0 引言

分数阶微分方程已经广泛应用于分数物理学、分子动力学、自动控制、电化学等各个科学研究领域^[1-3].关于分数阶微分方程的边值问题也一直是分数阶微积分理论的一个重要研究课题，其在模拟工程、物理和生命科学等应用科学领域的许多现象中具有很大的优势，如非线性扩散、气体的燃烧和热交换、布朗运动等问题^[4-5].此外，诸多学者都独立地探讨了各类分数阶微分方程初值问题^[6-9].

本文主要讨论一类Caputo型分数阶微分方程边值问题：

$ \left\{ \begin{array}{l} ^cD_a^\alpha y\left( x \right) = f\left( {x,y\left( x \right)} \right),x \in \left[ {a,b} \right],1 < \alpha < 2\\ \mathit{\boldsymbol{A}}\left( \begin{array}{l} y\left( a \right)\\ y'\left( a \right) \end{array} \right) + \mathit{\boldsymbol{B}}\left( \begin{array}{l} y\left( b \right)\\ y'\left( b \right) \end{array} \right) = \mathit{\boldsymbol{C}} \end{array} \right. $

(1)

其中：A=$\left( {\begin{array}{*{20}{c}} {{a_{11}}}&{{a_{12}}}\\ {{a_{21}}}&{{a_{22}}} \end{array}} \right)$; B=$\left( {\begin{array}{*{20}{c}} {{b_{11}}}&{{b_{12}}}\\ {{b_{21}}}&{{b_{22}}} \end{array}} \right)$; C=$\left( {\begin{array}{*{20}{c}} {{c_1}}\\ {{c_2}} \end{array}} \right)$; ^cD_a^α是Caputo分数阶导数，核心思想是将其转化为等价的Fredholm积分方程，利用Schauder不动点定理和Banach不动点定理证明其解的存在唯一性.

1 预备知识

首先，我们来介绍几个基本概念和一些Caputo分数阶导数的性质以及相关引理.

定义1^[1]   设Ω=[a, b]是R中的有限区间，∀α∈R⁺，则连续函数f(x)的α阶Riemann-Liouville分数阶积分定义为

$ I_a^\alpha f\left( x \right) = \frac{1}{{\Gamma \left( \alpha \right)}}\int_a^x {{{\left( {x - t} \right)}^{\alpha - 1}}f\left( t \right){\rm{d}}t,a \le x \le b,} $

右端项在[a, b]上有定义. α=0时，令I_a⁰=I为恒等算子.

定义2^[1]  设Ω=[a, b]是R中的有限区间，∀α≥0，且n=[α]+1，则α阶Riemann-Liouville分数阶导数定义为

$ D_a^\alpha f\left( x \right) = \frac{{{{\rm{d}}^n}}}{{{\rm{d}}{x^n}}}I_a^{n - \alpha }f\left( x \right) = \frac{{{{\rm{d}}^n}}}{{{\rm{d}}{x^n}}}\frac{1}{{\Gamma \left( {n - \alpha } \right)}}\int_a^x {{{\left( {x - t} \right)}^{n - \alpha - 1}}f\left( t \right){\rm{d}}t,a \le x \le b.} $

当α$ \notin $N时，n=[α]+1；当α∈N时，n=α.

定义3^[1]  设Ω=[a, b]是R中的有限区间，∀α∈R⁺，则连续函数f(x)的α阶Caputo分数阶导数定义为

$ ^cD_a^\alpha f\left( x \right) = D_a^\alpha \left[ {f\left( x \right) - \sum\limits_{k = 0}^{n - 1} {\frac{{{f^{\left( k \right)}}\left( a \right)}}{{k!}}{{\left( {x - a} \right)}^k}} } \right],a \le x \le b, $

当α$ \notin $N时，n=[α]+1；当α∈N时，n=α.

性质1  常数的Caputo分数阶导数为0，即^cD_a^αC=I_a^n-α0=0.

性质2  设f₁, f₂是两个连续函数，∀α≥0，且^cD_a^αf₁，^cD_a^αf₂几乎处处存在.设λ₁, λ₂∈R，则^cD_a^α(λ₁f₁+λ₂f₂)几乎处处存在，且^cD_a^α(λ₁f₁+λ₂f₂)=λ₁^cD_a^αf₁+λ₂^cD_a^αf₂.

