近年来, 自治或非自治种群生态系统的动力学行为得到了广泛的研究.目前许多重要的结果包括系统的平衡点, 全局渐近或指数稳定, 持久性或绝灭性, 存在周期或概周期解的全局吸引等, 在文献[1-16]中都有分析研究.例如, 在文献[17]中, Tan等考虑了如下带有脉冲扰动的竞争系统

$\begin{eqnarray} \left\{ \begin{array}{ll} \dot{x}_1(t)=x_1(t)[a_1(t)-b_1(t)x_1(t)-c_1(t)x_2(t)-d_1(t)x_{1}^{2}(t)],& \hbox{$t\neq\tau_k,$} \\ \dot{x}_2(t)=x_2(t)[a_2(t)-b_2(t)x_2(t)-c_2(t)x_1(t)-d_2(t)x_{2}^{2}(t)],& \hbox{$t\neq\tau_k,$}\\ x_1({\tau_{k}^+})=(1+\gamma_{1k})x_1(\tau_k),& \\\hbox{$t=\tau_k,k\in{\mathbb N},$}\\ x_2({\tau_{k}^+})=(1+\gamma_{2k})x_2(\tau_k),& \\\hbox{$t=\tau_k,k\in{\mathbb N},$} \end{array} \right. \end{eqnarray}$

(1.1)

其中$x_1(t)$和$x_2(t)$分别表示两个竞争物种关于时间$t$的种群密度, $a_1(t)$和$a_2(t)$分别表示两个物种的内禀增长率, $b_1(t),d_1(t),b_2(t)$和$d_2(t)$分别表示种群内部特定竞争影响系数, $c_1(t)$与$c_2(t)$是种群间活动影响强度, $\gamma_{1k}>-1$和$\gamma_{2k}>-1$且都是常数, $0<\tau_1<\tau_2<\cdots<\tau_k<\tau_{k+1}<\cdots$和$\lim\limits_{k\rightarrow+\infty}\tau_k=+\infty$.作者通过分析得到了系统$(1.1)$存在唯一渐近稳定概周期解的充分条件.

当种群间没有代级重叠时, 离散时间系统的差分方程比连续方程更适合研究种群动力系统, 因为离散时间系统的差分方程可以提供有效的数值模拟, 因此研究离散时间系统的差分方程是有意义的.近来, 有许多学者对带有反馈控制的连续或离散系统概周期解的存在性问题做了许多研究工作, 更多研究结果可以参看文献[18-24]和其中的参考文献.除此之外, 为了将差分方程与微分方程统一起来研究, Hilger于1988年在他的博士论文^[25]中提出了时间尺度理论.时间尺度${\mathbb T}$就是实数集${\mathbb R}$的一个非空闭子集, 其上拓扑是由实数集${\mathbb R}$的标准拓扑诱导的拓扑.时间尺度理论的建立, 统一了同一问题要分别对其离散系统和连续系统进行研究的情形.随着人们对时间尺度理论的进一步研究, 发现时间尺度上的微分方程不仅可以统一差分方程和微分方程, 还可以用来更精确地描述一些系统, 比如一种季节性世代不重叠的种群繁殖模型.由于环境等因素的改变, 此类模型不能单一的用连续系统或离散系统完全准确地来描述, 若将这个模型利用时间尺度来讨论, 当取到合适的时间尺度时, 就可以将种群模型精确地建立起来.因此, 时间尺度理论的提出, 丰富和完善了动力系统的研究内容, 尤其是种群动力学模型的研究.它不仅为我们研究更多种群动力学模型提供了新的理论依据, 也使我们可以更深刻的理解离散系统和连续系统的关系. 2001年, Bohner M和Peterson A出版了时标上的动力系统专著[26], 且在2年之后, 他们又共同完成了专著[27], 为进一步的研究时标理论打下了坚实的理论基础.近年来, 越来越多的研究者对时标上动力系统进行了研究, 研究结果也越来越丰富.时标上动力系统的研究己成为非常活跃的研究领域, 研究内容覆盖越来越广泛, 尤其是关于它们的周期性.而时标上的概周期动力方程目前已经被许多数学家所认可^[28-30].特别地, 时标上的概周期函数, 是本文讨论的核心问题, 最早在文献[28]中由Li和Wang提出, 这是一种更一般的概周期.有关时标概周期动力系统的一些结果, 我们可参看文献[31-33]和它们的参考文献.尽管许多学者研究了带有反馈控制的竞争系统, 大多都是讨论带有反馈控制的连续或离散系统.据我们所知, 还没有工作讨论时标上带有反馈控制的两种群竞争系统概周期解的存在性问题.

受以上文献的启发和提示, 本文我们将研究时标${\mathbb T}$上带有反馈控制的竞争系统

$\begin{eqnarray} \left\{ \begin{array}{ll} x_{1}^{\Delta}(t)=a_1(t)-b_1(t)\exp\{x_1(t)\}-c_1(t)\exp\{x_2(t)\}\\ -d_{1}(t)\exp\{{2x_{1}(t)}\} -e_{1}(t)u_{1}(t),& \hbox{} \\ x_{2}^{\Delta}(t)=a_2(t)-b_2(t)\exp\{x_2(t)\}-c_2(t)\exp\{x_1(t)\}\\ -d_{2}(t)\exp\{{2x_{2}(t)}\} -e_{2}(t)u_{2}(t),\hbox{} \\ u_{1}^{\Delta}(t)=h_{1}(t)-f_1(t)u_{1}(t)+g_1(t)\exp\{x_1(t)\},& \hbox{}\\ u_{2}^{\Delta}(t)=h_2(t)-f_2(t)u_{2}(t)+g_2(t)\exp\{x_2(t)\},& \hbox{} \end{array} \right. \end{eqnarray}$

(1.2)

其中$u_{i}(t)$是关于种群$x_i(t)$的控制变量, $a_{i}(t)$, $b_i(t),c_i(t),d_i(t),e_i(t),h_i(t),f_i(t)$和$g_i(t)$都是非负概周期函数, 且$g_{i}(\cdot):{\mathbb T}\rightarrow(0,1)(i=1,2)$.

本文的主要目的是讨论系统$(1.2)$的持久性, 依据持久性结果, 使用微分比较原理和构造适合的Lyapunov函数, 得到存在唯一渐近正概周期解存在的充分条件.

为方便行文, 我们引入以下记号:

$f^M=\sup\limits_{t\in {\mathbb T}}f(t),\;\;\;f^l=\inf\limits_{t\in {\mathbb T}}f(t),$

其中$f(t)$是非负有界函数.

在整篇文章中, 假设如下条件成立:

$(H_1)$ $a_{i}(t),b_{i}(t),c_{i}(t),d_{i}(t),e_{i}(t),h_{i}(t),f_{i}(t)$和$g_{i}(t)(i=1,2)$都是时标${\mathbb T}$上的概周期函数并且使得

$\begin{eqnarray*} &&0<a_{i}^l,0<b_{i}^l\leq b_{i}^M,0<c_{i}^l\leq c_{i}^M,0<d_{i}^l\leq d_{i}^M,0<e_{i}^l\leq e_{i}^M,\\&&0<h_{i}^l\leq h_{i}^M, -f_{i}^l,-f_{i}^M\in{\cal R}^+,i=1,2,\end{eqnarray*}$

其中${\cal R}^+$是集合${\mathbb T}$到${\mathbb R}$的正回归函数;

$(H_2)$ $a_{i}^M>b_{i}^l (i=1,2)$;

$(H_3)$ $\frac{a_{1}^l-c_{1}^M\exp\{x_{2}^*\}-e_{1}^Mu_{1}^*}{b_{1}^M+d_{1}^Mx_{1}^*}>1$;

$(H_4)$ $\frac{a_{2}^l-c_{2}^M\exp\{x_{1}^*\}-e_{2}^Mu_{2}^*}{b_{2}^M+d_{2}^Mx_{2}^*}>1$.

因为考虑到系统的生物意义, 系统(1.2) 的初始条件定义为

$\begin{eqnarray} {x}_i(t_0)=x_{i0}, {u}_i(t_0)=u_{i0},t_0\in{\mathbb T},x_{i0}>0,u_{i0}>0,\forall i=1,2. \end{eqnarray}$

(1.3)

注1.1 令$y_{i}(t)=\exp\{x_{i}(t)\}\ (i=1,2)$, 若${\mathbb T}={\mathbb R}$, 可得到系统$(1.2)$的连续模型如下

$\begin{eqnarray} \left\{ \begin{array}{ll} \dot{y}_1(t)=y_1(t)[a_1(t)-b_1(t)y_1(t)-c_1(t)y_2(t)-d_1(t)y_{1}^{2}(t) -e_{1}(t)v_{1}(t)],& \hbox{} \\ \dot{y}_2(t)=y_2(t)[a_2(t)-b_2(t)y_2(t)-c_2(t)y_1(t)-d_2(t)y_{2}^{2}(t) -e_2(t)v_{2}(t)],& \hbox{}\\ \dot{v}_1(t)=h_1(t)-f_1(t)v_1(t)+g_1(t)y_1(t),& \hbox{}\\ \dot{v}_2(t)=h_2(t)-f_2(t)v_2(t)+g_2(t)y_2(t). & \hbox{} \end{array} \right. \end{eqnarray}$

(1.4)

若${\mathbb T}={\mathbb Z}$, 系统$(1.2)$可化简为如下带有反馈控制的非自治离散竞争系统

$\begin{eqnarray} \left\{ \begin{array}{ll} y_1(n+1)=y_{1}(n)\exp\{a_1(n)-b_1(n)y_1(n)-c_1(n)y_2(n)-d_1(n)y_{1}^2(n)\\ -e_1(n)v_1(n)\},& \hbox{} \\ y_2(n+1)=y_2(n)\exp\{a_2(n)-b_2(n)y_2(n)-c_2(n)y_1(n)-d_2(n)y_{2}^2(n)\\ -e_2(n)v_2(n)\},& \hbox{} \\ \Delta v_1(n)=h_1(n)-f_1(n)v_1(n)+g_1(n)y_1(n),& \hbox{}\\ \Delta v_2(n)=h_2(n)-f_2(n)v_2(n)+g_2(n)y_2(n),n\in{\mathbb Z^+},& \hbox{} \end{array} \right. \end{eqnarray}$

(1.5)

其中$\Delta v_i(n)=v_{i}(n+1)-v_{i}(n)\ (i=1,2)$是一阶前差微分算子.借助微分比较原理和构造合适的Lyapunov泛函, 许多学者给出了离散种群系统存在正概周期解的充分条件^{[22-23, 25]}.系统$(1.5)$正概周期解的存在性问题与文献[22-23, 25]的研究方法非常类似.显然, 系统$(1.4)$和$(1.5)$是系统$(1.2)$的特殊情况.因此由系统$(1.2)$的持久性和概周期解的存在性与一致渐近稳定性, 进而可以得到系统$(1.4)$以及系统$(1.5)$的持久性和正概周期解的存在性和一致渐近稳定性, 这样就将连续与离散统一了起来.

