Langevin方程描述了布朗运动由于流体分子的碰撞, 颗粒在流体中做无规则运动的动力学行为, 具体模型如下

(1.1) $ \begin{equation} \frac{{{\rm d}^2}x}{{\rm d}{t^2}} + \gamma \frac{{{\rm d}x}}{{{\rm d}t}} = \Gamma (t), \end{equation} $

其中$ x $是时刻$ t $颗粒的坐标, $ \gamma = ({\alpha \mathord{\left/ {\vphantom {\alpha m}} \right. } m}) $表示单位质量的阻力系数, $ m $为颗粒的质量, 阻力来自介质分子对颗粒的碰撞, $ \Gamma (t) $表示为单位质量的涨落力, 也称为郎之万力^[1]. 模型(1) 的建立是基于布朗运动是马尔可夫过程, 不同时刻的$ \Gamma (t) $是相互独立的, 不具有时间相关性和记忆性. 然而, 当布朗颗粒在稠密的黏性流体、具有内部自由度的流体以及在湍流集团中运动时, 郎之万力$ \Gamma (t) $是具有时间相关性, 即布朗运动的过程依赖于历史过程, 具有记忆性, 是非马尔可夫的时间相关过程^[2]. 由此, 基于分数阶微分算子的特性(分数阶微分算子具有长程相关性和记忆性), Lutz和Burov等^[3-4]引入分数阶微分算子描述布朗颗粒运动过程中记忆性的阻力, 从而得到如下分数阶Langevin方程

(1.2) $ \begin{equation} x'' + \gamma {}^CD_{0 + }^\alpha x = \Gamma (t), \;0 < \alpha < 1, \end{equation} $

其中$ {}^CD_{0 + }^\alpha $是$ \alpha $阶Caputo型分数阶微分算子.

基于分数阶Langevin方程的应用背景, 使得对分数阶Langevin方程初值与边值问题的研究受到学者们广泛的关注^[5-16]. 2012年, Ahmad等^[7]讨论了如下分数阶Langevin方程三点边值问题

(1.3) $ \begin{equation} \left\{ \begin{array}{l} {}^CD_{0{\rm{ + }}}^\beta ({}^CD_{0{\rm{ + }}}^\alpha + \lambda )x(t) = f(t, x(t)), \;t \in (0, 1), \\ x(0) = 0, \;\;x(\eta ) = 0, \;\;x(1) = 0, \end{array} \right. \end{equation} $

其中$ 1 < \beta \le 2 $, $ 0 < \alpha \le 1 $, $ {}^CD_{0 + }^\rho $是$ \rho (\rho = \alpha , \beta ) $阶Caputo型分数阶微分算子, $ 0 < \eta < 1 $, $ f \in C([0, 1] \times {{\Bbb R}} , {{\Bbb R}} ) $, $ \lambda \in {{\Bbb R}} $. 作者运用Krasnoselskii不动点定理以及Banach压缩映射定理分别给出了(3)式解的存在性与唯一性的充分性条件.

2017年, Zhou等^[8]讨论了带$ p $-Laplacian算子的分数阶Langevin方程反周期边值问题

(1.4) $ \begin{equation} \left\{ \begin{array}{l} {}^CD_{0{\rm{ + }}}^\beta {\varphi _p}[({}^CD_{0{\rm{ + }}}^\alpha + \lambda )x(t)] = f(t, x(t), {}^CD_{0{\rm{ + }}}^\alpha x(t)), \;t \in (0, 1), \\ x(0) = - x(1), \;\;{}^CD_{0{\rm{ + }}}^\alpha x(0) = - {}^CD_{0{\rm{ + }}}^\alpha x(1), \end{array} \right. \end{equation} $

其中$ 0 < \alpha , \beta \le 1 $, $ {}^CD_{0 + }^\rho $是$ \rho (\rho = \alpha , \beta ) $阶Caputo型分数阶微分算子, $ \lambda \ge 0, $$ {\varphi _p}(s) = |s{|^{p - 2}}s $, $ p>1 $是$ p $-Laplacian算子, $ f:[0, 1] \times {{{\Bbb R}} ^2} \to {{\Bbb R}} $满足Carathéodory条件. 作者运用Leray-Schaefer不动点定理给出了(1.4) 式解的存在性的充分条件.