引理1^[1]   设α>0，f∈C[a, b]，则有^cD_a^α(I_a^αf)=f几乎处处成立.

引理2^[1]   设n=[α]+1，f∈C[a, b]，I_a^n-αf(x)∈Cⁿ[a, b]，则有

$ I_a^\alpha D_a^\alpha f\left( x \right) = f\left( x \right) - \sum\limits_{j = 1}^n {\frac{{{D^{n - j}}\left( {I_a^{n - \alpha }f} \right)\left( a \right)}}{{\Gamma \left( {\alpha - j + 1} \right)}}{{\left( {x - a} \right)}^{\alpha - j}}} . $

2 主要结果

定理1  设1＜α＜2，a_ij，b_ij，c_i∈R (i, j=1, 2)，f:[a, b]×R→R连续，且矩阵G=A+BE_T可逆，其中E_T=$\left( {\begin{array}{*{20}{c}} 1&{b - a}\\ 0&1 \end{array}} \right)$.则Caputo型分数阶微分方程边值问题(1) 有解y∈C[a, b], 当且仅当它是Fredholm积分方程

$ y\left( x \right) = {\lambda _1} + {\lambda _2}\left( {x - a} \right) + \frac{1}{{\Gamma \left( \alpha \right)}}\int_a^x {{{\left( {x - t} \right)}^{\alpha - 1}}f\left( {t,y\left( t \right)} \right){\rm{d}}t} $

(2)

的解，即式(1) 与(2) 等价.其中λ₁，λ₂由线性方程

$ \mathit{\boldsymbol{G}}\left( \begin{array}{l} {\lambda _1}\\ {\lambda _2} \end{array} \right) = \mathit{\boldsymbol{C}} - \int_a^b {\mathit{\boldsymbol{B}}\left( \begin{array}{l} \frac{{{{\left( {b - t} \right)}^{\alpha - 1}}}}{{\Gamma \left( \alpha \right)}}\\ \frac{{{{\left( {b - t} \right)}^{\alpha - 2}}}}{{\Gamma \left( {\alpha - 1} \right)}} \end{array} \right)} f\left( {t,y\left( t \right)} \right){\rm{d}}t $

决定.

证明  首先证明必要性.由定义2和3可知

$ ^cD_a^\alpha y\left( x \right) = D_a^\alpha \left[ {y\left( x \right) - \sum\limits_{k = 0}^{n - 1} {\frac{{{y^{\left( k \right)}}\left( a \right)}}{{k!}}{{\left( {x - a} \right)}^k}} } \right] = {D^n}\left( {I_a^{n - \alpha }\left[ {y\left( x \right) - \sum\limits_{k = 0}^{n - 1} {\frac{{{y^{\left( k \right)}}\left( a \right)}}{{k!}}{{\left( {x - a} \right)}^k}} } \right]} \right). $

由y∈C[a, b]是Caputo型分数阶微分方程边值问题(1) 的解，且f(x, y)∈C[a, b], 得^cD_a^αy(x)∈C[a, b]，则有I_a^n-α[y(x)-$\sum\limits_{k = 0}^{n - 1} {\frac{{{y^{\left( k \right)}}\left( a \right)}}{{k!}}} $(x-a)^k]∈Cⁿ[a, b].

在^cD_a^αy(x)=f(x, y(x))两边作用I_a^α，可得I_a^αcD_a^αy(x)=I_a^αf(x, y).由引理2，得到

$ \begin{array}{l} I_a^\alpha {\;^c}D_a^\alpha y\left( x \right) = I_a^\alpha D_a^\alpha \left[ {y\left( x \right) - \sum\limits_{k = 0}^{n - 1} {\frac{{{y^{\left( k \right)}}\left( a \right)}}{{k!}}{{\left( {x - a} \right)}^k}} } \right] = y\left( x \right) - \sum\limits_{k = 0}^{n - 1} {\frac{{{y^{\left( k \right)}}\left( a \right)}}{{k!}}{{\left( {x - a} \right)}^k}} - \\ \sum\limits_{j = 1}^n {\frac{{{D^{n - j}}\left( {I_a^{n - \alpha }g} \right)\left( a \right)}}{{\Gamma \left( {\alpha - j + 1} \right)}}{{\left( {x - a} \right)}^{\alpha - j}},} \end{array} $

