具有积分边界条件的非线性耦合分数微分方程组边值问题解的唯一性

Uniqueness of Solutions for Boundary Value Problems of Nonlinear Coupled Fractional Differential Equations with Integral Boundary Conditions

摘要: 分数阶微分方程模型具有深刻的物理背景和丰富的理论内涵，在诸多领域应用广泛，如血液流动问题、化学工程、热弹性、地下水流动、人口动力学等。目前关于带有积分边界条件的非线性耦合分数微分方程边值问题的求解相对较少，本文就是针对非线性耦合分数微分方程边值问题解的唯一性展开的研究，本文的非线性项中含有未知函数的导数项，使得研究的Banach空间更加复杂。首先，得到非线性系统对应的线性系统的Green函数，其次，分析Green函数的性质，构造积分算子，再次，利用Banach不动点定理得到边值问题解的唯一性结果，最后，给出一个示例。

Abstract: Fractional differential equation models have profound physical backgrounds and rich theoretical connotations, and are widely used in many fields, such as blood flow problems, chemical engineer-ing, thermoelasticity, groundwater flow, population dynamics, etc. At present, there is relatively lit-tle research on solving nonlinear coupled fractional differential equation boundary value problems with integral boundary conditions. This paper focuses on the uniqueness of solutions to nonlinear coupled fractional differential equation boundary value problems. The nonlinear term in this paper contains derivative terms of unknown functions, making the Banach studied more complex: Firstly, obtain the Green function of the linear system corresponding to the nonlinear system; secondly, an-alyze the properties of the Green function and construct an integral operator; thirdly, use Banach’s fixed point theorem to obtain the uniqueness result of the solution to the boundary value problem; finally, provide an example.

1. 引言

自21世纪初以来，分数阶微积分建模方法和理论已成功应用于粒子物理，异常扩散，复杂粘弹性材料的力学本构关系，系统控制，流变学，地球物理，生物医学工程，经济学等诸多领域，凸显了其独特的优势和不可替代性，其理论与应用研究已成为国际上研究的热点 [1] [2] [3] 关于分数阶微分方程边值问题解的存在性，唯一性和性质的理论研究一直是一个热门话题。对于一些研究历史和研究结果，我们可以参考文献 [4] [5] [6] [7] 。

因为具有积分边界的边值问题具有广泛的应用背景，另一方面，耦合系统的研究涉及分数微分方程也很重要，因为，这种系统出现在应用自然的各种问题中，生物学，化学和物理学可以方程组的形式进行建模 [8] [9] 。所以本文想要研究一类具有积分边界条件的非线性耦合分数阶微分方程边值问题的解。近年来，分数阶微分方程耦合系统的研究还是较少的，首先给出几个研究的具体例子：

2017年，利用基于具有递增或递减性质的凹型算子的一类不动点定理，Shah [10] 等人得到了下面系统

解存在的适当条件，其中 $J=\left[0,1\right]$ ， $f,g:J\times R\times R\to R$ 是连续函数，所用到的导数和积分类型为 Riemann-Liouville型导数和积分。

2018年，Chalishajar [11] 等人研究了下式

$\{\begin{array}{l}{}^{c}D{}^{q}u\left(t\right)=f\left(t,v\left(t\right)\right),q\in \left(1,2\right],t\in \left[0,1\right],\\ {}^{c}D{}^{p}v\left(t\right)=g\left(t,u\left(t\right)\right),p\in \left(1,2\right],t\in \left[0,1\right],\\ \alpha u\left(0\right)+\beta u^{\prime }\left(0\right)={\displaystyle {\int }_0^1{a}_1\left(u\left(s\right)\right)\text{d}s},\alpha u\left(1\right)+\beta u^{\prime }\left(1\right)={\displaystyle {\int }_0^1{a}_2\left(u\left(s\right)\right)\text{d}s,}\\ \tilde{\alpha }v\left(0\right)+\tilde{\beta }{v}^{\prime }\left(0\right)={\displaystyle {\int }_0^1{\tilde{a}}_1\left(v\left(s\right)\right)\text{d}s},\tilde{\alpha }v\left(1\right)+\tilde{\beta }{v}^{\prime }\left(1\right)={\displaystyle {\int }_0^1{\tilde{a}}_2\left(v\left(s\right)\right)\text{d}s,}\end{array}$

这一类具有积分边界条件的，考虑Caputo导数的耦合系统解的存在性，唯一性以及Ulam型稳定性，其中 $f,g:\left[0,1\right]\times R\to R$ 是连续函数， $a_1,a_2,{\tilde{a}}_1,{\tilde{a}}_2:R\to R$ 是连续函数，并且 $\alpha ,\tilde{\alpha }>0$ ， $\beta ,\tilde{\beta }\ge 0$ 的实数。

受到上述工作的启发，本文想要研究下面具有积分边界条件，非线性项含有导数项的耦合系统

$\{\begin{array}{l}{}^{c}D{}^{q}u\left(t\right)+f\left(t,v\left(t\right),{}^{c}D{}^{\delta }v\left(t\right),{}^{c}D{}^{q-1}u\left(t\right)\right)=0,t\in \left[0,1\right],\\ {}^{c}D{}^{p}v\left(t\right)+g\left(t,u\left(t\right),{}^{c}D{}^{\delta }u\left(t\right),{}^{c}D{}^{p-1}v\left(t\right)\right)=0,t\in \left[0,1\right],\\ \alpha u\left(0\right)-\beta u^{\prime }\left(0\right)={\displaystyle {\int }_0^1{h}_1\left(u\left(s\right)\right)\text{d}s},\alpha u\left(1\right)+\beta u^{\prime }\left(1\right)={\displaystyle {\int }_0^1{h}_2\left(u\left(s\right)\right)\text{d}s},u^{″}\left(0\right)=0,\\ \alpha v\left(0\right)-\beta v^{\prime }\left(0\right)={\displaystyle {\int }_0^1{\tilde{h}}_1\left(v\left(s\right)\right)\text{d}s},\alpha v\left(1\right)+\beta v^{\prime }\left(1\right)={\displaystyle {\int }_0^1{\tilde{h}}_2\left(v\left(s\right)\right)\text{d}s},v^{″}\left(0\right)=0\end{array}$ (1)

解的唯一性。在本文的剩余部分，总是假设其中 $2<q,p<3$ ， $0<\delta \le 1$ ， $\alpha ,\beta \in R$ 并且有 $\alpha >0$ ， $\beta \ge 0$ 。 $f,g:\left[0,1\right]\times R^3\to R$ 是连续函数， $h_1,h_2,{\tilde{h}}_1,{\tilde{h}}_2:R\to R$ 是连续函数， ${}^{c}D{}^q$ ， ${}^{C}D{}^p$ ， ${}^{C}D{}^{\delta }$ ， ${}^{C}D{}^{q-1}$ ， ${}^{C}D{}^{p-1}$ 都是Caputo型分数阶导数。

2. 预备知识

本节给出了Riemann-Liouville分数积分，分数导数以及Caputo分数导数的一些基本定义和引理，并给出了本文主要运用到的两个不动点定理。

在本节中，总是假设 $\mathbb{N}=\left\{1,2,3,\cdots \right\}$ ， $\eta >0$ 并且 $\left[\eta \right]$ 表示 $\eta$ 的整数部分。

2.1. 定义1 [1]

$\left[0,1\right]$ 上 $\eta$ 阶的Riemann-Liouville分数积分 $I_{0+}^{\eta }u$ 的定义如下

$\left(I_{0+}^{\eta }u\right)\left(t\right):=\frac{1}{\Gamma \left(\eta \right)}{\displaystyle {\int }_0^t\frac{u\left(s\right)}{{\left(t-s\right)}^{1-\eta }}\text{d}s}.$

2.2. 定义2 [1]

