基于方向导数的非线性资产投资模型解的性质

A Nonlinear Asset Investment Model Based on Directional Derivatives and the Properties of Its Solutions

摘要: 研究了基于方向导数的非线性资产投资模型，该模型为含有非线性和非局部边界条件的系统。考虑了依赖资产总量的折旧率函数，运用泛函分析和积分方程的理论，证明了系统解的存在唯一性。

Abstract: This paper discusses a nonlinear asset investment model based on directional derivative, the model is a system with nonlinear and non-local boundary conditions. We consider the depreciation rate function that depends on the total assets, applying the theory of functional analysis and integral equation, we obtain the existence and uniqueness of the system solutions.

1. 引言

经济系统的定量研究，无论在理论上还是应用中都有着重要意义。长期以来，这一领域的经济学家和自然科学家密切合作，大大推动了数量经济研究的进展，先后产生了数理经济学、经济计量学、经济控制论等理论。经济系统是一个不断演化着的复杂系统。许多学者运用系统科学的理论和思想对此进行了研究，并已取得了一些成果 [1] - [8]，由于经济系统从功能上最终归结为生产和消费，因此从这个意义上讲，可以将经济系统看作为生灭过程，生灭过程在客观世界中广泛存在，如生物种群的繁衍等，对这类过程的研究也已取得了大量的成果 [9] - [12]，从而也为经济系统的研究提供了思路，本文在已有模型的基础上，考虑依赖资产总量的资本折旧率函数的投资控制模型，其模型如下：

$\{\begin{array}{l}Du\left(r,t\right)+\mu \left(r,t,U\left(t\right)\right)u\left(r,t\right)=g\left(r,t\right),\left(r,t\right)\in Q_T,\\ u\left(r,0\right)=u_0\left(r\right),r\in \left(0,a_m\right),\\ u\left(0,t\right)=\phi \left(t\right)=r\left(t\right)A\left(t\right)L^{\beta }\left(t\right){\left[{\displaystyle {\int }_0^{a_m}u\left(r,t\right)\text{d}r}\right]}^{\alpha },t\in \left(0,T\right),\\ U\left(t\right)={\displaystyle {\int }_0^{a_m}u\left(r,t\right)\text{d}r},t\in \left(0,T\right)\end{array}$ (1)

其中 $Q_T=\left(0,a_m\right)\times \left(0,T\right)$， $a_m,T\in \left(0,+\infty \right)$， $Du\left(r,t\right)=\underset{\epsilon \to 0}{\lim }\frac{u\left(r+\epsilon ,t+\epsilon \right)-u\left(r,t\right)}{\epsilon }$

$u\left(r,t\right)$ ——t时刻资产存量按役龄的分布密度函数；

$U\left(t\right)$ ——t时刻资产总量；

$\mu \left(r,t,U\left(t\right)\right)$ ——t时刻依赖资产总量按役龄的相对折旧率；

$g\left(r,t\right)$ ——t时刻由进出口差额形成的资产按役龄分布密度函数；

$u_0\left(r\right)$ ——初始时刻资产按役龄的分布密度函数；

$\phi \left(t\right)$ ——t时刻新增资产；

$r\left(t\right)$ ——t时刻生产性积累率；

$A\left(t\right)$ ——t时刻综合要素生产率；

$L\left(t\right)$ ——t时刻劳动力函数；

$\alpha$ ——资产弹性系数；

$\beta$ ——劳动力弹性系数。

2. 模型的求解

为方便讨论，令 $\Gamma \left(t\right)=r\left(t\right)A\left(t\right)L^{\beta }\left(t\right)$ 这里本文作如下假设：

(H1)对任意 $\left(r,t,U\right)\in Q_T\times \left(0,+\infty \right)$， $\mu \left(r,t,U\right)\ge 0$ ；对固定U， $\mu \in L_{loc}^1\left(\left[0,a_m\right)\times \left[0,T\right]\right)$， ${\displaystyle {\int }_0^{a_m}\mu \left(r,t-a_m+r,U\right)\text{d}r}=+\infty$， $\mu$ 关于U局部Lipschitz连续，即对任意 $M>0$，存在 $L\left(M\right)$，满足： $\left|\mu \left(r,t,U_1\right)-\mu \left(r,t,U_2\right)\right|\le L\left(M\right)\cdot \left|U_1-U_2\right|$，对任意的 $U_1,U_2\in \left[0,M\right]$。

(H2) $\Gamma \left(t\right)\in L^{\infty }\left(0,T\right)$，且存在常数 $\Gamma >0$，满足 $0\le \Gamma \left(t\right)\le \Gamma$，几乎处处成立；

