+{\text{e}}^{-b_{q}t}{\displaystyle {\int }_0^t{\text{e}}^{b_{q}s}}[{\displaystyle \sum _{p=1}^n{d}_{1qp}^R}{g}_p^R\left({\upsilon }_p\left(s\right)\right)-{\displaystyle \sum _{p=1}^n{d}_{1qp}^I}{g}_p^I\left({\upsilon }_p\left(s\right)\right)\\ +{\displaystyle \sum _{p=1}^n{d}_{2qp}^R}{g}_p^R\left({\upsilon }_p\left({\sigma }_{p}s\right)\right)-{\displaystyle \sum _{p=1}^n{d}_{2qp}^I}{g}_p^I\left({\upsilon }_p\left({\sigma }_{p}s\right)\right)]\text{d}s,\\ \Lambda \left({\omega }_q^I\left(t\right)\right)={\text{e}}^{-b_{q}t}{\eta }_q^I\left(0\right)+{\text{e}}^{-b_{q}t}{\displaystyle {\int }_0^t{\text{e}}^{b_{q}s}}[{\displaystyle \sum _{p=1}^n{d}_{1qp}^R}{g}_p^I\left({\upsilon }_p\left(s\right)\right)+{\displaystyle \sum _{p=1}^n{d}_{1qp}^I}{g}_p^R\left({\upsilon }_p\left(s\right)\right)\\ +{\displaystyle \sum _{p=1}^n{d}_{2qp}^R}{g}_p^I\left({\upsilon }_p\left({\sigma }_{p}s\right)\right)+{\displaystyle \sum _{p=1}^n{d}_{2qp}^I}{g}_p^R\left({\upsilon }_p\left({\sigma }_{p}s\right)\right)]\text{d}s,\end{array}$ (4.1)

其中，$t\in \left[0,+\infty \right)$ 。初始条件是：

$\Lambda {\left({\upsilon }_p^R\left(t\right),{\upsilon }_p^I\left(t\right),{\omega }_q^R\left(t\right),{\omega }_q^I\left(t\right)\right)}^{\text{T}}={\left({\xi }_p^R\left(t\right),{\xi }_p^I\left(t\right),{\eta }_q^R\left(t\right),{\eta }_q^I\left(t\right)\right)}^{\text{T}},$ (4.2)

其中，$t\in \left[\rho t_0-t_0,0\right]$ 。

2) 证明$\Lambda$ 是自映射。

即证明对于$\forall {\left({\upsilon }_p^R\left(t\right),{\upsilon }_p^I\left(t\right),{\omega }_q^R\left(t\right),{\omega }_q^I\left(t\right)\right)}^{\text{T}}\in \Xi$ ，有$\Lambda {\left({\upsilon }_p^R\left(t\right),{\upsilon }_p^I\left(t\right),{\omega }_q^R\left(t\right),{\omega }_q^I\left(t\right)\right)}^{\text{T}}\in \Xi$ 。也就是要证明$\Lambda {\left({\upsilon }_p^R\left(t\right),{\upsilon }_p^I\left(t\right),{\omega }_q^R\left(t\right),{\omega }_q^I\left(t\right)\right)}^{\text{T}}$ 满足条件1)~3)。显然条件1)~2)可以满足，而条件3)满足，只要${\text{e}}^{\gamma t}\Lambda {\left({\upsilon }_p^R\left(t\right),{\upsilon }_p^I\left(t\right),{\omega }_q^R\left(t\right),{\omega }_q^I\left(t\right)\right)}^{\text{T}}\in \Xi$ 成立，即：

${\text{e}}^{\gamma t}{\left(J_1,J_2,J_3,J_4\right)}^{\text{T}}\to {\left(0,0,0,0\right)}^{\text{T}}\in {ℝ}^{2n+2m}$ (4.3)

其中

$\begin{array}{c}{J}_1={\text{e}}^{-a_{p}t}{\xi }_p^R\left(0\right)+{\text{e}}^{-a_{p}t}{\displaystyle {\int }_0^t{\text{e}}^{a_{p}s}}[{\displaystyle \sum _{q=1}^m{c}_{1pq}^R}{f}_q^R\left({\omega }_q\left(s\right)\right)-{\displaystyle \sum _{q=1}^m{c}_{1pq}^I}{f}_q^I\left({\omega }_q\left(s\right)\right)\\ +{\displaystyle \sum _{q=1}^m{c}_{2pq}^R}{f}_q^R\left({\omega }_q\left({\theta }_{q}s\right)\right)-{\displaystyle \sum _{q=1}^m{c}_{2pq}^I}{f}_q^I\left({\omega }_q\left({\theta }_{q}s\right)\right)]\text{d}s,\end{array}$

