0 引言

分数阶微分方程在工程、化学、物理、生物等领域有着广泛应用，例如热传导领域和流体学领域^[1-3], 而且分数阶导数模型克服了经典整数阶微分模型理论与实验结果不吻合的缺点^[4], 因此研究分数阶微分方程边值问题有着重要的意义.近年来，大量文献报道微分方程^[5-6]和分数阶微分方程^{[4, 7-10]}边值问题解的存在性.2005年，当1 < α≤2时，Bai等^[7]推导了分数阶微分方程边值问题

$ \left\{ {\begin{array}{*{20}{l}} {D_{0 + }^\alpha u\left( t \right) + f\left( {t, u\left( t \right)} \right) = 0, t \in \left( {0, 1} \right), }\\ {u\left( 0 \right) = u\left( 1 \right) = 0, } \end{array}} \right. $

的格林函数及重要的性质(f:[0, +∞)×$ \mathbb{R} $→[0, +∞)是连续的，D₀₊^α是标准的Riemann-Liouville微分)，并运用锥拉伸与锥压缩不动点原理，研究其正解的存在性.2009年，当2 < α≤3时，Bai等^[8]利用Leray-Schauder不动点定理和Krasnoselskii不动点定理证明Caputo分数阶微分方程边值问题

$ \left\{ {\begin{array}{*{20}{l}} {D_{0 + }^\alpha u\left( t \right) + f\left( {t, u\left( t \right)} \right) = 0, t \in \left( {0, 1} \right), }\\ {u\left( 0 \right) = u'\left( 1 \right) = u''\left( 0 \right) = 0, } \end{array}} \right. $

至少存在一个正解，其中f:[0, +∞)×$ \mathbb{R} $→[0, +∞)是连续的，D₀₊^α是标准的Caputo微分.2015年，Abdulkadir Dogan^[9]利用Leggett-Williams不动点定理证明二阶微分方程边值问题

$ \left\{ {\begin{array}{*{20}{l}} {u''\left( t \right) + f\left( {u\left( t \right)} \right) = 0, t \in \left[{0, 1} \right], }\\ {u'\left( 0 \right) = u\left( 1 \right) = 0, } \end{array}} \right. $

存在正解，其中f:$ \mathbb{R} $→[0, +∞)是连续的.

目前研究分数阶微分方程边值问题的主要工具有锥拉伸与锥压缩不动点原理、Krasnoselskii不动点原理、Schauder不动点原理上下解等.本文利用Leggett-Williams不动点定理, 参照文献[9]中的方法研究Caputo分数阶微分方程边值问题

$ \left\{ {\begin{array}{*{20}{l}} {D_{0 + }^\alpha u\left( t \right) + f\left( {t,u\left( t \right)} \right) = 0,t \in \left( {0,1} \right),}\\ {u'\left( 0 \right) = u\left( 1 \right) = 0,} \end{array}} \right. $

(0.1)

正解的存在性，其中1 < α≤2，f:[0, +∞)×$ \mathbb{R} $→[0, +∞)是连续的，D₀₊^α是标准的Caputo微分.一方面，边值问题(1)包含了文献[9]的整数阶微分方程边值问题，推广了文献[9]的结果；另一方面，非线性项f范围有所扩大.

1 预备知识

定义1.1^[11]一个连续函数u:(0, +∞)→$ \mathbb{R} $的α阶Caputo导数定义为

$ {}^CD_{0 + }^\alpha u\left( t \right) = \frac{1}{{\mathit{\Gamma} \left( {n-\alpha } \right)}}\int_0^t {\frac{{{u^{\left( n \right)}}\left( s \right)}}{{{{\left( {t-s} \right)}^{\alpha-n + 1}}}}} {\rm{d}}s, $

其中α>0，n=[α]+1，[α]代表实数α的整数部分。上式右边在(0, +∞)内逐点有定义.