其中: g(x)=y(x)-$\sum\limits_{k = 0}^{n - 1} {\frac{{{y^{\left( k \right)}}\left( a \right)}}{{k!}}} $(x-a)^k.此外，有

$ \begin{array}{l} {D^{n - j}}\left( {I_a^{n - \alpha }g} \right)\left( x \right) = {D^{n - j - 1}}D\left( {I_a^{n - \alpha }\left[ {y\left( x \right) - \sum\limits_{k = 0}^{n - 1} {\frac{{{y^{\left( k \right)}}\left( a \right)}}{{k!}}{{\left( {x - a} \right)}^k}} } \right]} \right) = {D^{n - j - 1}}\left( {I_a^{n - \alpha }\left[ {y'\left( x \right) - } \right.} \right.\\ \left. {\left. {\sum\limits_{k = 1}^{n - 1} {\frac{{{y^{\left( k \right)}}\left( a \right)}}{{\Gamma \left( k \right)}}{{\left( {x - a} \right)}^{k - 1}}} } \right]} \right) = I_a^{n - \alpha }\left[ {{y^{\left( {n - j} \right)}}\left( x \right) - \sum\limits_{k = n - j}^{n - 1} {\frac{{{y^{\left( k \right)}}\left( a \right)}}{{\Gamma \left( {k - n + j + 1} \right)}}{{\left( {x - a} \right)}^{k - n + j}}} } \right] = \\ \frac{{{{\left( {x - a} \right)}^{n - \alpha }}}}{{\Gamma \left( {n - \alpha } \right)}}\int_0^1 {{{\left( {1 - s} \right)}^{n - \alpha - 1}}\left[ {{y^{\left( {n - j} \right)}}\left( {a + s\left( {x - a} \right)} \right) - \sum\limits_{k = n - j}^{n - 1} {\frac{{{y^{\left( k \right)}}\left( a \right)}}{{\Gamma \left( {k - n + j + 1} \right)}}{{\left( {s\left( {x - a} \right)} \right)}^{k - n + j}}} } \right]} {\rm{d}}s. \end{array} $

因为n>α，则D^n-j(I_a^n-αg)(x)∈C[a, b]，j=1, …, n，由上式可得D^n-j(I_a^n-αg)(a)=0，j=1, …, n.

因为1＜α＜2，n=[α]+1=2，于是有

$ \begin{array}{l} I_a^\alpha {\;^c}D_a^\alpha y\left( x \right) = y\left( x \right) - \sum\limits_{k = 0}^{n - 1} {\frac{{{y^{\left( k \right)}}\left( a \right)}}{{k!}}{{\left( {x - a} \right)}^k}} - \sum\limits_{j = 1}^n {\frac{{{D^{n - j}}\left( {I_a^{n - \alpha }g} \right)\left( a \right)}}{{\Gamma \left( {\alpha - j + 1} \right)}}{{\left( {x - a} \right)}^{\alpha - j}}} = \\ y\left( x \right) - \sum\limits_{k = 0}^{n - 1} {\frac{{{y^{\left( k \right)}}\left( a \right)}}{{k!}}{{\left( {x - a} \right)}^k}} = y\left( x \right) - y\left( a \right) - y'\left( a \right)\left( {x - a} \right), \end{array} $

则y(x)=y(a)+y′(a)(x-a)+I_a^αf(x, y)，y′(x)=y′(a)+I_a^α-1f(x, y).进一步可以得到

$ y\left( b \right) = y\left( a \right) + y'\left( a \right)\left( {b - a} \right) + \frac{1}{{\Gamma \left( \alpha \right)}}\int_a^b {{{\left( {b - t} \right)}^{\alpha - 1}}f\left( {t,y\left( t \right)} \right){\rm{d}}t,} $

$ y'\left( b \right) = y'\left( a \right) + \frac{1}{{\Gamma \left( {\alpha - 1} \right)}}\int_a^b {{{\left( {b - t} \right)}^{\alpha - 2}}f\left( {t,y\left( t \right)} \right){\rm{d}}t,} $

代入边值条件A$\left( {\begin{array}{*{20}{c}} {y\left( a \right)}\\ {y'\left( a \right)} \end{array}} \right) + \mathit{\boldsymbol{B}}\left( {\begin{array}{*{20}{c}} {y\left( b \right)}\\ {y'\left( b \right)} \end{array}} \right)$=C，可得

