一类非线性时滞微分系统的稳定性

The Stabilities for a Class of Nonlinear Differential Systems with Time-Delay

摘要: 由一类积分不等式推导给出一类非线性时滞微分系统Lipschitz稳定性判断准则。

Abstract: In this paper, we discuss the stability for a class of nonlinear differential systems by using integral inequalities.

1. 准备工作

定义1 [1] ：若时滞微分方程中未知函数的最高阶导数含有两个不同的变元值，称之为中立型时滞微分方程；

考虑时滞微分系统： $\frac{\text{d}x}{\text{d}t}=f\left(t,x\left(t\right),x\left(t-\tau \right),\stackrel{\dot{}}{x}\left(t-\tau \right)\right)$ (1.1)

其中 $x\in R^n$， $y=col\left(x_1,x_2,\cdots ,x_m\right)$， $z=col\left(x_{m+1},x_{m+2},\cdots ,x_n\right)$， $x=col\left(y,z\right)$， $f\left(t,0,0,0\right)\equiv 0$， $\tau \ge 0$ 为常数，给定初始函数：

$x\left(t\right)=\phi \left(t\right)$， $\stackrel{\dot{}}{x}\left(t\right)=\stackrel{\dot{}}{\phi }\left(t\right)$ $\forall t\in E_{t_0}=\left[t_0-\tau ,t_0\right]$ 连续可微 (1.2)

我们总假定(1.1)满足初始条件(1.2)的解存在唯一，并用 $x\left(t,t_0,\phi \right)$ 表示(1.1)满足条件(1.2)的解。

定义2 [2] ：称(1.1)的平凡解关于部分变元y是Lipschitz稳定的，如果存在常数 $M\left(t_0\right)>0$ 和 $\delta \left(t_0\right)>0$，使当 $\Vert \phi \Vert +\Vert \stackrel{\dot{}}{\phi }\Vert <\delta$ (对 $\forall t\in E_{t_0}$ )时，有： $\Vert y\left(t;t_0,\phi \right)\Vert +\Vert \stackrel{\dot{}}{y}\left(t;t_0,\phi \right)\Vert \le M\left(t_0\right)\left(\Vert \phi \Vert +\Vert \stackrel{\dot{}}{\phi }\Vert \right)$ 于 $t\ge t_0\ge 0$ 成立，简记为LS；

定义3 [3] [4] ：称(1.1)的平凡解关于部分变元y是一致Lipschitz稳定的，若定义2中的M和 $\delta$ 均与 $t_0$ 无关，简记为ULS；

引理1：设如下条件于 $t\ge 0$ 成立

(I)：常数 $c>0$ ；

(II)： $\begin{array}{c}u\left(t\right)\le c+{\displaystyle {\int }_0^{t}a\left(s\right)u\left(s\right)\text{d}s}+{\displaystyle \underset{i=1}{\overset{I}{\sum }}{\displaystyle {\int }_0^t{b}_i\left(s\right)u^{{\alpha }_i+1}\left(s\right)\text{d}s}}\\ +{\displaystyle \underset{j=1}{\overset{J}{\sum }}{\displaystyle {\int }_0^t{c}_j\left(s\right){\displaystyle {\int }_0^s{d}_j\left(\sigma \right)u\left(\sigma \right)\text{d}\sigma \text{d}s}}}\\ +{\displaystyle \underset{k=1}{\overset{K}{\sum }}{\displaystyle {\int }_0^t{e}_k\left(s\right){\displaystyle {\int }_0^s{f}_k\left(\sigma \right)u^{{\beta }_k+1}\left(\sigma \right)\text{d}\sigma \text{d}s}}}\\ +{\displaystyle {\int }_0^t{\displaystyle \underset{i=1}{\overset{I}{\sum }}{b}_i\left(s\right)u^{{\alpha }_i}\left(s\right)}\left[{\displaystyle \underset{l=1}{\overset{L}{\sum }}{\displaystyle {\int }_0^s{g}_l\left(\sigma \right){\displaystyle {\int }_0^{\sigma }{h}_l\left(\tau \right)u\left(\tau \right)\text{d}\tau \text{d}\sigma }}}\right]\text{d}s}\\ +{\displaystyle {\int }_0^t{\displaystyle \underset{i=1}{\overset{I}{\sum }}{b}_i\left(s\right)u^{{\alpha }_i}\left(s\right)}\left[{\displaystyle \underset{m=1}{\overset{M}{\sum }}{\displaystyle {\int }_0^s{p}_m\left(\sigma \right){\displaystyle {\int }_0^{\sigma }{q}_m\left(\tau \right)u^{{\gamma }_m+1}\left(\tau \right)\text{d}\tau \text{d}\sigma }}}\right]\text{d}s}\end{array}$

其中： $u\left(t\right)$， $a\left(t\right)$， $b_i\left(t\right)\left(i=1,2,\cdots ,I\right)$， $c_j\left(t\right),d_j\left(t\right)\left(j=1,2,\cdots ,J\right)$，