(H3) $u_0\left(r\right)\in L^1\left(0,a_m\right)$，且存在 $\eta ,\delta ,T_{\delta }>0$，当 $0\le t\le T_{\delta }$ 时，有 ${\displaystyle {\int }_0^{a_m-t-\delta }{u}_0\left(r\right)\text{d}r}\ge {\eta }_1$， $u_0\left(r\right)\ge 0$，几乎处处成立；

(H4) $g\left(r,t\right)\in L^1\left(Q_T\right)$， ${\Vert g\Vert }_{L^1\left(Q_{T_{\delta }}\right)}\ge {\eta }_2$， $Q_{T_{\delta }}=\left(0,a_m-\delta \right)\times \left(0,T_{\delta }\right)$， $g\left(r,t\right)\ge 0$，几乎处处成立。

定义2.1 所谓系统(1)的解是指函数 $u\left(r,t\right)\in L^{\infty }\left(0,T;L^1\left(0,a_m\right)\right)$，沿着每条特征线( $r-t=k,\left(r,t\right)\in \overline{Q_T},k\in R$ )绝对连续，满足：

$\{\begin{array}{l}Du\left(r,t\right)+\mu \left(r,t,U\left(t\right)\right)u\left(r,t\right)=g\left(r,t\right),a.e.\left(r,t\right)\in Q_T,\hfill \\ \underset{\epsilon \to 0^{+}}{\lim }u\left(r+\epsilon ,\epsilon \right)=u_0\left(r\right),a.e.r\in \left(0,a_m\right),\hfill \\ \underset{\epsilon \to 0^{+}}{\lim }u\left(\epsilon ,t+\epsilon \right)=\phi \left(t\right)=\Gamma \left(t\right){\left[{\displaystyle {\int }_0^{a_m}u\left(r,t\right)\text{d}r}\right]}^{\alpha },a.e.t\in \left(0,T\right),\hfill \\ U\left(t\right)={\displaystyle {\int }_0^{a_m}u\left(r,t\right)\text{d}r},a.e.t\in \left(0,T\right).\hfill \end{array}$ (2)

其中， $Du\left(r,t\right)=\underset{\epsilon \to 0}{\lim }\frac{u\left(r+\epsilon ,t+\epsilon \right)-u\left(r,t\right)}{\epsilon }$

第一步：固定函数 $U\in L^{\infty }\left(0,T\right),U\left(t\right)\ge 0a.e.t\in \left(0,T\right)$，应用特征线法可将系统(1)的解表示为：

$u\left(r,t;U\right)=\{\begin{array}{l}{u}_0\left(r-t\right)E\left(r,t,t;U\right)+{\displaystyle {\int }_0^{\text{t}}g\left(r-s,t-s\right)E}\left(r,t,s;U\right)\text{d}s,r\ge t,\\ \phi \left(t-r;U\right)E\left(r,t,r;U\right)+{\displaystyle {\int }_0^{\text{r}}g\left(r-s,t-s\right)E}\left(r,t,s;U\right)\text{d}s,r<t\end{array}$ (3)

其中：

$E\left(r,t,s;U\right)=\exp \left\{-{\displaystyle {\int }_0^s\mu \left(r-\tau ,t-\tau ,U\left(t-\tau \right)\right)\text{d}\tau }\right\}$

$\begin{array}{c}\phi \left(t;U\right)=\Gamma \left(t\right)[{\displaystyle {\int }_0^t\phi \left(t-r;U\right)E\left(r,t,r;U\right)\text{d}r}+{\displaystyle {\int }_0^t{\displaystyle {\int }_0^{r}g\left(r-s,t-s\right)E\left(r,t,s;U\right)\text{d}s\text{d}r}}\\ +{\displaystyle {\int }_t^{a_m}{u}_0\left(r-t\right)E\left(r,t,t;U\right)\text{d}r}+{{\displaystyle {\int }_t^{a_m}{\displaystyle {\int }_0^{t}g\left(r-s,t-s\right)E\left(r,t,s;U\right)\text{d}s\text{d}r}}]}^{\alpha }\end{array}$

对上式做变量替换得：

$\begin{array}{c}\phi \left(t;U\right)=\Gamma \left(t\right)[{\displaystyle {\int }_0^t\phi \left(r;U\right)E\left(t-r,t,t-r;U\right)\text{d}r}+{\displaystyle {\int }_0^t{\displaystyle {\int }_0^{r}g\left(r-s,t-s\right)E\left(r,t,s;U\right)\text{d}s\text{d}r}}\\ +{\displaystyle {\int }_0^{a_m-t}{u}_0\left(r\right)E\left(t+r,t,t;U\right)\text{d}r}+{{\displaystyle {\int }_t^{a_m}{\displaystyle {\int }_0^{t}g\left(r-s,t-s\right)E\left(r,t,s;U\right)\text{d}s\text{d}r}}]}^{\alpha }\end{array}$