$\left[0,1\right]$ 上 $\eta$ 阶的Riemann-Liouville分数导数 $D_{0+}^{\eta }u$ 的定义如下

$\left(D_{0+}^{\eta }u\right)\left(t\right):={\left(\frac{\text{d}}{\text{d}t}\right)}^n\left(I_{0+}^{n-\eta }u\right)\left(t\right){\displaystyle {\int }_0^t\frac{u\left(s\right)}{{\left(t-s\right)}^{\eta -n+1}}\text{d}s},$

其中 $n=\left[\eta \right]+1$ 。

2.3. 定义3 [1]

$\left[0,1\right]$ 上 $\eta$ 阶的Caputo分数导数 $D_{0+}^{\eta }u$ 的定义如下

$\left({}^{C}D{}_{0+}^{\eta }u\right)\left(t\right):=\left(D_{0+}^{\eta }\left[u\left(s\right)-{\displaystyle \underset{k=0}{\overset{n-1}{\sum }}\frac{u^{\left(k\right)}\left(0\right)}{k!}{s}^k}\right]\right)\left(t\right),$

其中

$n=\{\begin{array}{l}\left[\eta \right]+1,\eta \notin \mathbb{N}\text{,}\\ \eta ,\eta \in \mathbb{N}\text{.}\end{array}$ (2)

2.4. 引理1 [1]

设 $s>\eta$ ，当 $t\in \left[0,1\right]$ 时，若 $u\in C\left[0,1\right]$ ，则有 $\left({}^{C}D{}_{0+}^{\eta }{I}_{0+}^{s}u\right)\left(t\right)=\left(I_{0+}^{s-\eta }u\right)\left(t\right)$ 成立。

2.5. 引理2 [2]

设n由(2)给出，那么有以下关系式成立：

(1) 当 $k\in \left\{0,1,2,\cdots ,n-1\right\}$ 时， ${}^{C}D{}_{0+}^{\eta }{t}^k=0$ ；

(2) 如果 $v>n$ ，那么 ${}^{C}D{}_{0+}^{\eta }{t}^{v-1}=\frac{\Gamma \left(v\right)}{\Gamma \left(v-\eta \right)}{t}^{v-\eta -1}$ 。

2.6. 引理3 [3]

设n由(2)给出，如果 $u\in A{C}^n\left[0,1\right]$ 或者 $u\in C^n\left[0,1\right]$ ，则有

$\left(I_{0+}^{\eta }{}^{C}D{}_{0+}^{\eta }u\right)\left(t\right)=u\left(t\right)-{\displaystyle \underset{k=0}{\overset{n-1}{\sum }}\frac{u^{\left(k\right)}\left(0\right)}{k!}{t}^k}.$

2.7. 引理3 [3]

(Banach不动点定理)设F是从完备度量空间 $F\left(X,d\right)$ 到其自身的压缩映射，那么F有一个唯一的不动点 $x\in X$ 。

2.8. 引理4

设 $y,{\gamma }_1,{\gamma }_2\in C\left[0,1\right]$ ，则下面的线性微分方程边值问题

$\{\begin{array}{l}{}^{c}D{}^{q}u\left(t\right)+y\left(t\right)=0,t\in \left[0,1\right],\\ u^{″}\left(0\right)=0,\\ \alpha u\left(0\right)-\beta u^{\prime }\left(0\right)={\displaystyle {\int }_0^1{\gamma }_1\left(s\right)\text{d}s,}\\ \alpha u\left(1\right)+\beta u^{\prime }\left(1\right)={\displaystyle {\int }_0^1{\gamma }_2\left(s\right)\text{d}s}\end{array}$

有唯一解

$u\left(t\right)={\displaystyle {\int }_0^1{G}_q}\left(t,s\right)y\left(s\right)\text{d}s+\frac{\alpha +\beta -\alpha t}{2\alpha \beta +{\alpha }^2}{\displaystyle {\int }_0^1{\gamma }_1\left(s\right)\text{d}s}+\frac{\beta +\alpha t}{2\alpha \beta +{\alpha }^2}{\displaystyle {\int }_0^1{\gamma }_2\left(s\right)\text{d}s},$

其中，

$G_q\left(t,s\right)=\{\begin{array}{l}\frac{-{\left(t-s\right)}^{q-1}}{\Gamma \left(q\right)}+\frac{\left(\beta +\alpha t\right){\left(1-s\right)}^{q-1}}{\left(2\beta +\alpha \right)\Gamma \left(q\right)}+\frac{\left(\beta +\beta t\right){\left(1-s\right)}^{q-2}}{\left(2\alpha \beta +{\alpha }^2\right)\Gamma \left(q-1\right)},0\le s\le t\le 1,\\ \frac{\left(\beta +\alpha t\right){\left(1-s\right)}^{q-1}}{\left(2\beta +\alpha \right)\Gamma \left(q\right)}+\frac{\left(\beta +\beta t\right){\left(1-s\right)}^{q-2}}{\left(2\alpha \beta +{\alpha }^2\right)\Gamma \left(q-1\right)},0\le t\le s\le 1,\end{array}$

证明 对边值问题方程项的左右两边同时进行积分，得到

$u\left(t\right)=-\left(I^{q}y\right)\left(t\right)-c_0-c_{1}t-c_2{t}^2,$

对其进行求导，得到

$u^{\prime }\left(t\right)=-\left(I^{q-1}y\right)\left(t\right)-c_1-2c_{2}t,$

接着求二阶导，可以有

$u^{″}\left(t\right)=-\left(I^{q-2}y\right)\left(t\right)-2c_2,$

由 $u^{″}\left(0\right)=0$ ，可得 $c_2=0$ ，则

$u\left(t\right)=-\left(I^{q}y\right)\left(t\right)-c_0-c_{1}t.$

根据边值问题的边界条件，代入相应的值之后，可以得到

$\begin{array}{l}{c}_0=-\frac{\alpha +\beta }{2\alpha \beta +{\alpha }^2}{\displaystyle {\int }_0^1{\gamma }_1}\left(s\right)\text{d}s-\frac{\beta }{2\alpha \beta +{\alpha }^2}{\displaystyle {\int }_0^1{\gamma }_2}\left(s\right)\text{d}s\\ -\frac{\beta }{2\beta +\alpha }\left(I^{q}y\right)\left(1\right)-\frac{{\beta }^2}{2\alpha \beta +{\alpha }^2}\left(I^{q-1}y\right)\left(1\right),\end{array}$

$\begin{array}{l}{c}_1=\frac{1}{2\beta +\alpha }{\displaystyle {\int }_0^1{\gamma }_1}\left(s\right)\text{d}s-\frac{1}{2\beta +\alpha }{\displaystyle {\int }_0^1{\gamma }_2}\left(s\right)\text{d}s\\ -\frac{\alpha }{2\beta \text{+}\alpha }\left(I^{q}y\right)\left(1\right)-\frac{\beta }{2\beta +\alpha }\left(I^{q-1}y\right)\left(1\right).\end{array}$

因此，代入相应的值之后，可以得到

$\begin{array}{c}u\left(t\right)=-\left(I^{q}y\right)\left(t\right)-c_0-c_{1}t\\ =-\frac{1}{\Gamma \left(q\right)}{\displaystyle {\int }_0^t{\left(t-s\right)}^{q-1}y\left(s\right)ds}+\frac{\beta +\alpha t}{\left(2\beta +\alpha \right)\Gamma \left(q\right)}{\displaystyle {\int }_0^1{\left(1-s\right)}^{q-1}y\left(s\right)\text{d}s}\\ +\frac{\beta +\beta t}{\left(2\alpha \beta +{\alpha }^2\right)\Gamma \left(q-1\right)}{\displaystyle {\int }_0^1{\left(1-s\right)}^{q-2}y\left(s\right)\text{d}s}+\frac{\alpha +\beta -\alpha t}{2\alpha \beta +{\alpha }^2}{\displaystyle {\int }_0^1{\gamma }_1}\left(s\right)\text{d}s+\frac{\beta +\alpha t}{2\alpha \beta +{\alpha }^2}{\displaystyle {\int }_0^1{\gamma }_2}\left(s\right)\text{d}s\\ ={\displaystyle {\int }_0^1{G}_q}\left(t,s\right)y\left(s\right)ds+\frac{\alpha +\beta -\alpha t}{2\alpha \beta +{\alpha }^2}{\displaystyle {\int }_0^1{\gamma }_1}\left(s\right)\text{d}s+\frac{\beta +\alpha t}{2\alpha \beta +{\alpha }^2}{\displaystyle {\int }_0^1{\gamma }_2}\left(s\right)\text{d}s,\end{array}$

