ABSTRACT

In recent years, people pay more and more attention to the problem of environmental pollution, and it is very important to study the law of population development in polluted environment. In this paper, a predator-prey population model with size-structured in polluted environment is proposed. The formal solution of the model is obtained by establishing the corresponding assumptions and using the characteristic line method. Then the existence and uniqueness of the solution are proved by inequality estimation and Banach fixed point theorem.

Keywords:Size-Structured, Existence and Uniqueness, Environmental Pollution, Banach Fixed Point Theorem

污染环境下具有尺度结构的捕食种群模型解的存在唯一性

曹雪靓

兰州交通大学数学系，甘肃 兰州

收稿日期：2018年6月17日；录用日期：2018年7月5日；发布日期：2018年7月12日

摘 要

近年来，环境污染问题受到人们的广泛关注，同时研究污染环境下种群的发展规律也显得尤为重要。本文提出一个污染环境下具有尺度结构的捕食种群模型，通过建立相应的假设并运用特征线法得到了系统的形式解，然后通过不等式估计和Banach不动点定理证明了模型解的存在唯一性。

关键词 :尺度结构，存在唯一性，环境污染，Banach不动点定理

Copyright © 2018 by author and Hans Publishers Inc.

This work is licensed under the Creative Commons Attribution International License (CC BY).

http://creativecommons.org/licenses/by/4.0/

1. 建立模型

随着工业的发展，环境污染日趋严重，研究毒素对生物种群的影响显得尤重要。对这一问题的研究始于T.G.Hallam和他同事发表的论文 [1] [2] [3] ，随后国内外学者对这方面的问题进行了深入研究。2014年，雒志学首次将毒素种群模型和具有年龄结构的种群模型结合进行了研究 [4] [5] 。基于上述文献启发，本文提出一类具有尺度结构的捕食-被捕食者种群模型并研究其解的存在唯一性。模型如下：

$\{\begin{array}{l}\frac{\partial p\left(s,t\right)}{\partial t}+\frac{\partial \left[g\left(s\right)p\left(s,t\right)\right]}{\partial s}=-\left[{\mu }_1\left(s,c_{10}\left(t\right)\right)+{\lambda }_1\left(t\right)q\left(t\right)\right]p\left(s,t\right),\left(s,t\right)\in Q,\\ \frac{\text{d}q\left(t\right)}{\text{d}t}=-\left[{\mu }_2\left(c_{20}\left(t\right)\right)+{\lambda }_2\left(t\right)P\left(t\right)\right]q\left(t\right),t\in \left(0,+\infty \right),\\ \frac{\text{d}{c}_{10}\left(t\right)}{\text{d}t}=k{c}_e\left(t\right)-l{c}_{10}\left(t\right)-n{c}_{10}\left(t\right),t\in \left(0,+\infty \right),\\ \frac{\text{d}{c}_{20}\left(t\right)}{\text{d}t}=k{c}_e\left(t\right)-l{c}_{20}\left(t\right)-n{c}_{20}\left(t\right),t\in \left(0,+\infty \right),\\ \frac{\text{d}{c}_e\left(t\right)}{\text{d}t}=-k_1{c}_e\left(t\right)\left[P\left(t\right)+q\left(t\right)\right]+l_1\left[c_{10}\left(t\right)P\left(t\right)+c_{20}\left(t\right)q\left(t\right)\right]-h{c}_e\left(t\right)+v\left(t\right),\\ g\left(0\right)p\left(0,t\right)={\lambda }_3\left(t\right)q\left(t\right){\displaystyle {\int }_0^m{\beta }_1\left(s,c_{10}\left(t\right)\right)p\left(s,t\right)\text{d}s,}\\ p\left(s,0\right)=p_0\left(s\right),P\left(t\right)={\displaystyle {\int }_0^{m}p\left(s,t\right)\text{d}s},q\left(0\right)=q_0,s\in \left(0,m\right),\\ 0\le c_0\left(0\right)\le 1,0\le c_e\left(0\right)\le 1.\end{array}$ (1)

其中 $Q=(0,m)\times (0,+\infty )$ ，m为捕食者种群个体的最大尺度。模型中其他参数表示含义如下:

$p_1\left(s,t\right)$ ：t时刻尺度为s的捕食者种群个体密度；

$q\left(t\right)$ ：t时刻尺度为s的食饵种群个体密度；

$g\left(s\right)$ ：捕食者种群个体的尺度增长函数；

$c_{10}\left(t\right),c_{20}\left(t\right)$ ：分别表示t时刻捕食者种群个体、食饵种群个体体内的毒素浓度；

$c_e\left(t\right)$ ：t时刻环境中的毒素浓度；

$v\left(t\right)$ ：t时刻外界向环境中输入的毒素浓度；

${\lambda }_1\left(t\right),{\lambda }_2\left(t\right)$ ：t时刻捕食者、食饵的相互作用因子；

${\beta }_1\left(s,c_{10}\left(t\right)\right),{\mu }_1\left(s,c_{10}\left(t\right)\right)$ ：分别表示尺度为s，体内毒素浓度为 $c_{10}\left(t\right)$ 的捕食者种群的出生率和死亡率；