$e_k\left(t\right),f_k\left(t\right)\left(k=1,2,\cdots ,K\right)$， $g_l\left(t\right),h_l\left(t\right)\left(l=1,2,\cdots ,L\right)$，

$p_m\left(t\right),q_m\left(t\right)\left(m=1,2,\cdots ,M\right)$ 均为 $R_{+}$ 上非负连续函数，

且： ${\alpha }_i\left(i=1,2,\cdots ,I\right)$， ${\beta }_k\left(k=1,2,\cdots ,K\right)$， ${\gamma }_m\left(m=1,2,\cdots ,M\right)$ 均为大于1的常数， $1\le {\alpha }_1\le {\alpha }_2\le \cdots \le {\alpha }_I$， $1\le {\beta }_1\le {\beta }_2\le \cdots \le {\beta }_K$， $1\le {\gamma }_1\le {\gamma }_2\le \cdots \le {\gamma }_M$

令： $\bar{\alpha }=\max \left({\alpha }_I,{\beta }_K,{\gamma }_M\right)$， $\underset{\_}{\alpha }=\min \left({\alpha }_1,{\beta }_1,{\gamma }_1\right)$

(III) 设 $\begin{array}{c}A\left(t\right)=a\left(t\right)+{\displaystyle \underset{i=1}{\overset{I}{\sum }}{b}_i\left(t\right)}+{\displaystyle \underset{j=1}{\overset{J}{\sum }}{c}_j\left(t\right)}{\displaystyle {\int }_0^t{d}_j\left(s\right)\text{d}s}+2{\displaystyle \underset{k=1}{\overset{K}{\sum }}{e}_k\left(t\right)}{\displaystyle {\int }_0^t{f}_k\left(s\right)\text{d}s}\\ +{\displaystyle \underset{i=1}{\overset{I}{\sum }}{b}_i\left(t\right)}{\displaystyle {\int }_0^t{\displaystyle \underset{l=1}{\overset{L}{\sum }}{g}_l\left(s\right)}{\displaystyle {\int }_0^s{h}_l\left(\sigma \right)\text{d}\sigma }\text{d}s}+{\displaystyle \underset{i=1}{\overset{I}{\sum }}{b}_i\left(t\right)}{\displaystyle {\int }_0^t{\displaystyle \underset{m=1}{\overset{M}{\sum }}{p}_m\left(s\right)}{\displaystyle {\int }_0^s{q}_m\left(\sigma \right)\text{d}\sigma }\text{d}s}\end{array}$

$\begin{array}{c}B\left(t\right)={\displaystyle \underset{i=1}{\overset{I}{\sum }}{b}_i\left(t\right)}+{\displaystyle \underset{k=1}{\overset{K}{\sum }}{e}_k\left(t\right)}{\displaystyle {\int }_0^t{f}_k\left(s\right)\text{d}s}+{\displaystyle \underset{i=1}{\overset{I}{\sum }}{b}_i\left(t\right)}{\displaystyle {\int }_0^t{\displaystyle \underset{l=1}{\overset{L}{\sum }}{g}_l\left(s\right)}{\displaystyle {\int }_0^s{h}_l\left(\sigma \right)\text{d}\sigma }\text{d}s}\\ +2{\displaystyle \underset{i=1}{\overset{I}{\sum }}{b}_i\left(t\right)}{\displaystyle {\int }_0^t{\displaystyle \underset{m=1}{\overset{M}{\sum }}{p}_m\left(s\right)}{\displaystyle {\int }_0^s{q}_m\left(\sigma \right)\text{d}\sigma }\text{d}s}\end{array}$

$C\left(t\right)={\displaystyle {\int }_0^t{\displaystyle \underset{m=1}{\overset{M}{\sum }}{p}_m\left(s\right)}{\displaystyle {\int }_0^s{q}_m\left(\sigma \right)\text{d}\sigma }\text{d}s}$

$E_1\left(t\right)=1-\bar{\alpha }{c}^{\bar{\alpha }}{\displaystyle {\int }_0^t\left[B\left(\tau \right)+C\left(\tau \right)\right]\exp \left(\bar{\alpha }{\displaystyle {\int }_0^{\tau }A\left(\sigma \right)\text{d}\sigma }\right)\text{d}\tau }$

$F\left(t\right)=a\left(t\right)+{\displaystyle \underset{j=1}{\overset{J}{\sum }}{c}_j\left(t\right){\displaystyle {\int }_0^t{d}_j\left(s\right)\text{d}s}}$