$\begin{array}{c}{J}_2={\text{e}}^{-a_{p}t}{\xi }_p^I\left(0\right)+{\text{e}}^{-a_{p}t}{\displaystyle {\int }_0^t{\text{e}}^{a_{p}s}}[{\displaystyle \sum _{q=1}^m{c}_{1pq}^R}{f}_q^I\left({\omega }_q\left(s\right)\right)+{\displaystyle \sum _{q=1}^m{c}_{1pq}^I}{f}_q^R\left({\omega }_q\left(s\right)\right)\\ +{\displaystyle \sum _{q=1}^m{c}_{2pq}^R}{f}_q^I\left({\omega }_q\left({\theta }_{q}s\right)\right)+{\displaystyle \sum _{q=1}^m{c}_{2pq}^I}{f}_q^R\left({\omega }_q\left({\theta }_{q}s\right)\right)]\text{d}s,\end{array}$

$\begin{array}{c}{J}_3={\text{e}}^{-b_{q}t}{\eta }_q^R\left(0\right)+{\text{e}}^{-b_{q}t}{\displaystyle {\int }_0^t{\text{e}}^{b_{q}s}}[{\displaystyle \sum _{p=1}^n{d}_{1qp}^R}{g}_p^R\left({\upsilon }_p\left(s\right)\right)-{\displaystyle \sum _{p=1}^n{d}_{1qp}^I}{g}_p^I\left({\upsilon }_p\left(s\right)\right)\\ +{\displaystyle \sum _{p=1}^n{d}_{2qp}^R}{g}_p^R\left({\upsilon }_p\left({\sigma }_{p}s\right)\right)-{\displaystyle \sum _{p=1}^n{d}_{2qp}^I}{g}_p^I\left({\upsilon }_p\left({\sigma }_{p}s\right)\right)]\text{d}s,\end{array}$

$\begin{array}{c}{J}_4={\text{e}}^{-b_{q}t}{\eta }_q^I\left(0\right)+{\text{e}}^{-b_{q}t}{\displaystyle {\int }_0^t{\text{e}}^{b_{q}s}}[{\displaystyle \sum _{p=1}^n{d}_{1qp}^R}{g}_p^I\left({\upsilon }_p\left(s\right)\right)+{\displaystyle \sum _{p=1}^n{d}_{1qp}^I}{g}_p^R\left({\upsilon }_p\left(s\right)\right)\\ +{\displaystyle \sum _{p=1}^n{d}_{2qp}^R}{g}_p^I\left({\upsilon }_p\left({\sigma }_{p}s\right)\right)+{\displaystyle \sum _{p=1}^n{d}_{2qp}^I}{g}_p^R\left({\upsilon }_p\left({\sigma }_{p}s\right)\right)]\text{d}s.\end{array}$

当$t\to +\infty$ 时，显然有

$\begin{array}{ll}\left|{\text{e}}^{\gamma t}{\text{e}}^{-a_{p}t}{\xi }_p^R\left(0\right)\right|\to 0,\hfill & \left|{\text{e}}^{\gamma t}{\text{e}}^{-a_{p}t}{\xi }_p^I\left(0\right)\right|\to 0,\hfill \\ \left|{\text{e}}^{\gamma t}{\text{e}}^{-b_{q}t}{\eta }_q^R\left(0\right)\right|\to 0,\hfill & \left|{\text{e}}^{\gamma t}{\text{e}}^{-b_{q}t}{\eta }_q^I\left(0\right)\right|\to 0.\hfill \end{array}$ (4.4)

此外，由$\left|{\text{e}}^{\gamma t}{\omega }_p^R\left(t\right)\right|\to 0$ 和$\left|{\text{e}}^{\gamma t}{\omega }_p^I\left(t\right)\right|\to 0$ 可知，对任意给定的$\epsilon >0$ ，存在相应的正数$t^{\ast }$ ，使得当$t>t^{\ast }$ 时，有

$\left|{\text{e}}^{\gamma t}{\omega }_p^R\left(t\right)\right|<\epsilon ,\left|{\text{e}}^{\gamma t}{\omega }_p^I\left(t\right)\right|<\epsilon .$ (4.5)

那么当$t>t^{\ast }$ 时，也有

${\text{e}}^{\gamma t}\Vert {\omega }_q\left(t\right)\Vert <\epsilon .$ (4.6)