$ \mathit{\boldsymbol{G}}\left( \begin{array}{l} y\left( a \right)\\ y'\left( a \right) \end{array} \right) = \mathit{\boldsymbol{C}} - \int_a^b {\mathit{\boldsymbol{B}}\left( \begin{array}{l} \frac{{{{\left( {b - t} \right)}^{\alpha - 1}}}}{{\Gamma \left( \alpha \right)}}\\ \frac{{{{\left( {b - t} \right)}^{\alpha - 2}}}}{{\Gamma \left( {\alpha - 1} \right)}} \end{array} \right)} f\left( {t,y\left( t \right)} \right){\rm{d}}t. $

由此可知y∈C[a, b]是Fredholm积分方程(2) 的解.

下面证明充分性. y∈C[a, b]是Fredholm积分方程(2) 的解，方程(2) 两边作用^cD_a^α，由性质1和2以及引理1, 可得^cD_a^αy(x)=^cD_a^αλ₁+^cD_a^α(λ₂(x-a))+^cD_a^α(I_a^αf(x, y))=f(x, y)，且有

$ y\left( a \right) = {\lambda _1},y\left( b \right) = {\lambda _1} + {\lambda _2}\left( {b - a} \right) + \frac{1}{{\Gamma \left( \alpha \right)}}\int_a^b {{{\left( {b - t} \right)}^{\alpha - 1}}f\left( {t,y\left( t \right)} \right){\rm{d}}t,} $

$ y'\left( a \right) = {\lambda _2},y'\left( b \right) = {\lambda _2} + \frac{1}{{\Gamma \left( {\alpha - 1} \right)}}\int_a^b {{{\left( {b - t} \right)}^{\alpha - 2}}f\left( {t,y\left( t \right)} \right){\rm{d}}t.} $

于是可以得到

$ \mathit{\boldsymbol{A}}\left( \begin{array}{l} y\left( a \right)\\ y'\left( a \right) \end{array} \right) + \mathit{\boldsymbol{B}}\left( \begin{array}{l} y\left( b \right)\\ y'b \end{array} \right) = \mathit{\boldsymbol{A}}\left( \begin{array}{l} {\lambda _1}\\ {\lambda _2} \end{array} \right) + \mathit{\boldsymbol{B}}\;{\mathit{\boldsymbol{E}}_T}\left( \begin{array}{l} {\lambda _1}\\ {\lambda _2} \end{array} \right) + \int_a^b {\mathit{\boldsymbol{B}}\left( \begin{array}{l} \frac{{{{\left( {b - t} \right)}^{\alpha - 1}}}}{{\Gamma \left( \alpha \right)}}\\ \frac{{{{\left( {b - t} \right)}^{\alpha - 2}}}}{{\Gamma \left( {\alpha - 1} \right)}} \end{array} \right)} f\left( {t,y\left( t \right)} \right){\rm{d}}t \\= \mathit{\boldsymbol{C}}. $

因此，y∈C[a, b]是Caputo型分数阶微分方程边值问题(1) 的解.

定理2  在定理1的假设条件下，若f:[a, b]×R→R一致有界，则Caputo型分数阶微分方程边值问题(1) 至少有一个解y∈C[a, b].

证明  由假设f:[a, b]×R→R一致有界，则存在常数M>0，使得f(x, y)≤M.

另外，由G=A+BE_T可逆，令G^-1C=$\left( {\begin{array}{*{20}{c}} {{e_1}}\\ {{e_2}} \end{array}} \right)$，G^-1B=$\left( {\begin{array}{*{20}{c}} {{d_1}}&{{d_2}}\\ {{d_3}}&{{d_4}} \end{array}} \right)$，可以得到

$ \left( \begin{array}{l} {\lambda _1}\\ {\lambda _2} \end{array} \right) = {\mathit{\boldsymbol{G}}^{ - 1}}\mathit{\boldsymbol{C}} - \int_a^b {\left( {{\mathit{\boldsymbol{G}}^{ - 1}}\mathit{\boldsymbol{B}}\left( \begin{array}{l} \frac{{{{\left( {b - t} \right)}^{\alpha - 1}}}}{{\Gamma \left( \alpha \right)}}\\ \frac{{{{\left( {b - t} \right)}^{\alpha - 2}}}}{{\Gamma \left( {\alpha - 1} \right)}} \end{array} \right)} \right)} f\left( {t,y\left( t \right)} \right){\rm{d}}t = \\ \left( \begin{array}{l} {e_1}\\ {e_2} \end{array} \right) - \int_a^b {\left( {\begin{array}{*{20}{c}} {{d_1}}&{{d_2}}\\ {{d_3}}&{{d_4}} \end{array}} \right)\left( \begin{array}{l} \frac{{{{\left( {b - t} \right)}^{\alpha - 1}}}}{{\Gamma \left( \alpha \right)}}\\ \frac{{{{\left( {b - t} \right)}^{\alpha - 2}}}}{{\Gamma \left( {\alpha - 1} \right)}} \end{array} \right)} f\left( {t,y\left( t \right)} \right){\rm{d}}t. $