令 $\varphi \left(t;U\right)={\left(\frac{\phi \left(t;U\right)}{\Gamma \left(t\right)}\right)}^{1/\alpha }$，则 $\phi \left(t;U\right)={\varphi }^{\alpha }\left(t;U\right)\Gamma \left(t\right)$，从而上式变为：

$\begin{array}{c}\varphi \left(t;U\right)={\displaystyle {\int }_0^t\Gamma \left(r\right){\varphi }^{\alpha }\left(r;U\right)E\left(t-r,t,t-r;U\right)\text{d}r}+{\displaystyle {\int }_0^t{\displaystyle {\int }_0^{r}g\left(r-s,t-s\right)E\left(r,t,s;U\right)\text{d}s\text{d}r}}\\ +{\displaystyle {\int }_0^{a_m-t}{u}_0\left(r\right)E\left(t+r,t,t;U\right)\text{d}r}+{\displaystyle {\int }_t^{a_m}{\displaystyle {\int }_0^{t}g\left(r-s,t-s\right)E\left(r,t,s;U\right)\text{d}s\text{d}r}}\end{array}$

则：

$\varphi \left(t;U\right)=F\left(t;U\right)+{\displaystyle {\int }_0^{t}K\left(t,t-r;U\right){\varphi }^{\alpha }\left(r;U\right)\text{d}r}$ (4)

$\{\begin{array}{l}F\left(t;U\right)={\displaystyle {\int }_0^t{\displaystyle {\int }_0^{r}g\left(r-s,t-s\right)E\left(r,t,s;U\right)\text{d}s\text{d}r}}+{\displaystyle {\int }_0^{a_m-t}{u}_0\left(r\right)E\left(t+r,t,t;U\right)\text{d}r}\\ +{\displaystyle {\int }_t^{a_m}{\displaystyle {\int }_0^{t}g\left(r-s,t-s\right)E\left(r,t,s;U\right)\text{d}s\text{d}r}}\\ K\left(t,r;U\right)=\Gamma \left(t-r\right)E\left(r,t,r;U\right)\end{array}$

由标准证明，易得以下引理。

引理2.1：在假设(H1)-(H4)成立情况下，非线性Volterra积分方程(4)的解在 $L^{\infty }\left(0,T\right)$ 上存在且唯一。

引理2.2：对任意给定的T，假设 $u\left(r,t\right)$ 是模型(1)的解，由假设(H1)知，

$0\le \underset{\_}{\mu }=\underset{\begin{array}{c}\left(r,t\right)\in Q_T\\ U\ge 0\end{array}}{\inf }\mu \left(r,t;U\right)$ 那么，对 $0\le t\le T$，有：

$\begin{array}{l}\phi \left(t;U\right)\le \Gamma M_1{\text{e}}^{\left(\Gamma -\underset{\_}{\mu }\right)t}\le \Gamma M_1{\text{e}}^{\Gamma T}\triangleq \Gamma M_{1T},\\ U(t)\le M_1{\text{e}}^{\left(\Gamma -\underset{\_}{\mu }\right)t}\le M_1{\text{e}}^{\Gamma T}=M_{1T}.\end{array}$

证明：

$\begin{array}{c}\phi \left(t;U\right)=\Gamma \left(t\right)[{\displaystyle {\int }_0^t\phi \left(r;U\right)E\left(t-r,t,t-r;U\right)\text{d}r}+{\displaystyle {\int }_0^t{\displaystyle {\int }_0^{r}g\left(r-s,t-s\right)E\left(r,t,s;U\right)\text{d}s\text{d}r}}\\ +{\displaystyle {\int }_0^{a_m-t}{u}_0\left(r\right)E\left(t+r,t,t;U\right)\text{d}r}+{{\displaystyle {\int }_t^{a_m}{\displaystyle {\int }_0^{t}g\left(r-s,t-s\right)E\left(r,t,s;U\right)\text{d}s\text{d}r}}]}^{\alpha }\\ \le \Gamma \left(t\right){\left[{\displaystyle {\int }_0^t\phi \left(r;U\right){\text{e}}^{-\underset{\_}{\mu }\left(t-r\right)}}\text{d}r+{\Vert g\Vert }_{L^1\left(Q_T\right)}{\text{e}}^{-\underset{\_}{\mu }t}+{\text{e}}^{-\underset{\_}{\mu }t}{\displaystyle {\int }_0^{a_m}\left(u_0\left(r\right)+\frac{1}{a_m}\right)\text{d}r}\right]}^{\alpha }\\ \le \Gamma \left(t\right){\left[{\displaystyle {\int }_0^t\phi \left(r;U\right){\text{e}}^{-\underset{\_}{\mu }\left(t-r\right)}\text{d}r}+{\Vert g\Vert }_{L^1\left(Q_T\right)}{\text{e}}^{-\underset{\_}{\mu }t}+{\text{e}}^{-\underset{\_}{\mu }t}\left({\displaystyle {\int }_0^{a_m}{u}_0\left(r\right)\text{d}r}+1\right)\right]}^{\alpha }\\ \le {\text{e}}^{-\underset{\_}{\mu }t}\Gamma \left(t\right)\left[{\displaystyle {\int }_0^t\phi \left(r;U\right){\text{e}}^{\underset{\_}{\mu }r}}\text{d}r+{\Vert g\Vert }_{L^1\left(Q_T\right)}+{\Vert u_0\Vert }_{L^1\left(0,a_m\right)}+1\right]\end{array}$