本文的主要结构如下:第2部分, 我们先为得到本文的一些主要结果做准备工作; 第3部分, 在持久性的基础上, 应用微分比较原理和构造合适的Lyapunov泛函, 我们得到系统(1.2) 存在正概周期解的充分条件; 一个数值算例在第4部分中给出.

现在我们给出主要结果中需要的一些定义和有用的引理.

定义2.1 ^[26] 令${\mathbb T}$为实数集${\mathbb R}$的一个非空闭子集(时标).前跃和后跃算子$\sigma,\rho:{\mathbb T}\rightarrow{\mathbb T}$以及graininess算子$\mu:{\mathbb T}\rightarrow{\mathbb R}_+$分别定义如下

$\begin{eqnarray*} \sigma(t)=\inf\{s\in{\mathbb T}:s>t\},\;\;\;\rho(t)=\sup\{s\in{\mathbb T}:s<t\}\;\; \mbox{且}\;\; \mu(t)=\sigma(t)-t. \end{eqnarray*}$

$t\in{\mathbb T}$称作是左稠密的, 如果$t>\inf{\mathbb T}$且$\rho(t)=t$; 称作是左离散的, 如果$\rho(t)<t$; 称作是右稠密的, 如果$t<\sup{\mathbb T}$且$\sigma(t)=t$; 称作是右离散的, 如果$\sigma(t)>t$.如果${\mathbb T}$有一个左离散的最小值$m$, 那么${\mathbb T}^k={\mathbb T}-{m}$; 否则${\mathbb T}^k={\mathbb T}.$如果${\mathbb T}$有一个右离散的最小值$m$, 那么${\mathbb T}_k={\mathbb T}-{m}$; 否则${\mathbb T}_k={\mathbb T}.$

一个函数$f:{\mathbb T}\rightarrow{\mathbb R}$称作是右稠密连续的, 如果它在${\mathbb T}$中的右稠密点是连续的, 且在${\mathbb T}$中的左稠密点的左极限是存在的.如果$f$在每一个右稠密点处处连续, 则称$f$是在${\mathbb T}$上的连续函数.我们定义$C[J, {\Bbb R}]=\{u(t):u(t)$在$J$上连续$\}$, 以及$C^1[J, {\Bbb R}]=\{u(t):u^{\Delta}(t)$在$J$上连续$\}$.

设函数$y:{\mathbb T}\rightarrow{\mathbb R}$且$t\in{\mathbb T}^k$, 我们称$y^{\Delta}(t)$为$y$在$t$处的delta导数, 且具有如下性质

$\begin{eqnarray*} |[y(\sigma(t))-y(s)]-y^{\Delta[\sigma(t)-s]}|<\epsilon|\sigma(t)-s|,\forall s\in U. \end{eqnarray*}$

如果$y$是右稠密连续的, 令$Y^{\Delta}(t)=y(t)$, 则我们定义delta积分如下

$\begin{eqnarray*} \int_{a}^ty(s)\Delta s=Y(t)-Y(a). \end{eqnarray*}$

一个函数$p:{\mathbb T}\rightarrow{\mathbb R}$称作是回归的, 如果

$1+\mu(t)p(t)\neq0,\forall t\in{\mathbb T}^k$

均成立.

所有回归右连续函数的集合$p:{\mathbb T}\rightarrow{\mathbb R}$记作${\cal R}={\cal R}({\mathbb T},{\mathbb R})$.如果

$1+\mu(t)p(t)>0, \forall t\in{\mathbb T},$

那么$p:{\mathbb T}\rightarrow{\mathbb R}$是正回归函数.定义集合

${\cal R}^+ = {\cal R}^+({\mathbb T}, {\mathbb R})=\{p\in{\cal R}:1+\mu(t)p(t)>0,\forall t\in{\mathbb T}\}.$

如果$r$是一个回归函数, 那么广义指数函数$e_r$定义如下

$\begin{eqnarray*} e_{r}(t,s)=\exp\bigg\{\int_{s}^t\xi_{\mu(\tau)}(r(\tau))\Delta\tau\bigg\},s,t\in{\mathbb T}, \end{eqnarray*}$

其柱变换如下

$\begin{eqnarray*} \xi_h(z)=\left\{ \begin{array}{ll} \frac{\log(1+hz)}{h},& \hbox{ $h\neq0,$} \\ z,& \hbox{ $h=0.$} \end{array} \right. \end{eqnarray*}$

定义2.2 ^[26] 设$p,q:{\mathbb T}\rightarrow{\mathbb R}$是两个回归函数, 定义

$p\oplus q=p+q+\mu pq,\ominus p=-\frac{p}{1+\mu p},p\ominus q=p\oplus(\ominus q). $

引理2.1 ^[26] 假设$p,q:{\mathbb T}\rightarrow{\mathbb R}$是两个回归函数, 那么

(1) $e_0(t,s)\equiv1$且$e_{p}(t,t)\equiv1$;

(2) $e_p(\sigma(t),s)=(1+\mu(t))p(t)e_{p}(t,t)$;

(3) $e_p(t,s)=\frac{1}{e_p(s,t)}=e_{\ominus p}(s,t)$;

(4) $e_p(t,s)e_{p}(s,r)=e_p(t,r)$;

(5) $e_p(t,s)e_{p}(s,r)=e_{p}(t,r)$;

(6) 如果$a,b,c\in{\mathbb T}$, 那么$\int_{a}^be_p(c,\sigma(t))\Delta t =e_{p}(c,a)-e_p(c,b)$.

引理2.2 ^[26] 以下结论成立:

(1) $(\nu_1f+\nu_2g)^\Delta=\nu_1f^\Delta+\nu_2g^\Delta$, 对任意常数$\nu_1$和$\nu_2$;

(2) $(fg)^\Delta(t)=f^\Delta(t)g(t)+f(\sigma(t))g^\Delta(t)=f(t)g^\Delta(t)+f^\Delta(t)g(\sigma(t))$;

(3) 如果$f^\Delta\geq0$, 那么$f$是增函数.

定义2.3 ^[26] 一个函数是$f:{\mathbb T}\rightarrow{\mathbb R}$正回归的, 如果$1+\mu(t)f(t)>0,\forall t\in {\mathbb T}$.

记${\cal R}^+$是集合${\mathbb T}$到${\mathbb R}$的正回归函数.

引理2.3 ^[26] 假设$p\in{\cal R}^+$, 那么

(1) $e_{p}(t,s)>0,$ $\forall s,t\in{\mathbb T}$;

(2) 如果$p(t)\leq q(t)$, $\forall t\geq s,t,s\in{\mathbb T}$, 那么$e_{p}(t,s)\leq e_{q}(t,s)$, $\forall t\geq s$.

定义2.4 ^[28] ${\mathbb T}$称作是概周期时标, 函数$f:{\mathbb T}\rightarrow{\mathbb R}^n$称为概周期函数, 如果$\forall\epsilon>0$, 集合$ E(\epsilon,f)=\{\tau\in\Pi:|f(t+\tau)-f(t)|<\epsilon,\forall t\in{\mathbb T}\}\neq\{0\} $在${\mathbb T}$中相对紧, 也就是说, 对$\forall\epsilon>0$, 存在实数$l(\epsilon)>0$使得每一个长度为$l(\epsilon)$的区间至少包含一个$\tau\in E(\epsilon,f)$, 有$|f(t+\tau)-f(t)|<\epsilon,\forall t\in{\mathbb T}.$则集合$E(\epsilon,f)$称作$f(t)$的$\epsilon$ -不变集, $l(\epsilon)$称作$f(t)$的$\epsilon$ -不变数.

引理2.4 ^[28] 如果$f\in C({\mathbb T},{\mathbb R})$是一个概周期函数, 那么函数$f$在${\mathbb T}$上是有界的.

引理2.5 ^[28] 如果$f,g\in C({\mathbb T},{\mathbb R})$是概周期函数, 那么$f+g,fg$也是概周期的.

定义2.5 称系统$(1.2)$是持久的, 如果对于系统$(1.2)$的每一个解$(x_1(t),x_2(t),u_1(t),$ $u_2(t))^T$, 存在正常数$m_i,q_i,M_i$和$Q_i(i=1,2)$使得

$ m_i\leq\liminf\limits_{t\rightarrow\infty}x_i(t)\leq\limsup\limits_{ t\rightarrow\infty}x_i(t)\leq M_i, $

$q_i\leq\liminf\limits_{t\rightarrow\infty}u_i(t)\leq\limsup\limits_{t\rightarrow\infty}u_i(t)\leq Q_i. $

引理2.6 ^[30] 令$-a\in {\cal R}^+.$

(1) 如果$x^\Delta(t)\leq b-ax^{\alpha}(t)$, 那么

$x(t)\leq x(t_0)e_{(-a)}(t,t_0)+\bigg(\frac{b}{a}\bigg)^\alpha(1-e_{(-a)}(t,t_0)). $

特别地, 如果$a>0,b>0$, 则对$t>t_0$有$\limsup\limits_{t\rightarrow+\infty}x(t)\leq(\frac{b}{a})^\alpha,$其中$\alpha$是正常数.