$\begin{array}{c}G\left(t\right)={\displaystyle \underset{i=1}{\overset{I}{\sum }}{b}_i\left(t\right){Щ}^{{\alpha }_i-\underset{\_}{\alpha }}}+{\displaystyle \underset{k=1}{\overset{K}{\sum }}{e}_k\left(t\right){\displaystyle {\int }_0^t{f}_k\left(s\right){Щ}^{{\beta }_k-\underset{\_}{\alpha }}\text{d}s}}\\ +{\displaystyle \underset{i=1}{\overset{I}{\sum }}{b}_i\left(t\right)N^{{\alpha }_i-\underset{\_}{\alpha }}{\displaystyle {\int }_0^t{\displaystyle \underset{l=1}{\overset{L}{\sum }}{g}_l\left(s\right){\displaystyle {\int }_0^s{h}_l\left(\sigma \right)\text{d}\sigma }}}}\end{array}$

$H\left(t\right)={\displaystyle \underset{m=1}{\overset{M}{\sum }}{p}_m\left(t\right){\displaystyle {\int }_0^t{q}_m\left(s\right){Щ}^{{\gamma }_m-\underset{\_}{\alpha }}\text{d}s}}$

$E_2\left(t\right)=1-\underset{\_}{\alpha }{c}^{\underset{\_}{\alpha }}{\displaystyle {\int }_0^t\left[G\left(\tau \right)+H\left(\tau \right)\right]\exp \left(\underset{\_}{\alpha }{\displaystyle {\int }_0^{\tau }F\left(\sigma \right)\text{d}\sigma }\right)\text{d}\tau }$

且 ${\displaystyle {\int }_0^{+\infty }A\left(s\right)\text{d}s}<+\infty$，

$E_1\left(t\right)>0$， $\left[1-\left(\bar{\alpha }-1\right)c^{\bar{\alpha }}{\displaystyle {\int }_0^{t}B\left(s\right)E_1^{-\frac{1}{\bar{\alpha }}}\left(s\right)\exp \left(\bar{\alpha }{\displaystyle {\int }_0^{s}A\left(\tau \right)\text{d}\tau }\right)\text{d}s}\right]>0$

$E_2\left(t\right)>0$， $\left[1-\left(\underset{\_}{\alpha }-1\right)c^{\underset{\_}{\alpha }}{\displaystyle {\int }_0^{t}G\left(s\right)E_2^{-\frac{1}{\underset{\_}{\alpha }}}\left(s\right)\exp \left(\underset{\_}{\alpha }{\displaystyle {\int }_0^{s}F\left(\tau \right)\text{d}\tau }\right)\text{d}s}\right]>0$

则有： $u\left(t\right)\le c\exp \left({\displaystyle {\int }_0^{t}F\left(s\right)\text{d}s}\right)\cdot {\left[1-\left(\underset{\_}{\alpha }-1\right)c^{\underset{\_}{\alpha }}{\displaystyle {\int }_0^{t}G\left(s\right)E_2^{-\frac{1}{\underset{\_}{\alpha }}}\left(s\right)\exp \left(\underset{\_}{\alpha }{\displaystyle {\int }_0^{s}F\left(\tau \right)\text{d}\tau }\right)\text{d}s}\right]}^{-\frac{1}{\underset{\_}{\alpha }-1}}$

注：该引理已在文献 [5] 中证明。

2. 一类非线性时滞微分系统的稳定性

考虑如下微分系统：

$\frac{\text{d}x}{\text{d}t}=A\left(t\right)x\left(t\right)+f\left(t,x\left(t-\delta \left(t\right)\right),{\displaystyle {\int }_0^{t}h\left\{s,x\left(s-\delta \left(s\right)\right),{\displaystyle {\int }_0^{s}g\left[\sigma ,x\left(\sigma -\delta \left(\sigma \right)\right)\right]\text{d}\sigma }\right\}\text{d}s}\right)$ (2.1)

其中： $A\left(t\right)={\left(a_{ij}\left(t\right)\right)}_{n\times n}$， $f\in C\left[I\times R^n\times R^n\times R^n,R^n\right]$

$h\in C\left[I\times R^n\times R^n\times R^n,R^n\right]$， $g\in C\left[I\times R^n\times R^n,R^n\right]$

$0<{\delta }_0\le \delta \left(t\right)\le \delta$， $A\left(t\right)$ 有界， $f\left(t,0,0,0\right)=0$， $x\left(t\right)=\phi \left(t\right)$，于 $-\delta \le t\le 0$

由引理1，可得与系统(2.1)的解等价的积分系统的解：

$\begin{array}{c}x\left(t\right)=Y\left(t,0\right)\phi \left(0\right)\\ +{\displaystyle {\int }_0^{t}Y\left(t,s\right)f\left(s,x\left(s-\delta \left(s\right)\right),{\displaystyle {\int }_0^{s}h\left\{\tau ,x\left(\tau -\delta \left(\tau \right)\right),{\displaystyle {\int }_0^{\tau }g\left[\sigma ,x\left(\sigma -\delta \left(\sigma \right)\right)\right]\text{d}\sigma }\right\}\text{d}\tau }\right)\text{d}s}\end{array}$ (2.2)