${\mu }_2\left(c_{20}\left(t\right)\right)$ ：体内毒素浓度为 $c_{20}\left(t\right)$ 的食饵种群的死亡率；

本文作如下假设：

(H₁) ${\mu }_1\left(s,c_{10}\left(t\right)\right)\in L_{\text{loc}}^1\left(Q\right)$ ， $0<{\mu }_1\left(s,c_{10}\left(t\right)\right)\le {\mu }^0$ ， ${\displaystyle {\int }_0^m{\mu }_1\left(s,c_{10}\left(t\right)\right)\text{d}s=+\infty }$ ；

(H₂) ${\beta }_1\left(s,c_{10}\left(t\right)\right)\in L_{\text{loc}}^1\left(Q\right)$ ， $0<{\beta }_1\left(s,c_{10}\left(t\right)\right)\le {\beta }^0$ ；

(H₃) ${\lambda }_i\left(t\right)\in L^{\infty }\left(0,T\right)$ ， $0<{\lambda }_i\left(t\right)<{\lambda }_i^0$ ， $i=1,2,3$ ；

(H₄) $g\in C^1\left(0,m\right)$ ， $0<g\left(s\right)\le g^{*}$ ， $s\in \left(0,m\right)$ ；

(H₅) $v(\cdot )\in L^2\left(0,T\right)$ ， $0<v(t)<v^0<+\infty$ ；

(H₆) $\left|{\beta }_1\left(s,x_1\right)-{\beta }_1\left(s,x_2\right)\right|\le L_{{\beta }_1}\left|x_1-x_2\right|$ ， $\left|{\mu }_1\left(s,x_1\right)-{\mu }_1\left(s,x_2\right)\right|\le L_{{\mu }_1}\left|x_1-x_2\right|$ ， $\left|{\beta }_2\left(x_1\right)-{\beta }_2\left(x_2\right)\right|\le L_{{\beta }_2}\left|x_1-x_2\right|$ ， $\left|{\mu }_2\left(x_1\right)-{\mu }_2\left(x_2\right)\right|\le L_{{\mu }_2}\left|x_1-x_2\right|$ ；

(H₇) $0<p\left(s,t\right)<p^{*}$ ， $0<q\left(t\right)<q^{\ast }$ ；

(H₈) $l<k<l+n$ ， $v^0<h$ [6] 。

2. 模型解的存在唯一性

定义1.1：称向量 $\left(p\left(s,t\right),q\left(t\right),c_{10}\left(t\right),c_{20}\left(t\right),c_e\left(t\right)\right)$ 为模型(1)沿特征线 $\Gamma \left(s\right)={\displaystyle {\int }_0^{s}1/g\left(r\right)\text{d}r}$ 的解如果满足如下方程：

$\{\begin{array}{l}\frac{\partial p\left(s,t\right)}{\partial t}+\frac{\partial \left[g\left(s\right)p\left(s,t\right)\right]}{\partial s}=-\left[{\mu }_1\left(s,c_{10}\left(t\right)\right)+{\lambda }_1\left(t\right)q\left(t\right)\right]p\left(s,t\right),\text{a}\text{.e}\text{.}\left(s,t\right)\in Q,\\ \frac{\text{d}q\left(t\right)}{\text{d}t}=-\left[{\mu }_2\left(c_{20}\left(t\right)\right)+{\lambda }_2\left(t\right)P\left(t\right)+u_2\left(t\right)\right]q\left(t\right),\text{a}\text{.e}\text{.}t\in \left(0,+\infty \right),\\ \frac{\text{d}{c}_{10}\left(t\right)}{\text{d}t}=k{c}_e\left(t\right)-l{c}_{10}\left(t\right)-n{c}_{10}\left(t\right),\text{a}\text{.e}\text{.}t\in \left(0,+\infty \right),\\ \frac{\text{d}{c}_{20}\left(t\right)}{\text{d}t}=k{c}_e\left(t\right)-l{c}_{20}\left(t\right)-n{c}_{20}\left(t\right),\text{a}\text{.e}\text{.}t\in \left(0,+\infty \right),\\ \frac{\text{d}{c}_e\left(t\right)}{\text{d}t}=-k_1{c}_e\left(t\right)\left[P\left(t\right)+q\left(t\right)\right]+l_1\left[c_{10}\left(t\right)P\left(t\right)+c_{20}\left(t\right)q\left(t\right)\right]-h{c}_e\left(t\right)+v\left(t\right),\\ \underset{\epsilon \to 0^{+}}{\lim }g\left(0\right)p\left({\Gamma }^{-1}\left(\epsilon \right),t+\epsilon \right)={\lambda }_3\left(t\right)q\left(t\right){\displaystyle {\int }_0^m{\beta }_1\left(s,c_{10}\left(t\right)\right)p\left(s,t\right)\text{d}s,}\\ \underset{\epsilon \to 0^{+}}{\lim }p\left({\Gamma }^{-1}\left(\Gamma \left(s\right)+\epsilon \right),\epsilon \right)=p_0\left(s\right),P\left(t\right)={\displaystyle {\int }_0^{m}p\left(s,t\right)\text{d}s},q\left(0\right)=q_0,\text{a}\text{.e}\text{.}s\in \left(0,m\right),\\ 0\le c_{10}\left(0\right)\le 1,0\le c_{20}\left(0\right)\le 1,0\le c_e\left(0\right)\le 1.\end{array}$ (2)