(2) 如果$x^\Delta(t)\geq b-ax^{\alpha}(t)$, 那么

$ x(t)\geq x(t_0)e_{(-a)}(t,t_0)+\bigg(\frac{b}{a}\bigg)^\alpha(1-e_{(-a)}(t,t_0)). $

特别地, 如果$a>0,b>0$, 则对$t>t_0$有$\liminf\limits_{t\rightarrow+\infty}x(t)\geq(\frac{b}{a})^\alpha$, 其中$\alpha$是正常数.

这一部分, 我们将给出系统(1.2) 持久性的充分条件.

命题 3.1 假设$(H_1)-(H_3)$成立.则系统$(1.2)$的每个解$(x_1(t),x_2(t),u_1(t),u_2(t))^T$都满足

$\begin{eqnarray*} \limsup_{t\rightarrow+\infty} x_{i}(t)\leq x_{i}^*,\;\;\limsup_{t\rightarrow+\infty} u_{i}(t)\leq u_{i}^*,i=1,2, \end{eqnarray*}$

其中

$x_{i}^*=\frac{a_{i}^M-b_{i}^l}{b_{i}^l},u_{i}^*=\frac{h_{i}^{M}+g_{i}^M\exp\{x_{i}^*\}}{f_{i}^{l}}.$

证 若$(x_1(t),x_2(t),u_1(t),u_2(t))^T$是系统$(1.2)$的任一个解, 由系统$(1.2)$的第一和第二个方程, 当$x\in {\mathbb R}$时, 应用Bernoulli不等式$\exp\{x\}\geq1+x$, 可得

$\begin{eqnarray*} x_{i}^{\Delta}(t)&\leq&a_i(t)-b_i(t)\exp\{x_{i}(t)\} \leq a_i(t)-b_i(t)(x_{i}(t)+1)\\ \leq (a_{i}^M-b_{i}^l)-b_{i}^lx_{i}(t). \end{eqnarray*}$

由引理$2.5$可得

$\begin{eqnarray} \limsup_{t\rightarrow+\infty} x_{i}(t) \leq\frac{a_{i}^M-b_{i}^l}{b_{i}^l}:=x_{i}^*. \end{eqnarray}$

(3.1)

$\forall\epsilon>0$, 存在一个$t_0\in{\mathbb T}$, 使得$x_{i}(t)\leq x_{i}^*+\epsilon,\forall t\geq t_0$.

类似地, 由系统$(1.2)$的第三和第四个方程, 可得

$\begin{eqnarray*} u_{i}^{\Delta}(t)&\leq&h_{i}(t)-f_{i}(t)u_i(t)+g_{i}(t)\exp\{x_{i}^{*}+\epsilon\}\\&\leq& h_{i}^M-f_{i}^l u_{i}(t)+g_{i}^M\exp\{x_{i}^{*}+\epsilon\},\;\;i=1,2. \end{eqnarray*}$

令$\epsilon\rightarrow0$, 由引理$2.5$可得

$\begin{eqnarray} \limsup_{t\rightarrow+\infty} u_i(t)\leq\frac{h_{i}^{M}+g_{i}^M\exp\{x_{i}^*\}}{f_{i}^{l}}:=u_{i}^*,\;\;i=1,2, \end{eqnarray}$

(3.2)

也即

$\begin{eqnarray*} u_i(t)\leq u_{i}^*+\epsilon,\;\;i=1,2. \end{eqnarray*}$

证毕.

命题 3.2 假设$(H_1),(H_3)$和$(H_4)$成立.则系统$(1.2)$的每一个解$(x_1(t),x_2(t),u_1(t),u_2(t))^T$都满足

$\begin{eqnarray*} \liminf_{t\rightarrow+\infty} x_{i}(t)\geq x_{i*},\;\;\liminf_{t\rightarrow+\infty} u_{i}(t)\geq u_{i*},i=1,2, \end{eqnarray*}$

其中$x_{1*}=\frac{a_{1}^l-c_{1}^M\exp\{x_{2}^*\}-e_{1}^Mu_{1}^*}{b_{1}^M+d_{1}^Mx_{1}^*},x_{2*}=\frac{a_{2}^l-c_{2}^M\exp\{x_{1}^* \}-e_{2}^Mu_{2}^*}{b_{2}^M+d_{2}^Mx_{2}^*},u_{i*}=\frac{h_{i}^{l}+g_{i}^l\exp\{x_{i*}\}}{f_{i}^{M}}.$

证 根据命题$3.1$, 存在一个$t_1\in{\mathbb T}$使得$x_{i}(t)\leq x_{i}^*+\epsilon,u_{i}(t)\leq u_{i}^*+\epsilon,\forall t>t_1,i=1,2$.则当$t>t_1$时, 由系统$(1.2)$的第一个方程, 可得

$\begin{eqnarray*} x_{1}^{\Delta}(t) \geq a_1(t)-b_1(t)\exp\{x_{1}(t)\}\\ -c_1(t)\exp\{x_2(t)\}-d_{1}(t)\exp\{2x_{1}(t)\}-e_1(t)u_{1}(t)\\ \geq a_{1}^l-c_{1}^M\exp\{x_{2}^*+\epsilon\}-e_{1}^M(u_{1}^*+\epsilon)\\ -[b_{1}^M+d_{1}^M(x_{1}^*+\epsilon)]\exp\{x_{1}(t)\}. \end{eqnarray*}$

假设$t>t_1$, 有

$\begin{eqnarray} a_{1}^l-c_{1}^M\exp\{x_{2}^*+\epsilon\}-e_{1}^M(u_{1}^*+\epsilon)-[b_{1}^M+d_{1}^M(x_{1}^*+\epsilon)]\exp\{x_{1}(t)\}\leq0. \end{eqnarray}$

(3.3)

另一方面, 假设存在$\tilde{t}\geq t_1$使得

$\begin{eqnarray*} a_{1}^l-c_{1}^M\exp\{x_{2}^*+\epsilon\}-e_{1}^M(u_{1}^*+\epsilon)-[b_{1}^M+d_{1}^M(x_{1}^*+\epsilon)]\exp\{x_{1}(\tilde{t})\}>0, \end{eqnarray*}$

$\forall t\in[t_1,\tilde{t})_{{\mathbb T}}$, 有

$\begin{eqnarray*} a_{1}^l-c_{1}^M\exp\{x_{2}^*+\epsilon\}-e_{1}^M(u_{1}^*+\epsilon)-[b_{1}^M+d_{1}^M(x_{1}^*+\epsilon)]\exp\{x_{1}(t)\}\leq0. \end{eqnarray*}$

因此, 有

$\begin{eqnarray*} x_{1}(\tilde{t})<\ln\frac{a_{1}^l-c_{1}^M\exp\{x_{2}^*+\epsilon\}-e_{1}^M(u_{1}^*+\epsilon)}{b_{1}^M+d_{1}^M(x_{1}^*+\epsilon)}, \end{eqnarray*}$

同理, $\forall t\in[t_1,\tilde{t})_{\mathbb T}$, 有

$\begin{eqnarray*} x_{1}(\tilde{t})>\ln\frac{a_{1}^l-c_{1}^M\exp\{x_{2}^*+\epsilon\}-e_{1}^M(u_{1}^*+\epsilon)}{b_{1}^M+d_{1}^M(x_{1}^*+\epsilon)}, \end{eqnarray*}$

可推出$x_{1}^\Delta(\tilde{t})<0$.这是矛盾的.因此, 当$t\geq t_1$时, 系统$(1.2)$成立, 即

$\begin{eqnarray} x_{1}(t)\geq\ln\frac{a_{1}^l-c_{1}^M\exp\{x_{2}^*+\epsilon\}-e_{1}^M(u_{1}^*+\epsilon)}{b_{1}^M+d_{1}^M(x_{1}^*+\epsilon)}, \end{eqnarray}$

(3.4)

令$\epsilon\rightarrow0$, 则

$\begin{eqnarray*} \liminf_{t\rightarrow+\infty} x_{1}(t)\geq \ln\frac{a_{1}^l-c_{1}^M\exp\{x_{2}^*\}-e_{1}^Mu_{1}^*}{b_{1}^M+d_{1}^Mx_{1}^*}=\ln x_{1*}. \end{eqnarray*}$

$\forall\epsilon>0$, 可得$\liminf\limits_{t\rightarrow+\infty} x_{1}(t)\geq \ln (x_{1*}-\epsilon)$.即$\forall\epsilon_1>0$, 存在$t_1\in{\mathbb T}$, 使得$x_{1}(t)\geq\ln( x_{1*}-\epsilon_1)$对$\forall t\geq t_1$都成立.

类似地, 由系统$(1.2)$的第二个方程, 可得

$\begin{eqnarray*} x_{2}^{\Delta}(t) \geq a_2(t)-b_2(t)\exp\{x_{2}(t)\}\\ -c_2(t)\exp\{x_1(t)\}-d_{2}(t)\exp\{2x_{2}(t)\}-e_2(t)u_{2}(t)\\ \geq a_{2}^l-c_{2}^M\exp\{x_{1}^*+\epsilon\}-e_{2}^M(u_{2}^*+\epsilon)-[b_{2}^M+d_{2}^M(x_{2}^*+\epsilon)]\exp\{x_{2}(t)\}. \end{eqnarray*}$

则当$t>t_2$时, 下式成立

$\begin{eqnarray} a_{2}^l-c_{2}^M\exp\{x_{1}^*+\epsilon\}-e_{2}^M(u_{2}^*+\epsilon)-[b_{2}^M+d_{2}^M(x_{2}^*+\epsilon)]\exp\{x_{2}(t)\}\leq0. \end{eqnarray}$

(3.5)

否则, 假设存在$\tilde{t}\geq t_2$, 使得

$ a_{2}^l-c_{2}^M\exp\{x_{1}^*+\epsilon\}-e_{2}^M(u_{2}^*+\epsilon)-[b_{2}^M+d_{2}^M(x_{2}^*+\epsilon)]\exp\{x_{2}(\tilde{t})\}>0,\\ \forall t\in[t_2,\tilde{t})_{{\mathbb T}}, $

可得

$ a_{2}^l-c_{2}^M\exp\{x_{1}^*+\epsilon\}-e_{2}^M(u_{2}^*+\epsilon)-[b_{2}^M+d_{2}^M(x_{2}^*+\epsilon)]\exp\{x_{2}(t)\}\leq0. $