$\phi \left(t;U\right){\text{e}}^{\underset{\_}{\lambda }t}\le \Gamma \left[{\displaystyle {\int }_0^t\phi \left(r;U\right){\text{e}}^{\underset{\_}{\lambda }r}}\text{d}r+{\Vert g\Vert }_{L^1\left(Q_T\right)}+{\Vert u_0\Vert }_{L^1\left(0,a_m\right)}+1\right]$

由Gronwall不等式得：

$\phi \left(t;U\right)\le \Gamma \left({\Vert g\Vert }_{L^1\left(Q_T\right)}+{\Vert u_0\Vert }_{L^1\left(0,a_m\right)}+1\right){\text{e}}^{\left(\Gamma -\underset{\_}{\mu }\right)t}=\Gamma M_1{\text{e}}^{\left(\Gamma -\underset{\_}{\mu }\right)t}\le \Gamma M_1{\text{e}}^{\Gamma t}$

$\begin{array}{c}U\left(t\right)={\displaystyle {\int }_0^t\phi \left(t-r\right)E\left(r,t,r;U\right)\text{d}r}+{\displaystyle {\int }_0^t{\displaystyle {\int }_0^{r}g\left(r-s,t-s\right)E\left(r,t,s;U\right)\text{d}s\text{d}r}}\\ +{\displaystyle {\int }_t^{a_m}{u}_0\left(r-t\right)E\left(r,t,t;U\right)\text{d}r}+{\displaystyle {\int }_t^{a_m}{\displaystyle {\int }_0^{t}g\left(r-s,t-s\right)E\left(r,t,s;U\right)\text{d}s\text{d}r}}\\ ={\displaystyle {\int }_0^t\phi \left(t-r\right)E\left(r,t,r;U\right)\text{d}r}+{\displaystyle {\int }_0^t{\displaystyle {\int }_0^{r}g\left(r-s,t-s\right)E\left(r,t,s;U\right)\text{d}s\text{d}r}}\\ +{\displaystyle {\int }_0^{a_m-t}{u}_0\left(r\right)E\left(r+t,t,t;U\right)\text{d}r}+{\displaystyle {\int }_t^{a_m}{\displaystyle {\int }_0^{t}g\left(r-s,t-s\right)E\left(r,t,s;U\right)\text{d}s\text{d}r}}\end{array}$

$\begin{array}{c}U\left(t\right)\le {\displaystyle {\int }_0^t\phi \left(t-r\right){\text{e}}^{-\underset{\_}{\lambda }r}\text{d}r}+{\text{e}}^{-\underset{\_}{\lambda }t}\left({\Vert g\Vert }_{L^1\left(Q_T\right)}+{\Vert u_0\Vert }_{L^1\left(0,a_{+}\right)}+1\right)\\ \le {\displaystyle {\int }_0^t\Gamma M_1{\text{e}}^{\left(\Gamma -\underset{\_}{\lambda }\right)\left(t-r\right)}{\text{e}}^{-\underset{\_}{\lambda }r}\text{d}r}+{\text{e}}^{-\underset{\_}{\lambda }t}{M}_1\\ \le \Gamma M_1{\text{e}}^{\left(\Gamma -\underset{\_}{\lambda }\right)t}{\displaystyle {\int }_0^t{\text{e}}^{-\Gamma r}\text{d}r}+{\text{e}}^{-\underset{\_}{\lambda }t}{M}_1\\ =M_1{\text{e}}^{\left(\Gamma -\underset{\_}{\lambda }\right)t}\end{array}$

令 $W=\left\{v\in L^{\infty }\left(0,T;L^1\left(0,a_m\right)\right);v\left(r,t\right)\ge 0,{\Vert v\left(\cdot ,t\right)\Vert }_{L^1\left(0,a_m\right)}\le M_{1T}\right\}$，

$W_1=\left\{h\in L^{\infty }\left(0,T\right);0\le h\left(t\right)\le M_{1T},t\in \left(0,T\right)\right\}$