定义1.2：模型(1)的解空间为：

$\begin{array}{l}X=\{\left(p,q,c_{10},c_{20},c_e\right)\in L^{\infty }\left(0,T;L^1\left(0,m\right)\right)\times {\left[L^{\infty }\left(0,T\right)\right]}^4|p\left(s,t\right)>0,q\left(t\right)>0,\\ 0\le {\displaystyle {\int }_0^{m}p\left(s,t\right)\text{d}s}\le M,\text{a}\text{.e}\text{.}\left(s,t\right)\in Q\}.\end{array}$

定理1.1：模型(1)解的形式为

$p\left(s,t\right)=\{\begin{array}{l}\exp \{-{\displaystyle {\int }_0^t[{\mu }_1\left({\Gamma }^{-1}\left(\Gamma \left(s\right)+\tau -t\right),c_{10}\left(\tau \right)\right)+g_s\left({\Gamma }^{-1}\left(\Gamma \left(s\right)+\tau -t\right)\right)}\\ {{\displaystyle }}_{}^{}+{\lambda }_1\left(\tau \right)q\left(\tau \right)]\text{d}\tau \}{p}_0\left({\Gamma }^{-1}\left(\Gamma \left(s\right)-t\right)\right),\Gamma \left(s\right)>t,\\ \exp \{-{\displaystyle {\int }_0^{\Gamma \left(s\right)}[{\mu }_1\left({\Gamma }^{-1}\left(\tau \right),c_{10}\left(t+\tau -\Gamma \left(s\right)\right)\right)+{\lambda }_2\left(t+\tau -\Gamma \left(s\right)\right)P\left(t+\tau -\Gamma \left(s\right)\right)}\\ +g_s\left({\Gamma }^{-1}\left(\tau \right)\right)]\text{d}\tau \}B\left(t-\Gamma \left(s\right)\right),\Gamma \left(s\right)<t,\end{array}$

其中 $B\left(t\right)=p\left(0,t\right)$ 。

$q\left(t\right)=q_0\cdot \exp \left\{-{\displaystyle {\int }_0^t\left[{\mu }_2\left(c_{20}\left(\tau \right)\right)+{\lambda }_2\left(\tau \right)P\left(\tau \right)\right]\text{d}\tau }\right\}$ ；

$c_{i0}\left(t\right)=c_{i0}\left(0\right)\exp \left\{-\left(l+n\right)t\right\}+k{\displaystyle {\int }_0^t{c}_e\left(\tau \right)\exp \left\{\left(\tau -t\right)\left(l+n\right)\right\}\text{d}\tau }$ ， $i=1,2$ ；

$\begin{array}{l}{c}_e\left(t\right)={\displaystyle {\int }_0^t\left[l_1{c}_{10}\left(\tau \right)P\left(\tau \right)+l_1{c}_{20}\left(\tau \right)q\left(\tau \right)+v\left(\tau \right)\right]\exp \left\{{\displaystyle {\int }_t^{\tau }\left[k_{1}P\left(r\right)+k_{1}q\left(r\right)+h\right]\text{d}r}\right\}\text{d}\tau }\\ +c_e\left(0\right)\exp \left\{-{\displaystyle {\int }_0^t\left(k_{1}P\left(r\right)+k_{1}q\left(r\right)+h\right)\text{d}r}\right\}.\end{array}$

证明：模型(1)的解由特征线法和常数变易法即可导出 [7] [8] [9] 。

定理1.2：如果假设(H₁)~(H₈)成立，模型(1)存在唯一解

$\left(p\left(s,t\right),q\left(t\right),c_{10}\left(t\right),c_{20}\left(t\right),c_e\left(t\right)\right)\in X$ 。

证明：首先，定义映射 $F:X\to X$ ，

$F\left(p,q,c_{10},c_{20},c_e\right)=\left(F_1\left(p,q,c_{10},c_{20},c_e\right),F_2\left(p,q,c_{10},c_{20},c_e\right),\cdot \cdot \cdot ,F_5\left(p,q,c_{10},c_{20},c_e\right)\right),$