因此

$ x_{2}(\tilde{t})<\ln\frac{a_{2}^l-c_{2}^M\exp\{x_{1}^*+\epsilon\}-e_{2}^M(u_{2}^*+\epsilon)}{b_{2}^M+d_{2}^M(x_{2}^*+\epsilon)}, $

因此, $\forall t\in[t_2,\tilde{t})_{\mathbb T}$, 有

$ x_{2}(\tilde{t})>\ln\frac{a_{2}^l-c_{2}^M\exp\{x_{1}^*+\epsilon\}-e_{2}^M(u_{2}^*+\epsilon)}{b_{2}^M+d_{2}^M(x_{2}^*+\epsilon)}, $

故有$x_{2}^\Delta(\tilde{t})<0$, 这与假设是矛盾的.因此, 当$t\geq t_2$时, 系统$(1.2)$成立, 故有下列不等式成立

$\begin{eqnarray} x_{2}(t)\geq\ln\frac{a_{2}^l-c_{2}^M\exp\{x_{1}^*+\epsilon\}-e_{2}^M(u_{2}^*+\epsilon)}{b_{2}^M+d_{2}^M(x_{2}^*+\epsilon)}. \end{eqnarray}$

(3.6)

令$\epsilon\rightarrow0$, 则

$ \liminf\limits_{t\rightarrow+\infty} x_{2}(t)\geq \ln\frac{a_{2}^l-c_{2}^M\exp\{x_{1}^* \}-e_{2}^Mu_{2}^*}{b_{2}^M+d_{2}^Mx_{2}^*}. $

$\forall\epsilon>0$, 可得$\liminf\limits_{t\rightarrow+\infty}x_{2}(t)\geq\ln (x_{2*}-\epsilon)$.因此, $\forall\epsilon_2>0$, 存在一个$t_3\in{\mathbb T}$使得$x_{2}(t)\geq\ln( x_{2*}-\epsilon_2),\forall t\geq t_3$.

由系统$(1.2)$的第三和第四个方程可得

$\begin{eqnarray*} u_{i}^\Delta(t)&\geq&h_{i}(t)-f_{i}(t)u_i(t)+g_{i}(t)\exp\{x_{i*}-\epsilon_i\}\\&\geq& h_{i}^l-f_{i}^M u_{i}(t)+g_{i}^l\exp\{x_{i*}-\epsilon_i\},\;\;i=1,2. \end{eqnarray*}$

由引理$2.5$可得

$\begin{eqnarray} \liminf_{t\rightarrow+\infty} u_i(t)\geq\frac{h_{i}^{l}+g_{i}^l\exp\{x_{i*}-\epsilon_i\}}{f_{i}^{M}},\;\;i=1,2. \end{eqnarray}$

(3.7)

令$\epsilon_i\rightarrow0(i=1,2)$, 则有

$ \liminf\limits_{t\rightarrow+\infty} u_i(t)\geq\frac{h_{i}^{l}+g_{i}^l\exp\{x_{i*}\}}{f_{i}^{M}}=u_{i*},\;\;i=1,2. $

证毕.

由命题3.1和3.2, 下面给出这一部分的主要结果:

定理 3.1 假设$(H_1)-(H_4)$成立.则系统$(1.2)$是持久的.

考虑时标上非线性概周期微分系统

$\begin{eqnarray} x^{\Delta}(t)=f(t,x),\;t\in{\mathbb T}^+, \end{eqnarray}$

(4.1)

其中$f:{\mathbb T}\times{\mathbb S}_B\rightarrow{\mathbb R},{\mathbb S}_B=\{x\in{\mathbb R}:\|x\|_0<B\},\|x\|_0=\sup_{t\rightarrow{\mathbb T}}|x(t)|,$当$x\in{\mathbb S}_B$时, 函数$f(t,x)$在$t$上一致收敛且是连续概周期的.为了寻找系统$(1.2)$的解, 考虑系统$(1.2)$的乘积系统

$ x^{\Delta}(t)=f(t,x),\;\;y^{\Delta}(t)=f(t,y). $

引理 4.1 假设对于给定的函数$f(t,x)$, 定义在${\mathbb T}^+\times {\mathbb S}_B\times{\mathbb S}_B$上的Lyapunov函数$V(t,x,y)$满足:

(ⅰ) $a(\|x-y\|_0)\leq V(t,x,y)\leq b(\|x-y\|_0)$, 其中$a,b\in \kappa,\kappa=\{a\in C({\mathbb R}^+,{\mathbb R}^+):a(0)=0,\;\;\mbox{且}\; a\;\mbox{是增函数}\}$.

(ⅱ) $|V(t,x,y)-V(t,x_1,y_1)|\leq L(\|x-x_1\|_0+\|y-y_0\|_0)$, 其中$L>0.$

(ⅲ) $D^+V^{\Delta}_{(4.1)}(t,x,y)\leq -cV(t,x,y)$, 其中$c>0$, $-c\in {\cal R}^+$.

进一步, 当$t\in {\mathbb T}^+$时, 如果存在系统$(4.1)$的一个解$x(t)\in{\mathbb S}$, 其中${\mathbb S}\subset{\mathbb S}_B$是一个紧集, 则$f(t,x)$存在唯一的一致渐近稳定概周期解$p(t)\in{\mathbb S}$.当$x\in{\mathbb S}_B$时, $f(t,x)$收敛于$x$, 那么$p(t)$也是周期的.

命题 4.1 若$(H_1)-(H_4)$成立, 则$\Omega\neq\emptyset$.

证 由于$a_i(t),b_i(t),c_i(t),d_i(t),e_i(t),f_i(t),g_i(t)$和$h_i(t),i=1,2$都是概周期函数, 则存在一个序列$\tau=\{\tau_p\}\subseteq{\mathbb T}$当$p\rightarrow+\infty$有$\tau_p\rightarrow+\infty$, 使得

$a_i(t+\tau_p)=a_i(t),\;\;\;b_i(t+\tau_p)=b_i(t),\\\;\;\;c_i(t+\tau_p)=c_i(t),\;\;\;d_i(t+\tau_p)=d_i(t),$

$e_i(t+\tau_p)=e_i(t),\;\;\;f_i(t+\tau_p)=f_i(t),\\\;\;\;g_i(t+\tau_p)=g_i(t),\;\;\;h_i(t+\tau_p)=h_i(t),$

当$p\rightarrow+\infty$, $\forall\epsilon>0$.由命题3.1和命题3.2可得, 存在$t_0\in{\mathbb T}$使得

$ x_{i*}-\epsilon\leq x_{i}(t)\leq x_{i}^*+\epsilon,u_{i*}-\epsilon\leq u_{i}(t)\leq u_{i}^*+\epsilon,\;\;\;\;t\geq t_0,i=1,2. $

当$t\geq t_0-\tau_p,p=1,2,\cdots$时, 记$x_{ip}(t)=x_{i}(t+\tau_p))$和$u_{ip}(t)=u_{i}(t+\tau_p)$.对任意的正整数$q$, 容易验证存在一个序列$\{x_{ip}(t):p\geq q \}$和$\{u_{ip}(t):p\geq q\}$使得序列$\{x_{ip}(t)\}$和$\{u_{ip}(t)\}$有子序列, 再次记作$\{x_{ip}(t)\}$和$\{u_{ip}(t)\}$, 当$p\rightarrow+\infty$时, 在${\mathbb T}$的任意区间分别都收敛.因此, 有序列$\{y_{i}(t)\}$和$\{v_{i}(t)\}$使得

$\begin{eqnarray*} x_{ip}(t)\rightarrow y_{i}(t),\;\;u_{ip}(t)\rightarrow v_{i}(t),\;\;\forall t\in{\mathbb T},\;\;p\rightarrow+\infty,i=1,2, \end{eqnarray*}$

联立方程组

$\begin{eqnarray} \left\{ \begin{array}{ll} x_{1p}^{\Delta}(t)=a_1(t+\tau_p)-b_1(t+\tau_p)\exp\{x_{1p}(t+\tau_p)\}\\-c_1(t+\tau_p)\exp\{x_{2p}(t+\tau_p)\}\\ \;\;\;\;\;\;\;\;\;\;\;\;\;-d_{1}(t+\tau_p)\exp\{{2x_{1p}(t+\tau_p)}\}-e_{1}(t+\tau_p)u_{1p}(t+\tau_p),& \hbox{} \\ x_{2p}^{\Delta}(t)=a_2(t+\tau_p)-b_2(t+\tau_p)\exp\{x_{2p}(t+\tau_p)\}\\-c_2(t+\tau_p)\exp\{x_{1p}(t+\tau_p)\}\\\;\;\;\;\;\;\;\;\;\;\;\;\;-d_{2}(t+\tau_p)\exp\{{2x_{2p}(t+\tau_p)}\}-e_{2}(t+\tau_p)u_{2p}(t+\tau_p),\hbox{} \\ u_{1p}^{\Delta}(t)=h_{1}(t+\tau_p)-f_1(t+\tau_p)u_{1p}(t+\tau_p)+g_1(t+\tau_p)\exp\{x_{1p}(t+\tau_p)\},& \hbox{}\\ u_{2p}^{\Delta}(t)=h_2(t+\tau_p)-f_2(t+\tau_p)u_{2p}(t+\tau_p)+g_2(t+\tau_p)\exp\{x_{2p}(t+\tau_p)\} , \end{array} \right. \end{eqnarray}$

(4.2)

可得

$\begin{eqnarray} \left\{ \begin{array}{ll} y_{1}^{\Delta}(t)=a_1(t)-b_1(t)\exp\{y_1(t)\}-c_1(t)\exp\{y_2(t)\}\\-d_{1}(t)\exp\{{2y_{1}(t)}\} -e_{1}(t)v_{1}(t),& \hbox{} \\ y_{2}^{\Delta}(t)=a_2(t)-b_2(t)\exp\{y_2(t)\}-c_2(t)\exp\{y_1(t)\}\\ -d_{2}(t)\exp\{{2y_{2}(t)}\} -e_{2}(t)v_{2}(t),\hbox{} \\ v_{1}^{\Delta}(t)=h_{1}(t)-f_1(t)v_{1}(t)+g_1(t)\exp\{y_1(t)\},& \hbox{}\\ v_{2}^{\Delta}(t)=h_2(t)-f_2(t)v_{2}(t)+g_2(t)\exp\{y_2(t)\}. & \hbox{} \end{array} \right. \end{eqnarray}$

(4.3)

容易看出$(Y(t),V(t))$是系统(1.2) 的一个解, $\forall \epsilon>0$时, 当$t\in{\mathbb T},i=1,2$时有$x_{i*}-\epsilon\leq y_{i}(t)\leq x_{i}^*+\epsilon,u_{i*}-\epsilon\leq v_{i}(t)\leq u_{i}^*+\epsilon,$ $x_{i*}\leq y_{i}(t)\leq x_{i}^*,u_{i*}\leq v_{i}(t)\leq u_{i}^*$.