显然W是 $L^{\infty }\left(0,T;L^1\left(0,a_m\right)\right)$ 的闭子集。

引理2.3：存在常数 $M_{2T}>0$ (依赖于T)，对任意的 $U_1,U_2\in W_1$，有：

$\left|\phi \left(t;U_1\right)-\phi \left(t;U_2\right)\right|\le M_{2T}\left|U_1\left(t\right)-U_2\left(t\right)\right|$

证明：

$\begin{array}{l}\phi \left(t;U_1\right)-\phi \left(t;U_2\right)\\ =\Gamma \left(t\right)[{\displaystyle {\int }_0^t\phi \left(r;U_1\right)\Pi \left(t-r,t,t-r;U_1\right)\text{d}r}+{\displaystyle {\int }_0^t{\displaystyle {\int }_0^{r}g\left(r-s,t-s\right)\Pi \left(r,t,s;U_1\right)\text{d}s\text{d}r}}\\ +{\displaystyle {\int }_0^{a_{+}-t}{u}_0\left(r\right)\Pi \left(t+r,t,t;U_1\right)\text{d}r}+{{\displaystyle {\int }_t^{a_{+}}{\displaystyle {\int }_0^{t}g\left(r-s,t-s\right)\Pi \left(r,t,s;U_1\right)\text{d}s\text{d}r}}]}^{\alpha }\\ -\Gamma \left(t\right)[{\displaystyle {\int }_0^t\phi \left(r;U_2\right)\Pi \left(t-r,t,t-r;U_2\right)\text{d}r}+{\displaystyle {\int }_0^t{\displaystyle {\int }_0^{r}g\left(r-s,t-s\right)\Pi \left(r,t,s;U_1\right)\text{d}s\text{d}r}}\\ +{\displaystyle {\int }_0^{a_{+}-t}{u}_0\left(r\right)\Pi \left(t+r,t,t;U_2\right)\text{d}r}+{{\displaystyle {\int }_t^{a_{+}}{\displaystyle {\int }_0^{t}g\left(r-s,t-s\right)\Pi \left(r,t,s;U_2\right)\text{d}s\text{d}r}}]}^{\alpha }\end{array}$

$\begin{array}{l}\le \Gamma \left(t\right)\alpha {\left(\gamma \rho \right)}^{\alpha -1}[{\displaystyle {\int }_0^t\phi \left(r;U_1\right)\Pi \left(t-r,t,t-r;U_1\right)\text{d}r}+{\displaystyle {\int }_0^t{\displaystyle {\int }_0^{r}g\left(r-s,t-s\right)\Pi \left(r,t,s;U_1\right)\text{d}s\text{d}r}}\\ +{\displaystyle {\int }_0^{a_m-t}{u}_0\left(r\right)\Pi \left(t+r,t,t;U_1\right)\text{d}r}+{\displaystyle {\int }_t^{a_m}{\displaystyle {\int }_0^{t}g\left(r-s,t-s\right)\Pi \left(r,t,s;U_1\right)\text{d}s\text{d}r}}\\ -{\displaystyle {\int }_0^t\phi \left(r;U_2\right)\Pi \left(t-r,t,t-r;U_2\right)\text{d}r}+{\displaystyle {\int }_0^t{\displaystyle {\int }_0^{r}g\left(r-s,t-s\right)\Pi \left(r,t,s;U_1\right)\text{d}s\text{d}r}}\\ +{\displaystyle {\int }_0^{a_{+}-t}{u}_0\left(r\right)\Pi \left(t+r,t,t;U_2\right)\text{d}r}+{\displaystyle {\int }_t^{a_{+}}{\displaystyle {\int }_0^{t}g\left(r-s,t-s\right)\Pi \left(r,t,s;U_2\right)\text{d}s\text{d}r}}]\\ =\Gamma \left(t\right)\alpha {\left(\gamma \rho \right)}^{\alpha -1}{\displaystyle {\int }_0^{a_{+}}\left(u\left(r,t;U_1\right)-u\left(r,t;U_2\right)\right)\text{d}r}\end{array}$

从而有：

$\left|\phi \left(t;U_1\right)-\phi \left(t;U_2\right)\right|\le \Gamma \alpha {\left(\gamma \rho \right)}^{\alpha -1}\left|{\displaystyle {\int }_0^{a_{+}}\left(u\left(r,t;U_1\right)-u\left(r,t;U_2\right)\right)\text{d}r}\right|=M_{2T}\left|U_1\left(t\right)-U_2\left(t\right)\right|$

令 $M_{2T}=\Gamma \alpha {\left(\gamma \rho \right)}^{\alpha -1}$，则有： $\left|\phi \left(t;U_1\right)-\phi \left(t;U_2\right)\right|\le M_{2T}\left|U_1\left(t\right)-U_2\left(t\right)\right|$