由假设3，有

$\begin{array}{l}\left|{\text{e}}^{\gamma t}{\text{e}}^{-a_{p}t}{\displaystyle {\int }_0^t{\text{e}}^{a_{p}s}}{\displaystyle \sum _{q=1}^m{c}_{1qp}^R}{f}_q^R\left({\omega }_q\left(s\right)\right)\text{d}s\right|\le {\text{e}}^{-\left(a_p-\gamma \right)t}{\displaystyle {\int }_0^t{\text{e}}^{a_{p}s}}{\displaystyle \sum _{q=1}^m\left|c_{1pq}^R\right|\cdot F_q\cdot \Vert {\omega }_q\left(s\right)\Vert \text{d}s}\\ \le {\displaystyle \sum _{q=1}^m\left|c_{1pq}^R\right|\cdot F_q\cdot \left[{\text{e}}^{-\left(a_p-\gamma \right)t}{\displaystyle {\int }_0^{t^{\ast }}{\text{e}}^{a_{p}s}}\Vert {\omega }_q\left(s\right)\Vert \text{d}s+{\text{e}}^{-\left(a_p-\gamma \right)t}{\displaystyle {\int }_{t^{\ast }}^t{\text{e}}^{a_{p}s}}\Vert {\omega }_q\left(s\right)\Vert \text{d}s\right]}\\ \le {\displaystyle \sum _{q=1}^m\left|c_{1pq}^R\right|\cdot F_q\cdot }\left[{\text{e}}^{-\left(a_p-\gamma \right)t}{\text{e}}^{a_p{t}^{\ast }}\cdot t^{\ast }\cdot \underset{s\in \left[0,t^{\ast }\right]}{\sup }\Vert {\omega }_q\left(s\right)\Vert +{\text{e}}^{-\left(a_p-\gamma \right)t}{\displaystyle {\int }_{t^{\ast }}^t{\text{e}}^{\left(a_p-\gamma \right)s}}{\text{e}}^{\gamma s}\Vert {\omega }_q\left(s\right)\Vert \text{d}s\right].\end{array}$

根据(4.6)，可得

$\left|{\text{e}}^{\gamma t}{\text{e}}^{-a_{p}t}{\displaystyle {\int }_0^t{\text{e}}^{a_{p}s}}{\displaystyle \sum _{q=1}^m{c}_{1qp}^R}{f}_q^R\left({\omega }_q\left(s\right)\right)\text{d}s\right|\le {\displaystyle \sum _{q=1}^m\left|c_{1pq}^R\right|F_q}\left[{\text{e}}^{-\left(a_p-\gamma \right)t}{\text{e}}^{a_p{t}^{\ast }}{t}^{\ast }\cdot \underset{s\in \left[0,t^{\ast }\right]}{\sup }\Vert {\omega }_q\left(s\right)\Vert +\frac{\epsilon }{a_p-\gamma }\right].$ (4.7)

同理可得

$\left|{\text{e}}^{\gamma t}{\text{e}}^{-a_{p}t}{\displaystyle {\int }_0^t{\text{e}}^{a_{p}s}}{\displaystyle \sum _{q=1}^m{c}_{1pq}^I}{f}_q^I\left({\omega }_q\left(s\right)\right)\text{d}s\right|\le {\displaystyle \sum _{q=1}^m\left|c_{1pq}^I\right|F_q}\left[{\text{e}}^{-\left(a_p-\gamma \right)t}{\text{e}}^{a_p{t}^{\ast }}{t}^{\ast }\cdot \underset{s\in \left[0,t^{\ast }\right]}{\sup }\Vert {\omega }_q\left(s\right)\Vert +\frac{\epsilon }{a_p-\gamma }\right].$ (4.8)

由$\epsilon$ 的任意性，可得当$t\to +\infty$ 时，

$\begin{array}{l}\left|{\text{e}}^{\gamma t}{\text{e}}^{-a_{p}t}{\displaystyle {\int }_0^t{\text{e}}^{a_{p}s}}{\displaystyle \sum _{q=1}^m{c}_{1pq}^R}{f}_q^R\left({\omega }_q\left(s\right)\right)\text{d}s\right|\to 0,\\ \left|{\text{e}}^{\gamma t}{\text{e}}^{-a_{p}t}{\displaystyle {\int }_0^t{\text{e}}^{a_{p}s}}{\displaystyle \sum _{q=1}^m{c}_{1pq}^I}{f}_q^I\left({\omega }_q\left(s\right)\right)\text{d}s\right|\to 0.\end{array}$ (4.9)

由$\left|{\text{e}}^{\gamma t}{\omega }_p^R\left(t\right)\right|\to 0$ 和$\left|{\text{e}}^{\gamma t}{\omega }_p^I\left(t\right)\right|\to 0$ 可知，对任意给定的$\epsilon >0$ ，存在相应的正数$T$ ，使得当$t>\rho T$ 时，有

$\left|{\text{e}}^{\gamma t}{\omega }_p^R\left(t\right)\right|<\epsilon ,\left|{\text{e}}^{\gamma t}{\omega }_p^I\left(t\right)\right|<\epsilon .$ (4.10)