2018年, Fazli和Nieto^[9]讨论了如下分数阶Langevin方程反周期边值问题

(1.5) $ \begin{equation} \left\{ \begin{array}{l} {}^CD_{0{\rm{ + }}}^\beta ({}^CD_{0{\rm{ + }}}^\alpha + \lambda )x(t) = f(t, x(t)), \;t \in (0, 1), \;0 < \alpha \le 1, \;1 < \beta \le 2, \\ x(0) = - x(1), \;{}^CD_{0{\rm{ + }}}^\alpha x(0) = - {}^CD_{0{\rm{ + }}}^\alpha x(1), \;{}^CD_{0{\rm{ + }}}^\alpha {}^CD_{0{\rm{ + }}}^\alpha x(0) = - {}^CD_{0{\rm{ + }}}^\alpha {}^CD_{0{\rm{ + }}}^\alpha x(1), \end{array} \right. \end{equation} $

其中$ {}^CD_{0 + }^\rho $是$ \rho (\rho = \alpha , \beta ) $阶Caputo型分数阶微分算子, $ \lambda \in {{\Bbb R}} , $$ f \in C([0, 1] \times {{\Bbb R}} , {{\Bbb R}} ). $作者运用混合单调算子耦合不动点定理给出了(1.5)式解的存在性以及唯一性的充分条件.

2020年, Salem等^[16]讨论了如下分数阶Langevin方程多点边值问题

(1.6) $ \begin{equation} \left\{ \begin{array}{l} {}^CD_{0{\rm{ + }}}^\beta ({}^CD_{0{\rm{ + }}}^\alpha + \lambda )u(t) = f(t, u(t)), \;t \in (0, 1), \\ u(0) = 0, \;{}^CD_{0{\rm{ + }}}^\alpha u(0) = 0, \;u(1) = \mu u(\eta ), \end{array} \right. \end{equation} $

其中$ 0 < \alpha \le 1 $, $ 1 < \beta \le 2 $, $ {}^CD_{0 + }^\rho $是$ \rho (\rho = \alpha , \beta ) $阶Caputo型分数阶微分算子, $ \mu , \lambda \in {\rm{{{\Bbb R}} }} $, $ 0 < \eta < 1 $满足$ \mu {\eta ^{\alpha + 1}} \ne 1 $, $ f \in {C^1}([0, 1] \times {{\Bbb R}} , {{\Bbb R}} ) $. 作者运用Banach压缩映射定理以及算子不动点定理给出了(1.6) 式解的存在性以及唯一性的充分条件.

近年来, 另一种形式的分数阶微分方程边值问题受到了人们的关注, 即考虑带变系数Laplacian算子的分数阶微分方程边值问题^[17-22]. 2016年, Shen和Liu^[17]讨论了如下带$ p(t) $-Laplacian算子的分数阶微分方程边值问题

(1.7) $ \begin{equation} \left\{ \begin{array}{l} D_{0{\rm{ + }}}^\beta {\varphi _{p(t)}}(D_{0{\rm{ + }}}^\alpha x(t)) + f(t, x(t)) = 0, \;t \in (0, 1), \\ x(0) = 0, \;D_{0{\rm{ + }}}^{\alpha - 1}x(1) = \gamma I_{0{\rm{ + }}}^{\alpha - 1}x(\eta ), \;D_{0{\rm{ + }}}^\alpha x(0) = 0, \end{array} \right. \end{equation} $

其中$ 1 < \alpha \le 2 $, $ 0 < \beta \le 1 $, $ D_{0 + }^\rho $是$ \rho (\rho = \alpha , \beta ) $阶Riemann-Liouville型分数阶微分算子, $ \gamma > 0 $, $ 0 < \eta < 1 $, $ f \in C([0, 1] \times {{\Bbb R}} , {{\Bbb R}} ) $, $ {\varphi _{p(t)}}( \cdot ) $是$ p(t) $-Laplacian算子, $ p(t) > 1 $, $ p(t) \in {C^1}[0, 1] $ (具体定义参见预备知识). 作者运用Schaefer不动点定理以及Mawhin连续性定理分别给出了问题(1.7)在共振与非共振情形下解存在的充分性条件.