引理1.1^[11]令α>0，若u∈ACⁿ[0,1]或u∈Cⁿ[0,1]，则

$ I_{0 + }^\alpha {}^CD_{0 + }^\alpha u\left( t \right) = u\left( t \right)-\sum\limits_{k = 0}^{n-1} {\frac{{{u^{\left( k \right)}}\left( 0 \right)}}{{k!}}{t^k}}, $

其中n=[α]+1，I₀₊^α代表α阶Riemann-Liouville型积分.

引理1.2令α∈(1, 2]，给定h∈C[0,1]，则

$ {}^CD_{0 + }^\alpha u\left( t \right) + h\left( t \right) = 0,t \in \left( {0,1} \right), $

(1.1)

$ 2u'\left( 0 \right) = u\left( 1 \right) = 0, $

(1.2)

的唯一解为$ u\left( t \right) = \int_0^1 {G\left( {t, s} \right)h\left( s \right){\rm{d}}s}, $其中$ G\left( {t, s} \right) = \left\{ {\begin{array}{*{20}{l}} {\frac{{{{\left( {1-s} \right)}^{\alpha-1}}-{{\left( {t - s} \right)}^{\alpha - 1}}}}{{\mathit{\Gamma} \left( \alpha \right)}}, 0 \le s \le t \le 1}\\ {\frac{{{{\left( {1 - s} \right)}^{\alpha - 1}}}}{{\mathit{\Gamma} \left( \alpha \right)}}, 0 \le s \le t \le 1} \end{array}} \right. $是格林函数.

证明由引理1.1可得，(1.1)式等价于方程u(t)=-I₀₊^αh(t)+C₁+C₂t，其中C₁, C₂∈$ \mathbb{R} $.从而u′(t)=-I₀₊^α-1h(t)+C₂.由(1.2)式可知C₂=0, $ {C_1} = \frac{1}{{\mathit{\Gamma} \left( \alpha \right)}}\int_0^1 {{{\left( {1-s} \right)}^{\alpha-1}}} h\left( s \right){\rm{d}}s. $

因此，(1.1)~(1.2)式的唯一解是

$ \begin{array}{*{20}{l}} {\;\;\;\;\;\;\;u\left( t \right) = \frac{{- 1}}{{\mathit{\Gamma} \left( \alpha \right)}}\int_0^t {{{\left( {t- s} \right)}^{\alpha- 1}}h\left( s \right){\rm{d}}} s + \frac{1}{{\mathit{\Gamma} \left( \alpha \right)}}\int_0^1 {\left( {1 - } \right.} }\\ {{{\left. s \right)}^{\alpha - 1}}h\left( s \right){\rm{d}}s = \frac{1}{{\mathit{\Gamma} \left( \alpha \right)}}\int_0^t {\left[{{{\left( {1-s} \right)}^{\alpha-1}}-{{\left( {t - s} \right)}^{\alpha - 1}}} \right]h\left( s \right){\rm{d}}} s + }\\ {\frac{1}{{\mathit{\Gamma} \left( \alpha \right)}}\int_t^1 {{{\left( {t -s} \right)}^{\alpha -1}}h\left( s \right){\rm{d}}} s = \int_0^1 {G\left( {t, s} \right)h\left( s \right)} {\rm{d}}s.} \end{array} $

引理1.3引理1.2中的G(t, s)有下列性质：

(ⅰ)G(t, s)∈C([0,1]×[0,1], $ \mathbb{R} $)且G(t, s)>0, t, s∈(0, 1)；

(ⅱ)$ \mathit{\Gamma} \left( \alpha \right)\mathop {\max }\limits_{0 \le t \le 1} G\left( {t, s} \right) = {\left( {1- s} \right)^{\alpha- 1}}, t, s \in \left[{0, 1} \right]. $

(ⅲ)$ \int_0^1 {G\left( {s, s} \right){\rm{d}}s = \frac{1}{{\alpha \mathit{\Gamma} \left( \alpha \right)}}, \int_0^1 {G\left( {t, s} \right){\rm{d}}s} = \frac{{1-{t^\alpha }}}{{\alpha \mathit{\Gamma} \left( \alpha \right)}}, } \\ 0 \le t \le 1, $