定义算子： $\begin{array}{l}K:W\to L^{\infty }\left(0,T;L^1\left(0,a_m\right)\right),\\ \left(Kv\right)\left(r,t\right)=u\left(r,t,V\left(t\right)\right),V\left(t\right)={\displaystyle {\int }_0^{a_m}v\left(r,t\right)\text{d}r}\end{array}$

引理2.4：算子K满足： $\forall v\in W,Kv\in W$，且存在常数 $M_T^{*}>0$，对任意的 $u_1,u_2\in H$，

有： ${\Vert \left(K{u}_1\right)\left(t\right)-\left(K{u}_2\right)\left(t\right)\Vert }_{L^1\left(0,a_m\right)}\le M_T^{*}{{\displaystyle {\int }_0^t\Vert u_1\left(\cdot ,s\right)-u_2\left(\cdot ,s\right)\Vert }}_{L^1\left(0,a_m\right)}\text{d}s$。

证明：对任意的 $v\in W$，令 $V\left(t\right)={\displaystyle {\int }_0^{a_m}v\left(r,t\right)\text{d}r}$，显然， $V\in L^{\infty }\left(0,T\right),V\left(t\right)\ge 0$

令 $\phi \left(t;V\right)$ 为积分方程(4)的解，则由引理2.2知，

$\phi \left(t;V\right)\le \Gamma M_{1T},{\displaystyle {\int }_0^{a_m}u\left(r,t;V\left(t\right)\right)\text{d}r}\le M_{1T}$

从而 $u\left(r,t;V\left(t\right)\right)=\left(Kv\right)\left(r,t\right)\in W$。

对任意的 $u_1,u_2\in W,U_1\left(t\right)={\displaystyle {\int }_0^{a_m}{u}_1\left(r,t\right)\text{d}r}\in W_1,U_2\left(t\right)={\displaystyle {\int }_0^{a_m}{u}_2\left(r,t\right)\text{d}r}\in W_1$，

由方程(3)、(4)知：

$\begin{array}{l}{\Vert \left(K{u}_1\right)\left(t\right)-\left(K{u}_2\right)\left(t\right)\Vert }_{L^1\left(0,a_m\right)}\\ ={\Vert u\left(\cdot ,t;U_1\right)-u\left(\cdot ,t;U_2\right)\Vert }_{L^1\left(0,a_m\right)}={\displaystyle {\int }_0^{a_m}\left|u\left(r,t;U_1\right)-u\left(r,t;U_2\right)\right|\text{d}r}\\ ={\displaystyle {\int }_0^t\left|u\left(r,t;U_1\right)-u\left(r,t;U_2\right)\right|\text{d}r}+{\displaystyle {\int }_t^{a_m}\left|u\left(r,t;U_1\right)-u\left(r,t;U_2\right)\right|\text{d}r}\end{array}$

$\begin{array}{l}={\displaystyle {\int }_0^t|[\phi \left(t-r;U_1\right)E\left(r,t,r;U_1\right)-\phi \left(t-r;U_2\right)E{\left(r,t,r;U_2\right)}_{}^{}}\\ +{\displaystyle {\int }_0^{r}g\left(r-s,t-s\right)\left(E\left(r,t,s;U_1\right)-E\left(r,t,s;U_2\right)\right)\text{d}s}]|\text{d}r\\ +{\displaystyle {\int }_t^{a_m}|[u_0\left(r-t\right){\left(E\left(r,t,t;U_1\right)-E\left(r,t,t;U_2\right)\right)}_{}^{}}\\ +{\displaystyle {\int }_0^{t}g\left(r-s,t-s\right)\left(E\left(r,t,s;U_1\right)-E\left(r,t,s;U_2\right)\right)\text{d}s}]|\text{d}r\end{array}$

$\begin{array}{l}\le {\displaystyle {\int }_0^t\left|\phi \left(t-r;U_1\right)-\phi \left(t-r;U_2\right)\right|E\left(r,t,r;U_1\right)\text{d}r}\\ +{\displaystyle {\int }_0^t\phi \left(t-r;U_2\right)\left|E\left(r,t,r;U_1\right)-E\left(r,t,r;U_2\right)\right|\text{d}r}\\ +{\displaystyle {\int }_0^t{\displaystyle {\int }_0^{r}g\left(r-s,t-s\right)\left|E\left(r,t,s;U_1\right)-E\left(r,t,s;U_2\right)\right|\text{d}s\text{d}r}}\\ +{\displaystyle {\int }_t^{a_m}{u}_0\left(r-t\right)\left|E\left(r,t,t;U_1\right)-E\left(r,t,t;U_2\right)\right|\text{d}r}\\ +{\displaystyle {\int }_t^{a_m}{\displaystyle {\int }_0^{t}g\left(r-s,t-s\right)\left|E\left(r,t,s;U_1\right)-E\left(r,t,s;U_2\right)\right|\text{d}s\text{d}r}}\end{array}$