证明结束。

3. 主要结果

本节主要求解问题(1)唯一解的存在性。下面先给出一些假设：

(A1) 存在三个非负函数 $a_i\in C\left[0,1\right]$ ， $i=1,2,3$ ，有下面不等式

$\left|f\left(t,x,y,z\right)\right|\le a_1\left(t\right)+a_2\left(t\right)\left(\left|x\right|+\left|y\right|\right)+a_3\left|z\right|,$

对于任意的 $t\in \left[0,1\right]$ ， $x,y,z\in R$ 都成立。

类似地，存在三个非负函数 ${\tilde{a}}_i\in C\left[0,1\right]$ ， $i=1,2,3$ ，有下面不等式

$\left|g\left(t,x,y,z\right)\right|\le {\tilde{a}}_1\left(t\right)+{\tilde{a}}_2\left(t\right)\left(\left|x\right|+\left|y\right|\right)+{\tilde{a}}_3\left|z\right|,$

对于任意的 $t\in \left[0,1\right]$ ， $x,y,z\in R$ 都成立。

(A2) 存在大于0的数 $L_{h_1}$ ， $L_{h_2}$ ，使得

$\left|h_1\left(x\left(t\right)\right)\right|\le L_{h_1}\left|x\left(t\right)\right|,\left|h_2\left(x\left(t\right)\right)\right|\le L_{h_2}\left|x\left(t\right)\right|,$

对于任意的 $t\in \left[0,1\right]$ ， $x\left(t\right)\in C\left[0,1\right]$ 都成立。

类似地，存在大于0的数 ${\tilde{L}}_{{\tilde{h}}_1}$ ， ${\tilde{L}}_{{\tilde{h}}_2}$ ，使得

$\left|{\tilde{h}}_1\left(x\left(t\right)\right)\right|\le {\tilde{L}}_{{\tilde{h}}_1}\left|x\left(t\right)\right|,\left|h_2\left(x\left(t\right)\right)\right|\le {\tilde{L}}_{{\tilde{h}}_2}\left|x\left(t\right)\right|,$

对于任意的 $t\in \left[0,1\right]$ ， $x\left(t\right)\in C\left[0,1\right]$ 都成立。

(A3)设

${\eta }_{q1}={\displaystyle {\int }_0^1\left|G_q\left(t,s\right)\right|a_1\left(s\right)\text{d}s}<\infty ,$

${\theta }_{q1}=2{\displaystyle {\int }_0^1\left|G_q\left(t,s\right)\right|a_2\left(s\right)\text{d}s}+{\displaystyle {\int }_0^1\left|G_q\left(t,s\right)\right|a_3\left(s\right)\text{d}s}+\left|\frac{\left(\alpha +\beta \right)\left(L_{h_1}+L_{h_2}\right)}{2\alpha \beta +{\alpha }^2}\right|<\frac{1}{2};$

${\eta }_{q2}={\displaystyle {\int }_0^1\left|\frac{\partial }{\partial t}{G}_q\left(t,s\right)\right|a_1\left(s\right)\text{d}s}<\infty ,$

${\theta }_{q2}=2{\displaystyle {\int }_0^1\left|\frac{\partial }{\partial t}{G}_q\left(t,s\right)\right|a_2\left(s\right)\text{d}s}+{\displaystyle {\int }_0^1\left|\frac{\partial }{\partial t}{G}_q\left(t,s\right)\right|a_3\left(s\right)\text{d}s}+\left|\frac{\alpha \left(L_{h_1}+L_{h_2}\right)}{2\alpha \beta +{\alpha }^2}\right|<\frac{\Gamma \left(2-\delta \right)}{2};$

${\eta }_{q3}=\frac{1}{\Gamma \left(q-2\right)}{\displaystyle {\int }_0^t{\left(t-s\right)}^{q-3}{a}_1\left(s\right)\text{d}s}<\infty ,$

${\theta }_{q3}=\frac{2}{\Gamma \left(q-2\right)}{\displaystyle {\int }_0^t{\left(t-s\right)}^{q-3}{a}_2\left(s\right)\text{d}s}+\frac{1}{\Gamma \left(q-2\right)}{\displaystyle {\int }_0^t{\left(t-s\right)}^{q-3}{a}_3\left(s\right)\text{d}s}<\frac{\Gamma \left(3-q\right)}{2}.$

类似地，设

${\eta }_{p1}={\displaystyle {\int }_0^1\left|G_p\left(t,s\right)\right|{\tilde{a}}_1\left(s\right)\text{d}s}<\infty ,$

${\theta }_{p1}=2{\displaystyle {\int }_0^1\left|G_p\left(t,s\right)\right|{\tilde{a}}_2\left(s\right)\text{d}s}+{\displaystyle {\int }_0^1\left|G_p\left(t,s\right)\right|{\tilde{a}}_3\left(s\right)\text{d}s}+\left|\frac{\left(\alpha +\beta \right)\left({\tilde{L}}_{{\tilde{h}}_1}+{\tilde{L}}_{{\tilde{h}}_2}\right)}{2\alpha \beta +{\alpha }^2}\right|<\frac{1}{2};$

${\eta }_{p2}={\displaystyle {\int }_0^1\left|\frac{\partial }{\partial t}{G}_p\left(t,s\right)\right|{\tilde{a}}_1\left(s\right)\text{d}s}<\infty ,$

${\theta }_{p2}=2{\displaystyle {\int }_0^1\left|\frac{\partial }{\partial t}{G}_p\left(t,s\right)\right|{\tilde{a}}_2\left(s\right)\text{d}s}+{\displaystyle {\int }_0^1\left|\frac{\partial }{\partial t}{G}_p\left(t,s\right)\right|{\tilde{a}}_3\left(s\right)\text{d}s}+\left|\frac{\alpha \left({\tilde{L}}_{{\tilde{h}}_1}+{\tilde{L}}_{{\tilde{h}}_2}\right)}{2\alpha \beta +{\alpha }^2}\right|<\frac{\Gamma \left(2-\delta \right)}{2};$

${\eta }_{p3}=\frac{1}{\Gamma \left(p-2\right)}{\displaystyle {\int }_0^t{\left(t-s\right)}^{p-3}{\tilde{a}}_1\left(s\right)\text{d}s}<\infty ,$

${\theta }_{p3}=\frac{2}{\Gamma \left(p-2\right)}{\displaystyle {\int }_0^t{\left(t-s\right)}^{p-3}{\tilde{a}}_2\left(s\right)\text{d}s}+\frac{1}{\Gamma \left(p-2\right)}{\displaystyle {\int }_0^t{\left(t-s\right)}^{p-3}{\tilde{a}}_3\left(s\right)\text{d}s}<\frac{\Gamma \left(3-p\right)}{2}.$

(A4)存在一个连续函数 $s:\left[0,1\right]\to \left[0,+\infty \right)$ ，一个关于每一个变量都非减的函数 $\psi :\left[0,+\infty \right)\times \left[0,+\infty \right)\times \left[0,+\infty \right)\to \left[0,+\infty \right)$ 并且对于任意的 $w\in \left[0,+\infty \right)$ ，都有 $\psi \left(w,w,w\right)\le w$ ，则下面不等式

$\left|f\left(t,x,y,z\right)-f\left(t,\bar{x},\bar{y},\bar{z}\right)\right|\le s\left(t\right)\psi \left(\left|x-\bar{x}\right|,\left|y-\bar{y}\right|,\left|z-\bar{z}\right|\right),$