主要结论：对于系统(2.1)而言，假设下列条件于 $t\ge 0$ 时成立：

1) $\Vert Y\left(t,s\right)\Vert \le \exp \left(r\left(t-s\right)\right)$， $\left(t\ge s\ge 0\right)$， $r\le 0$

2) $\begin{array}{l}\Vert f\left(t,x\left(t-\delta \left(t\right)\right),{\displaystyle {\int }_0^{t}h\left\{s,x\left(s-\delta \left(s\right)\right),{\displaystyle {\int }_0^{s}g\left[\sigma ,x\left(\sigma -\delta \left(\sigma \right)\right)\right]\text{d}\sigma }\right\}\text{d}s}\right)\Vert \\ \le a\left(t\right)\Vert x\left(t-\delta \left(t\right)\right)\Vert +{\displaystyle \underset{i=1}{\overset{I}{\sum }}{b}_i\left(t\right){\Vert x\left(t-\delta \left(t\right)\right)\Vert }^{{\alpha }_i+1}}\\ +{\displaystyle \underset{j=1}{\overset{J}{\sum }}{c}_j\left(t\right){\displaystyle {\int }_0^t\exp \left(r\left(t-s\right)\right)d_j\left(s\right)\Vert x\left(s-\delta \left(s\right)\right)\Vert \text{d}s}}\\ +{\displaystyle \underset{k=1}{\overset{K}{\sum }}{e}_k\left(t\right){\displaystyle {\int }_0^t\exp \left(r\left(t-s\right)\right)f_k\left(s\right){\Vert x\left(s-\delta \left(s\right)\right)\Vert }^{{\beta }_k+1}\text{d}s}}\\ +{\displaystyle \underset{i=1}{\overset{I}{\sum }}{b}_i\left(t\right){\Vert x\left(t-\delta \left(t\right)\right)\Vert }^{{\alpha }_i}}{\displaystyle {\int }_0^t\exp \left(r\left(t-s\right)\right){\displaystyle \underset{l=1}{\overset{L}{\sum }}{g}_l\left(s\right){\displaystyle {\int }_0^s{h}_l\left(\sigma \right)\Vert x\left(\sigma -\delta \left(\sigma \right)\right)\Vert \text{d}\sigma \text{d}s}}}\\ +{\displaystyle \underset{i=1}{\overset{I}{\sum }}{b}_i\left(t\right){\Vert x\left(t-\delta \left(t\right)\right)\Vert }^{{\alpha }_i}}{\displaystyle {\int }_0^t{\displaystyle \underset{m=1}{\overset{M}{\sum }}\exp \left(r\left(t-s\right)\right)p_m\left(s\right){\displaystyle {\int }_0^s{q}_m\left(\sigma \right){\Vert x\left(\sigma -\delta \left(\sigma \right)\right)\Vert }^{{\gamma }_m+1}\text{d}\sigma \text{d}s}}}\end{array}$

3) 其中 $a\left(t\right)$， $b_i\left(t\right)\left(i=1,2,\cdots ,I\right)$， $c_j\left(t\right),d_j\left(t\right)\left(j=1,2,\cdots ,J\right)$

$e_k\left(t\right),f_k\left(t\right)\left(k=1,2,\cdots ,K\right)$， $g_l\left(t\right),h_l\left(t\right)\left(l=1,2,\cdots ,L\right)$，

$p_m\left(t\right),q_m\left(t\right)\left(m=1,2,\cdots ,M\right)$ 如 [6] 中定理 3.1.1 所设，且单调不增。

4) $F\left(t\right)=M_{2}a\left(t\right)+M_2{\displaystyle \underset{j=1}{\overset{J}{\sum }}{c}_j\left(t\right){\displaystyle {\int }_0^t{d}_j\left(t\right)\text{d}t}}$ 且 ${\displaystyle {\int }_0^{+\infty }F\left(s\right)\text{d}s<+\infty }$

$\begin{array}{c}G\left(t\right)={\displaystyle \underset{i=1}{\overset{I}{\sum }}{M}_2{b}_i\left(t\right){\text{e}}^{{\alpha }_{i}rt}{Щ}^{{\alpha }_i-\underset{\_}{\alpha }}}+{\displaystyle \underset{k=1}{\overset{K}{\sum }}{M}_2{e}_k\left(t\right){\displaystyle {\int }_0^t{f}_k\left(s\right){\text{e}}^{{\beta }_{k}rs}{Щ}^{{\beta }_k-\underset{\_}{\alpha }}\text{d}s}}\\ +{\displaystyle \underset{i=1}{\overset{I}{\sum }}{M}_2{b}_i\left(t\right){\text{e}}^{{\alpha }_{i}rt}{N}^{{\alpha }_i-\underset{\_}{\alpha }}{\displaystyle {\int }_0^t{\displaystyle \underset{l=1}{\overset{L}{\sum }}{g}_l\left(s\right){\displaystyle {\int }_0^s{h}_l\left(\sigma \right)\text{d}\sigma }}}}\end{array}$