定理 4.1 假设$(H_1)-(H_4)$成立.则系统$(1.2)$至少存在一个唯一渐近稳定概周期解$X(t)=(x_1(t),x_2(t),u_1(t),u_2(t))^T\in\Omega$.

证 由命题$4.1$, 存在$X(t)$使得$x_{i*}\leq x_{i}(t)\leq x_{i}^*,u_{i*}\leq u_{i}(t)\leq u_{i}^*,i=1,2,\forall t\in {\mathbb T}.$因此, $|x_{i}(t)|<A_i,|u_i(t)|<B_i$, 其中$A_i=\max\{|x_{i*}|,|x_{i}^*|\},B_i=\max\{|u_{i*}|,|u_{i}^*|\},i=1,2.$定义$\|X\|=\sup\limits_{t\in{\mathbb T}}\sum\limits \limits_{i=1}^2|x_{i}(t)|+\sup\limits_{t\in{\mathbb T}} \sum\limits_{i=1}^2|u_{i}(t)|,(u,v)\in {\mathbb R}^{2\times2}$.假若$X_1=(x(t),u(t)),X_2=(y(t),v(t))$是系统$(1.2)$任意两个解, 则可得

$\begin{eqnarray} \left\{ \begin{array}{ll} x_{1}^{\Delta}(t)=a_1(t)-b_1(t)\exp\{x_1(t)\}-c_1(t)\exp\{x_2(t)\}-d_{1}(t)\exp\{{2x_{1}(t)}\}\\ -e_{1}(t)u_{1}(t),& \hbox{} \\ x_{2}^{\Delta}(t)=a_2(t)-b_2(t)\exp\{x_2(t)\}-c_2(t)\exp\{x_1(t)\}\\ -d_{2}(t)\exp\{{2x_{2}(t)}\} -e_{2}(t)u_{2}(t),\hbox{} \\ u_{1}^{\Delta}(t)=h_{1}(t)-f_1(t)u_{1}(t)+g_1(t)\exp\{x_1(t)\},& \hbox{}\\ u_{2}^{\Delta}(t)=h_2(t)-f_2(t)u_{2}(t)+g_2(t)\exp\{x_2(t)\},& \hbox{}\\ y_{1}^{\Delta}(t)=a_1(t)-b_1(t)\exp\{y_1(t)\}-c_1(t)\exp\{y_2(t)\}\\ -d_{1}(t)\exp\{{2y_{1}(t)}\} -e_{1}(t)v_{1}(t),& \hbox{} \\ y_{2}^{\Delta}(t)=a_2(t)-b_2(t)\exp\{y_2(t)\}-c_2(t)\exp\{y_1(t)\}\\ -d_{2}(t)\exp\{{2y_{2}(t)}\} -e_{2}(t)v_{2}(t),\hbox{} \\ v_{1}^{\Delta}(t)=h_{1}(t)-f_1(t)v_{1}(t)+g_1(t)\exp\{y_1(t)\},& \hbox{}\\ v_{2}^{\Delta}(t)=h_2(t)-f_2(t)v_{2}(t)+g_2(t)\exp\{y_2(t)\}. & \hbox{} \end{array} \right. \end{eqnarray}$

(4.4)

考虑在${\mathbb T}^+\times\Omega\times\Omega$上的Lyapunov函数$V(t,X_1,X_2)$定义为

$\begin{eqnarray*} V(t,X_1,X_2)&=&\sum\limits_{i=1}^2(x_i(t)-y_i(t))^2+\sum\limits_{i=1}^2(u_i(t)-v_i(t))^2\\ &=&(x_1-y_1)^2+(x_2-y_2)^2+(u_1-v_1)^2+(u_2-v_2)^2. \end{eqnarray*}$

容易知道范数$\|X_1-X_2\|=\sup\limits_{t\in{\mathbb T}}\sum\limits_{i=1}^2|x_i(t)-y_i(t)|+\sup\limits_{t\in{\mathbb T}}\sum\limits_{i=1}^2|u_i(t)-v_i(t)|$和范数$\|X_1-X_2\|_*=\sup\limits_{t\in{\mathbb T}}\sqrt{\sum\limits_{i=1}^2(x_i(t)-y_i(t))^2+(u_i(t)-v_i(t))^2}$是等价的, 也即, 存在两个常数$C_1>0,C_2>0$使得$C_1\|X_1-X_2\|\leq\|X_1-X_2\|_*\leq C_2\|X_1-X_2\|$.因此, $(C_1\|X_1-X_2\|)^2\leq V(t,X_1,X_2)\leq(C_2\|X_1-X_2\|)^2$.若$a,b\in C({\mathbb R}^+,{\mathbb R}^+),a(x)=C_{1}^2x^2, b(x)=C_{2}^2x^2$, 因此, 引理$4.1$中的条件(ⅰ)是满足的.除此之外, 有

$\begin{equation} |V(t,X_1,X_2)-V(t,X_{1}^*,X_{2}^*)|\\ =\left|\sum\limits_{i=1}^2[(x_{i}(t)-y_{i}(t))^2+(u_{i}(t)-v_{i}(t))^2]-\sum\limits_{i=1}^2[(x_{i}^*(t)-y_{i}^*(t))^2\\+(u_{i}^*(t)-v_{i}^*(t))^2]\right|\\ \leq\left|\sum\limits_{i=1}^2[(x_{i}(t)-y_{i}(t))^2-(x_{i}^*(t)-y_{i}^*(t))^2]+\sum\limits_{i=1}^2[(u_{i}(t)-v_{i}(t))^2\\-(u_{i}^*(t)-v_{i}^*(t))^2]\right|\\ \leq\sum\limits_{i=1}^2\left|(x_i(t)-y_i(t))-(x_{i}^*(t)-y_{i}^*(t))\right|\left|(x_{i}(t)-y_i(t))+(x_{i}^*(t)-y_{i}^*(t))\right|\\ +\sum\limits_{i=1}^2\left|(u_i(t)-v_i(t))-(u_{i}^*(t)-v_{i}^*(t))\\\right|\left|(u_{i}(t)-v_i(t))+(u_{i}^*(t)-v_{i}^*(t))\right|\\ \leq\sum\limits_{i=1}^2\left|(x_i(t)-y_i(t))-(x_{i}^*(t)-y_{i}^*(t))\right|(\left|x_{i}(t)\right|+\left|y_{i}(t)\right|+\left|x_{i}^*(t)\right|+\left|y_{i}^*(t)\right|)\\ +\sum\limits_{i=1}^2\left|(u_i(t)-v_i(t))-(u_{i}^*(t)-v_{i}^*(t))\\\right|\big(\left|u_{i}(t)\right|+\left|v_{i}(t)\right|+\left|u_{i}^*(t)\right|+\left|v_{i}^*(t)\right|\big)\\ \leq L\sum\limits_{i=1}^2\{\left|x_{i}(t)-x_{i}^*(t)\right|+\left|u_{i}(t)-u_{i}^*(t)\right|+\left|y_{i}(t)-y_{i}^*(t)\right|+\left|v_{i}(t)-v_{i}^*(t)\right|\}\\ \leq L\big(\|X_1-X_{1}^*\|+\|X_2-X_{2}^*\|\big), \end{equation}$

其中$(X_{1}^*,X_{2}^*)=(x_{1}^*,x_{2}^*,u_{1}^*,u_{2}^*),L=4\max\{A_i,B_i:i=1,2\}.$故引理$4.1$的条件(ⅱ)满足.计算$V(t,X_1,X_2)$沿系统$(4.6)$的右上导数$D^+V^{\Delta}:$

$\begin{equation} D^+V^{\Delta}(t,X_1,X_2) =\sum\limits_{i=1}^2[2(x_{i}(t)-y_{i}(t))\\+\mu(x_{i}(t)-y_{i}(t))^{\Delta}](x_{i}(t)-y_{i}(t))^{\Delta}\\ +\sum\limits_{i=1}^2[2(u_{i}(t)-v_{i}(t))+\mu(u_{i}(t)-v_{i}(t))^{\Delta}](u_{i}(t)-v_{i}(t))^{\Delta}\\ =V_1+V_2+V_3+V_4,\end{equation}$

其中

$\begin{eqnarray*} &&V_1=[2(x_{1}(t)-y_{1}(t))+\mu(x_{1}(t)-y_{1}(t))^{\Delta}](x_{1}(t)-y_{1}(t))^{\Delta},\\ &&V_2=[2(x_{2}(t)-y_{2}(t))+\mu(x_{2}(t)-y_{2}(t))^{\Delta}](x_{2}(t)-y_{2}(t))^{\Delta},\\ &&V_3=[2(u_{1}(t)-v_{1}(t))+\mu(u_{1}(t)-v_{1}(t))^{\Delta}](u_{1}(t)-v_{1}(t))^{\Delta},\\ &&V_2=[2(u_{2}(t)-v_{2}(t))+\mu(u_{2}(t)-v_{2}(t))^{\Delta}](u_{2}(t)-v_{2}(t))^{\Delta}. \end{eqnarray*}$