对于任意的 $t\in \left[0,1\right]$ ， $x,y,z,\bar{x},\bar{y},\bar{z}\in R$ 都成立。

类似地，存在一个连续函数 $\tilde{s}:\left[0,1\right]\to \left[0,+\infty \right)$ ，一个关于每一个变量都非减的函数 $\tilde{\psi }:\left[0,+\infty \right)\times \left[0,+\infty \right)\times \left[0,+\infty \right)\to \left[0,+\infty \right)$ 并且对于任意的 $w\in \left[0,+\infty \right)$ ，都有 $\tilde{\psi }\left(w,w,w\right)\le w$ ，则下面不等式

$\left|g\left(t,x,y,z\right)-g\left(t,\bar{x},\bar{y},\bar{z}\right)\right|\le \tilde{s}\left(t\right)\tilde{\psi }\left(\left|x-\bar{x}\right|,\left|y-\bar{y}\right|,\left|z-\bar{z}\right|\right),$

对于任意的 $t\in \left[0,1\right]$ ， $x,y,z,\bar{x},\bar{y},\bar{z}\in R$ 都成立。

(A5) 存在大于0的数 $K_{h_1}$ ， $K_{h_2}$ 使得

$\left|h_1\left(x\right)-h_1\left(\bar{x}\right)\right|\le K_{h_1}\left|x-\bar{x}\right|,\left|h_2\left(x\right)-h_2\left(\bar{x}\right)\right|\le K_{h_2}\left|x-\bar{x}\right|,$

对于任意的 $t\in \left[0,1\right]$ ， $x,\bar{x}\in R$ 都成立。

类似地，存在大于0的数 ${\tilde{K}}_{{\tilde{h}}_1}$ ， ${\tilde{K}}_{{\tilde{h}}_2}$ 使得

$\left|{\tilde{h}}_1\left(x\right)-{\tilde{h}}_1\left(\bar{x}\right)\right|\le {\tilde{K}}_{{\tilde{h}}_1}\left|x-\bar{x}\right|,\left|{\tilde{h}}_2\left(x\right)-{\tilde{h}}_2\left(\bar{x}\right)\right|\le {\tilde{K}}_{{\tilde{h}}_2}\left|x-\bar{x}\right|,$

对于任意的 $t\in \left[0,1\right]$ ， $x,\bar{x}\in R$ 都成立。

(A6)通过 $G_q\left(t,s\right)$ 的表达式可以知道 $G_q\left(t,s\right)$ 关于t和s是连续的，经过计算，同样地可以得到 $\frac{\partial }{\partial t}{G}_q\left(t,s\right)$ 和 $\frac{{\partial }^2}{\partial t^2}{G}_q\left(t,s\right)$ 关于t和s是连续的，则可设

$b_q=\underset{t\in \left[0,1\right]}{\max }\left\{{\displaystyle {\int }_0^1\left|G_q\left(t,s\right)\right|\text{d}s},{\displaystyle {\int }_0^1\left|\frac{\partial }{\partial t}{G}_q\left(t,s\right)\right|\text{d}s,}{\displaystyle {\int }_0^1\left|\frac{{\partial }^2}{\partial t^2}{G}_q\left(t,s\right)\right|\text{d}s}\right\},$

类似地，可设

$b_p=\underset{t\in \left[0,1\right]}{\max }\left\{{\displaystyle {\int }_0^1\left|G_p\left(t,s\right)\right|\text{d}s},{\displaystyle {\int }_0^1\left|\frac{\partial }{\partial t}{G}_p\left(t,s\right)\right|\text{d}s,}{\displaystyle {\int }_0^1\left|\frac{{\partial }^2}{\partial t^2}{G}_p\left(t,s\right)\right|\text{d}s}\right\},$

设

${\rho }_1=\max \left\{b_p\cdot {\Vert \tilde{s}\Vert }_{\infty }+\frac{\left(\alpha +\beta \right)\left(K_{h_1}+K_{h_2}\right)}{2\alpha \beta +{\alpha }^2},b_q\cdot {\Vert s\Vert }_{\infty }+\frac{\left(\alpha +\beta \right)\left({\tilde{K}}_{{\tilde{h}}_1}+{\tilde{K}}_{{\tilde{h}}_2}\right)}{2\alpha \beta +{\alpha }^2}\right\},$

${\rho }_2=\max \left\{\frac{b_p\cdot {\Vert \tilde{s}\Vert }_{\infty }+\frac{2\alpha \left(K_{h_1}+K_{h_2}\right)}{2\alpha \beta +{\alpha }^2}}{\Gamma \left(2-\delta \right)},\frac{b_q\cdot {\Vert s\Vert }_{\infty }+\frac{2\alpha \left({\tilde{K}}_{{\tilde{h}}_1}+{\tilde{K}}_{{\tilde{h}}_2}\right)}{2\alpha \beta +{\alpha }^2}}{\Gamma \left(2-\delta \right)}\right\},$

和

${\rho }_3=\max \left\{\frac{b_p\cdot {\Vert \tilde{s}\Vert }_{\infty }}{\Gamma \left(4-p\right)},\frac{b_q\cdot {\Vert s\Vert }_{\infty }}{\Gamma \left(4-q\right)}\right\}.$

为了方便，定义空间 $B=\left\{u|u\in C^1\left[0,1\right],{}^{C}D{}^{q-1}u\in C\left[0,1\right]\right\}$ 和 $\tilde{B}=\left\{v|v\in C^1\left[0,1\right],{}^{C}D{}^{p-1}v\in C\left[0,1\right]\right\}$ ，两个空间的范数表达式分别为：

${\Vert u\Vert }_B=\max \left\{{\Vert u\Vert }_{\infty },{\Vert {}^{C}D{}^{\delta }u\Vert }_{\infty },{\Vert {}^{C}D{}^{q-1}u\Vert }_{\infty }\right\}$ 和 ${\Vert v\Vert }_{\tilde{B}}=\max \left\{{\Vert v\Vert }_{\infty },{\Vert {}^{C}D{}^{\delta }v\Vert }_{\infty },{\Vert {}^{C}D{}^{p-1}v\Vert }_{\infty }\right\},$

则空间 $B$ ， $\tilde{B}$ 都是Banach [12] 空间。因此乘积空间上的范数定义为 ${\Vert \left(u,v\right)\Vert }_{B\times \tilde{B}}\text{=}{\Vert u\Vert }_B\text{+}{\Vert v\Vert }_{\tilde{B}}$ 。则空间 $\left(B\times \tilde{B},{\Vert \left(\cdot ,\cdot \right)\Vert }_{B\times \tilde{B}}\right)$ 是一个Banach空间。

定义算子 $T:B\times \tilde{B}\to B\times \tilde{B}$ 为

$\begin{array}{c}T\left(u,v\right)\left(t\right)=\left(\begin{array}{c}{\displaystyle {\int }_0^1{G}_q\left(t,s\right)f\left(s,v\left(s\right),{}^{C}D{}^{\delta }v\left(s\right),{}^{C}D{}^{q-1}u\left(s\right)\right)\text{d}s}+\frac{\alpha +\beta -\alpha t}{2\alpha \beta +{\alpha }^2}{\displaystyle {\int }_0^1{h}_1\left(u\left(s\right)\right)\text{d}s+\frac{\beta +\alpha t}{2\alpha \beta +{\alpha }^2}{\displaystyle {\int }_0^1{h}_2\left(u\left(s\right)\right)\text{d}s}}\\ {\displaystyle {\int }_0^1{G}_p\left(t,s\right)f\left(s,u\left(s\right),{}^{C}D{}^{\delta }u\left(s\right),{}^{C}D{}^{p-1}v\left(s\right)\right)\text{d}s}+\frac{\alpha +\beta -\alpha t}{2\alpha \beta +{\alpha }^2}{\displaystyle {\int }_0^1{\tilde{h}}_1\left(v\left(s\right)\right)\text{d}s+\frac{\beta +\alpha t}{2\alpha \beta +{\alpha }^2}{\displaystyle {\int }_0^1{\tilde{h}}_2\left(v\left(s\right)\right)\text{d}s}}\end{array}\right)\\ =\left(\begin{array}{l}{T}_q\left(v,u\right)\left(t\right)\\ T_p\left(u,v\right)\left(t\right)\end{array}\right).\end{array}$ (3)