那么也有当$t>T$ 时，

${\text{e}}^{\gamma t}\Vert {\omega }_q\left(t\right)\Vert <\epsilon .$ (4.11)

由假设3，有

$\begin{array}{l}\left|{\text{e}}^{\gamma t}{\text{e}}^{-a_{p}t}{\displaystyle {\int }_0^t{\text{e}}^{a_{p}s}}{\displaystyle \sum _{q=1}^m{c}_{2pq}^R}{f}_q^R\left({\omega }_q\left({\theta }_{q}s\right)\right)\text{d}s\right|\le {\text{e}}^{-\left(a_p-\gamma \right)t}{\displaystyle {\int }_0^t{\text{e}}^{a_{p}s}}{\displaystyle \sum _{q=1}^m\left|c_{2pq}^R\right|\cdot F_q\Vert {\omega }_q\left({\theta }_{q}s\right)\Vert \text{d}s}\\ \le {\displaystyle \sum _{q=1}^m\left|c_{2pq}^R\right|\cdot F_q\left[{\text{e}}^{-\left(a_p-\gamma \right)t}{\displaystyle {\int }_0^T{\text{e}}^{a_{p}s}}\Vert {\omega }_q\left({\theta }_{q}s\right)\Vert \text{d}s+{\text{e}}^{-\left(a_p-\gamma \right)t}{\displaystyle {\int }_T^t{\text{e}}^{a_{p}s}}\Vert {\omega }_q\left({\theta }_{q}s\right)\Vert \text{d}s\right]}\\ \le {\displaystyle \sum _{q=1}^m\left|c_{2pq}^R\right|\cdot F_q\left[{\text{e}}^{-\left(a_p-\gamma \right)t}{\text{e}}^{a_{p}T}\cdot T\cdot \underset{s\in \left[0,\rho T\right]}{\sup }\Vert {\omega }_q\left(s\right)\Vert +{\text{e}}^{-\left(a_p-\gamma \right)t}\underset{s\in \left[\rho T,t\right]}{\sup }\Vert {\omega }_q\left(s\right)\Vert {\displaystyle {\int }_T^t{\text{e}}^{a_{p}s}}\text{d}s\right]}\\ \le {\displaystyle \sum _{q=1}^m\left|c_{2pq}^R\right|}\cdot F_q\left[{\text{e}}^{-\left(a_p-\gamma \right)t}{\text{e}}^{a_{p}T}\cdot T\cdot \underset{s\in \left[0,\rho T\right]}{\sup }\Vert {\omega }_q\left(s\right)\Vert +\frac{1}{a_p}\cdot {\text{e}}^{\gamma t}\cdot \underset{s\in \left[\rho T,t\right]}{\sup }\Vert {\omega }_q\left(s\right)\Vert \right].\end{array}$

又根据(4.11)，可得

$\left|{\text{e}}^{\gamma t}{\text{e}}^{-a_{p}t}{\displaystyle {\int }_0^t{\text{e}}^{a_{p}s}}{\displaystyle \sum _{q=1}^m{c}_{2pq}^R}{f}_q^R\left({\omega }_q\left({\theta }_{q}s\right)\right)\text{d}s\right|\le {\displaystyle \sum _{q=1}^m\left|c_{2pq}^R\right|\cdot F_q\left[{\text{e}}^{-\left(a_p-\gamma \right)t}{\text{e}}^{a_{p}T}\cdot T\cdot \underset{s\in \left[0,\rho T\right]}{\sup }\Vert {\omega }_q\left(s\right)\Vert +\frac{\epsilon }{a_p}\right]}。$ (4.12)

同理可得

$\left|{\text{e}}^{\gamma t}{\text{e}}^{-a_{p}t}{\displaystyle {\int }_0^t{\text{e}}^{a_{p}s}}{\displaystyle \sum _{q=1}^m{c}_{2pq}^I}{f}_q^I\left({\omega }_q\left({\theta }_{q}s\right)\right)\text{d}s\right|\le {\displaystyle \sum _{q=1}^m\left|c_{2pq}^I\right|\cdot F_q\cdot \left[{\text{e}}^{-\left(a_p-\gamma \right)t}{\text{e}}^{a_{p}T}\cdot T\cdot \underset{s\in \left[0,\rho T\right]}{\sup }\Vert {\omega }_q\left(s\right)\Vert +\frac{\epsilon }{a_p}\right]}.$ (4.13)