令U={y∈C[a, b]:‖y‖≤R}，其中:

$ R = \left| {{e_1}} \right| + \left| {{e_2}} \right|\left( {b - a} \right) + M\left[ {\left( {\left| {{d_1}} \right| + 1} \right)\frac{{{{\left( {b - a} \right)}^\alpha }}}{{\Gamma \left( {\alpha + 1} \right)}} + \left| {{d_2}} \right|\frac{{{{\left( {b - a} \right)}^{\alpha - 1}}}}{{\Gamma \left( \alpha \right)}} + \left| {{d_3}} \right|\frac{{{{\left( {b - a} \right)}^{\alpha + 1}}}}{{\Gamma \left( {\alpha + 1} \right)}} + \left| {{d_4}} \right|\frac{{{{\left( {b - a} \right)}^\alpha }}}{{\Gamma \left( \alpha \right)}}} \right], $

显然U是C[a, b]中的有界闭凸子集.

在U上定义算子F:Fy(x)=λ₁+λ₂(x-a)+$\frac{1}{{\Gamma \left( \alpha \right)}}\int_\alpha ^x {} $(x-t)^α-1f(t, y(t))dt.显然算子F的不动点就是边值问题(1) 的解.我们分以下几步来证明：

第一步：算子F是U→U的.

$ \begin{array}{l} \left| {Fy\left( x \right)} \right| = \left| {{\lambda _1} + {\lambda _2}\left( {x - a} \right) + \frac{1}{{\Gamma \left( \alpha \right)}}\int_a^x {{{\left( {x - 1} \right)}^{\alpha - 1}}f\left( {t,y\left( t \right)} \right){\rm{d}}t} } \right| \le \left| {{\lambda _1}} \right| + \left| {{\lambda _2}} \right|\left( {b - a} \right) + \\ \frac{{M{{\left( {b - a} \right)}^\alpha }}}{{\Gamma \left( {\alpha + 1} \right)}} \le M\left[ {\left( {\left| {{d_1}} \right| + 1} \right)\frac{{{{\left( {b - a} \right)}^\alpha }}}{{\Gamma \left( {\alpha + 1} \right)}} + \left| {{d_2}} \right|\frac{{{{\left( {b - a} \right)}^{\alpha - 1}}}}{{\Gamma \left( \alpha \right)}} + \left| {{d_3}} \right|\frac{{{{\left( {b - a} \right)}^{\alpha + 1}}}}{{\Gamma \left( {\alpha + 1} \right)}} + } \right.\\ \left. {\left| {{d_4}} \right|\frac{{{{\left( {b - a} \right)}^\alpha }}}{{\Gamma \left( \alpha \right)}}} \right] + \left| {{e_1}} \right| + \left| {{e_2}} \right|\left( {b - a} \right) = R, \end{array} $

又由算子F的定义以及f的连续性知Fy∈C[a, b]，则F:U→U.