$H\left(t\right)={\displaystyle \underset{m=1}{\overset{M}{\sum }}{M}_2{p}_m\left(t\right){\displaystyle {\int }_0^t{q}_m\left(s\right){\text{e}}^{{\gamma }_{m}rs}{Щ}^{{\gamma }_m-\underset{\_}{\alpha }}\text{d}s}}$

$E\left(t\right)=1-\underset{\_}{\alpha }{c}^{\underset{\_}{\alpha }}{\displaystyle {\int }_0^t\left[G\left(\tau \right)+H\left(\tau \right)\right]\exp \left(\underset{\_}{\alpha }{\displaystyle {\int }_0^{\tau }F\left(\sigma \right)\text{d}\sigma }\right)\text{d}\tau }$

$E\left(t\right)>0$，且， $\left[1-\left(\underset{\_}{\alpha }-1\right)c^{\underset{\_}{\alpha }}{\displaystyle {\int }_0^{t}G\left(s\right)E^{-\frac{1}{\underset{\_}{\alpha }}}\left(s\right)\exp \left(\underset{\_}{\alpha }{\displaystyle {\int }_0^{s}F\left(\tau \right)\text{d}\tau }\right)\text{d}s}\right]>0$

${\displaystyle {\int }_0^{\infty }G\left(s\right)E^{-\frac{1}{\underset{\_}{\alpha }}}\left(s\right)\exp \left(\underset{\_}{\alpha }{\displaystyle {\int }_0^{s}F\left(\tau \right)\text{d}\tau }\right)\text{d}s}<\infty$ 其中 $Щ$ 如 [7] [8] [9] 中引理所设

则系统(2.1)的零解在 $C_1$ 中一致Lipschitz渐近稳定。

证明：由系统(2.1)的等价系统(2.2)：

$x\left(t\right)=Y\left(t,0\right)\phi \left(0\right)+{\displaystyle {\int }_0^{t}Y\left(t,s\right)f\left(s,x\left(s-\delta \left(s\right)\right),{\displaystyle {\int }_0^{s}h\left\{\tau ,x\left(\tau -\delta \left(\tau \right)\right),{\displaystyle {\int }_0^{\tau }g\left[\sigma ,x\left(\sigma -\delta \left(\sigma \right)\right)\right]\text{d}\sigma }\right\}\text{d}\tau }\right)\text{d}s}$

由条件1)，2)，得到：

$\begin{array}{c}\Vert x\left(t\right)\Vert \le \Vert \phi \left(0\right)\Vert \exp \left(rt\right)+{\displaystyle {\int }_0^t\exp \left(r\left(t-s\right)\right)a\left(s\right)\Vert x\left(s-\delta \left(s\right)\right)\Vert \text{d}s}\\ +{\displaystyle {\int }_0^t{\displaystyle \underset{i=1}{\overset{I}{\sum }}{b}_i\left(s\right)\exp \left(r\left(t-s\right)\right){\Vert x\left(s-\delta \left(s\right)\right)\Vert }^{{\alpha }_i+1}}}\text{d}s\\ +{\displaystyle {\int }_0^t{\displaystyle \underset{j=1}{\overset{J}{\sum }}{c}_j\left(s\right)\exp \left(r\left(t-s\right)\right){\displaystyle {\int }_0^s\exp \left(r\left(s-\theta \right)\right)d_j\left(\theta \right)\Vert x\left(\theta -\delta \left(\theta \right)\right)\Vert \text{d}\theta \text{d}s}}}\\ +{\displaystyle {\int }_0^t{\displaystyle \underset{k=1}{\overset{K}{\sum }}{e}_k\left(s\right)\exp \left(r\left(t-s\right)\right){\displaystyle {\int }_0^s\exp \left(r\left(s-\theta \right)\right)f_k\left(\theta \right){\Vert x\left(\theta -\delta \left(\theta \right)\right)\Vert }^{{\beta }_k+1}\text{d}\theta \text{d}s}}}\end{array}$

$\begin{array}{l}+{\displaystyle {\int }_0^t{\displaystyle \underset{i=1}{\overset{I}{\sum }}{b}_i\left(s\right){\Vert x\left(s-\delta \left(s\right)\right)\Vert }^{{\alpha }_i}}}\exp \left(r\left(t-s\right)\right)\\ \times {\displaystyle {\int }_0^s\exp \left(r\left(s-\theta \right)\right){\displaystyle \underset{l=1}{\overset{L}{\sum }}{g}_l\left(\theta \right){\displaystyle {\int }_0^{\theta }{h}_l\left(\sigma \right)\Vert x\left(\sigma -\delta \left(\sigma \right)\right)\Vert \text{d}\sigma \text{d}\theta \text{d}s}}}\\ +{\displaystyle {\int }_0^t{\displaystyle \underset{i=1}{\overset{I}{\sum }}{b}_i\left(s\right){\Vert x\left(s-\delta \left(s\right)\right)\Vert }^{{\alpha }_i}}}\exp \left(r\left(t-s\right)\right)\\ \times {\displaystyle {\int }_0^s{\displaystyle \underset{m=1}{\overset{M}{\sum }}\exp \left(r\left(s-\theta \right)\right)p_m\left(\theta \right)}}{\displaystyle {\int }_0^{\theta }{q}_m\left(\sigma \right){\Vert x\left(\sigma -\delta \left(\sigma \right)\right)\Vert }^{{\gamma }_m+1}\text{d}\sigma \text{d}\theta \text{d}s}\end{array}$ (2.3)