由于$\epsilon$ 的任意性，当$t\to +\infty$ 时，有

$\begin{array}{l}\left|{\text{e}}^{\gamma t}{\text{e}}^{-a_{p}t}{\displaystyle {\int }_0^t{\text{e}}^{a_{p}s}}{\displaystyle \sum _{q=1}^m{c}_{2pq}^R}{f}_q^R\left({\omega }_q\left({\theta }_{q}s\right)\right)\text{d}s\right|\to 0,\\ \left|{\text{e}}^{\gamma t}{\text{e}}^{-a_{p}t}{\displaystyle {\int }_0^t{\text{e}}^{a_{p}s}}{\displaystyle \sum _{q=1}^m{c}_{2pq}^I}{f}_q^I\left({\omega }_q\left({\theta }_{q}s\right)\right)\text{d}s\right|\to 0.\end{array}$ (4.14)

由(4.4)、(4.9)和(4.14)可得$\left|{\text{e}}^{\gamma t}\cdot J_1\right|\to 0$ ，类似可证$\left|{\text{e}}^{\gamma t}\cdot J_2\right|\to 0$ ，$\left|{\text{e}}^{\gamma t}\cdot J_3\right|\to 0$ ，$\left|{\text{e}}^{\gamma t}\cdot J_4\right|\to 0$ ，即(4.3)成立，也就证明了$\Lambda$ 是$\Xi$ 上的自映射。

3) 证明$\Lambda :\Xi \to \Xi$ 是压缩映射。

对任意的$z\left(t\right)={\left(z_1\left(t\right),z_2\left(t\right),z_3\left(t\right),z_4\left(t\right)\right)}^{\text{T}}={\left(z^{\left(1\right)}\left(t\right),z^{\left(2\right)}\left(t\right),\cdots ,{\widehat{z}}^{\left(2n+2m\right)}\left(t\right)\right)}^{\text{T}}$，$\overline{z}\left(t\right)={\left({\overline{z}}_1\left(t\right),{\overline{z}}_2\left(t\right),{\overline{z}}_3\left(t\right),{\overline{z}}_4\left(t\right)\right)}^{\text{T}}={\left({\overline{z}}^{\left(1\right)}\left(t\right),{\overline{z}}^{\left(2\right)}\left(t\right),\cdots ,{\overline{z}}^{\left(2n+2m\right)}\left(t\right)\right)}^{\text{T}}\in \Xi$，有

$\Lambda \left(z\left(t\right)\right)-\Lambda \left(\overline{z}\left(t\right)\right)={\left(Q_1,Q_2,Q_3,Q_4\right)}^{\text{T}}.$

其中

$\begin{array}{c}{Q}_1={\text{e}}^{-a_{p}t}{\displaystyle {\int }_0^t{\text{e}}^{a_{p}s}}\{{\displaystyle \sum _{q=1}^m{c}_{1pq}^R}\left[f_q^R\left({\omega }_q\left(s\right)\right)-f_q^R\left({\overline{\omega }}_q\left(s\right)\right)\right]-{\displaystyle \sum _{q=1}^m{c}_{1pq}^I}\left[f_q^I\left({\omega }_q\left(s\right)\right)-f_q^I\left({\overline{\omega }}_q\left(s\right)\right)\right]\\ +{\displaystyle \sum _{q=1}^m{c}_{2pq}^R}\left[f_q^R\left({\omega }_q\left({\theta }_{q}s\right)\right)-f_q^R\left({\overline{\omega }}_q\left({\theta }_{q}s\right)\right)\right]-{\displaystyle \sum _{q=1}^m{c}_{2pq}^I}\left[f_q^I\left({\omega }_q\left({\theta }_{q}s\right)\right)-f_q^I\left({\overline{\omega }}_q\left({\theta }_{q}s\right)\right)\right]\}\text{d}s,\end{array}$

$\begin{array}{c}{Q}_2={\text{e}}^{-a_{p}t}{\displaystyle {\int }_0^t{\text{e}}^{a_{p}s}}\{{\displaystyle \sum _{q=1}^m{c}_{1pq}^R}\left[f_q^I\left({\omega }_q\left(s\right)\right)-f_q^I\left({\overline{\omega }}_q\left(s\right)\right)\right]+{\displaystyle \sum _{q=1}^m{c}_{1pq}^I}\left[f_q^R\left({\omega }_q\left(s\right)\right)-f_q^R\left({\overline{\omega }}_q\left(s\right)\right)\right]\\ +{\displaystyle \sum _{q=1}^m{c}_{2pq}^R}\left[f_q^I\left({\omega }_q\left({\theta }_{q}s\right)\right)-f_q^I\left({\overline{\omega }}_q\left({\theta }_{q}s\right)\right)\right]+{\displaystyle \sum _{q=1}^m{c}_{2pq}^I}\left[f_q^R\left({\omega }_q\left({\theta }_{q}s\right)\right)-f_q^R\left({\overline{\omega }}_q\left({\theta }_{q}s\right)\right)\right]\}\text{d}s,\end{array}$