根据假设(H3)、(H4)、引理2.2、引理2.3得：

$\begin{array}{l}{\Vert \left(K{u}_1\right)\left(t\right)-\left(K{u}_2\right)\left(t\right)\Vert }_{L^1\left(0,a_m\right)}\le M_{2T}{\displaystyle {\int }_0^t\left|U_1\left(t-r\right)-U_2\left(t-r\right)\right|\text{d}r}\\ +{\displaystyle {\int }_0^t\Gamma M_{1T}\left|\exp \left\{-{\displaystyle {\int }_0^r\mu \left(r-\tau ,t-\tau ,U_1\left(t-\tau \right)\right)\text{d}\tau }\right\}-\exp \left\{-{\displaystyle {\int }_0^r\mu \left(r-\tau ,t-\tau ,U_2\left(t-\tau \right)\right)\text{d}\tau }\right\}\right|\text{d}r}\\ +{\displaystyle {\int }_0^t{\displaystyle {\int }_0^{r}g\left(r-s,t-s\right)\left|\exp \left\{-{\displaystyle {\int }_0^s\mu \left(r-\tau ,t-\tau ,U_1\left(t-\tau \right)\right)\text{d}\tau }\right\}-\exp \left\{-{\displaystyle {\int }_0^s\mu \left(r-\tau ,t-\tau ,U_2\left(t-\tau \right)\right)\text{d}\tau }\right\}\right|\text{d}s\text{d}r}}\\ +{\displaystyle {\int }_t^{a_m}{u}_0\left(r-t\right)\left|\exp \left\{-{\displaystyle {\int }_0^s\mu \left(r-\tau ,t-\tau ,U_1\left(t-\tau \right)\right)\text{d}\tau }\right\}-\exp \left\{-{\displaystyle {\int }_0^s\mu \left(r-\tau ,t-\tau ,U_2\left(t-\tau \right)\right)\text{d}\tau }\right\}\right|\text{d}r}\\ +{\displaystyle {\int }_t^{a_m}{\displaystyle {\int }_0^{t}g\left(r-s,t-s\right)\left|\exp \left\{-{\displaystyle {\int }_0^s\mu \left(r-\tau ,t-\tau ,U_1\left(t-\tau \right)\right)\text{d}\tau }\right\}-\exp \left\{-{\displaystyle {\int }_0^s\mu \left(r-\tau ,t-\tau ,U_2\left(t-\tau \right)\right)\text{d}\tau }\right\}\right|\text{d}s\text{d}r}}\end{array}$

$\begin{array}{l}\le M_{2T}{\displaystyle {\int }_0^t\left|U_1\left(s\right)-U_2\left(s\right)\right|\text{d}s}+{\displaystyle {\int }_0^t\Gamma M_{1T}{\displaystyle {\int }_0^r\left|\mu \left(r-\tau ,t-\tau ,U_1\left(t-\tau \right)\right)-\mu \left(r-\tau ,t-\tau ,U_2\left(t-\tau \right)\right)\right|\text{d}\tau \text{d}r}}\\ +{\displaystyle {\int }_0^t{\displaystyle {\int }_0^{r}g\left(r-s,t-s\right){\displaystyle {\int }_0^s\left|\mu \left(r-\tau ,t-\tau ,U_1\left(t-\tau \right)\right)-\mu \left(r-\tau ,t-\tau ,U_2\left(t-\tau \right)\right)\right|\text{d}\tau \text{d}s\text{d}r}}}\\ +{\displaystyle {\int }_t^{a_m}{u}_0\left(r-t\right){\displaystyle {\int }_0^s\left|\mu \left(r-\tau ,t-\tau ,U_1\left(t-\tau \right)\right)-\mu \left(r-\tau ,t-\tau ,U_2\left(t-\tau \right)\right)\right|\text{d}\tau \text{d}r}}\\ +{\displaystyle {\int }_t^{a_m}{\displaystyle {\int }_0^{t}g\left(r-s,t-s\right){\displaystyle {\int }_0^s\left|\mu \left(r-\tau ,t-\tau ,U_1\left(t-\tau \right)\right)-\mu \left(r-\tau ,t-\tau ,U_2\left(t-\tau \right)\right)\right|\text{d}\tau \text{d}s\text{d}r}}}\end{array}$