定理5

如果假设(A1)~(A6)成立，并且当 $\rho =\max \left\{{\rho }_1,{\rho }_2,{\rho }_3\right\}<1$ 时，问题有唯一解。

证明 关于该定理的证明，本文主要运用的是Banach不动点定理。首先，给出两个非负数

${\lambda }_1=\max \left\{\frac{2{\eta }_{q1}}{1-2{\theta }_{q1}},\frac{2{\eta }_{q2}}{\Gamma \left(2-\delta \right)-2{\theta }_{q2}},\frac{2{\eta }_{q3}}{\Gamma \left(3-q\right)-2{\theta }_{q3}}\right\},$

${\lambda }_2=\max \left\{\frac{2{\eta }_{p1}}{1-2{\theta }_{p1}},\frac{2{\eta }_{p2}}{\Gamma \left(2-\delta \right)-2{\theta }_{p2}},\frac{2{\eta }_{p3}}{\Gamma \left(3-q\right)-2{\theta }_{p3}}\right\},$

设

$\lambda =\max \left\{{\lambda }_1,{\lambda }_2\right\},$

定义一个集合 $\omega =\left\{\left(u,v\right)\in B\times \tilde{B}:{\Vert u\Vert }_B\le \frac{\lambda }{2},{\Vert v\Vert }_{\tilde{B}}\le \frac{\lambda }{2},{\Vert \left(u,v\right)\Vert }_{B\times \tilde{B}}\le \lambda \right\}$ ，为了证明由(3)式定义的算子T是从 $\omega$ 映射到自身的，可以先得到

$\begin{array}{l}\left|T_q\left(v,u\right)\left(t\right)\right|\\ =|{\displaystyle {\int }_0^1{G}_q\left(t,s\right)f\left(s,v\left(s\right),{}^{C}D{}^{\delta }v\left(s\right),{}^{C}D{}^{q-1}u\left(s\right)\right)\text{d}s}+\frac{\alpha +\beta -\alpha t}{2\alpha \beta +{\alpha }^2}{\displaystyle {\int }_0^1{h}_1\left(u\left(s\right)\right)\text{d}s}+\frac{\beta +\alpha t}{2\alpha \beta +{\alpha }^2}{\displaystyle {\int }_0^1{h}_2\left(u\left(s\right)\right)\text{d}s}|\\ \le {\displaystyle {\int }_0^1\left|G_q\left(t,s\right)\right|}{a}_1\left(s\right)\text{d}s+{\displaystyle {\int }_0^1\left|G_q\left(t,s\right)\right|a_2\left(s\right)\left(\left|v\left(s\right)+{}^{C}D{}^{\delta }v\left(s\right)\right|\right)\text{d}s}\\ +{\displaystyle {\int }_0^1\left|G_q\left(t,s\right)\right|a_3\left(s\right)\left|{}^{C}D{}^{q-1}u\left(s\right)\right|\text{d}s}+\left|\frac{\alpha +\beta }{2\alpha \beta +{\alpha }^2}\right|\left[{\displaystyle {\int }_0^1\left|h_1\left(u\left(s\right)\right)\right|\text{d}s}+{\displaystyle {\int }_0^1\left|h_2\left(u\left(s\right)\right)\right|\text{d}s}\right]\end{array}$

$\begin{array}{l}\le {\displaystyle {\int }_0^1\left|G_q\left(t,s\right)\right|a_1\left(s\right)\text{d}s}+\lambda \left[2{\displaystyle {\int }_0^1\left|G_q\left(t,s\right)\right|a_2\left(s\right)\text{d}s}+{\displaystyle {\int }_0^1\left|G_q\left(t,s\right)\right|a_3\left(s\right)\text{d}s}+\left|\frac{\alpha +\beta \left(L_{h_1}+L_{h_2}\right)}{2\alpha \beta +{\alpha }^2}\right|\right]\\ \le {\eta }_{q_1}+\lambda {\theta }_{q_1}\le \frac{\lambda }{2},\end{array}$

接着，逐次求导可得

$\begin{array}{l}\left|{T^{\prime }}_q\left(v,u\right)\left(t\right)\right|\\ =|{\displaystyle {\int }_0^1\frac{\partial }{\partial t}{G}_q\left(t,s\right)f\left(s,v\left(s\right),{}^{C}D{}^{\delta }v\left(s\right),{}^{C}D{}^{q-1}u\left(s\right)\right)\text{d}s}+\frac{-\alpha }{2\alpha \beta +{\alpha }^2}{\displaystyle {\int }_0^1{h}_1\left(u\left(s\right)\right)\text{d}s}+\frac{\alpha }{2\alpha \beta +{\alpha }^2}{\displaystyle {\int }_0^1{h}_2\left(u\left(s\right)\right)\text{d}s}|\\ \le {\displaystyle {\int }_0^1\left|\frac{\partial }{\partial t}{G}_q\left(t,s\right)\right|a_1\left(s\right)\text{d}s}+{\displaystyle {\int }_0^1\left|\frac{\partial }{\partial t}{G}_q\left(t,s\right)\right|a_2\left(s\right)\left(\left|v\left(s\right)+{}^{C}D{}^{\delta }v\left(s\right)\right|\right)\text{d}s}\\ +{\displaystyle {\int }_0^1\left|\frac{\partial }{\partial t}{G}_q\left(t,s\right)\right|a_3\left(s\right)\left|{}^{C}D{}^{q-1}u\left(s\right)\right|\text{d}s}+\left|\frac{\alpha }{2\alpha \beta +{\alpha }^2}\right|\left[{\displaystyle {\int }_0^1\left|h_1\left(u\left(s\right)\right)\right|\text{d}s}+{\displaystyle {\int }_0^1\left|h_2\left(u\left(s\right)\right)\right|\text{d}s}\right]\\ \le {\displaystyle {\int }_0^1\left|\frac{\partial }{\partial t}{G}_q\left(t,s\right)\right|a_1\left(s\right)\text{d}s}+\lambda \left[2{\displaystyle {\int }_0^1\left|\frac{\partial }{\partial t}{G}_q\left(t,s\right)\right|a_2\left(s\right)\text{d}s}+{\displaystyle {\int }_0^1\left|\frac{\partial }{\partial t}{G}_q\left(t,s\right)\right|a_3\left(s\right)\text{d}s}+\left|\frac{\alpha \left(L_{h_1}+L_{h_2}\right)}{2\alpha \beta +{\alpha }^2}\right|\right]\\ \le {\eta }_{q_2}+\lambda {\theta }_{q_2}\end{array}$

和

$\begin{array}{l}\left|{T^{″}}_q\left(v,u\right)\left(t\right)\right|\\ =\left|{\displaystyle {\int }_0^1\frac{{\partial }^2}{\partial t^2}{G}_q\left(t,s\right)f\left(s,v\left(s\right),{}^{C}D{}^{\delta }v\left(s\right),{}^{C}D{}^{q-1}u\left(s\right)\right)\text{d}s}\right|\\ =\left|{\displaystyle {\int }_0^t\frac{-1}{\Gamma \left(q-2\right)}{\left(t-s\right)}^{q-3}f\left(s,v\left(s\right),{}^{C}D{}^{\delta }v\left(s\right),{}^{C}D{}^{q-1}u\left(s\right)\right)\text{d}s}\right|\\ \le \frac{1}{\Gamma \left(q-2\right)}{\displaystyle {\int }_0^t{\left(t-s\right)}^{q-3}\left[a_1\left(s\right)+a_2\left(s\right)\left(\left|v\left(s\right)\right|+\left|{}^{C}D{}^{\delta }v\left(s\right)\right|\right)+a_3\left(s\right)\left|{}^{C}D{}^{q-1}u\left(s\right)\right|\right]\text{d}s}\\ \le {\eta }_{q_3}+\lambda {\theta }_{q_3}.\end{array}$