$\begin{array}{c}{Q}_3={\text{e}}^{-b_{q}t}{\displaystyle {\int }_0^t{\text{e}}^{b_{q}s}}\{{\displaystyle \sum _{p=1}^n{d}_{1qp}^R}\left[g_p^R\left({\upsilon }_p\left(s\right)\right)-g_p^R\left({\overline{\upsilon }}_p\left(s\right)\right)\right]-{\displaystyle \sum _{p=1}^n{d}_{1qp}^I}\left[g_p^I\left({\upsilon }_p\left(s\right)\right)-g_p^I\left({\overline{\upsilon }}_p\left(s\right)\right)\right]\\ +{\displaystyle \sum _{p=1}^n{d}_{2qp}^R}\left[g_p^R\left({\upsilon }_p\left({\sigma }_{p}s\right)\right)-g_p^R\left({\overline{\upsilon }}_p\left({\sigma }_{p}s\right)\right)\right]-{\displaystyle \sum _{p=1}^n{d}_{2qp}^I}\left[g_p^I\left({\upsilon }_p\left({\sigma }_{p}s\right)\right)-g_p^I\left({\overline{\upsilon }}_p\left({\sigma }_{p}s\right)\right)\right]\}\text{d}s,\end{array}$

$\begin{array}{c}{Q}_4={\text{e}}^{-b_{q}t}{\displaystyle {\int }_0^t{\text{e}}^{b_{q}s}}\{{\displaystyle \sum _{p=1}^n{d}_{1qp}^R}\left[g_p^I\left({\upsilon }_p\left(s\right)\right)-g_p^I\left({\overline{\upsilon }}_p\left(s\right)\right)\right]+{\displaystyle \sum _{p=1}^n{d}_{1qp}^I}\left[g_p^R\left({\upsilon }_p\left(s\right)\right)-g_p^R\left({\overline{\upsilon }}_p\left(s\right)\right)\right]\\ +{\displaystyle \sum _{p=1}^n{d}_{2qp}^R}\left[g_p^I\left({\upsilon }_p\left({\sigma }_{p}s\right)\right)-g_p^I\left({\overline{\upsilon }}_p\left({\sigma }_{p}s\right)\right)\right]+{\displaystyle \sum _{p=1}^n{d}_{2qp}^I}\left[g_p^R\left({\upsilon }_p\left({\sigma }_{p}s\right)\right)-g_p^R\left({\overline{\upsilon }}_p\left({\sigma }_{p}s\right)\right)\right]\}\text{d}s.\end{array}$

由假设3，我们可以对$Q_1$ 进行估计：

$\begin{array}{c}\left|Q_1\right|={\text{e}}^{-a_{p}t}{\displaystyle {\int }_0^t{\text{e}}^{a_{p}s}}\{{\displaystyle \sum _{q=1}^m{c}_{1pq}^R}\left[f_q^R\left({\omega }_q\left(s\right)\right)-f_q^R\left({\overline{\omega }}_q\left(s\right)\right)\right]-{\displaystyle \sum _{q=1}^m{c}_{1pq}^I}\left[f_q^I\left({\omega }_q\left(s\right)\right)-f_q^I\left({\overline{\omega }}_q\left(s\right)\right)\right]\\ +{\displaystyle \sum _{q=1}^m{c}_{2pq}^R}\left[f_q^R\left({\omega }_q\left({\theta }_{q}s\right)\right)-f_q^R\left({\overline{\omega }}_q\left({\theta }_{q}s\right)\right)\right]-{\displaystyle \sum _{q=1}^m{c}_{2pq}^I}\left[f_q^I\left({\omega }_q\left({\theta }_{q}s\right)\right)-f_q^I\left({\overline{\omega }}_q\left({\theta }_{q}s\right)\right)\right]\}\text{d}s\\ \le {\text{e}}^{-a_{p}t}{\displaystyle {\int }_0^t{\text{e}}^{a_{p}s}}\{{\displaystyle \sum _{q=1}^m\left|c_{1pq}^R\right|\cdot F_q\cdot \Vert {\omega }_q\left(s\right)-{\overline{\omega }}_q\left(s\right)\Vert }+{\displaystyle \sum _{q=1}^m\left|c_{1pq}^I\right|\cdot F_q\cdot \Vert {\omega }_q\left(s\right)-{\overline{\omega }}_q\left(s\right)\Vert }\\ +{\displaystyle \sum _{i=1}^n\left|c_{2pq}^R\right|\cdot F_q\cdot \Vert {\omega }_q\left({\theta }_{q}s\right)-{\overline{\omega }}_q\left({\theta }_{q}s\right)\Vert +{\displaystyle \sum _{i=1}^n\left|c_{2pq}^I\right|\cdot F_q\cdot \Vert {\omega }_q\left({\theta }_{q}s\right)-{\overline{\omega }}_q\left({\theta }_{q}s\right)\Vert }}\}\text{d}s.\end{array}$