$\begin{array}{l}\le M_{2T}{\displaystyle {\int }_0^t\left|U_1\left(s\right)-U_2\left(s\right)\right|\text{d}s}+\Gamma M_{1T}TL\left(M_{1T}\right){\displaystyle {\int }_0^r\left|U_1\left(t-\tau \right)-U_2\left(t-\tau \right)\right|\text{d}\tau }\\ +2{\Vert g\Vert }_{L^1\left(0,a_m\right)}L\left(M_{1T}\right){\displaystyle {\int }_0^s\left|U_1\left(t-\tau \right)-U_2\left(t-\tau \right)\right|\text{d}\tau }+{\Vert u_0\Vert }_{L^1\left(0,a_m\right)}L\left(M_{1T}\right){\displaystyle {\int }_0^s\left|U_1\left(t-\tau \right)-U_2\left(t-\tau \right)\right|\text{d}\tau }\\ \le M_{2T}{\displaystyle {\int }_0^t\left|U_1\left(s\right)-U_2\left(s\right)\right|\text{d}s}+\Gamma M_{1T}TL\left(M_{1T}\right){\displaystyle {\int }_0^t\left|U_1\left(s\right)-U_2\left(s\right)\right|\text{d}s}\\ +2{\Vert g\Vert }_{L^1\left(0,a_m\right)}L\left(M_{1T}\right){\displaystyle {\int }_0^s\left|U_1\left(s\right)-U_2\left(s\right)\right|\text{d}s}+{\Vert u_0\Vert }_{L^1\left(0,a_m\right)}L\left(M_{1T}\right){\displaystyle {\int }_0^s\left|U_1\left(s\right)-U_2\left(s\right)\right|\text{d}s}\end{array}$

$\begin{array}{l}=\left\{M_{2T}+\Gamma M_{1T}TL\left(M_{1T}\right)+2{\Vert g\Vert }_{L^1\left(0,a_m\right)}L\left(M_{1T}\right)+{\Vert u_0\Vert }_{L^1\left(0,a_m\right)}L\left(M_{1T}\right)\right\}{\displaystyle {\int }_0^t\left|U_1\left(s\right)-U_2\left(s\right)\right|\text{d}s}\\ \le M_T^{*}{\displaystyle {\int }_0^t\left|U_1\left(s\right)-U_2\left(s\right)\right|\text{d}s}\le M_T^{*}{\displaystyle {\int }_0^t{\Vert u_1\left(\cdot ,s\right)-u_2\left(\cdot ,s\right)\Vert }_{L^1\left(0,a_m\right)}\text{d}s}\end{array}$

其中 $M_T^{*}=M_{2T}+\Gamma M_{1T}TL\left(M_{1T}\right)+2{\Vert g\Vert }_{L^1\left(0,a_m\right)}L\left(M_{1T}\right)+{\Vert u_0\Vert }_{L^1\left(0,a_m\right)}L\left(M_{1T}\right)$

定理2.1：模型(1)有唯一非负解。

证明：任取 $\tau >M_T^{*}$，我们定义 $L^{\infty }\left(0,T;L^1\left(0,a_{+}\right)\right)$ 上的等价范数：

${\Vert u\Vert }_{\ast }=\text{Ess}{\sup }_{t\in \left(0,T\right)}\left\{{\text{e}}^{-\tau t}{\Vert u\left(\cdot ,t\right)\Vert }_{L^1\left(0,a_m\right)}\right\},$

由引理2.4知： $\begin{array}{c}{\Vert K{u}_1-K{u}_2\Vert }_{\ast }=\text{Ess}{\sup }_{t\in \left(0,T\right)}\left\{{\text{e}}^{-\tau t}{\Vert K{u}_1\left(\cdot ,t\right)-K{u}_2\left(\cdot ,t\right)\Vert }_{L^1\left(0,a_m\right)}\right\}\\ \le M_T^{*}\text{Ess}{\sup }_{t\in \left(0,T\right)}\left\{{\text{e}}^{-\tau t}{\displaystyle {\int }_0^t{\Vert u_1\left(\cdot ,s\right)-u_2\left(\cdot ,s\right)\Vert }_{L^1\left(0,a_m\right)}}\text{d}s\right\}\\ \le M_T^{*}\text{Ess}{\sup }_{t\in \left(0,T\right)}\left\{{\text{e}}^{-\tau t}{\displaystyle {\int }_0^t{\text{e}}^{\tau s}{\text{e}}^{-\tau s}{\Vert u_1\left(\cdot ,s\right)-u_2\left(\cdot ,s\right)\Vert }_{L^1\left(0,a_m\right)}}\text{d}s\right\}\\ \le \frac{M_T^{*}}{\tau }{\Vert u_1-u_2\Vert }_{\ast }<{\Vert u_1-u_2\Vert }_{\ast }\end{array}$

所以算子K是W上的严格压缩映射，由Banach不动点定理可得K在W上有唯一不动点。从而模型(1)有唯一非负解。

基金项目

新疆工程学院科研育人课题(2019xgy672112)。