(ⅳ)$ \frac{{\mathop {\max }\limits_{t \in \left[{\frac{1}{r}, 1-\frac{1}{r}} \right]} G\left( {t, s} \right)}}{{\mathop {\min }\limits_{t \in \left[{\frac{1}{r}, 1-\frac{1}{r}} \right]} G\left( {t, s} \right)}} \le M, \frac{{\mathop {\min }\limits_{t \in \left[{{t_1}, {t_2}} \right] \cup \left[{1-{t_2}, 1-{t_1}} \right]} G\left( {t, s} \right)}}{{\mathop {\max }\limits_{t \in \left[{{t_1}, {t_2}} \right] \cup \left[{1-{t_2}, 1-{t_1}} \right]} G\left( {t, s} \right)}} \ge m, $其中$ M = \frac{{{{\left( {1-\frac{1}{r}} \right)}^{\alpha-1}}}}{{{{\left( {1-\frac{1}{r}} \right)}^{\alpha - 1}} - {{\left( {1 - \frac{1}{{2r}}} \right)}^{\alpha - 1}}}}, $$ m = \frac{{1-{{\left( {1-{t_1}} \right)}^{\alpha-1}}}}{{1 - t_1^{\alpha - 1}}} $.

证明(i)~(iii)显然可得，只需证明(iv).

令$ {g_1}\left( {t, s} \right) = \frac{1}{{\mathit{\Gamma} \left( \alpha \right)}}\left( {{{\left( {1-s} \right)}^{\alpha-1}}-{{\left( {t - s} \right)}^{\alpha - 1}}} \right), {g_2}\left( {t, s} \right) = \frac{1}{{\mathit{\Gamma} \left( \alpha \right)}}{\left( {1 - s} \right)^{\alpha - 1}}, $则

$ \mathop {\max }\limits_{t \in \left[{\frac{1}{r}, 1-\frac{1}{r}} \right]} G\left( {t, s} \right) = \left\{ {\begin{array}{*{20}{l}} {{g_1}\left( {\frac{1}{r}, s} \right), s \in \left( 0 \right., \left. {\frac{1}{r}} \right], }\\ {{g_2}\left( {t, s} \right), s \in \left[{\frac{1}{r}, 1-\frac{1}{r}} \right], }\\ {{g_2}\left( {t, s} \right), s \in \left[{1-\frac{1}{r}} \right., \left. 1 \right), } \end{array}} \right. $

$ \mathop {\min }\limits_{t \in \left[{\frac{1}{r}, 1-\frac{1}{r}} \right]} G\left( {t, s} \right) = \left\{ {\begin{array}{*{20}{l}} {{g_1}\left( {1 -\frac{1}{r}, s} \right), s \in \left( 0 \right., \left. {\frac{1}{r}} \right], }\\ {{g_1}\left( {1 - \frac{1}{r}, s} \right), s \in \left[{\frac{1}{r}, 1-\frac{1}{r}} \right], }\\ {{g_2}\left( {t, s} \right), s \in \left[{1-\frac{1}{r}} \right., \left. 1 \right), } \end{array}} \right. $

从而$ \frac{{\mathop {\max }\limits_{t \in \left[{\frac{1}{r}, 1-\frac{1}{r}} \right]} G\left( {t, s} \right)}}{{\mathop {\min }\limits_{t \in \left[{\frac{1}{r}, 1-\frac{1}{r}} \right]} G\left( {t, s} \right)}} \le M $, 其中

$ M = \frac{{{{\left( {1-\frac{1}{r}} \right)}^{\alpha-1}}}}{{{{\left( {1-\frac{1}{r}} \right)}^{\alpha - 1}} - {{\left( {1 - \frac{1}{{2r}}} \right)}^{\alpha - 1}}}}. $