于是有：

$\begin{array}{c}\Vert x\left(t\right)\Vert {\text{e}}^{-rt}\le \Vert \phi \left(0\right)\Vert {\displaystyle {\int }_0^{t}a\left(s\right)\Vert x\left(s-\delta \left(s\right)\right)\Vert {\text{e}}^{-rs}}\text{d}s+{\displaystyle {\int }_0^t{\displaystyle \underset{i=1}{\overset{I}{\sum }}{b}_i\left(s\right){\Vert x\left(s-\delta \left(s\right)\right)\Vert }^{{\alpha }_i+1}{\text{e}}^{-rs}}}\text{d}s\\ +{\displaystyle {\int }_0^t{\displaystyle \underset{j=1}{\overset{J}{\sum }}{c}_j\left(s\right){\text{e}}^{-rs}{\displaystyle {\int }_0^s{\text{e}}^{r\left(s-\theta \right)}{d}_j\left(\theta \right)}\Vert x\left(\theta -\delta \left(\theta \right)\right)\Vert \text{d}\theta }}\text{d}s\\ +{\displaystyle {\int }_0^t{\displaystyle \underset{k=1}{\overset{K}{\sum }}{e}_k\left(s\right){\text{e}}^{-rs}{\displaystyle {\int }_0^s{\text{e}}^{r\left(s-\theta \right)}{f}_k\left(\theta \right)}{\Vert x\left(\theta -\delta \left(\theta \right)\right)\Vert }^{{\beta }_k+1}\text{d}\theta }}\text{d}s\\ +{\displaystyle {\int }_0^t{\displaystyle \underset{i=1}{\overset{I}{\sum }}{b}_i\left(s\right){\Vert x\left(s-\delta \left(s\right)\right)\Vert }^{{\alpha }_i}{\text{e}}^{-rs}}}{\displaystyle {\int }_0^s{\text{e}}^{r\left(s-\theta \right)}{\displaystyle \underset{l=1}{\overset{L}{\sum }}{g}_l\left(\theta \right){\displaystyle {\int }_0^{\theta }{h}_l\left(\sigma \right)\Vert x\left(\sigma -\delta \left(\sigma \right)\right)\Vert }\text{d}\sigma \text{d}\theta \text{d}s}}\\ +{\displaystyle {\int }_0^t{\displaystyle \underset{i=1}{\overset{I}{\sum }}{b}_i\left(s\right){\Vert x\left(s-\delta \left(s\right)\right)\Vert }^{{\alpha }_i}{\text{e}}^{-rs}}{\displaystyle {\int }_0^s{\text{e}}^{r\left(s-\theta \right)}{\displaystyle \underset{m=1}{\overset{M}{\sum }}{p}_m\left(\theta \right){\displaystyle {\int }_0^{\theta }{q}_m\left(\sigma \right){\Vert x\left(\sigma -\delta \left(\sigma \right)\right)\Vert }^{{\gamma }_m+1}}\text{d}\sigma \text{d}\theta \text{d}s}}}\end{array}$ (2.4)

令 $u\left(t\right)=\Vert x\left(t\right)\Vert {\text{e}}^{-rt}$，由 ${\text{e}}^{-rt}={\text{e}}^{-r\left(t-\delta \left(t\right)\right)}\cdot {\text{e}}^{-r\delta \left(t\right)}$，注意到 $\delta \left(t\right)\le \delta$，故可设 $\underset{0\le t<+\infty }{\sup }{\text{e}}^{-r\delta \left(t\right)}=M_1<+\infty$，由此可得：