再根据积分计算以及空间$\Xi$ 上距离的定义，有

$\begin{array}{c}\left|Q_1\right|\le \frac{1}{a_p}\cdot {\displaystyle \sum _{q=1}^m\left(\left|c_{1pq}^R\right|+\left|c_{1pq}^I\right|+\left|c_{2pq}^R\right|+\left|c_{2pq}^I\right|\right)}\cdot F_q\cdot \underset{s\in [0,t]}{\sup }\Vert {\omega }_q\left(s\right)-{\overline{\omega }}_q\left(s\right)\Vert \\ \le \frac{1}{a_p}\cdot {\displaystyle \sum _{q=1}^m\left(\left|c_{1pq}^R\right|+\left|c_{1pq}^I\right|+\left|c_{2pq}^R\right|+\left|c_{2pq}^I\right|\right)}\cdot F_q\cdot dist\left(z\left(t\right),\overline{z}\left(t\right)\right).\end{array}$

根据同样的处理，能够得到

$\begin{array}{l}\left|Q_2\right|\le \frac{1}{a_p}\cdot {\displaystyle \sum _{q=1}^m\left(\left|c_{1pq}^R\right|+\left|c_{1pq}^I\right|+\left|c_{2pq}^R\right|+\left|c_{2pq}^I\right|\right)}\cdot F_q\cdot dist\left(z\left(t\right),\overline{z}\left(t\right)\right),\\ \left|Q_3\right|\le \frac{1}{b_q}\cdot {\displaystyle \sum _{q=1}^m\left(\left|d_{1qp}^R\right|+\left|d_{1qp}^I\right|+\left|d_{2qp}^R\right|+\left|d_{2qp}^I\right|\right)}\cdot G_p\cdot dist\left(z\left(t\right),\overline{z}\left(t\right)\right),\end{array}$

$\left|Q_4\right|\le \frac{1}{b_q}\cdot {\displaystyle \sum _{q=1}^m\left(\left|d_{1qp}^R\right|+\left|d_{1qp}^I\right|+\left|d_{2qp}^R\right|+\left|d_{2qp}^I\right|\right)}\cdot G_p\cdot dist\left(z\left(t\right),\overline{z}\left(t\right)\right).$

从而有$dist\left(\Lambda \left(z\left(t\right)\right)-\Lambda \left(\overline{z}\left(t\right)\right)\right)\le k\cdot dist\left(z\left(t\right)-\overline{z}\left(t\right)\right)$ ，所以$\Lambda :\Xi \to \Xi$ 是压缩映射。因此，$\Lambda$ 在$\Xi$ 上存在唯一的不动点${\left({\upsilon }_p^R\left(t\right),{\upsilon }_p^I\left(t\right),{\omega }_q^R\left(t\right),{\omega }_q^I\left(t\right)\right)}^{\text{T}}$ ，这也就是系统(3.5)的解，满足${\text{e}}^{\gamma t}{\left({\upsilon }_p^R\left(t\right),{\upsilon }_p^I\left(t\right),{\omega }_q^R\left(t\right),{\omega }_q^I\left(t\right)\right)}^{\text{T}}\to 0\in {ℂ}^{2n+2m}$ ，于是系统(3.5)是全局指数稳定的，这就证明了定理1。

Table 1. Comparation of the Lyapunov function method

表1. 与Lyapunov函数法对比

	Lyapunov函数法	Banach不动点定理方法
是否要求激活函数有界	是	否
主要使用的方法	Lyapunov函数法	Banach不动点定理方法
时滞类型	常时滞	比例时滞
是否是BAM神经网络	否	是

表1是文献[15]与本文的对比结果。从表1可知，本文不要求激活函数是有界的，减弱了对激活函数的限制。同时，本文利用Bnanch不动点定理得出的系统稳定的充分条件与利用Lyapunov函数法得到系统稳定的充分条件是不同的，故本文通过Banach不动点定理研究比例时滞复值BAM神经网络的稳定性是对已有结果的补充。