考虑系统$(4.4)$, 当$i=1,2$时有

$\begin{eqnarray} \left\{ \begin{array}{ll} (x_{1}(t)-y_1(t))^{\Delta}=-b_1(t)(\exp\{x_{1}(t)\}-\exp\{y_1(t)\})-c_1(t)(\exp\{x_2(t)\}\\ -\exp\{y_2(t)\})\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;-d_1(t)\exp\{2x_1(t)\}-\exp\{2y_1(t)\} -e_1(t)(u_1(t)-v_1(t)) ,& \hbox{} \\ (x_{2}(t)-y_2(t))^{\Delta}=-b_2(t)(\exp\{x_{2}(t)\}-\exp\{y_2(t)\})-c_2(t)(\exp\{x_1(t)\}\\ -\exp\{y_1(t)\})\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;-d_2(t)\exp\{2x_2(t)\}-\exp\{2y_2(t)\} -e_2(t)(u_2(t)-v_2(t)),& \hbox{} \\ (u_{1}(t)-v_1(t))^{\Delta}=-f_1(t)(u_{1}(t)-v_1(t))+g_1(t)(\exp\{x_{1}(t)\}\\-\exp\{y_1(t)\}),& \hbox{}\\ (u_{2}(t)-v_2(t))^{\Delta}=-f_2(t)(u_{2}(t)-v_2(t))+g_2(t)(\exp\{x_{2}(t)\}\\-\exp\{y_2(t)\}).& \hbox{} \end{array} \right. \end{eqnarray}$

(4.5)

应用微分中值定理可得$\exp\{x_i(t)\}-\exp\{y_i(t)\}=\xi_i(t)(x_i(t)-y_i(t)),$ $\exp\{2x_i(t)\}-\exp\{2y_i(t)\}$ $=2\eta_i(t)(x_i(t)-y_i(t)),$其中$\xi_i$和$\eta_i$分别介于$x_i(t)$和$y_i(t)$, $2x_i(t)$和$2y_i(t)$之间, 且$\xi_i=\exp\{\xi_i(t)\},\eta_i=\exp\{2\eta_i(t)\}$, $i=1,2$.若$\bar{x}_{i}^{\Delta}(t)=(x_{i}(t)-y_i(t))^{\Delta},\bar{u}_{i}^{\Delta}(t)=(u_{i}(t)-v_i(t))^{\Delta}$ $(i=1,2),$则系统$(4.5)$化简为

$\begin{eqnarray} \left\{ \begin{array}{ll} \bar{x}_{1}^{\Delta}(t)=-b_1(t)\xi_{1}(t)\bar{x}_1(t)-c_1(t)\xi_2(t)\bar{x}_{2}(t)-2d_1(t)\eta_1(t)\bar{x}_1(t)\\ -e_1(t)\bar{u}_1(t),& \hbox{} \\ \bar{x}_{2}^{\Delta}(t)=-b_2(t)\xi_{2}(t)\bar{x}_2(t)-c_2(t)\xi_1(t)\bar{x}_{1}(t)-2d_2(t)\eta_2(t)\bar{x}_2(t)\\ -e_2(t)\bar{u}_2(t), & \hbox{} \\ \bar{u}_{1}^{\Delta}(t)=-f_1(t)\bar{u}_{1}(t)+g_1(t)\xi_{1}(t)\bar{x}_1(t),& \hbox{}\\ \bar{u}_{2}^{\Delta}(t)=-f_2(t)\bar{u}_{2}(t)+g_2(t)\xi_{2}(t)\bar{x}_2(t).& \hbox{} \end{array} \right. \end{eqnarray}$

(4.6)

因此

$\begin{eqnarray*} V_1&=&[2(x_{1}(t)-y_{1}(t))+\mu(t)(x_{1}(t)-y_{1}(t))^{\Delta}](x_{1}(t)-y_{1}(t))^{\Delta} \\&= &[2\bar{x}_1(t)+\mu(t)(-b_1(t)\xi_{1}(t)\bar{x}_1(t) -c_1(t)\xi_2(t)\bar{x}_{2}(t)-2d_1(t)\eta_1(t)\bar{x}_1(t)\\ & &-e_1(t)\bar{u}_1(t))][-b_1(t)\xi_{1}(t)\bar{x}_1(t) -c_1(t)\xi_2(t)\bar{x}_{2}(t)-2d_1(t)\eta_1(t)\bar{x}_1(t)\\ & & -e_1(t)\bar{u}_1(t)\big] \\ &=&[\mu(t)b_{1}^2(t)\xi_{1}^2(t)+4\mu(t) b_1(t)d_1(t)\xi_1(t)\eta_1(t)+4\mu(t) d_{1}^2(t)\eta_{1}^2(t) \\ & &-2b_1(t)\xi_1(t)-4d_1(t)\eta_1(t)]\bar{x}_1^2(t)+\mu(t) c_{1}^2(t)\xi_{2}^2(t)\bar{x}_{2}^2(t)+[-2c_{1}(t)\xi_2(t) \\ & &+2\mu(t) b_1(t)c_1(t)\xi_1(t)\xi_2(t)+4\mu(t) c_1(t)d_1(t)\xi_2(t)\eta_1(t)]\bar{x}_1(t)\bar{x}_2(t) \\ & & +[2\mu b_1(t)e_1(t)\xi_1(t)-2e_1(t)+4\mu(t) d_1(t)e_1(t)\eta_1(t)]\bar{x}_1(t)\bar{u}_1(t) \\ & &+2\mu(t)c_1(t)e_1(t)\xi_2(t)\bar{x}_2(t)\bar{u}_1(t)+\mu(t)e_{1}^2(t)\bar{u}_{1}^2 \\& \leq&[\mu(t)b_{1}^2(t)\xi_{1}^2(t)+4\mu(t) b_1(t)d_1(t)\xi_1(t)\eta_1(t)+4\mu(t) d_{1}^2(t)\eta_{1}^2(t) \\ & &-2b_1(t)\xi_1(t)-4d_1(t)\eta_1(t)]\bar{x}_{1}^2(t)+\mu(t) c_{1}^2(t)\xi_{2}^2(t)\bar{x}_{2}^2(t) \\ & & +|\mu(t) b_1(t)c_1(t)\xi_1(t)\xi_2(t)-c_{1}(t)\xi_2(t)+2\mu(t) c_1(t)d_1(t)\xi_2(t)\eta_1(t)|(\bar{x}_{1}^2(t) \\ & &+\bar{x}_{2}^2(t))+ |\mu(t)b_1(t)e_1(t)\xi_1(t)-e_1(t)+2\mu(t) d_1(t)e_1(t)\eta_1(t)|(\bar{x}_{1}^2(t) \\ & &+\bar{u}_{1}^2(t))+|\mu(t)c_1(t)e_1(t)\xi_2(t)|(\bar{x}_{2}^2(t)+\bar{u}_{1}^2(t))+\mu(t)e_{1}^2(t)\bar{u}_{1}^2\\ & \leq&|\mu(t)b_{1}^2(t)\xi_{1}^2(t)+4\mu(t) b_1(t)d_1(t)\xi_1(t)\eta_1(t)+4\mu(t) d_{1}^2(t)\eta_{1}^2(t)-2b_1(t) \\ & &\times\xi_1(t)-4d_1(t)\eta_1(t)+\mu(t) b_1(t)c_1(t)\xi_1(t)\xi_2(t) +2\mu(t) c_1(t)d_1(t)\xi_2(t)\\ & & \times\eta_1(t)-c_{1}(t)\xi_2(t)-e_1(t)+2\mu(t) d_1(t)e_1(t)\eta_1(t) +\mu(t)b_1(t)e_1(t)\xi_1(t)| \\ & & \times\bar{x}_{1}^2(t)+|\mu(t) c_{1}^2(t)\xi_{2}^2(t)+\mu(t) b_1(t)c_1(t)\xi_1(t)\xi_2(t)-c_{1}(t)\xi_2(t) \\ & & +2\mu(t) c_1(t)d_1(t)\xi_2(t)\eta_1(t)+\mu(t)c_1(t)e_1(t)\xi_2(t)|\bar{x}_{2}^2(t)+ |\mu(t)e_{1}^2(t) \\ & &+\mu(t)b_1(t)e_1(t)\xi_1(t)-e_1(t)+2\mu(t) d_1(t)e_1(t)\eta_1(t)\\ &&+\mu(t)c_1(t)e_1(t)\xi_2(t)|\bar{u}_{1}^2(t)\\ &\leq&|A_1|\bar{x}_{1}^2+|B_1|\bar{x}_{2}^2+|C_1|\bar{u}_{1}^2, \end{eqnarray*}$

其中

$\begin{eqnarray*} A_1&=&\mu(t)b_{1}^{M2}\xi_{1}^{M2}+4\mu(t) b_{1}^Md_{1}^M\xi_{1}^M\eta_{1}^M+4\mu(t) d_{1}^{M2}\eta_{1}^{M2}-2b_{1}^l\xi_{1}^l -4d_{1}^l\eta_{1}^l \\& &+\mu(t) b_{1}^Mc_{1}^M\xi_{1}^M\xi_{2}^M-c_{1}^l\xi_{2}^l +2\mu(t) c_{1}^Md_{1}^M\xi_{2}^M\eta_{1}^M-e_{1}^l+2\mu(t) d_{1}^Me_{1}^M\eta_{1}^M\\& &+\mu(t)b_{1}^Me_{1}^M\xi_{1}^M,\\ B_1&=&\mu(t) c_{1}^{M2}\xi_{2}^{M2}+\mu(t) b_{1}^Mc_{1}^M\xi_{1}^M\xi_{2}^M-c_{1}^l\xi_{2}^l +2\mu(t) c_{1}^Md_{1}^M\xi_{2}^M\eta_{1}^M \\& &+\mu(t)c_{1}^Me_{1}^M\xi_{2}^M,\\ C_1&=&\mu(t)e_{1}^{M2}+\mu(t)b_{1}^Me_{1}^M\xi_{1}^M-e_{1}^l +2\mu(t) d_{1}^Me_{1}^M\eta_{1}^M+\mu(t)c_{1}^Me_{1}^M\xi_{2}^M. \end{eqnarray*}$