又

$ \begin{array}{*{20}{l}} {\mathop {\max }\limits_{t \in \left[{{t_1}, {t_2}} \right] \cup \left[{1-{t_2}, 1-{t_1}} \right]} G\left( {t, s} \right) =\\ \left\{ {\begin{array}{*{20}{l}} {{g_1}\left( {{t_1}, s} \right), s \in \left( 0 \right., \left. {{t_1}} \right]}\\ {{g_2}\left( {t, s} \right), s \in \left[{1- {t_2}, \left. 1 \right)} \right.} \end{array}, } \right.}\\ {\mathop {\min }\limits_{t \in \left[{{t_1}, {t_2}} \right] \cup \left[{1-{t_2}, 1-{t_1}} \right]} G\left( {t, s} \right) =\\ \left\{ {\begin{array}{*{20}{l}} {{g_1}\left( {1 -{t_1}, s} \right), s \in \left( 0 \right., \left. {1 -{t_2}} \right]}\\ {{g_2}\left( {t, s} \right), s \in \left[{1-{t_2}, \left. 1 \right)} \right.} \end{array}, } \right.} \end{array} $

从而$ \frac{{\mathop {\min }\limits_{t \in \left[{{t_1}, {t_2}} \right] \cup \left[{1-{t_2}, 1-{t_1}} \right]} G\left( {t, s} \right)}}{{\mathop {\max }\limits_{t \in \left[{{t_1}, {t_2}} \right] \cup \left[{1-{t_2}, 1-{t_1}} \right]} G\left( {t, s} \right)}} \ge m $其中$ m = \frac{{1-{{\left( {1-{t_1}} \right)}^{\alpha-1}}}}{{1 - t_1^{\alpha - 1}}}. $

令γ, β, θ是锥P上的非负连续凸函数，α, ψ是锥P上的非负连续凹函数，那么对非负实数h, a, b, d和c，定义下列凸集：

P(γ, c)={u∈P:γ(u) < c}，P(γ, α, a, c)={u∈P:a≤α(u), γ(u)≤c}，

Q(γ, β, d, c)={u∈P:β(u)≤d, γ(u)≤c}，

P(γ, θ, α, a, b, c)={u∈P:a≤α(u), θ(u)≤b, γ(u)≤c}，

Q(γ, β, ψ, h, d, c)={u∈P:h≤ψ(u), β(u)≤d, γ(u)≤c}.

定理1.1^[12]令E是一个实Banach空间，且P⊂E是一个锥.假设存在正数c和M，使锥P上的非负连续凹函数α, ψ及非负连续凸函数γ, β, θ满足

α(u)≤β(u), ‖u‖≤γ(u), u∈P(γ, c).

若F:P(γ, c)→P(γ, c)是一个全连续算子且存在非负实数h, d, a, b, 0 < d < a使得

(B1){u∈P(γ, θ, α, a, b, c):α(u)>a}≠∅且α(F(u)) > a, u∈P(γ, θ, α, a, b, c)；

(B2){u∈Q(γ, β, ψ, h, d, c):β(u) < d}≠∅且β(F(u)) < d, u∈Q(γ, β, ψ, h, d, c)；

(B3)若u∈P(γ, α, a, c)且θ(F(u))>b，则α(F(u))>a；

(B4)若u∈Q(γ, β, d, c)且ψ(F(u)) < h，则β(F(u)) < d.

那么F至少有3个不动点u₁, u₂, u₃∈P(γ, c), 使得

β(u₁) < d, a < α(u₂), d < β(u₃), α(u₃) < a.

2 主要结果

令E=C[0,1]，其范数为$ \left\| u \right\| = \mathop {\max }\limits_{t \in \left[{0, 1} \right]} \left| {u\left( t \right)} \right|. $当$ 0 < {t_3} < \frac{1}{2} $时，定义E中锥P为P={u∈E:u在[0,1]上是非负的凹函数，$ {\mathop {\min }\limits_{t \in \left[ {{t_3},1 - {t_3}} \right]} u\left( t \right) \ge \left\| u \right\|} $}又定义P上的非负连续凹函数α, ψ和非负连续凸函数γ, β, θ为