$\begin{array}{c}u\left(t\right)\le \Vert \phi \left(0\right)\Vert +M_1{\displaystyle {\int }_0^{t}a\left(s\right)u\left(s-\delta \left(s\right)\right)\text{d}s}+{\displaystyle {\int }_0^t{\displaystyle \underset{i=1}{\overset{I}{\sum }}{M}_1^{{\alpha }_i+1}{b}_i\left(s\right){\text{e}}^{{\alpha }_{i}rs}{u}^{{\alpha }_i+1}\left(s-\delta \left(s\right)\right)\text{d}s}}\\ +M_1{\displaystyle {\int }_0^t{\displaystyle \underset{j=1}{\overset{J}{\sum }}{c}_j\left(s\right){\displaystyle {\int }_0^s{d}_j\left(\theta \right)u\left(\theta -\delta \left(\theta \right)\right)\text{d}\theta }}}\text{d}s\\ +{\displaystyle {\int }_0^t{\displaystyle \underset{k=1}{\overset{K}{\sum }}{M}_1^{{\beta }_k+1}{e}_k\left(s\right){\displaystyle {\int }_0^s{\text{e}}^{{\beta }_{k}r\theta }{f}_k\left(\theta \right)u^{{\beta }_k+1}\left(\theta -\delta \left(\theta \right)\right)\text{d}\theta }}}\text{d}s\\ +{\displaystyle {\int }_0^t{\displaystyle \underset{i=1}{\overset{I}{\sum }}{M}_1^{{\alpha }_i+1}{b}_i\left(s\right)u^{{\alpha }_i}\left(s-\delta \left(s\right)\right){\text{e}}^{{\alpha }_{i}rs}}}{\displaystyle {\int }_0^s{\displaystyle \underset{l=1}{\overset{L}{\sum }}{g}_l\left(\theta \right){\displaystyle {\int }_0^{\theta }{h}_l\left(\sigma \right)u\left(\sigma -\delta \left(\sigma \right)\right)}\text{d}\sigma \text{d}\theta \text{d}s}}\\ +{\displaystyle {\int }_0^t{\displaystyle \underset{i=1}{\overset{I}{\sum }}{M}_1^{{\alpha }_i+1}{b}_i\left(s\right)u^{{\alpha }_i}\left(s-\delta \left(s\right)\right){\text{e}}^{{\alpha }_{i}rs}}}{\displaystyle {\int }_0^s{\displaystyle \underset{m=1}{\overset{M}{\sum }}{p}_m\left(\theta \right){\displaystyle {\int }_0^{\theta }{M}_1^{{\gamma }_m}{\text{e}}^{{\gamma }_{m}r\sigma }{q}_m\left(\sigma \right)u^{{\gamma }_m+1}\left(\sigma -\delta \left(\sigma \right)\right)}\text{d}\sigma \text{d}\theta \text{d}s}}\end{array}$ (2.5)

令： ${\phi }_1=\underset{-\delta \le t\le 0}{\sup }\Vert \phi \left(t\right)\Vert$， ${\phi }_2=\underset{-\delta \le t\le 0}{\sup }{\Vert \phi \left(t\right)\Vert }^{{\alpha }_i+1}$

${\phi }_3=\underset{-\delta \le t\le 0}{\sup }{\Vert \phi \left(t\right)\Vert }^{{\beta }_k+1}$， ${\phi }_4=\underset{-\delta \le t\le 0}{\sup }{\Vert \phi \left(t\right)\Vert }^{{\gamma }_m+1}$

$\phi =\max \left\{{\phi }_1,{\phi }_2,{\phi }_3,{\phi }_4,{\phi }_2{\phi }_4\right\}$

令： $\Vert u\left(t\right)\Vert =\{\begin{array}{l}\max \left\{{\phi }_1,\max \left(u\left(\xi \right)\right)\right\},0\le \xi \le t\\ {\phi }_1-\delta \le t\le 0\end{array}$

显然 $\Vert u\left(t\right)\Vert$ 单调不减，且由 $u\left(t\right)$ 的定义，有 $u\left(t-\delta \left(t\right)\right)\le {\phi }_1$，因此：

$\begin{array}{c}u\left(t\right)\le \phi +M_2{\displaystyle {\int }_0^{t}a\left(s\right)\Vert u\left(s\right)\Vert \text{d}s}+{\displaystyle {\int }_0^t{\displaystyle \underset{i=1}{\overset{I}{\sum }}{M}_2{b}_i\left(s\right){\text{e}}^{{\alpha }_{i}rs}{\Vert u\left(s\right)\Vert }^{{\alpha }_i+1}}}\text{d}s\\ +M_2{\displaystyle {\int }_0^t{\displaystyle \underset{j=1}{\overset{J}{\sum }}{c}_j\left(s\right){\displaystyle {\int }_0^s{d}_j\left(\theta \right)}\Vert u\left(\theta \right)\Vert \text{d}\theta }}\text{d}s\\ +{\displaystyle {\int }_0^t{\displaystyle \underset{k=1}{\overset{K}{\sum }}{M}_2{e}_k\left(s\right){\displaystyle {\int }_0^s{\text{e}}^{{\beta }_{k}r\theta }{f}_k\left(\theta \right)}{\Vert u\left(\theta \right)\Vert }^{{\beta }_k+1}\text{d}\theta }}\text{d}s\\ +{\displaystyle {\int }_0^t{\displaystyle \underset{i=1}{\overset{I}{\sum }}{M}_2{b}_i\left(s\right){\Vert u\left(s\right)\Vert }^{{\alpha }_i}{\text{e}}^{{\alpha }_{i}rs}}}{\displaystyle {\int }_0^s{\displaystyle \underset{l=1}{\overset{L}{\sum }}{g}_l\left(\theta \right){\displaystyle {\int }_0^{\theta }{h}_l\left(\sigma \right)\Vert u\left(\sigma \right)\Vert \text{d}\sigma \text{d}\theta \text{d}s}}}\\ +{\displaystyle {\int }_0^t{\displaystyle \underset{i=1}{\overset{I}{\sum }}{M}_2{b}_i\left(s\right){\Vert u\left(s\right)\Vert }^{{\alpha }_i}{\text{e}}^{{\alpha }_{i}rs}}}{\displaystyle {\int }_0^s{\displaystyle \underset{m=1}{\overset{M}{\sum }}{p}_m\left(\theta \right){\displaystyle {\int }_0^{\theta }{\text{e}}^{{\gamma }_{m}r\sigma }{q}_m\left(\sigma \right){\Vert u\left(\sigma \right)\Vert }^{{\gamma }_m+1}\text{d}\sigma \text{d}\theta \text{d}s}}}\end{array}$ (2.6)