$\begin{eqnarray*} V_2&=&[2(x_{2}(t)-y_{2}(t))+\mu(x_{2}(t)-y_{2}(t))^{\Delta}](x_{2}(t)-y_{2}(t))^{\Delta}\\ & =&[2\bar{x}_2(t)+\mu(t)(-b_2(t)\xi_2(t)\bar{x}_2(t)-c_2(t)\xi_1(t) -2d_2(t)\eta_2(t)\bar{x}_2(t)\\& &-e_2(t)\bar{u}_2(t))](-b_2(t)\xi_2(t)\bar{x}_2(t)-c_2(t)\xi_1(t) -2d_2(t)\eta_2(t)\bar{x}_2(t)\\& &-e_2(t)\bar{u}_2(t)) \\&=&[-2b_2(t)\xi_2(t)-4d_2(t)\eta_2(t)+\mu(t)b_{2}^2(t)\xi_{2}^2(t) +4\mu(t)b_{2}(t)d_2(t)\xi_2(t)\\ & &\times\eta_2(t)+4\mu(t)d_{2}^{2}(t)\eta_{2}^2(t)]\bar{x}_{2}^2(t) +\mu(t)c_{2}^2(t)\xi_{1}^2(t)\bar{x}_{1}^2(t)+\mu(t)e_{2}^2(t)\bar{u}_{2}^2(t)\\ & &+[4\mu(t)c_{2}(t)d_2(t)\xi_1(t)\eta_2(t)+2\mu(t)b_2(t)c_2(t)\xi_1(t)\xi_2(t) \\ & &-2c_2(t)\xi_1(t)]\bar{x}_1(t)\bar{x}_2(t)+2\mu(t)c_2(t)e_2(t)\xi_1(t)\bar{x}_1(t)\bar{u}_2(t) \\ & &+[4\mu(t)d_{2}(t)e_2(t)\eta_2(t)+2\mu(t)b_2(t)e_2(t)\xi_2(t)-2e_{2}(t)]\bar{x}_2(t)\bar{u}_2(t) \\&\leq&\mu(t)c_{2}^2(t)\xi_{1}^2(t)\bar{x}_{1}^2(t)+ [-2b_2(t)\xi_2(t)-4d_2(t)\eta_2(t)+\mu(t)b_{2}^2(t)\xi_{2}^2(t)\\ & &+4\mu(t)b_{2}(t)d_2(t)\xi_2(t)\eta_2(t)+4\mu(t)d_{2}^{2}(t)\eta_{2}^2(t)]\bar{x}_{2}^2(t)+\mu(t)e_{2}^2(t)\bar{u}_{2}^2(t)\\ & &+|2\mu(t)c_{2}(t)d_2(t)\xi_1(t)\eta_2(t) +\mu(t)b_2(t)c_2(t)\xi_1(t)\xi_2(t)\\ & &-c_2(t)\xi_1(t)|(\bar{x}_{1}^2(t)+\bar{x}_{2}^2(t)) +|\mu(t)c_{2}(t)e_{2}(t)\xi_{1}(t)|(\bar{x}_{1}^2(t)+\bar{u}_{2}^2(t))\\ & & +|2\mu(t)d_{2}(t)e_2(t)\eta_2(t)+\mu(t)b_2(t)\xi_2(t)-e_{2}(t)|(\bar{x}_{2}^2(t)+\bar{u}_{2}^2(t)) \\ &\leq&|A_2|\bar{x}_{1}^2(t)+ |B_2|\bar{x}_{2}^2(t)+|D_1|\bar{u}_2^2(t),\end{eqnarray*}$

其中

$\begin{eqnarray*} A_2&=&\mu(t)c_{2}^{M2}\xi_{1}^{M2}+2\mu(t)c_{2}^Md_{2}^M\xi_{1}^M\eta_{2}^M+\mu(t)b_{2}^Mc_{2}^M\xi_{1}^M\xi_{2}^M-c_{2}^l\xi_{1}^l\\ &&+\mu(t)c_{2}^Me_{2}^M\xi_{1}^M,\\ B_2&=&2\mu(t)c_{2}^Md_{2}^M\xi_{1}^M\eta_{2}^M+\mu(t)b_{2}^Mc_{2}^M\xi_{1}^M\xi_{2}^M-c_{2}^l\xi_{1}^l-2b_{2}^l\xi_{2}^l-4d_{2}^l\eta_{2}^l\\ &&+\mu(t)b_{2}^{M2}\xi_{2}^{M2}\\ & &+4\mu(t)b_{2}^Md_{2}^M\xi_{2}^M\eta_{2}^M+4\mu(t)d_{2}^{M2}\eta_{2}^{M2}+2\mu(t)d_{2}^Me_{2}^M\eta_{2}^M\\ &&+\mu(t)b_{2}^Me_{2}^M\xi_{2}^M-e_{2}^l,\\ D_1&=&\mu(t)c_{2}^Me_{2}^M\xi_{1}^M+2\mu(t)d_{2}^Me_{2}^M\eta_{2}^M+\mu(t)b_{2}^Me_{2}^M\xi_{2}^M-e_{2}^l+\mu(t)e_{2}^{M2}. \end{eqnarray*}$

$\begin{eqnarray*} V_3&=&[2(u_{1}(t)-v_{1}(t))+\mu(u_{1}(t)-v_{1}(t))^{\Delta}](u_{1}(t)-v_{1}(t))^{\Delta}\\ &=&[2\bar{u}_1(t)+\mu(t)(-h_1(t)\bar{u}_1(t)+f_1(t)\xi_{1}(t)\bar{x}_1(t))]\\& &\times(-h_1(t)\bar{u}_1(t)+f_1(t)\xi_{1}(t)\bar{x}_1(t)) \\&=&-2h_1(t)\bar{u}_{1}^2(t)+\mu(t)h_{1}^2(t)\bar{u}_{1}^2(t)-\mu(t)f_1(t)\xi_1(t)h_1(t)\bar{x}_1(t)\bar{u}_1(t)\\& &+2f_{1}(t)\xi_{1}(t)\bar{x}_1(t)\bar{u}_1(t)-\mu(t)f_1(t)\xi_1(t)h_1(t)\bar{x}_1(t)\bar{u}_1(t)\\& & +\mu(t)f_{1}^2(t)\xi_{1}^2(t)\bar{x}_{1}^2(t) \\ &=&(\mu(t)h_{1}^2(t)-2h_1(t))\bar{u}_{1}^2(t)+\mu(t)f_{1}^2(t)\xi_{1}^2(t)\bar{x}_{1}^2(t)\\& &+(2f_{1}(t)\xi_{1}(t)-2\mu(t)f_1(t)\xi_1(t)h_1(t))\bar{x}_1(t)\bar{u}_1(t)\\ &\leq& (\mu(t)h_{1}^2(t)-2h_{1}(t))\bar{u}_{1}^2(t)+\mu(t)f_{1}^2(t){\xi_{1}}^2(t)\bar{x}_{1}^2(t)\\& &+|f_{1}(t)\xi_{1}(t)-\mu(t)f_1(t)\xi_{1}(t)h_1(t)|(\bar{x}_{1}^2(t)+\bar{u}_{1}^2(t)) \\&=&|A_3|\bar{x}_{1}^2(t)+|C_2|\bar{u}_{1}^2(t),\end{eqnarray*}$

其中

$\begin{eqnarray*} A_3&=&\mu(t){f_{1}^M}^2{\xi_{1}^M}^2+f_{1}^M\xi_{1}^M-\mu(t)f_1^l\xi_{1}^lh_1^l,\\ C_2&=&\mu(t)h_{1}^{M2}-2h_{1}^l+f_{1}^M\xi_{1}^M-\mu(t)f_1^l\xi_{1}^lh_1^l. \end{eqnarray*}$

$\begin{eqnarray*} V_4&=&[2(u_{2}(t)-v_{2}(t))+\mu(u_{2}(t)-v_{2}(t))^{\Delta}](u_{2}(t)-v_{2}(t))^{\Delta}\\ &=&[2\bar{u}_2(t)+\mu(t)(-h_2(t)\bar{u}_2(t)+f_2(t)\xi_{2}(t)\bar{x}_2(t))(-h_2(t)\bar{u}_2(t)\\ &&+f_2(t)\xi_{2}(t)\bar{x}_2(t)) \\&=&-2h_2(t)\bar{u}_{2}^2(t)+\mu(t)h_{2}^2(t)\bar{u}_{2}^2(t)-\mu(t)f_2(t)\xi_2(t)h_2(t)\bar{x}_2(t)\bar{u}_2(t)\\ & &+2f_{2}(t)\xi_{2}(t)\bar{x}_2(t)\bar{u}_2(t)-\mu(t)f_2(t)\xi_2(t)h_2(t)\bar{x}_2(t)\bar{u}_2(t)\\ &&+\mu(t)f_{2}^2(t)\xi_{2}^2(t)\bar{x}_{2}^2(t) \\ &=&(\mu(t)h_{2}^2(t)-2h_2(t))\bar{u}_{2}^2(t)+\mu(t)f_{2}^2(t)\xi_{2}^2(t)\bar{x}_{2}^2(t)\\& &+(2f_{2}(t)\xi_{2}(t)-2\mu(t)f_2(t)\xi_2(t)h_2(t))\bar{x}_2(t)\bar{u}_2(t)\\ &\leq& (\mu(t){h_{2}}^2(t)-2h_{2}(t))\bar{u}_{2}^2(t) +\mu(t){f_{2}}^2(t){\xi_{2}}^2(t)\bar{x}_{2}^2(t) +|f_{2}(t)\xi_{2}(t)\\& &-\mu(t)f_2(t)\xi_{2}(t)h_2(t)|(\bar{x}_{2}^2(t)+\bar{u}_{2}^2(t))\\ &=& |B_3|\bar{x}_{2}^2(t)+|D_2|\bar{u}_{2}^2(t),\end{eqnarray*}$

其中

$\begin{eqnarray*} B_3&=&\mu(t){f_{2}^M}^2{\xi_{2}^M}^2+f_{2}^M\xi_{2}^M-\mu(t)f_2^l\xi_{2}^lh_2^l, \\D_2&=&\mu(t){h_{2}^M}^2-2h_{2}^l+f_{2}^M\xi_{2}^M-\mu(t)f_2^l\xi_{2}^lh_2^l. \end{eqnarray*}$