$ \begin{array}{*{20}{l}} {\alpha \left( u \right) = \mathop {\min }\limits_{t \in \left[ {{t_1},{t_2}} \right] \cup \left[ {1 - {t_2},1 - {t_1}} \right]} u\left( t \right),\beta \left( u \right) = }\\ {\mathop {\max }\limits_{t \in \left[ {1/r,1 - 1/r} \right]} u\left( t \right)\gamma \left( u \right) = \mathop {\max }\limits_{t \in \left[ {0,{t_3}} \right] \cup \left[ {1 - {t_3},1} \right]} u\left( t \right),\theta \left( u \right) = }\\ {\mathop {\max }\limits_{t \in \left[ {{t_1},{t_2}} \right] \cup \left[ {1 - {t_2},1 - {t_1}} \right]} u\left( t \right),\psi \left( u \right) = \mathop {\min }\limits_{t \in \left[ {1/r,1 - 1/r} \right]} u\left( t \right).} \end{array} $

其中t₁, t₂和r是非负的实数满足$ 0 < \frac{1}{r} < {t_1} < {t_2} \le \frac{1}{2}. $对任意u∈P, u是边值问题(1)的解等价于$ u\left( t \right) = \int_0^1 {G\left( {t, s} \right)f\left( {s, u\left( s \right)} \right){\rm{d}}s}, $且对任意u∈P有

$ \alpha \left( u \right) \le \beta \left( u \right), $

(2.1)

$ \begin{array}{*{20}{l}} {\;\;\;\;\;\gamma \left( u \right) = \mathop {\max }\limits_{t \in \left[ {0,{t_3}} \right] \cup \left[ {1 - {t_3},1} \right]} u\left( t \right) \ge \mathop {\max }\limits_{t \in \left[ {0,{t_3}} \right]} u\left( t \right) = }\\ {\mathop {\max }\limits_{t \in \left[ {0,{t_3}} \right]} \left( {\int_0^{{t_3}} {G\left( {t,s} \right)f\left( {s,u\left( s \right)} \right){\rm{d}}s + } \int_{{t_3}}^{1 - {t_3}} {G\left( {t,s} \right)f\left( {s,} \right.} } \right.}\\ {\left. {\left. {u\left( s \right)} \right){\rm{d}}s + \int_{1 - {t_3}}^1 {G\left( {t,s} \right)f\left( {s,u\left( s \right)} \right){\rm{d}}s} } \right) = }\\ {\frac{1}{{\mathit{\Gamma} \left( \alpha \right)}}\int_0^1 {{{\left( {1 - s} \right)}^{\alpha - 1}}} f\left( {s,u\left( s \right)} \right){\rm{d}}s = \left\| u \right\|.} \end{array} $

(2.2)

定理2.1假设存在非负实数a, b和c使得$ 0 < a < b \le \frac{{c{t_1}}}{{{t_2}}}, $若f满足下列的条件：

(H1)$ \begin{array}{*{20}{l}} {\;\;\;\;\;\;\;\;\;\;f\left( {t, u\left( t \right)} \right) < \frac{a}{{\mathit{\Gamma} \left( \alpha \right){{\left( {1- \frac{1}{r}} \right)}^{\alpha- 1}}}}, t \in \left[{\frac{1}{r}, 1-} \right.}\\ {\left. {\frac{1}{r}} \right], u\left( t \right) \in \left[{\frac{a}{M}, a} \right], } \end{array} $

(H2)$ \begin{array}{*{20}{l}} {\;\;\;\;\;\;\;\;\;\;\;\;\;f\left( {t, u\left( t \right)} \right) > \frac{{\alpha b\mathit{\Gamma} \left( \alpha \right)}}{{1- {{\left( {1- {t_1}} \right)}^\alpha }}}, t \in \left[{{t_1}, {t_2}} \right] \cup }\\ {\left[{1-{t_2}, 1-{t_1}} \right], u\left( t \right) \in \left[{b, \frac{b}{m}} \right], } \end{array} $