根据定义，可以有

$\begin{array}{l}\left|{}^{C}D{}^{\delta }\left(T_q\left(v,u\right)\right)\left(t\right)\right|\\ =\left|\frac{1}{\Gamma \left(1-\delta \right)}{\displaystyle {\int }_0^t{\left(t-s\right)}^{-\delta }}{T^{\prime }}_q\left(v,u\right)\left(s\right)\text{d}s\right|\\ \le \frac{1}{\Gamma \left(1-\delta \right)}{\displaystyle {\int }_0^t{\left(t-s\right)}^{-\delta }}\left|{T^{\prime }}_q\left(v,u\right)\left(s\right)\right|\text{d}s\\ \le \frac{1}{\Gamma \left(1-\delta \right)}{\displaystyle {\int }_0^t{\left(t-s\right)}^{-\delta }}\left({\eta }_{q_2}+\lambda {\theta }_{q_2}\right)\text{d}s\\ \le \frac{1}{\Gamma \left(2-\delta \right)}\left({\eta }_{q_2}+\lambda {\theta }_{q_2}\right)\le \frac{\lambda }{2}\end{array}$

和

$\begin{array}{l}\left|{}^{C}D{}^{q-1}\left(T_q\left(v,u\right)\right)\left(t\right)\right|\\ =\left|\frac{1}{\Gamma \left(3-q\right)}{\displaystyle {\int }_0^t{\left(t-s\right)}^{2-q}}{T^{″}}_q\left(v,u\right)\left(s\right)\text{d}s\right|\\ \le \frac{1}{\Gamma \left(3-q\right)}{\displaystyle {\int }_0^t{\left(t-s\right)}^{2-q}}\left|{T^{″}}_q\left(v,u\right)\left(s\right)\right|\text{d}s\\ \le \frac{1}{\Gamma \left(3-q\right)}{\displaystyle {\int }_0^t{\left(t-s\right)}^{2-q}}\left({\eta }_{q_3}+\lambda {\theta }_{q_3}\right)\text{d}s\\ \le \frac{1}{\Gamma \left(4-q\right)}\left({\eta }_{q_3}+\lambda {\theta }_{q_3}\right)\le \frac{\lambda }{2}.\end{array}$

则由范数的定义，可以得到

${\Vert T_q\left(v,u\right)\Vert }_B=\max \left\{{\Vert T_q\left(v,u\right)\Vert }_{\infty },{\Vert {}^{C}D{}^{\delta }\left(T_q\left(v,u\right)\right)\Vert }_{\infty },{\Vert {}^{C}D{}^{q-1}\left(T_q\left(v,u\right)\right)\Vert }_{\infty }\right\}\le \frac{\lambda }{2}$

类似地，可以有

${\Vert T_p\left(u,v\right)\Vert }_{\tilde{B}}=\max \left\{{\Vert T_p\left(u,v\right)\Vert }_{\infty },{\Vert {}^{C}D{}^{\delta }\left(T_p\left(u,v\right)\right)\Vert }_{\infty },{\Vert {}^{C}D{}^{p-1}\left(T_p\left(u,v\right)\right)\Vert }_{\infty }\right\}\le \frac{\lambda }{2}$

根据乘积空间上范数的定义，可得到 ${\Vert T\left(u,v\right)\Vert }_{B\times \tilde{B}}={\Vert T_q\left(v,u\right)\Vert }_B+{\Vert T_p\left(u,v\right)\Vert }_{\tilde{B}}\le \lambda$ 这就说明是自身T到自身的映射。

接下来，需要证明映射T是一个压缩映射。根据T的定义，相应范数的定义，假设(A4)~(A6)，可以有

$\begin{array}{l}\left|T_q\left(v,u\right)\left(t\right)-T_q\left(\bar{v},\bar{u}\right)\left(t\right)\right|\\ =|{\displaystyle {\int }_0^1{G}_q\left(t,s\right)\left(f\left(s,v\left(s\right),{}^{C}D{}^{\delta }v\left(s\right),{}^{C}D{}^{q-1}u\left(s\right)\right)-f\left(s,\bar{v}\left(s\right),{}^{C}D{}^{\delta }\bar{v}\left(s\right),{}^{C}D{}^{q-1}\bar{u}\left(s\right)\right)\right)}\text{d}s\\ \text{+}\frac{\alpha +\beta -\alpha t}{2\alpha \beta +{\alpha }^2}{\displaystyle {\int }_0^1\left(h_1\left(u\left(s\right)\right)-h_1\left(\bar{u}\left(s\right)\right)\right)\text{d}s+\frac{\beta +\alpha t}{2\alpha \beta +{\alpha }^2}{\displaystyle {\int }_0^1\left(h_2\left(u\left(s\right)\right)-h_2\left(\bar{u}\left(s\right)\right)\right)\text{d}s}}|\\ \le {\displaystyle {\int }_0^1\left|G_q\left(t,s\right)\right|s\left(t\right)\psi }\left(\left|v\left(s\right)-\bar{v}\left(s\right)\right|,\left|{}^{C}D{}^{\delta }v\left(s\right)-{}^{C}D{}^{\delta }\bar{v}\left(s\right)\right|,\left|{}^{C}D{}^{q-1}v\left(s\right)-{}^{C}D{}^{q-1}\bar{v}\left(s\right)\right|\right)\text{d}s\\ \text{+}\frac{\left(\alpha +\beta \right)\left(K_{h_1}+K_{h_2}\right)}{2\alpha \beta +{\alpha }^2}\left|u\left(s\right)-\bar{u}\left(s\right)\right|\end{array}$

$\begin{array}{l}\le {\Vert s\Vert }_{\infty }\cdot \max \left\{{\Vert v-\bar{v}\Vert }_{\tilde{B}},{\Vert u-\bar{u}\Vert }_B\right\}{\displaystyle {\int }_0^1\left|G_q\left(t,s\right)\right|\text{d}s}+\frac{\left(\alpha +\beta \right)\left(K_{h_1}+K_{h_2}\right)}{2\alpha \beta +{\alpha }^2}{\Vert u-\bar{u}\Vert }_B\\ \le {\Vert s\Vert }_{\infty }\cdot \left({\Vert v-\bar{v}\Vert }_{\tilde{B}}+{\Vert u-\bar{u}\Vert }_B\right){\displaystyle {\int }_0^1\left|G_q\left(t,s\right)\right|\text{d}s}+\frac{\left(\alpha +\beta \right)\left(K_{h_1}+K_{h_2}\right)}{2\alpha \beta +{\alpha }^2}{\Vert u-\bar{u}\Vert }_B\\ \le b_q\cdot {\Vert s\Vert }_{\infty }\cdot \left({\Vert v-\bar{v}\Vert }_{\tilde{B}}+{\Vert u-\bar{u}\Vert }_B\right)+\frac{\left(\alpha +\beta \right)\left(K_{h_1}+K_{h_2}\right)}{2\alpha \beta +{\alpha }^2}{\Vert u-\bar{u}\Vert }_B,\end{array}$