第二步：算子F是连续的.事实上，设{y_n}∈U，且存在y∈U, 使得y_n→y(n→∞)，则对任意的x∈[a, b]，有

$ \begin{array}{l} \left| {F{y_n}\left( x \right) - Fy\left( x \right)} \right| \le \int_a^b {\left( {\left| {{d_1}} \right|\;\;\;\;\left| {{d_2}} \right|} \right)} \left( \begin{array}{l} \frac{{{{\left( {b - t} \right)}^{\alpha - 1}}}}{{\Gamma \left( \alpha \right)}}\\ \frac{{{{\left( {b - t} \right)}^{\alpha - 2}}}}{{\Gamma \left( {\alpha - 1} \right)}} \end{array} \right)\left| {f\left( {t,{y_n}\left( t \right)} \right) - f\left( {t,y\left( t \right)} \right)} \right|{\rm{d}}t + \\ \left( {b - a} \right)\int_a^b {\left( {\left| {{d_3}} \right|\;\;\;\;\left| {{d_4}} \right|} \right)} \left( \begin{array}{l} \frac{{{{\left( {b - t} \right)}^{\alpha - 1}}}}{{\Gamma \left( \alpha \right)}}\\ \frac{{{{\left( {b - t} \right)}^{\alpha - 2}}}}{{\Gamma \left( {\alpha - 1} \right)}} \end{array} \right)\left| {f\left( {t,{y_n}\left( t \right)} \right) - f\left( {t,y\left( t \right)} \right)} \right|{\rm{d}}t + \\ \frac{1}{{\Gamma \left( \alpha \right)}}\int_a^x {{{\left( {x - t} \right)}^{\alpha - 1}}} \left| {f\left( {t,{y_n}\left( t \right)} \right) - f\left( {t,y\left( t \right)} \right)} \right|{\rm{d}}t \le \left[ {\left( {\left| {{d_1}} \right| + 1} \right)\frac{{{{\left( {b - a} \right)}^\alpha }}}{{\Gamma \left( {\alpha + 1} \right)}} + \left| {{d_2}} \right|\frac{{{{\left( {b - a} \right)}^{\alpha - 1}}}}{{\Gamma \left( \alpha \right)}} + } \right.\\ \left. {\left| {{d_3}} \right|\frac{{{{\left( {b - a} \right)}^{\alpha + 1}}}}{{\Gamma \left( {\alpha + 1} \right)}} + \left| {{d_4}} \right|\frac{{{{\left( {b - a} \right)}^\alpha }}}{{\Gamma \left( \alpha \right)}}} \right]\left\| {f\left( {t,{y_n}\left( t \right)} \right) - f\left( {t,y\left( t \right)} \right)} \right\|. \end{array} $

因为y_n→y(n→∞)，所以由f的连续性知‖f(t, y_n(t))-f(t, y(t))‖→0(n→∞)，故由‖Fy_n(x)-Fy(x)‖→0(n→∞)，由此说明算子F是连续的.

第三步：F(U)是一致有界的.由第一步的证明以及U的定义，显然F(U)是一致有界的.

第四步：F(U)是等度连续的.任意的x₁, x₂∈[a, b]，不妨设x₁＜x₂，对任意的y∈U，有

$ \begin{array}{l} \left| {Fy\left( {{x_2}} \right) - Fy\left( {{x_1}} \right)} \right| \le \left| {\frac{1}{{\Gamma \left( \alpha \right)}}\int_a^{{x_1}} {\left[ {{{\left( {{x_2} - t} \right)}^{\alpha - 1}}} - {{{\left( {{x_1} - t} \right)}^{\alpha - 1}}}\right]} } \right.f\left( {t,y\left( t \right)} \right){\rm{d}}t + \\ \left. {\frac{1}{{\Gamma \left( \alpha \right)}}\int_x^{{x_2}} {{{\left( {{x_2} - t} \right)}^{\alpha - 1}}f\left( {t,y\left( t \right)} \right){\rm{d}}t} } \right| + \left( {{x_2} - {x_1}} \right)\left| {{\lambda _2}} \right| \le \\ \frac{M}{{\Gamma \left( {\alpha + 1} \right)}}\left[ {x_2^\alpha - x_1^\alpha + {{\left( {{x_2} - {x_1}} \right)}^\alpha }} \right] + \left( {{x_2} - {x_1}} \right)\left| {{\lambda _2}} \right| \le \\ \left\{ {M\left[ {\left| {{d_3}} \right|\frac{{{{\left( {b - a} \right)}^{\alpha + 1}}}}{{\Gamma \left( {\alpha + 1} \right)}} + \left| {{d_4}} \right|\frac{{{{\left( {b - a} \right)}^\alpha }}}{{\Gamma \left( \alpha \right)}}} \right] + \left| {{e_2}} \right|} \right\}\left( {{x_2} - {x_1}} \right) + \\ \frac{M}{{\Gamma \left( {\alpha + 1} \right)}}\left[ {\alpha {\xi ^{\alpha - 1}}\left( {{x_2} - {x_1}} \right) + {{\left( {{x_2} - {x_1}} \right)}^\alpha }} \right] = \\ \left\{ {M\left[ {\left| {{d_3}} \right|\frac{{{{\left( {b - a} \right)}^{\alpha + 1}}}}{{\Gamma \left( {\alpha + 1} \right)}} + \left| {{d_4}} \right|\frac{{{{\left( {b - a} \right)}^\alpha }}}{{\Gamma \left( \alpha \right)}}} \right] + \left| {{e_2}} \right| + \frac{M}{{\Gamma \left( {\alpha + 1} \right)}}\alpha {\xi ^{\alpha - 1}}} \right\}\left( {{x_2} - {x_1}} \right) + \\ \frac{M}{{\Gamma \left( {\alpha + 1} \right)}}{\left( {{x_2} - {x_1}} \right)^\alpha }, \end{array} $