其中

$F_1\left(p,q,c_{10},c_{20},c_e\right)=\{\begin{array}{l}\exp \{-{\displaystyle {\int }_0^t[{\mu }_1\left({\Gamma }^{-1}\left(\Gamma \left(s\right)+\tau -t\right),c_{10}\left(\tau \right)\right)+g_s\left({\Gamma }^{-1}\left(\Gamma \left(s\right)+\tau -t\right)\right)}\\ {{\displaystyle }}_{}^{}+{\lambda }_1\left(\tau \right)q\left(\tau \right)]\text{d}\tau \}{p}_0\left({\Gamma }^{-1}\left(\Gamma \left(s\right)-t\right)\right),\Gamma \left(s\right)>t,\\ \exp \{-{\displaystyle {\int }_0^{\Gamma \left(s\right)}[{\mu }_1\left({\Gamma }^{-1}\left(\tau \right),c_{10}\left(t+\tau -\Gamma \left(s\right)\right)\right)+{\lambda }_2\left(t+\tau -\Gamma \left(s\right)\right)P\left(t+\tau -\Gamma \left(s\right)\right)}\\ +g_s\left({\Gamma }^{-1}\left(\tau \right)\right)]\text{d}\tau \}B\left(t-\Gamma \left(s\right)\right),\Gamma \left(s\right)<t,\end{array}$

$F_2\left(p,q,c_{10},c_{20},c_e\right)=q_0\cdot \exp \left\{-{\displaystyle {\int }_0^t\left[{\mu }_2\left(c_{20}\left(\tau \right)\right)+{\lambda }_2\left(\tau \right)P\left(\tau \right)\right]\text{d}\tau }\right\},$

$F_3\left(p,q,c_{10},c_{20},c_e\right)=c_{10}\left(0\right)\exp \left\{-\left(l+n\right)t\right\}+k{\displaystyle {\int }_0^t{c}_e\left(\tau \right)\exp \left\{\left(\tau -t\right)\left(l+n\right)\right\}\text{d}\tau },$

$F_4\left(p,q,c_{10},c_{20},c_e\right)=c_{20}\left(0\right)\exp \left\{-\left(l+n\right)t\right\}+k{\displaystyle {\int }_0^t{c}_e\left(\tau \right)\exp \left\{\left(\tau -t\right)\left(l+n\right)\right\}\text{d}\tau },$

$\begin{array}{c}{F}_5\left(p,q,c_{10},c_{20},c_e\right)={\displaystyle {\int }_0^t\left[l_1{c}_{10}\left(\tau \right)P\left(\tau \right)+l_1{c}_{20}\left(\tau \right)q\left(\tau \right)+v\left(\tau \right)\right]\exp \left\{{\displaystyle {\int }_t^{\tau }\left[k_{1}P\left(r\right)+k_{1}q\left(r\right)+h\right]\text{d}r}\right\}\text{d}\tau }\\ +c_e\left(0\right)\exp \left\{-{\displaystyle {\int }_0^t\left(k_{1}P\left(r\right)+k_{1}q\left(r\right)+h\right)\text{d}r}\right\}.\end{array}$

显然， $F\left(p,q,c_{10},c_{20},c_e\right)\in X$ 。

另外，

$\begin{array}{l}{\displaystyle {\int }_0^{m}p\left(s,t\right)\text{d}s}={\displaystyle {\int }_0^{{\Gamma }^{-1}\left(t\right)}p\left(s,t\right)\text{d}s}+{\displaystyle {\int }_{{\Gamma }^{-1}\left(t\right)}^{m}p\left(s,t\right)\text{d}s}\\ ={\displaystyle {\int }_0^{{\Gamma }^{-1}\left(t\right)}\exp \{-{\displaystyle {\int }_0^{\Gamma \left(s\right)}[{\mu }_1\left({\Gamma }^{-1}\left(\tau \right),c_{10}\left(t+\tau -\Gamma \left(s\right)\right)\right)+{\lambda }_2\left(t+\tau -\Gamma \left(s\right)\right)P\left(t+\tau -\Gamma \left(s\right)\right)}}\\ +g_s\left({\Gamma }^{-1}\left(\tau \right)\right)]\text{d}\tau \}B\left(t-\Gamma \left(s\right)\right)\text{d}s+{\displaystyle {\int }_{{\Gamma }^{-1}\left(t\right)}^m\exp \{-{\displaystyle {\int }_0^t[{\mu }_1}\left({\Gamma }^{-1}\left(\Gamma \left(s\right)+\tau -t\right),c_{10}\left(\tau \right)\right)}\\ +g_s\left({\Gamma }^{-1}\left(\Gamma \left(s\right)+\tau -t\right)\right)+{\lambda }_1\left(\tau \right)q\left(\tau \right)]\text{d}\tau \}{p}_0\left({\Gamma }^{-1}\left(\Gamma \left(s\right)-t\right)\right)\text{d}s\\ \le {\displaystyle {\int }_0^{{\Gamma }^{-1}\left(t\right)}B\left(t-\Gamma \left(s\right)\right)\text{d}s}+{\displaystyle {\int }_{{\Gamma }^{-1}\left(t\right)}^m{p}_0\left({\Gamma }^{-1}\left(\Gamma \left(s\right)-t\right)\right)\text{d}s}\\ \le g^{-1}\left(0\right){\lambda }_3^0{q}^{\ast }{\beta }^0{\displaystyle {\int }_0^t{\displaystyle {\int }_0^{m}p\left(s,\tau \right)\text{d}s\text{d}\tau }}+m{p}^{\ast }\end{array}$