(H3)$ \begin{array}{*{20}{l}} {\;\;\;\;\;\;\;f\left( {t, u\left( t \right)} \right) \le c\alpha \mathit{\Gamma} \left( \alpha \right), t \in \left[{0, {t_3}} \right] \cup \left[{1-{t_3}, } \right.}\\ {\left. 1 \right], u\left( t \right) \in \left[{0, c} \right].} \end{array} $

那么，边值问题(1)至少有3个正解u₁, u₂和u₃, 满足

$ \begin{array}{*{20}{l}} {\mathop {\max }\limits_{t \in \left[{0, {t_3}} \right] \cup \left[{1-{t_3}, 1} \right]} {u_i} \le c, i = 1, 2, 3, }\\ {\mathop {\min }\limits_{t \in \left[{{t_1}, {t_2}} \right] \cup \left[{1-{t_2}, 1-{t_1}} \right]} {u_1}\left( t \right) > b, \mathop {\max }\limits_{t \in \left[{1/r, 1-1/r} \right]} {u_2}\left( t \right) < a, }\\ {\mathop {\min }\limits_{t \in \left[{{t_1}, {t_2}} \right] \cup \left[{1-{t_2}, 1-{t_1}} \right]} {u_3}\left( t \right) < b, \mathop {\max }\limits_{t \in \left[{1/r, 1-1/r} \right]} {u_3}\left( t \right) > a.} \end{array} $

证明在锥P上定义算子A为

$ Au\left( t \right) = \int_0^1 {G\left( {t, s} \right)f\left( {s, u\left( s \right)} \right)} {\rm{d}}s. $

因为

$ \begin{array}{*{20}{l}} {\;\;\;\;\;\;Au\left( t \right) = \frac{1}{{\mathit{\Gamma} \left( \alpha \right)}}\int_0^t {\left( {{{\left( {1-s} \right)}^{\alpha-1}}-{{\left( {t - s} \right)}^{\alpha - 1}}} \right)} f\left( {s, } \right.}\\ {\left. {u\left( s \right)} \right){\rm{d}}s + \frac{1}{{\mathit{\Gamma} \left( \alpha \right)}}\int_t^1 {{{\left( {1 - s} \right)}^{\alpha - 1}}f\left( {s, u\left( s \right)} \right){\rm{d}}s} \ge }\\ {\frac{1}{{\mathit{\Gamma} \left( \alpha \right)}}\int_0^t {{{\left( {1 - s} \right)}^{\alpha - 1}}f\left( {s, u\left( s \right)} \right){\rm{d}}s} + \frac{1}{{\mathit{\Gamma} \left( \alpha \right)}}\int_t^1 {{{\left( {1 - s} \right)}^{\alpha - 1}}f\left( {s, } \right.} }\\ {\left. {u\left( s \right)} \right){\rm{d}}s = \left\| {Au} \right\|, } \end{array} $

所以A:P→P连续.由Arzela-Ascoli定理易证A:P→P是全连续的.

首先，对任意u∈P，由(2.1)式和(2.2)式可知α(u)≤β(u), ‖u‖≤γ(u).若u∈P(γ, c)，则‖u‖≤c.又由(H3)得

$ \begin{array}{*{20}{l}} {\;\;\;\;\;\;\gamma \left( {Au} \right) = \mathop {\max }\limits_{t \in \left[{0, {t_3}} \right] \cup \left[{1-{t_3}, 1} \right]} \int_0^1 {G\left( {t, s} \right)f\left( {s, u\left( s \right)} \right)} {\rm{d}}s \le }\\ {c\alpha \mathit{\Gamma} \left( \alpha \right)\int_0^1 {G\left( {s, s} \right){\rm{d}}s} = c, } \end{array} $

因此，A:P(γ, c)→P(γ, c).说明{u∈P(γ, θ, α, a, $ \frac{{{t_2}}}{{{t_1}}}a $, c):α(u)>a}≠∅，{u∈Q(γ, β, ψ, $ \frac{{2d}}{r} $, d, c):β(u) < d}≠∅.其次