相应地，逐次求导之后可得

$\begin{array}{l}\left|{T^{\prime }}_q\left(v,u\right)\left(t\right)-{T^{\prime }}_q\left(\bar{v},\bar{u}\right)\left(t\right)\right|\\ =|{\displaystyle {\int }_0^1\frac{\partial }{\partial t}{G}_q\left(t,s\right)\left(f\left(s,v\left(s\right),{}^{C}D{}^{\delta }v\left(s\right),{}^{C}D{}^{q-1}u\left(s\right)\right)-f\left(s,\bar{v}\left(s\right),{}^{C}D{}^{\delta }\bar{v}\left(s\right),{}^{C}D{}^{q-1}\bar{u}\left(s\right)\right)\right)}\text{d}s\\ +\frac{-\alpha }{2\alpha \beta +{\alpha }^2}{\displaystyle {\int }_0^1\left(h_1\left(u\left(s\right)\right)-h_1\left(\bar{u}\left(s\right)\right)\right)\text{d}s+\frac{\alpha }{2\alpha \beta +{\alpha }^2}{\displaystyle {\int }_0^1\left(h_2\left(u\left(s\right)\right)-h_2\left(\bar{u}\left(s\right)\right)\right)\text{d}s}}|\\ \le {\displaystyle {\int }_0^1\left|\frac{\partial }{\partial t}{G}_q\left(t,s\right)\right|s\left(t\right)\psi \left(\left|v\left(s\right)-\bar{v}\left(s\right)\right|,\left|{}^{C}D{}^{\delta }v\left(s\right)-{}^{C}D{}^{\delta }\bar{v}\left(s\right)\right|,\left|{}^{C}D{}^{q-1}v\left(s\right)-{}^{C}D{}^{q-1}\bar{v}\left(s\right)\right|\right)\text{d}s}\\ +\frac{2\alpha \left(K_{h_1}+K_{h_2}\right)}{2\alpha \beta +{\alpha }^2}\left|u\left(s\right)-\bar{u}\left(s\right)\right|\\ \le b_q\cdot {\Vert s\Vert }_{\infty }\cdot \left({\Vert v-\bar{v}\Vert }_{\tilde{B}}+{\Vert u-\bar{u}\Vert }_B\right)+\frac{2\alpha \left(K_{h_1}+K_{h_2}\right)}{2\alpha \beta +{\alpha }^2}{\Vert u-\bar{u}\Vert }_B\end{array}$

和

$\begin{array}{l}\left|{T^{″}}_q\left(v,u\right)\left(t\right)-{T^{″}}_q\left(\bar{v},\bar{u}\right)\left(t\right)\right|\\ =\left|{\displaystyle {\int }_0^1\frac{{\partial }^2}{\partial t^2}{G}_q\left(t,s\right)\left(f\left(s,v\left(s\right),{}^{C}D{}^{\delta }v\left(s\right),{}^{C}D{}^{q-1}u\left(s\right)\right)-f\left(s,\bar{v}\left(s\right),{}^{C}D{}^{\delta }\bar{v}\left(s\right),{}^{C}D{}^{q-1}\bar{u}\left(s\right)\right)\right)\text{d}s}\right|\\ \le {\displaystyle {\int }_0^1\left|\frac{{\partial }^2}{\partial t^2}{G}_q\left(t,s\right)\right|s\left(t\right)\psi \left(\left|v\left(s\right)-\bar{v}\left(s\right)\right|,\left|{}^{C}D{}^{\delta }v\left(s\right)-{}^{C}D{}^{\delta }\bar{v}\left(s\right)\right|,\left|{}^{C}D{}^{q-1}v\left(s\right)-{}^{C}D{}^{q-1}\bar{v}\left(s\right)\right|\right)\text{d}s}\\ \le b_q\cdot {\Vert s\Vert }_{\infty }\cdot \left({\Vert v-\bar{v}\Vert }_{\tilde{B}}+{\Vert u-\bar{u}\Vert }_B\right).\end{array}$

根据定义，可以得到

$\begin{array}{l}\left|{}^{C}D{}^{\delta }{T}_q\left(v,u\right)\left(t\right)-{}^{C}D{}^{\delta }{T}_q\left(\bar{v},\bar{u}\right)\left(t\right)\right|\\ =\left|\frac{1}{\Gamma \left(1-\delta \right)}{\displaystyle {\int }_0^t{\left(t-s\right)}^{-\delta }\left({T^{\prime }}_q\left(v,u\right)\left(t\right)-{T^{\prime }}_q\left(\bar{v},\bar{u}\right)\left(t\right)\right)\text{d}s}\right|\\ \le \frac{1}{\Gamma \left(2-\delta \right)}\left[b_q\cdot {\Vert s\Vert }_{\infty }\cdot \left({\Vert v-\bar{v}\Vert }_{\tilde{B}}+{\Vert u-\bar{u}\Vert }_B\right)+\frac{2\alpha \left(K_{h_1}+K_{h_2}\right)}{2\alpha \beta +{\alpha }^2}{\Vert u-\bar{u}\Vert }_B\right]\end{array}$

和

$\begin{array}{l}\left|{}^{C}D{}^{q-1}{T}_q\left(v,u\right)\left(t\right)-{}^{C}D{}^{q-1}{T}_q\left(\bar{v},\bar{u}\right)\left(t\right)\right|\\ =\left|\frac{1}{\Gamma \left(3-q\right)}{\displaystyle {\int }_0^t{\left(t-s\right)}^{2-q}\left({T^{″}}_q\left(v,u\right)\left(t\right)-{T^{″}}_q\left(\bar{v},\bar{u}\right)\left(t\right)\right)\text{d}s}\right|\\ \le \frac{1}{\Gamma \left(4-q\right)}{b}_q\cdot {\Vert s\Vert }_{\infty }\cdot \left({\Vert v-\bar{v}\Vert }_{\tilde{B}}+{\Vert u-\bar{u}\Vert }_B\right).\end{array}$

由范数的定义，可以得到 $T_q\left(v,u\right)\left(t\right)-T_q\left(\bar{v},\bar{u}\right)\left(t\right)$ 在空间B中的范数为

$\begin{array}{l}{\Vert T_q\left(v,u\right)-T_q\left(\bar{v},\bar{u}\right)\Vert }_B\\ =\max \left\{{\Vert T_q\left(v,u\right)-T_q\left(\bar{v},\bar{u}\right)\Vert }_{\infty },{\Vert {}^{C}D{}^{\delta }{T}_q\left(v,u\right)-{}^{C}D{}^{\delta }{T}_q\left(\bar{v},\bar{u}\right)\Vert }_{\infty },{\Vert {}^{C}D{}^{q-1}{T}_q\left(v,u\right)-{}^{C}D{}^{q-1}{T}_q\left(\bar{v},\bar{u}\right)\Vert }_{\infty }\right\},\end{array}$

类似地，可以有

$\begin{array}{l}{\Vert T_p\left(u,v\right)-T_p\left(\bar{u},\bar{v}\right)\Vert }_{\tilde{B}}\\ =\max \left\{{\Vert T_p\left(u,v\right)-T_p\left(\bar{u},\bar{v}\right)\Vert }_{\infty },{\Vert {}^{C}D{}^{\delta }{T}_p\left(u,v\right)-{}^{C}D{}^{\delta }{T}_p\left(\bar{u},\bar{v}\right)\Vert }_{\infty },{\Vert {}^{C}D{}^{p-1}{T}_p\left(u,v\right)-{}^{C}D{}^{p-1}{T}_p\left(\bar{u},\bar{v}\right)\Vert }_{\infty }\right\}.\end{array}$

根据乘积空间范数的定义及 $\rho =\max \left\{{\rho }_1,{\rho }_2,{\rho }_3\right\}<1$ ，可以得到

$\begin{array}{c}{\Vert T\left(v,u\right)-T\left(\bar{v},\bar{u}\right)\Vert }_{B\times \tilde{B}}={\Vert T_q\left(v,u\right)-T_q\left(\bar{v},\bar{u}\right)\Vert }_B+{\Vert T_p\left(u,v\right)-T_p\left(\bar{u},\bar{v}\right)\Vert }_{\tilde{B}}\\ \le \rho {\Vert T\left(v,u\right)-T\left(\bar{v},\bar{u}\right)\Vert }_{B\times \tilde{B}},\end{array}$