其中 $M_2=\max \left\{M_1,M_1^{{\alpha }_i+1},M_1^{{\beta }_k+1},M_1^{{\gamma }_m+1},M_1^{{\alpha }_i+{\gamma }_m+1}\right\}$

注意到定理条件，由引理可得;

$u\left(t\right)\le c\exp \left({\displaystyle {\int }_0^{t}F\left(s\right)\text{d}s}\right)\cdot {\left[1-\left(\underset{\_}{\alpha }-1\right)c^{\underset{\_}{\alpha }}{\displaystyle {\int }_0^{t}G\left(s\right)E^{-\frac{1}{\underset{\_}{\alpha }}}\left(s\right)\exp \left(\underset{\_}{\alpha }{\displaystyle {\int }_0^{s}F\left(\tau \right)\text{d}\tau }\right)\text{d}s}\right]}^{-\frac{1}{\underset{\_}{\alpha }-1}}$

其中： $c=\phi$，注意到 $u\left(t\right)$ 的定义，可得：

$\Vert x\left(t\right)\Vert \le \phi {\text{e}}^{rt}\exp \left({\displaystyle {\int }_0^{t}F\left(s\right)\text{d}s}\right){\left[1-\left(\underset{\_}{\alpha }-1\right)c^{\underset{\_}{\alpha }}{\displaystyle {\int }_0^{t}G\left(s\right)E^{-\frac{1}{\underset{\_}{\alpha }}}\left(s\right)\exp \left(\underset{\_}{\alpha }{\displaystyle {\int }_0^{s}F\left(\tau \right)\text{d}\tau }\right)\text{d}s}\right]}^{-\frac{1}{\underset{\_}{\alpha }-1}}$

注意到定理的条件，有如下事实：

$r<0$， ${\displaystyle {\int }_0^{+\infty }F\left(s\right)\text{d}s}<+\infty$， $E\left(t\right)>0$， $\underset{\_}{\alpha }\ge 1$

$\left[1-\left(\underset{\_}{\alpha }-1\right)c^{\underset{\_}{\alpha }}{\displaystyle {\int }_0^{t}G\left(s\right)E^{-\frac{1}{\underset{\_}{\alpha }}}\left(s\right)\exp \left(\underset{\_}{\alpha }{\displaystyle {\int }_0^{s}F\left(\tau \right)\text{d}\tau }\right)\text{d}s}\right]>0$，且有：

$\left[1-\left(\underset{\_}{\alpha }-1\right)c^{\underset{\_}{\alpha }}{\displaystyle {\int }_0^{t}G\left(s\right)E^{-\frac{1}{\underset{\_}{\alpha }}}\left(s\right)\exp \left(\underset{\_}{\alpha }{\displaystyle {\int }_0^{s}F\left(\tau \right)\text{d}\tau }\right)\text{d}s}\right]<1$，于是令：

$\begin{array}{c}M\left(t,0\right)={\text{e}}^{rt}\exp \left({\displaystyle {\int }_0^{t}F\left(s\right)\text{d}s}\right){\left[1-\left(\underset{\_}{\alpha }-1\right)c^{\underset{\_}{\alpha }}{\displaystyle {\int }_0^{t}G\left(s\right)E^{-\frac{1}{\underset{\_}{\alpha }}}\left(s\right)\exp \left(\underset{\_}{\alpha }{\displaystyle {\int }_0^{s}F\left(\tau \right)\text{d}\tau }\right)\text{d}s}\right]}^{-\frac{1}{\underset{\_}{\alpha }-1}}\\ \le {\text{e}}^{rt}\exp \left({\displaystyle {\int }_0^{t}F\left(s\right)\text{d}s}\right)\end{array}$

于是，对一切 $t\ge 0$ 时，有 $\underset{t\to \infty }{\lim }M\left(t,t_0\right)=0$ 一致的成立。于是，可得结论成立。

基金项目

内蒙古自治区高等学校科学研究项目(NJZY16141，NJZY17064)。

参考文献

NOTES

^*通讯作者。