其中: ξ介于x₁, x₂之间.则当x₁→x₂时，|Fy(x₂)-Fy(x₁)|→0.因此F(U)是等度连续的.

由Ascoli-Arzela定理知F(U)是相对紧集，因此F:U→U全连续.根据Schauder不动点定理知F在U中至少有一个不动点.

综上，我们就证明了Caputo型分数阶微分方程边值问题(1) 至少有一个解y∈C[a, b].

定理3  在定理1的假设条件下，若f:[a, b]×R→R对于第二个变元满足Lipschitz条件，即存在一个适合的Lipschitz常数L＜[(|d₁|+1)$\frac{{{{\left( {b - a} \right)}^\alpha }}}{{\Gamma \left( {\alpha + 1} \right)}} + \left| {{d_2}} \right|\frac{{{{\left( {b - a} \right)}^{\alpha - 1}}}}{{\Gamma \left( \alpha \right)}} + \left| {{d_3}} \right|\frac{{{{\left( {b - a} \right)}^{\alpha + 1}}}}{{\Gamma \left( {\alpha + 1} \right)}} + \left| {{d_4}} \right|\frac{{{{\left( {b - a} \right)}^\alpha }}}{{\Gamma \left( \alpha \right)}}$]^-1，使得

$ \left| {f\left( {x,{y_1}} \right) - f\left( {x,{y_2}} \right)} \right| \le L\left| {{y_1} - {y_2}} \right|, $

则Caputo型分数阶微分方程边值问题(1) 有唯一的解y∈C[a, b].

证明  在C[a, b]上定义算子F:C[a, b]→C[a, b],

$ Fy\left( x \right) = {\lambda _1} + {\lambda _2}\left( {x - a} \right) + \frac{1}{{\Gamma \left( \alpha \right)}}\int_a^x {{{\left( {x - t} \right)}^{\alpha - 1}}f} \left( {t,y\left( t \right)} \right){\rm{d}}t. $

在定理1的假设条件下，若|f(x, y₁)-f(x, y₂)|≤L|_y1-y2|，则有

$ \begin{array}{l} \left| {F{y_1}\left( x \right) - F{y_2}\left( x \right)} \right| \le \int_a^b {\left( {\left| {{d_1}} \right|\;\;\;\;\left| {{d_2}} \right|} \right)} \left( \begin{array}{l} \frac{{{{\left( {b - t} \right)}^{\alpha - 1}}}}{{\Gamma \left( \alpha \right)}}\\ \frac{{{{\left( {b - t} \right)}^{\alpha - 2}}}}{{\Gamma \left( {\alpha - 1} \right)}} \end{array} \right)\left| {f\left( {t,{y_1}\left( t \right)} \right) - f\left( {t,{y_2}\left( t \right)} \right)} \right|{\rm{d}}t + \\ \left( {b - a} \right)\int_a^b {\left( {\left| {{d_3}} \right|\;\;\;\;\left| {{d_4}} \right|} \right)} \left( \begin{array}{l} \frac{{{{\left( {b - t} \right)}^{\alpha - 1}}}}{{\Gamma \left( \alpha \right)}}\\ \frac{{{{\left( {b - t} \right)}^{\alpha - 2}}}}{{\Gamma \left( {\alpha - 1} \right)}} \end{array} \right)\left| {f\left( {t,{y_1}\left( t \right)} \right) - f\left( {t,{y_2}\left( t \right)} \right)} \right|{\rm{d}}t + \\ \frac{1}{{\Gamma \left( \alpha \right)}}\int_a^x {{{\left( {x - t} \right)}^{\alpha - 1}}\left| {f\left( {t,{y_1}\left( t \right)} \right) - f\left( {t,{y_2}\left( t \right)} \right)} \right|} {\rm{d}}t \le K\left| {{y_1}\left( x \right) - {y_2}\left( x \right)} \le {{y_1}\left( x \right) - {y_2}\left( x \right)} \right|, \end{array} $