则由Gronwall引理可得

$\begin{array}{l}{\displaystyle {\int }_0^{m}p\left(s,t\right)\text{d}s}\\ \le m{p}^{\ast }\cdot \exp \left\{{\displaystyle {\int }_0^t{g}^{-1}\left(0\right){\lambda }_3^0{q}^{\ast }{\beta }^0}\text{d}t\right\}\\ =M.\end{array}$

接下来证明解对变量的连续依赖性。

设 $x^i=\left(p^i,q^i,c_{10}^i,c_{20}^i,c_e^i\right)$ ， $i=1,\text{2}$ 。当 $\Gamma \left(s\right)<t$ 时，有如下不等式成立：

$\begin{array}{l}{\displaystyle {\int }_0^m\left|p^1\left(s,t\right)-p^2\left(s,t\right)\right|\text{d}s}\\ ={\displaystyle {\int }_0^{{\Gamma }^{-1}\left(t\right)}\left|p^1\left(s,t\right)-p^2\left(s,t\right)\right|\text{d}s}+{\displaystyle {\int }_{{\Gamma }^{-1}\left(t\right)}^m\left|p^1\left(s,t\right)-p^2\left(s,t\right)\right|\text{d}s}\\ ={\displaystyle {\int }_0^{{\Gamma }^{-1}\left(t\right)}|B^1\exp \{-{\displaystyle {\int }_0^{\Gamma \left(s\right)}[{\mu }_1\left({\Gamma }^{-1}\left(\tau \right),c_{10}^1\left(t+\tau -\Gamma \left(s\right)\right)\right)}}\\ +{\lambda }_2\left(t+\tau -\Gamma \left(s\right)\right)P^1\left(t+\tau -\Gamma \left(s\right)\right)+g_s\left({\Gamma }^{-1}\left(\tau \right)\right)]\text{d}\tau \}\\ -B^2\exp \{-{\displaystyle {\int }_0^{\Gamma \left(s\right)}[{\mu }_1\left({\Gamma }^{-1}\left(\tau \right),c_{10}^2\left(t+\tau -\Gamma \left(s\right)\right)\right)}\end{array}$

$\begin{array}{l}+g_s\left({\Gamma }^{-1}\left(\tau \right)\right)+{\lambda }_2\left(t+\tau -\Gamma \left(s\right)\right)P^2\left(t+\tau -\Gamma \left(s\right)\right)]\text{d}\tau \}|\text{d}s\\ +{\displaystyle {\int }_{{\Gamma }^{-1}\left(t\right)}^m|\exp \{-{\displaystyle {\int }_0^t[{\mu }_1\left({\Gamma }^{-1}\left(\Gamma \left(s\right)+\tau -t\right),c_{10}^1\left(\tau \right)\right)}+g_s\left({\Gamma }^{-1}\left(\Gamma \left(s\right)+\tau -t\right)\right)}\\ +{\lambda }_1\left(\tau \right)q^1\left(\tau \right)]\text{d}\tau \}-\exp \{-{\displaystyle {\int }_0^t[{\mu }_1\left({\Gamma }^{-1}\left(\Gamma \left(s\right)+\tau -t\right),c_{10}^2\left(\tau \right)\right)}\\ +g_s\left({\Gamma }^{-1}\left(\Gamma \left(s\right)+\tau -t\right)\right)+{\lambda }_1\left(\tau \right)q^2\left(\tau \right)]\text{d}\tau \}|p_0\left({\Gamma }^{-1}\left(\Gamma \left(s\right)-t\right)\right)\text{d}s\end{array}$

$\begin{array}{l}\le {\displaystyle {\int }_0^{{\Gamma }^{-1}\left(t\right)}\left|B^1-B^2\right|\text{d}s}+B^2\cdot {\displaystyle {\int }_0^{{\Gamma }^{-1}\left(t\right)}{\displaystyle {\int }_0^{\Gamma \left(s\right)}|{\mu }_1\left({\Gamma }^{-1}\left(\tau \right),c_{10}^1\left(t+\tau -\Gamma \left(s\right)\right)\right)}}\\ -{\mu }_1\left({\Gamma }^{-1}\left(\tau \right),c_{10}^2\left(t+\tau -\Gamma \left(s\right)\right)\right)|\text{d}\tau \text{d}s+B^2\cdot {\displaystyle {\int }_0^{{\Gamma }^{-1}\left(t\right)}{\displaystyle {\int }_0^{\Gamma \left(s\right)}{\lambda }_2\left(t+\tau -\Gamma \left(s\right)\right)}}\\ \cdot \left|P^1\left(t+\tau -\Gamma \left(s\right)\right)-P^2\left(t+\tau -\Gamma \left(s\right)\right)\right|\text{d}\tau \text{d}s\\ +{\displaystyle {\int }_{{\Gamma }^{-1}\left(t\right)}^m{p}_0\left({\Gamma }^{-1}\left(\Gamma \left(s\right)-t\right)\right)\cdot {\displaystyle {\int }_0^t|{\mu }_1\left({\Gamma }^{-1}\left(\tau \right),c_{10}^1\left(t+\tau -\Gamma \left(s\right)\right)\right)}}\\ -{\mu }_1\left({\Gamma }^{-1}\left(\tau \right),c_{10}^2\left(t+\tau -\Gamma \left(s\right)\right)\right)|\text{d}\tau \text{d}s\end{array}$