注意到, 变系数Laplacian算子在许多领域中有着广泛的应用, 例如: 非线性弹性力学、电流变流体、图像处理等^[23-26]. 显然, $ p(t) $-Laplacian算子是对$ p $-Laplacian算子的推广, 当取$ p(t) \equiv p > 1 $ ($ p $为常数), $ p(t) $-Laplacian算子将退化为$ p $-Laplacian算子. 基于上述工作的启发, 一个自然的想法是考虑带$ p(t) $-Laplacian算子的分数阶Langevin方程边值问题. 为此, 本文研究如下带$ p(t) $-Laplacian算子的分数阶Langevin方程反周期边值问题

(1.8) $ \begin{equation} \left\{ \begin{array}{l} {}^CD_{0 + }^\beta {\varphi _{p(t)}}[({}^CD_{0 + }^\alpha + \lambda )x(t)] = f(t, x(t), {}^CD_{0 + }^\alpha x(t)), \;t \in (0, 1), \\ x(0) = - x(1), \;\;{}^CD_{0 + }^\alpha x(0) = - {}^CD_{0 + }^\alpha x(1), \end{array} \right. \end{equation} $

其中$ 0 < \alpha , \beta \le 1 $, $ \lambda \ge 0 $, $ 1 < \alpha + \beta \le 2 $, $ {}^CD_{0 + }^\rho $是$ \rho (\rho = \alpha , \beta ) $阶Caputo型分数阶微分算子, $ f \in C([0, 1] \times {{\bf{{{\Bbb R}} }}^2}, {\bf{{{\Bbb R}} }}) $, $ {\varphi _{p(t)}}( \cdot ) $是$ p(t) $-Laplacian算子, $ p(0) = p(1) $, $ p(t) > 1 $. 注意到, 当$ p(t) \equiv p > 1 $($ p $为常数) 时, 文献[8] 研究的问题是本文的特例.

定义 2.1^[27-28]    函数$ f:(0, + \infty ) \to {{{\Bbb R}} } $的$ \alpha (\alpha > 0) $阶Riemann-Liouville型分数阶积分定义为

$ I_{0 + }^\alpha f(t) = \frac{1}{{\Gamma (\alpha )}}\int_0^t {{{(t - s)}^{\alpha - 1}}f(s){\rm d}s} , $

其中假设右式在$ (0, + \infty ) $上有定义.

定义 2.2^[27-28]    函数$ f:(0, + \infty ) \to {{{\Bbb R}} } $的$ \alpha (\alpha > 0) $阶Caputo型分数阶导数定义为

$ {}^CD_{0 + }^\alpha f(t) = I_{0 + }^{n - \alpha }\frac{{{{\rm d}^n}f(t)}}{{{\rm d}{t^n}}} = \frac{1}{{\Gamma (n - \alpha )}}\int_0^t {{{(t - s)}^{n - \alpha - 1}}{f^{(n)}}(s){\rm d}s} , $

其中$ n = [\alpha ] + 1 $, 假设右式在$ (0, + \infty ) $上有定义.

引理 2.1^[27-28]    令$ \alpha > 0 $. 假设$ f \in A{C^n}[0, 1] $, 则

$ I_{0 + }^\alpha {}^CD_{0 + }^\alpha f(t) = f(t) + {c_0} + c{}_1t + {c_2}{t^2} + \cdots + c{}_{n - 1}{t^{n - 1}}, $

其中$ {c_i} \in {{{\Bbb R}} }, \;i = 0, 1, 2, \cdot \cdot \cdot , n - 1, \;n = [\alpha ] + 1. $

定义 2.3^[26]    对任意的$ (t, x) \in [0, 1] \times {{{\Bbb R}} } $, $ {\varphi _{p(t)}}(x) = |x{|^{p(t) - 2}}x $是从$ {{{\Bbb R}} } $到$ {{{\Bbb R}} } $的同胚映射, 且当$ t $固定时$ {\varphi _{p(t)}}( \cdot ) $是严格单调增的, 其逆映射定义如下

$ \left\{ \begin{array}{ll} \varphi _{p(t)}^{ - 1}(x) = |x{|^{\frac{{2 - p(t)}}{{p(t) - 1}}}}x, \;&x \in {{{\Bbb R}} }\backslash \{ 0\} , \\ \varphi _{p(t)}^{ - 1}(x) = 0, \;&x = 0 \end{array} \right. $

是将有界集映成有界集的连续映射.