其中: K=L[(|d₁|+1)$\frac{{{{\left( {b - a} \right)}^\alpha }}}{{\Gamma \left( {\alpha + 1} \right)}} + \left| {{d_2}} \right|\frac{{{{\left( {b - a} \right)}^{\alpha - 1}}}}{{\Gamma \left( \alpha \right)}} + \left| {{d_3}} \right|\frac{{{{\left( {b - a} \right)}^{\alpha + 1}}}}{{\Gamma \left( {\alpha + 1} \right)}} + \left| {{d_4}} \right|\frac{{{{\left( {b - a} \right)}^\alpha }}}{{\Gamma \left( \alpha \right)}}$]≤1.

则在假设条件下算子F成为压缩算子，根据Banach不动点定理，Caputo型分数阶微分方程边值问题(1) 有唯一的解y∈C[a, b].

注：以上讨论了1＜α＜2时，Caputo型分数阶微分方程边值问题(1) 的解的存在唯一性问题，事实上，当n-1＜α＜n时，可以得到与(1) 类似的边值问题，用同样的方法可以讨论其解的存在唯一性.

3 例子

我们考虑如下Caputo型分数阶微分方程边值问题：

$ \left\{ \begin{array}{l} ^cD_0^{\frac{3}{2}}y\left( x \right) = \frac{{2{{\rm{e}}^{ - x}}y}}{{1 + 15{{\rm{e}}^x}\left( {1 + y} \right)}},x \in \left[ {0,1} \right],\\ \left( {\begin{array}{*{20}{c}} 1&2\\ 2&4 \end{array}} \right)\left( \begin{array}{l} y\left( 0 \right)\\ y'\left( 0 \right) \end{array} \right) + \left( {\begin{array}{*{20}{c}} 2&1\\ 0&3 \end{array}} \right)\left( \begin{array}{l} y\left( 1 \right)\\ y'\left( 1 \right) \end{array} \right) = \left( \begin{array}{l} 1\\ 2 \end{array} \right). \end{array} \right. $

通过简单的计算, 我们可以得到

$ \mathit{\boldsymbol{G}} = \mathit{\boldsymbol{A}} + \mathit{\boldsymbol{B}}\;{\mathit{\boldsymbol{E}}_T} = \left( {\begin{array}{*{20}{c}} 3&5\\ 1&7 \end{array}} \right),{\mathit{\boldsymbol{G}}^{ - 1}} = \left( {\begin{array}{*{20}{c}} {\frac{7}{{16}}}&{ - \frac{5}{{16}}}\\ { - \frac{1}{{16}}}&{\frac{3}{{16}}} \end{array}} \right),{\mathit{\boldsymbol{G}}^{ - 1}}\mathit{\boldsymbol{B}} = \left( {\begin{array}{*{20}{c}} {\frac{7}{8}}&{ - \frac{1}{2}}\\ { - \frac{1}{8}}&{\frac{1}{2}} \end{array}} \right). $

另外可以得到|f(x, y)|≤$\frac{1}{8}$‖y‖，|f(x, y₁)-f(x, y₂)|≤$\frac{1}{8}$|y₁-y₂|，即L=$\frac{1}{8}$，则

$ L < {\left[ {\frac{{15}}{8}\frac{1}{{\Gamma \left( {5/2} \right)}} + \frac{1}{2}\frac{1}{{\Gamma \left( {3/2} \right)}} + \frac{1}{8}\frac{1}{{\Gamma \left( {5/2} \right)}} + \frac{1}{2}\frac{1}{{\Gamma \left( {3/2} \right)}}} \right]^{ - 1}} = \frac{{3\sqrt \pi }}{{14}} \approx 0.3798. $

综上，由|f(x, y)|≤$\frac{1}{8}$‖y‖及定理2可知, 上述Caputo型分数阶微分方程边值问题有解y∈C[a, b].由L=$\frac{1}{8}$及定理3可以得到上述Caputo型分数阶微分方程边值问题有唯一的解y∈C[a, b].