$\begin{array}{l}+{\displaystyle {\int }_{{\Gamma }^{-1}\left(t\right)}^m{p}_0\left({\Gamma }^{-1}\left(\Gamma \left(s\right)-t\right)\right)}\cdot {\displaystyle {\int }_0^t{\lambda }_1\left(\tau \right)\left|q^1\left(\tau \right)-q^2\left(\tau \right)\right|\text{d}\tau \text{d}s}\\ \le \left(\frac{1}{g^{*}}{\lambda }_3^0\cdot {\beta }^0\cdot q^{\ast }+{\lambda }_2^0\cdot {\lambda }_3^0\cdot {\beta }^0\cdot q^{\ast }\cdot MT\right){\displaystyle {\int }_0^t{\displaystyle {\int }_0^m\left|p^1\left(s,\tau \right)-p^2\left(s,\tau \right)\right|\text{d}s\text{d}\tau }}\\ +\left(\frac{1}{g^{*}}{\lambda }_3^0\cdot {\beta }^0\cdot M+m\cdot {\lambda }_1^0{p}^{*}\right){\displaystyle {\int }_0^t\left|q^1\left(\tau \right)-q^2\left(\tau \right)\right|\text{d}\tau }\\ +\left(\frac{1}{g^{*}}{\lambda }_3^0\cdot q^{\ast }\cdot M{L}_{{\beta }_1}+T\cdot {\lambda }_3^0\cdot q^{\ast }\cdot M{L}_{{\mu }_1}+m\cdot p^{*}\cdot L_{{\mu }_1}\right){\displaystyle {\int }_0^t\left|c_{10}^1\left(\tau \right)-c_{10}^2\left(\tau \right)\right|\text{d}\tau }.\end{array}$

因此，

${\displaystyle {\int }_0^m\left|F_1\left(x^1\right)-F_1\left(x^2\right)\right|\text{d}s}\le M_1\left({\displaystyle {\int }_0^t{\displaystyle {\int }_0^m\left|p^1\left(s,\tau \right)-p^2\left(s,\tau \right)\right|\text{d}s\text{d}\tau }}+{\displaystyle {\int }_0^t\left|q^1\left(\tau \right)-q^2\left(\tau \right)\right|\text{d}\tau }+{\displaystyle {\int }_0^t\left|c_{10}^1\left(\tau \right)-c_{10}^2\left(\tau \right)\right|\text{d}}\tau \right),$ (3)

且

$\begin{array}{l}{M}_1=\max \{\frac{1}{g^{*}}{\lambda }_3^0\cdot {\beta }^0\cdot q^{\ast }+{\lambda }_2^0\cdot {\lambda }_3^0\cdot {\beta }^0\cdot q^{\ast }\cdot MT,\frac{1}{g^{*}}{\lambda }_3^0\cdot {\beta }^0\cdot M+m\cdot {\lambda }_1^0{p}^{*},\\ \frac{1}{g^{*}}{\lambda }_3^0\cdot q^{\ast }\cdot M{L}_{{\beta }_1}+T\cdot {\lambda }_3^0\cdot q^{\ast }\cdot M{L}_{{\mu }_1}+m\cdot p^{*}\cdot L_{{\mu }_1}\}.\end{array}$