引理 2.2(Schaefer不动点定理^[28])    设$ X $是Banach空间, 算子$ T:X \to X $为全连续算子, 若集合$ \Omega { = } \left\{ {x \in X|x { = } \mu Tx, \mu \in (0, 1)} \right\} $有界, 则算子$ T $在$ \Omega $中至少存在一个不动点.

定义空间$ X = \{ x:x, {}^CD_{0 + }^\alpha x \in C[0, 1]\} $, 赋予范数$ ||x|{|_X} = ||x|{|_\infty } + ||{}^CD_{0 + }^\alpha x|{|_\infty} $, 其中$ || \cdot |{|_\infty } = \mathop {\max }\limits_{t \in [0, 1]} | \cdot | $, 显然$ (X, || \cdot |{|_X}) $是Banach空间.

引理 3.1    设$ h \in C[0, 1] $, 则分数阶Langevin方程

(3.1) $ \begin{equation} {}^CD_{0 + }^\beta {\varphi _{p(t)}}[({}^CD_{0 + }^\alpha + \lambda )x(t)] = h(t), \;\;t \in (0, 1), \end{equation} $

在反周期边值条件

(3.2) $ \begin{equation} x(0) = - x(1), \;{}^CD_{0 + }^\alpha x(0) = - {}^CD_{0 + }^\alpha x(1) \end{equation} $

下, 有如下形式的解

$ \begin{eqnarray*} x(t) & = & I_{0 + }^\alpha \varphi _{p(t)}^{ - 1}(I_{0 + }^\beta h(t) + Ah(t)) + Bh(t) - \lambda I_{0 + }^\alpha x(t)\\ & = & \frac{1}{{\Gamma (\alpha )}}\int_0^t {{{(t - s)}^{\alpha - 1}}\varphi _{p(s)}^{ - 1}\left( {\frac{1}{{\Gamma (\beta )}}\int_0^s {{{(s - \tau )}^{\beta - 1}}h(\tau ){\rm{d}}\tau + } Ah(s)} \right){\rm d}s} + Bh(t)\\ & &{}- \frac{\lambda }{{\Gamma (\alpha )}}\int_0^t {{{(t - s)}^{\alpha - 1}}x(s){\rm d}s} , \end{eqnarray*} $

其中

$ \begin{eqnarray*} Ah(t)& = & - \frac{1}{2}{\left. {I_{0 + }^\beta h(t)} \right|_{t = 1}} = - \frac{1}{{2\Gamma (\beta )}}\int_0^1 {{{(1 - s)}^{\beta - 1}}h(s)} {\rm d}s, {\quad} \forall t \in [0, 1], \\ Bh(t) & = & - \frac{1}{2}I_{0 + }^\alpha \varphi _{p(t)}^{ - 1}{\left. {(I_{0 + }^\beta h(t) + Ah(t))} \right|_{t = 1}} + \frac{\lambda }{2}{\left. {I_{0 + }^\alpha x(t)} \right|_{t = 1}}\\ & = & - \frac{1}{{2\Gamma (\alpha )}}\int_0^1 {{{(1 - s)}^{\alpha - 1}}\varphi _{p(s)}^{ - 1}\left( {\frac{1}{{\Gamma (\beta )}}\int_0^s {{{(s - \tau )}^{\beta - 1}}h(\tau )} {\rm d}\tau + Ah(s)} \right)} {\rm d}s\\ &&{}+ \frac{\lambda }{{2\Gamma (\alpha )}}\int_0^1 {{{(1 - s)}^{\alpha - 1}}x(s){\rm d}s} , \;\;\forall t \in [0, 1]. \end{eqnarray*} $