5. 数值算例

在本节中，我们给出一个数值例子说明以上结果的有效性：令

$p=q=2$ ，$a_1=\text{2}\text{.623}$ ，$a_2=\text{3}\text{.442}$ ，$b_1=\text{4}\text{.651}$ ，$b_2=\text{5}\text{.137}$ ，$c_{111}^R=0.123$ ，$c_{111}^I=0.274$ ，$c_{112}^R=0.306$ ，$c_{112}^I=0.321$ ，$c_{211}^R=0.1$ ，$c_{211}^I=0.3176$ ，$c_{212}^R=0.244$ ，$c_{212}^I=0.401$ ，$c_{121}^R=0.113$ ，$c_{121}^I=0.277$ ，$c_{122}^R=0.316$ ，$c_{122}^I=0.331$ ，$c_{221}^R=0.15$ ，$c_{221}^I=0.317$ ，$c_{222}^R=0.233$ ，$c_{222}^I=0.422$ ，$d_{111}^R=0.121$ ，$d_{111}^I=0.271$ ，$d_{112}^R=0.333$ ，$d_{112}^I=0.421$ ，$d_{211}^R=0.136$ ，$d_{211}^I=0.316$ ，$d_{212}^R=0.241$ ，$d_{212}^I=0.402$ ，$d_{121}^R=0.124$ ，$d_{121}^I=0.264$ ，$d_{122}^R=0.324$ ，$d_{122}^I=0.455$ ，$d_{221}^R=0.138$ ，$d_{221}^I=0.346$ ，$d_{222}^R=0.238$ ，$d_{222}^I=0.372$ ，${\sigma }_1={\sigma }_2=0.5$ ，$f_1\left(t\right)=f_2\left(t\right)=1.2\tanh t$ ，$g_1\left(t\right)=g_2\left(t\right)=\frac{1}{20}\left(\left|t-1\right|-\left|t+1\right|\right)$ ，那么$F_1=F_2=1.2$ ，$G_1=G_2=0.1$ 。

同时，令$\varpi =0.96$ ，算得

$\begin{array}{l}\frac{1}{a_1}{\displaystyle \sum _{q=1}^2\left(\left|c_{11q}^R\right|+\left|c_{11q}^I\right|+\left|c_{21q}^R\right|+\left|c_{21q}^I\right|\right)}\cdot F_q=\frac{778}{815}<\varpi ,\\ \frac{1}{a_2}{\displaystyle \sum _{q=1}^2\left(\left|c_{12q}^R\right|+\left|c_{12q}^I\right|+\left|c_{22q}^R\right|+\left|c_{22q}^I\right|\right)}\cdot F_q=\frac{1184}{1573}<\varpi ,\\ \frac{1}{b_1}{\displaystyle \sum _{p=1}^2\left(\left|d_{11p}^R\right|+\left|d_{11p}^I\right|+\left|d_{21p}^R\right|+\left|d_{21p}^I\right|\right)}\cdot G_p=\frac{545}{11311}<\varpi ,\\ \frac{1}{b_2}{\displaystyle \sum _{p=1}^2\left(\left|d_{12p}^R\right|+\left|d_{12p}^I\right|+\left|d_{22p}^R\right|+\left|d_{22p}^I\right|\right)}\cdot G_p=\frac{1143}{25969}<\varpi ,\end{array}$

满足稳定性条件。接下来利用数值例子验证结果的有效性(见图1~8)。

Figure 1. Real part of $x_1$

图1. $x_1$ 的实部

Figure 2. Imaginary part of $x_1$

图2. $x_1$ 的虚部

Figure 3. Real part of $x_2$

图3. $x_2$ 的实部

Figure 4. Imaginary part of $x_2$

图4. $x_2$ 的虚部

Figure 5. Real part of $y_1$

图5. $y_1$ 的实部

Figure 6. Imaginary part of $y_1$

图6. $y_1$ 的虚部

Figure 7. Real part of $y_2$

图7. $y_2$ 的实部

Figure 8. Imaginary part of $y_2$

图8. $y_2$ 的虚部

6. 小结

本章主要使用Banach不动点定理讨论比例时滞复值BAM神经网络的全局指数稳定性，得到了神经网络满足稳定性的充分条件：$\frac{1}{a_p}{\displaystyle \sum _{q=1}^m\left(\left|c_{1pq}^R\right|+\left|c_{1pq}^I\right|+\left|c_{2pq}^R\right|+\left|c_{2pq}^I\right|\right)}\cdot F_q<\varpi$ ，$\frac{1}{b_q}{\displaystyle \sum _{p=1}^n\left(\left|d_{1qp}^R\right|+\left|d_{1qp}^I\right|+\left|d_{2qp}^R\right|+\left|d_{2qp}^I\right|\right)}\cdot G_p<\varpi$ $\left(p\in N,q\in M\right)$ ，并且所得结果是新的。与文献[15]不同，在本文中我们不要求激活函数是有界的，减弱了对激活函数的限制。

致 谢

这项研究得到云南财经大学研究生创新基金项目(2024YUFEYC083)、智慧口岸关键技术与创新应用研究(云南省重大科技专项，202402AD080005)、云南省跨境贸易与金融区块链国际联合研发中心、云南省科技厅建设面向南亚东南亚科技创新中心专项(202203AP140010)的资助，在此表示真诚的谢意。

NOTES

^*通讯作者。