当 $\Gamma \left(s\right)>t$ 时同理可得上述不等式成立。

$\begin{array}{c}\left|q^1-q^2\right|=q_0\cdot |\exp \left\{-{\displaystyle {\int }_0^t\left[{\mu }_2\left(c_{20}^1\left(\tau \right)\right)+{\lambda }_2\left(\tau \right)P^1\left(\tau \right)\right]\text{d}\tau }\right\}\\ -\exp \left\{-{\displaystyle {\int }_0^t\left[{\mu }_2\left(c_{20}^2\left(\tau \right)\right)+{\lambda }_2\left(\tau \right)P^2\left(\tau \right)\right]\text{d}\tau }\right\}|\\ \le q_0\cdot {\displaystyle {\int }_0^t\left|{\mu }_2\left(c_{20}^1\left(\tau \right)\right)-{\mu }_2\left(c_{20}^2\left(\tau \right)\right)\right|\text{d}\tau }+q_0\cdot {\displaystyle {\int }_0^t{\lambda }_2\left(\tau \right)\left|P^1\left(\tau \right)-P^2\left(\tau \right)\right|\text{d}\tau }\\ \le q_0{L}_{{\mu }_2}\cdot {\displaystyle {\int }_0^t\left|c_{20}^1\left(\tau \right)-c_{20}^2\left(\tau \right)\right|\text{d}\tau }+q_0{\lambda }_2^0{\displaystyle {\int }_0^t{\displaystyle {\int }_0^m\left|p^1\left(s,\tau \right)-p^2\left(s,\tau \right)\right|\text{d}s\text{d}\tau }}\\ \le M_2\left({\displaystyle {\int }_0^t{\displaystyle {\int }_0^m\left|p^1\left(s,\tau \right)-p^2\left(s,\tau \right)\right|\text{d}s\text{d}\tau }}+{\displaystyle {\int }_0^t\left|c_{20}^1\left(\tau \right)-c_{20}^2\left(\tau \right)\right|\text{d}}\tau \right)\end{array}$ (4)

其中， $M_2=\max \left\{q_0{L}_{{\mu }_2},q_0{\lambda }_2^0\right\}$ 。

$\begin{array}{l}\left|c_{i0}^1-c_{i0}^2\right|\\ =|c_{i0}\left(0\right)\exp \left\{-\left(l+n\right)t\right\}+k{\displaystyle {\int }_0^t{c}_e^1\left(\tau \right)\exp \left\{\left(\tau -t\right)\left(l+n\right)\right\}\text{d}\tau }\\ -c_{i0}\left(0\right)\exp \left\{-\left(l+n\right)t\right\}+k{\displaystyle {\int }_0^t{c}_e^2\left(\tau \right)\exp \left\{\left(\tau -t\right)\left(l+n\right)\right\}\text{d}\tau }|\\ =k{\displaystyle {\int }_0^t\left|c_e^1\left(\tau \right)-c_e^2\left(\tau \right)\right|\exp \left\{\left(\tau -t\right)\left(l+n\right)\right\}\text{d}\tau }\\ \le M_3{\displaystyle {\int }_0^t\left|c_e^1\left(\tau \right)-c_e^2\left(\tau \right)\right|\text{d}\tau }\left(M_3=k\right)\end{array}$ (5)

$\begin{array}{l}\left|c_e^1-c_e^2\right|\\ =|{\displaystyle {\int }_0^t\left[l_1{c}_{10}^1\left(\tau \right)P^1\left(\tau \right)+l_1{c}_{20}^1\left(\tau \right)q^1\left(\tau \right)+v\left(\tau \right)\right]\exp \left\{{\displaystyle {\int }_t^{\tau }\left[k_1{P}^1\left(r\right)+k_1{q}^1\left(r\right)+h\right]\text{d}r}\right\}}\text{d}\tau \\ +c_e\left(0\right)\exp \left\{-{\displaystyle {\int }_0^t\left(k_1{P}^1\left(r\right)+k_1{q}^1\left(r\right)+h\right)\text{d}r}\right\}\\ -{\displaystyle {\int }_0^t\left[l_1{c}_{10}^2\left(\tau \right)P^2\left(\tau \right)+l_1{c}_{20}^2\left(\tau \right)q^2\left(\tau \right)+v\left(\tau \right)\right]\exp \left\{{\displaystyle {\int }_t^{\tau }\left[k_1{P}^2\left(r\right)+k_1{q}^2\left(r\right)+h\right]\text{d}r}\right\}\text{d}\tau }\\ +c_e\left(0\right)\exp \left\{-{\displaystyle {\int }_0^t\left(k_1{P}^2\left(r\right)+k_1{q}^2\left(r\right)+h\right)\text{d}r}\right\}|\end{array}$

$\begin{array}{l}\le c_e\left(0\right)\cdot {\displaystyle {\int }_0^t\left|k_1\left(P^1\left(r\right)-P^2\left(r\right)\right)+k_1\left(q^1\left(r\right)-q^2\left(r\right)\right)\right|\text{d}r}\\ +{\displaystyle {\int }_0^t\left|l_1\left(c_{10}^1\left(\tau \right)P^1\left(\tau \right)-c_{10}^2\left(\tau \right)P^2\left(\tau \right)\right)+l_1\left(c_{20}^1\left(\tau \right)q^1\left(\tau \right)-c_{20}^2\left(\tau \right)q^2\left(\tau \right)\right)\right|}\text{d}\tau \\ +{\displaystyle {\int }_0^t\left[l_1{c}_{10}^2\left(\tau \right)P^2\left(\tau \right)+l_1{c}_{20}^2\left(\tau \right)q^2\left(\tau \right)+v\left(\tau \right)\right]}\cdot |\exp \left\{{\displaystyle {\int }_t^{\tau }\left[k_1{P}^1\left(r\right)+k_1{q}^1\left(r\right)+h\right]\text{d}r}\right\}\\ -\exp \left\{{\displaystyle {\int }_t^{\tau }\left[k_1{P}^2\left(r\right)+k_1{q}^2\left(r\right)+h\right]\text{d}r}\right\}|\text{d}\tau \end{array}$