(1)若u∈Q(γ, β, a, c)且$ \psi \left( {A\left( u \right)} \right) < \frac{a}{M} $, 则

$ \begin{array}{*{20}{l}} {\;\;\;\;\beta \left( {Au} \right) = \mathop {\max }\limits_{t \in \left[{1/r, 1-1/r} \right]} \int_0^1 {G\left( {t, s} \right)f\left( {s, u\left( s \right)} \right)} {\rm{d}}s \le }\\ {M\mathop {\min }\limits_{t \in \left[{1/r, 1-1/r} \right]} \int_0^1 {G\left( {t, s} \right)f\left( {s, u\left( s \right)} \right){\rm{d}}s = M\psi \left( {Au} \right)} < a;} \end{array} $

(2)若u∈Q(γ, β, ψ, $ \frac{a}{M} $, a, c)，由(H1)得

$ \begin{array}{*{20}{l}} {\;\;\;\;\;\;\beta \left( {Au} \right) = \mathop {\max }\limits_{t \in \left[{1/r, 1-1/r} \right]} \int_0^1 {G\left( {t, s} \right)f\left( {s, u\left( s \right)} \right)} {\rm{d}}s < }\\ {\frac{a}{{\mathit{\Gamma} \left( \alpha \right)\left( {1 - \frac{1}{r}} \right)\alpha - 1}}\mathop {\max }\limits_{t \in \left[{1/r, 1-1/r} \right]} \int_0^1 {G\left( {t, s} \right)} {\rm{d}}s = a;} \end{array} $

(3)若u∈Q(γ, α, b, c)，且θ(A(u))> $ \frac{b}{m} $，则

$ \begin{array}{*{20}{l}} {\;\;\;\;\alpha \left( {Au} \right) = \mathop {\min }\limits_{t \in \left[{{t_1}, {t_2}} \right] \cup \left[{1-{t_2}, 1-{t_1}} \right]} \int_0^1 {G\left( {t, s} \right)f\left( {s, u\left( s \right)} \right)} {\rm{d}}s \ge m}\\ {\mathop {\max }\limits_{t \in \left[{{t_1}, {t_2}} \right] \cup \left[{1-{t_2}, 1-{t_1}} \right]} \int_0^1 {G\left( {t, s} \right)f\left( {s, u\left( s \right)} \right)} {\rm{d}}s = m\theta \left( {Au} \right) > b;} \end{array} $

(4)若u∈Q(γ, θ, b, $ \frac{b}{m} $, c)，由(H2)得

$ \begin{array}{*{20}{l}} {\;\;\;\;\;\alpha \left( {Au} \right) = \mathop {\min }\limits_{t \in \left[{{t_1}, {t_2}} \right] \cup \left[{1-{t_2}, 1-{t_1}} \right]} \int_0^1 {G\left( {t, s} \right)f\left( {s, } \right.} }\\ {\left. {u\left( s \right)} \right){\rm{d}}s > \frac{{\alpha b\mathit{\Gamma} \left( \alpha \right)}}{{1 - {{\left( {1 - {t_1}} \right)}^\alpha }}}\mathop {\min }\limits_{t \in \left[{{t_1}, {t_2}} \right] \cup \left[{1-{t_2}, 1-{t_1}} \right]} \int_0^1 {G\left( {t, s} \right) \cdot } }\\ {{\rm{d}}s = b;} \end{array} $

最后，由定理1.1可得，边值问题(0.1)至少有3个正解u₁, u₂, u₃∈P(γ, c), 满足

α(u₁) > b, β(u₂) < a, α(u₃) < b, β(u₃) > a.

3 结论

本文研究了一类Caputo分数阶微分方程边值问题多解的存在性.证明时，将微分方程边值问题转化为积分方程，进一步转化为讨论积分算子不动点的问题，然后通过运用Leggett-Williams不动点定理该分数阶微分方程边值问题至少有3个正解存在的结果，其中格林函数的性质和非线性项的条件至关重要.