因此

$\begin{eqnarray*} V^{\Delta}&=&V_1+V_2+V_3+V_4\\ &\leq&|A_1|\bar{x}_{1}^2+|B_1|\bar{x}_{2}^2+|C_1|\bar{u}_{1}^2+|A_2|\bar{x}_{1}^2(t)+|B_2|\bar{x}_{2}^2(t)+|D_1|\bar{u}_2^2(t)\\ &&+|A_3|\bar{x}_{1}^2(t)\\ & &+|C_2|\bar{u}_{1}^2(t)+|B_3|\bar{x}_{2}^2(t)+|D_2|\bar{u}_{2}^2(t)\\ &\leq&|A_1+A_2+A_3|\bar{x}_1^2(t)+|B_1+B_2+B_3|\bar{x}_2^2(t)+|C_1+C_2|\bar{u}_1^2(t)+|D_1\\ &&+D_2|\bar{u}_2^2(t)\\&\leq& -[(E_1-\mu(t)F_1)\bar{x}_{1}^2(t)+(E_2-\mu(t)F_2)\bar{x}_{2}^2(t)+(E_3-\mu(t)F_3)\bar{u}_{1}^2(t)\\& &+(E_4-\mu(t)F_4)\bar{u}_{2}^2(t)] \\&\leq& -\Theta(\bar{x}_{1}^2(t)+\bar{x}_{2}^2(t)+\bar{u}_{1}^2(t)+\bar{u}_{2}^2(t))\\&=& -\Theta V(t,X_1,X_2),\end{eqnarray*}$

其中

$\begin{eqnarray*} \Theta&=&\min\{E_1-\mu F_1 ,E_2-\mu F_2 ,E_3-\mu F_3 ,E_4-\mu F_4 \},\\ E_1&=&2b_{1}^l\xi_{1}^l +4d_{1}^l\eta_{1}^l+c_{1}^l\xi_{2}^l +e_{1}^l+c_{2}^l\xi_{1}^l-f_{1}^M\xi_{1}^M,\\ F_1&=&b_{1}^{M2}\xi_{1}^{M2}+4 b_{1}^Md_{1}^M\xi_{1}^M\eta_{1}^M +4d_{1}^{M2}\eta_{1}^{M2} +b_{1}^Mc_{1}^M\xi_{1}^M\xi_{2}^M +2c_{1}^Md_{1}^M\xi_{2}^M\eta_{1}^M\\ & &+b_{1}^Me_{1}^M\xi_{1}^M +c_{2}^{M2}\xi_{1}^{M2}+2c_{2}^Md_{2}^M\xi_{1}^M\eta_{2}^M +b_{2}^Mc_{2}^M\xi_{1}^M\xi_{2}^M\\ & &+c_{2}^Me_{2}^M\xi_{1}^M+{f_{1}^M}^2{\xi_{1}^M}^2 \\ & &+2 d_{1}^Me_{1}^M\eta_{1}^M-f_1^l\xi_{1}^lh_1^l, \\ E_2&=&c_{1}^l\xi_{2}^l +c_{2}^l\xi_{1}^l+2b_{2}^l\xi_{2}^l+4d_{2}^l\eta_{2}^l +e_{2}^l-f_{2}^M\xi_{2}^M,\\ F_2&=&c_{1}^{M2}\xi_{2}^{M2}+ b_{1}^Mc_{1}^M\xi_{1}^M\xi_{2}^M +2c_{1}^Md_{1}^M\xi_{2}^M\eta_{1}^M +c_{1}^Me_{1}^M\xi_{2}^M+2c_{2}^Md_{2}^M\xi_{1}^M\eta_{2}^M \\ & &+b_{2}^Mc_{2}^M\xi_{1}^M\xi_{2}^M+b_{2}^{M2}\xi_{2}^{M2}+4b_{2}^Md_{2}^M\xi_{2}^M\eta_{2}^M+4d_{2}^{M2}\eta_{2}^{M2}+2d_{2}^Me_{2}^M\eta_{2}^M\\ & &+b_{2}^Me_{2}^M\xi_{2}^M\\ & &+{f_{2}^M}^2{\xi_{2}^M}^2-f_2^l\xi_{2}^lh_2^l,\\ E_3&=&e_{1}^l+2h_{1}^l-f_{1}^M\xi_{1}^M,\\ F_3&=&e_{1}^{M2}+b_{1}^Me_{1}^M\xi_{1}^M +2d_{1}^Me_{1}^M\eta_{1}^M+c_{1}^Me_{1}^M\xi_{2}^M+h_{1}^{M2}-f_1^l\xi_{1}^lh_1^l, \\ E_4&=&e_{2}^l+2h_{2}^l-f_{2}^M\xi_{2}^M, \end{eqnarray*}$

$\begin{eqnarray*} F_4&=&c_{2}^Me_{2}^M\xi_{1}^M+2d_{2}^Me_{2}^M\eta_{2}^M+b_{2}^Me_{2}^M\xi_{2}^M+e_{2}^{M2}+{h_{2}^M}^2 -f_2^l\xi_{2}^lh_2^l,\\ \xi_{i}^l&=&\exp\{x_{i*}\},\xi_{i}^M=\exp\{x_{i}^*\},\eta_{i}^l=\exp\{x_{i*}\},\eta_{i}^M=\exp\{x_{i}^*\}. \end{eqnarray*}$

证毕.

本部分, 我们给出一个例子证明主要结果的有效性.

例 考虑时标上带有反馈控制两种群竞争系统

$\begin{eqnarray} \left\{ \begin{array}{ll} x_{1}^{\Delta}(t)=a_1(t)-b_1(t)\exp\{x_1(t)\}-c_1(t)\exp\{x_2(t)\}-d_{1}(t)\exp\{{2x_{1}(t)}\}\\ -e_{1}(t)u_{1}(t),& \hbox{} \\ x_{2}^{\Delta}(t)=a_2(t)-b_2(t)\exp\{x_2(t)\}-c_2(t)\exp\{x_1(t)\} -d_{2}(t)\exp\{{2x_{2}(t)}\}\\ -e_{2}(t)u_{2}(t),\hbox{} \\ u_{1}^{\Delta}(t)=h_{1}(t)-f_1(t)u_{1}(t)+g_1(t)\exp\{x_1(t)\},& \hbox{}\\ u_{2}^{\Delta}(t)=h_2(t)-f_2(t)u_{2}(t)+g_2(t)\exp\{x_2(t)\},& \hbox{} \end{array} \right. \end{eqnarray}$

(5.1)

其中

$\begin{eqnarray*} &&a_1(t)=0.9+0.3\cos(2t), b_1(t)=0.6+0.02\sin(2t),\\&& a_2(t)=1.25-0.2\cos(2t),\,\, b_2(t)=0.8+0.06\sin(2t), \\&&c_1(t)=0.005+0.001\cos(2t),\;\; \,\,c_2(t)=0.005+0.002\sin(2t), \\&&d_1(t)=0.004+0.002\sin(2t), d_2(t)=0.0005+0.0001\cos(2t),\\&& e_1(t)=0.01+0.01\sin(2t), e_2(t)=0.0045+0.0001\cos(2t),\\&& h_1(t)=0.06+0.0001\sin(\sqrt2t),\, h_2(t)=0.05+0.0002\cos(\sqrt3t),\\&& f_1(t)=0.04+0.001\cos(\sqrt2t),\,\,\,f_2(t)=0.02-0.003\sin(\sqrt3t),\\&& g_1(t)=0.009+0.004\cos(\sqrt2t),\, g_2(t)=0.004+0.0004\sin(\sqrt3t). \end{eqnarray*}$

令${\mathbb T}=\bigcup\limits_{k=0}^\infty[k,k+0.85].$则系统$(5.1)$是持久的并且存在唯一渐近稳定概周期解.

证 根据已知条件验证假设, 利用数学软件计算可得

$\begin{eqnarray} \mu(t)=\left\{ \begin{array}{ll} 0,& t\in \bigcup\limits_{k=0}^\infty[k,k+0.85),\\ 0.15,& t\in \bigcup\limits_{k=0}^\infty\{k+0.85\}, \end{array} \right. \end{eqnarray}$

(5.2)

$x_{1*}=0.0903,x_{1}^*=0.6034,x_{2*}=0.0039,x_{2}^*=0.9595,$

$u_{1*}=1.5945,u_{1}^*=2.1505,u_{2*}=2.3224,u_{2}^*=3.6285,$

$a_{1}^M=0.93> 0.74=b_{1}^l,a_{2}^M=1.45>0.74=b_{2}^l,$

$a_{1}^l-c_{1}^M\exp\{x_{2}^*\}-e_{1}^Mu_{1}^*=0.8113>0,$

$a_{2}^l-c_{2}^M\exp\{x_{1}^*\}-e_{2}^Mu_{2}^*=1.0205>0.$

$\begin{eqnarray*} \Theta&=&\min\{E_1-\mu F_1 ,E_2-\mu F_2 ,E_3-\mu F_3 ,E_4-\mu F_4 \}\\&=&\min\{0.9813,0.6789,0.0411,0.0421\}\\&=&0.0411. \end{eqnarray*}$

则, $0<\Theta<1$.因此, 定理3.1和定理4.1的所有的条件都满足.故系统$(5.1)$是持久并且存在唯一渐近稳定概周期解.事实上, 利用Matlab软件作数值模拟, 当${\mathbb T}={\mathbb Z}$和${\mathbb T}={\mathbb R}$时, 系统$(5.1)$的数值模拟结果分别见图 1-8.由图 1-8, 我们能看出$(x_1(t),x_2(t),u_1(t),u_2(t))^T$是全局渐近稳定的.

众所周知, 连续和离散在实际应用中是非常重要, 而时标统一了连续与离散分析, 越来越多的学者开始关注并开始研究时标动力系统, 结果也会越来越丰富, 因此, 本文讨论带有反馈控制的非自治竞争种群模型的时标动力系统是有研究意义的, 而且我们得到了系统存在持久性和概周期解的一致渐近稳定性的充分条件并用Matlab做数值模拟验证了结果的有效性.