$\begin{array}{l}\le \left(k_1+l_1+TM{k}_1{l}_1+T{q}^{\ast }{k}_1{l}_1+T{k}_1{v}^0\right){\displaystyle {\int }_0^t{\displaystyle {\int }_0^m\left|p^1\left(s,\tau \right)-p^2\left(s,\tau \right)\right|\text{d}s\text{d}\tau }}\\ +\left(k_1+l_1+TM{k}_1{l}_1+T{q}^{\ast }{k}_1{l}_1+T{k}_1{v}^0\right){\displaystyle {\int }_0^t\left|q^1\left(\tau \right)-q^2\left(\tau \right)\right|\text{d}\tau }\\ +M{l}_1\cdot {\displaystyle {\int }_0^t\left|c_{10}^1\left(\tau \right)-c_{10}^2\left(\tau \right)\right|\text{d}\tau }+q^{\ast }{l}_1\cdot {\displaystyle {\int }_0^t\left|c_{20}^1\left(\tau \right)-c_{20}^2\left(\tau \right)\right|\text{d}\tau }\\ \le M_4\left({\displaystyle {\int }_0^t{\displaystyle {\int }_0^m\left|p^1\left(s,\tau \right)-p^2\left(s,\tau \right)\right|\text{d}s\text{d}\tau }}+{\displaystyle {\int }_0^t\left|q^1\left(\tau \right)-q^2\left(\tau \right)\right|\text{d}\tau }+{\displaystyle \underset{i=1}{\overset{2}{\sum }}{\displaystyle {\int }_0^t\left|c_{20}^1\left(\tau \right)-c_{20}^2\left(\tau \right)\right|\text{d}\tau }}\right)\end{array}$ (6)

且 $M_4=\max \left\{k_1+l_1+TM{k}_1{l}_1+T{q}^{\ast }{k}_1{l}_1+T{k}_1{v}^0,M{l}_1,q^{\ast }{l}_1\right\}$ 。

最后，定义解空间X上的范数为

${\Vert \left(p,q,c_{10},c_{20},c_e\right)\Vert }_X=Ess\underset{t\in \left(0,T\right)}{\sup }{\text{e}}^{-\lambda t}\left\{{\displaystyle {\int }_0^m\left|p\left(s,t\right)\right|\text{d}s}+\left|q\left(t\right)\right|+{\displaystyle \underset{i=1}{\overset{2}{\sum }}\left|c_{i0}\left(t\right)\right|}+\left|c_e\left(t\right)\right|\right\},$

$\lambda >0$ 足够大。

则由不等式(3)-(6)可得

$\begin{array}{l}{\Vert F\left(x^1\right)-F\left(x^2\right)\Vert }_X={\Vert F_1\left(x^1\right)-F_1\left(x^2\right),F_2\left(x^1\right)-F_2\left(x^2\right),\cdot \cdot \cdot ,F_5\left(x^1\right)-F_5\left(x^2\right)\Vert }_X\\ \le M_{5}Ess\underset{t\in \left(0,T\right)}{\sup }{\text{e}}^{-\lambda t}{\displaystyle {\int }_0^t\left\{{\displaystyle {\int }_0^m\left|p^1\left(s,\tau \right)-p^2\left(s,\tau \right)\right|\text{d}s+\left|q^1\left(\tau \right)-q^2\left(\tau \right)\right|+{\displaystyle \underset{i=1}{\overset{2}{\sum }}\left|c_{i0}^1\left(\tau \right)-c_{i0}^2\left(\tau \right)\right|+}\left|c_e^1\left(\tau \right)-c_e^2\left(\tau \right)\right|}\right\}}\\ =M_{5}Ess\underset{t\in \left(0,T\right)}{\sup }{\text{e}}^{-\lambda t}{\displaystyle {\int }_0^t{\text{e}}^{\lambda \tau }}\{{\text{e}}^{-\lambda \tau }[{\displaystyle {\int }_0^m\left|p^1\left(s,\tau \right)-p^2\left(s,\tau \right)\right|\text{d}s+\left|q^1\left(\tau \right)-q^2\left(\tau \right)\right|}\begin{array}{c}\\ \end{array}\\ +{\displaystyle \underset{i=1}{\overset{2}{\sum }}\left|c_{i0}^1\left(\tau \right)-c_{i0}^2\left(\tau \right)\right|}+\left|c_e^1\left(\tau \right)-c_e^2\left(\tau \right)\right|]\}\text{d}\tau \\ \le M_5{\Vert x^1-x^2\Vert }_{X}Ess\underset{t\in \left(0,T\right)}{\sup }\left\{{\text{e}}^{-\lambda t}{\displaystyle {\int }_0^t{\text{e}}^{\lambda \tau }}\text{d}\tau \right\}\le \frac{M_5}{\lambda }{\Vert x^1-x^2\Vert }_X\end{array}$