这就证明得到T是一个压缩映射，根据Banach不动点定理，可以证明问题(1)有唯一解。

4. 实例分析

考虑边值问题

$\{\begin{array}{l}{}^{C}D{}^{\frac{5}{2}}u\left(t\right)+\frac{{\text{e}}^{-\pi t}}{10}\cdot \frac{\left|v\left(t\right)\right|+\left|v^{\prime }\left(t\right)\right|+\left|{}^{C}D{}^{\frac{5}{2}}u\left(t\right)\right|}{4+\left|v\left(t\right)\right|+\left|v^{\prime }\left(t\right)\right|+\left|{}^{C}D{}^{\frac{5}{2}}u\left(t\right)\right|}=0,t\in \left[0,1\right],\\ {}^{C}D{}^{\frac{8}{3}}v\left(t\right)+\frac{{\text{e}}^{-t}+t^2+3}{40}\frac{\left|u\left(t\right)\right|+\left|u^{\prime }\left(t\right)\right|+\left|{}^{C}D{}^{\frac{5}{2}}v\left(t\right)\right|}{4+\left|u\left(t\right)\right|+\left|u^{\prime }\left(t\right)\right|+\left|{}^{C}D{}^{\frac{5}{2}}v\left(t\right)\right|}=0,t\in \left[0,1\right],\\ u\left(0\right)-u^{\prime }\left(0\right)={\displaystyle {\int }_0^1\frac{\left|u\left(s\right)\right|}{17+\left|u\left(s\right)\right|}\text{d}s,}u\left(1\right)+u^{\prime }\left(1\right)={\displaystyle {\int }_0^1\frac{\left|u\left(s\right)\right|}{19+\left|u\left(s\right)\right|}\text{d}s,u^{″}\left(0\right)=0,}\\ v\left(0\right)-v^{\prime }\left(0\right)={\displaystyle {\int }_0^1\frac{1}{34}\left[\cos v\left(s\right)+\sin v\left(s\right)\right]\text{d}s,}v\left(1\right)+v^{\prime }\left(1\right)={\displaystyle {\int }_0^1\frac{1}{38}\left[\cos v\left(s\right)+\sin v\left(s\right)\right]\text{d}s,v^{″}\left(0\right)=0.}\end{array}$

设 $h_1\left(u\left(s\right)\right)=\frac{\left|u\left(s\right)\right|}{17+\left|u\left(s\right)\right|}$ ， $h_2\left(u\left(s\right)\right)=\frac{\left|u\left(s\right)\right|}{19+\left|u\left(s\right)\right|}$ ， ${\tilde{h}}_1\left(u\left(s\right)\right)=\frac{1}{34}\left[\cos v\left(s\right)+\sin v\left(s\right)\right]$ ， ${\tilde{h}}_2\left(u\left(s\right)\right)=\frac{1}{38}\left[\cos v\left(s\right)+\sin v\left(s\right)\right]$ ，则 $h_1,h_2,{\tilde{h}}_1,{\tilde{h}}_2:R\to R$ 是连续函数。则可以得到

$\left|h_1\left(x\right)-h_1\left(\bar{x}\right)\right|=\frac{\left|x\right|}{17+\left|x\right|}-\frac{\left|\bar{x}\right|}{17+\left|\bar{x}\right|}\le \frac{1}{17}\left|x-\bar{x}\right|,$

同理，可得

$\left|h_2\left(x\right)-h_2\left(\bar{x}\right)\right|=\frac{1}{19}\left|x-\bar{x}\right|,$

由

$\left|{\tilde{h}}_1\left(x\right)-{\tilde{h}}_1\left(\bar{x}\right)\right|=\left|\frac{1}{34}\left[\cos x+\sin x\right]-\frac{1}{34}\left[\cos \bar{x}+\sin \bar{x}\right]\right|\le \frac{1}{17}\left|x-\bar{x}\right|,$

同理，可得

$\left|{\tilde{h}}_2\left(x\right)-{\tilde{h}}_2\left(\bar{x}\right)\right|\le \frac{1}{19}\left|x-\bar{x}\right|.$

因此，可知 $K_{h_1}={\tilde{K}}_{{\tilde{h}}_1}=\frac{1}{17}$ ， $K_{h_2}={\tilde{K}}_{{\tilde{h}}_2}=\frac{1}{19}$ 。

设 $f\left(t,x,y,z\right)=\frac{{\text{e}}^{-\pi t}}{10}\cdot \frac{\left|x\right|+\left|y\right|+\left|z\right|}{4+\left|x\right|+\left|y\right|+\left|z\right|}$ ， $g\left(t,x,y,z\right)=\frac{{\text{e}}^{-t}+t^2+3}{40}\cdot \frac{\left|x\right|+\left|y\right|+\left|z\right|}{4+\left|x\right|+\left|y\right|+\left|z\right|}$ ， $\left(t,x,y,z\right)\in \left[0,1\right]\times R\times R\times R$ ，那么 $f,g:\left[0,1\right]\times R^3\to R$ 是连续的。

由于

$\begin{array}{l}\left|f\left(t,x,y,z\right)-f\left(t,\bar{x},\bar{y},\bar{z}\right)\right|\\ =\frac{{\text{e}}^{-\pi t}}{10}\left|\frac{\left|x\right|+\left|y\right|+\left|z\right|}{4+\left|x\right|+\left|y\right|+\left|z\right|}-\frac{\left|\bar{x}\right|+\left|\bar{y}\right|+\left|\bar{z}\right|}{4+\left|\bar{x}\right|+\left|\bar{y}\right|+\left|\bar{z}\right|}\right|\\ \le \frac{{\text{e}}^{-\pi t}}{10}\frac{4\left(\left|x-\bar{x}\right|,\left|y-\bar{y}\right|,\left|z-\bar{z}\right|\right)}{\left(4+\left|x\right|+\left|y\right|+\left|z\right|\right)\left(4+\left|x\right|+\left|y\right|+\left|z\right|\right)}\\ \le \frac{{\text{e}}^{-\pi t}}{40}\left(\left|x-\bar{x}\right|,\left|y-\bar{y}\right|,\left|z-\bar{z}\right|\right)\end{array}$

同理，可得

$\left|g\left(t,x,y,z\right)-g\left(t,\bar{x},\bar{y},\bar{z}\right)\right|\le \frac{{\text{e}}^{-t}+t^2+3}{160}\left(\left|x-\bar{x}\right|,\left|y-\bar{y}\right|,\left|z-\bar{z}\right|\right)$

因此选择 $s\left(t\right)=\frac{{\text{e}}^{-\pi t}}{10}$ ， $\tilde{s}\left(t\right)=\frac{{\text{e}}^{-t}+t^2+3}{40}$ ， $\psi \left(x,y,z\right)=\tilde{\psi }\left(x,y,z\right)=\frac{\left|x\right|+\left|y\right|+\left|z\right|}{4}$ ，则可以得到 ${\Vert s\Vert }_{\infty }=\frac{1}{10}$ ， ${\Vert \tilde{s}\Vert }_{\infty }=\frac{1}{8}$ 。

又因为 $q=\frac{5}{2}$ ， $p=\frac{8}{3}$ ， $\alpha =\beta =1$ ，根据假设得到 $b_q=1.1284$ ， $b_p=1.1077$ 。通过计算，可以得到 $\rho =\max \left\{{\rho }_1,{\rho }_2,{\rho }_3\right\}<1$ 成立，即说明边值问题有唯一解。

5. 结论与展望

本文主要研究的是一类具有积分边界条件的非线性耦合分数微分方程组边值问题解的唯一性，全文的研究主要基于Banach压缩映射原理得到唯一解的存在性。

首先，针对耦合微分方程组中的单个非线性方程及相关的边界条件求解出非线性系统对应线性系统的Green函数，通过简单的观察和计算得到Green函数及其偏导函数的连续性，基于连续性以及使用Banach压缩映射原理的基本条件，本文给出了一些比较合理的假设，在这些假设成立的条件下运用Banach不动点定理得到问题(1)唯一解的存在性。

同参考文献 [10] [11] 相比，本文研究的非线性项里面含有未知函数的Caputo型分数阶导数项，这就使得所研究的问题所在的Banach空间特别复杂，相应地，耦合系统所在的乘积空间也更加复杂，使得研究的问题不仅仅在计算方面难度增大，而且在构造Banach空间和定义范数上具有更大的难度。

关于该耦合系统的求解，今后的研究方向为解的存在性定理和Ulam型稳定性，由于本文涉及的空间相对复杂，所以关于该方面的研究也是本文之后研究的重点和难点，非线性耦合分数阶微分方程边值问题是一个非常有意义的研究方向，以后我们将继续利用泛函分析的相关理论和知识研究这一类问题。