证    利用算子$ I_{0 + }^\beta $作用到(3.1) 式的两端, 则由引理2.1以及定义2.3知

(3.3) $ \begin{equation} ({}^CD_{0 + }^\alpha + \lambda )x(t) = \varphi _{p(t)}^{ - 1}\left( {{c_0} + I_{0 + }^\beta h(t)} \right), \;{c_0} \in {{{\Bbb R}} }, \end{equation} $

上式结合边值条件(3.2) 可得

$ {c_0} = - \frac{1}{2}{\left. {I_{0 + }^\beta h(t)} \right|_{t = 1}} = Ah(t), $

同理, 运用算子$ I_{0 + }^\alpha $作用到(3.3) 式的两端, 得到

(3.4) $ \begin{equation} x(t) = {c_1} + I_{0 + }^\alpha \varphi _{p(t)}^{ - 1}(Ah(t) + I_{0 + }^\beta h(t)) - \lambda I_{0 + }^\alpha x(t), \;{c_1} \in {{{\Bbb R}} }, \end{equation} $

注意到, 边值条件$ x(0) = - x(1) $, 可推出

(3.5) $ \begin{equation} {c_1} = - \frac{1}{2}I_{0 + }^\alpha \varphi _{p(t)}^{ - 1}{\left. {(Ah(t) + I_{0 + }^\beta h(t))} \right|_{t = 1}} + \frac{\lambda }{2}{\left. {I_{0 + }^\alpha x(t)} \right|_{t = 1}} = Bh(t), \end{equation} $

将(3.5) 式带入到(3.4) 式, 立知引理结论成立.

基于引理3.1, 定义算子$ T:C[0, 1] \to C[0, 1] $如下

$ \begin{eqnarray*} Tx(t)& = & I_{0 + }^\alpha \varphi _{p(t)}^{ - 1}(I_{0 + }^\beta Nx(t) + ANx(t)) + BNx(t) - \lambda I_{0 + }^\alpha x(t)\\ & = & \frac{1}{{\Gamma (\alpha )}}\int_0^t {{{(t - s)}^{\alpha - 1}}} \varphi _{p(s)}^{ - 1}\left( {\frac{1}{{\Gamma (\beta )}}\int_0^s {{{(s - \tau )}^{\beta - 1}}f(\tau , x(\tau ), {}^CD_{0 + }^\alpha x(\tau )){\rm d}\tau } } \right.\\ &&\left. { - \frac{1}{{2\Gamma (\beta )}}\int_0^1 {{{(1 - \tau )}^{\beta - 1}}} f(\tau , x(\tau ), {}^CD_{0 + }^\alpha x(\tau )){\rm d}\tau } \right){\rm d}s\\ & &- \frac{1}{{2\Gamma (\alpha )}}\int_0^1 {{{(1 - s)}^{\alpha - 1}}} \varphi _{p(s)}^{ - 1}\left( {\frac{1}{{\Gamma (\beta )}}\int_0^s {{{(s - \tau )}^{\beta - 1}}f(\tau , x(\tau ), {}^CD_{0 + }^\alpha x(\tau )){\rm d}\tau } } \right.\\ &&\left. { - \frac{1}{{2\Gamma (\beta )}}\int_0^1 {{{(1 - \tau )}^{\beta - 1}}} f(\tau , x(\tau ), {}^CD_{0 + }^\alpha x(\tau )){\rm d}\tau } \right){\rm d}s\\ & &+ \frac{\lambda }{{2\Gamma (\alpha )}}\int_0^1 {{{(1 - s)}^{\alpha - 1}}x(s){\rm d}s} - \frac{\lambda }{{\Gamma (\alpha )}}\int_0^t {{{(t - s)}^{\alpha - 1}}x(s){\rm d}s, } \end{eqnarray*} $

其中$ N:C[0, 1] \to C[0, 1] $是Nemytskii算子定义为

$ Nx(t) = f(t, x(t), {}^CD_{0 + }^\alpha x(t)), \;\;\forall t \in [0, 1], $

则由引理3.1, 可知边值问题(1.8) 的解即为算子$ T $的不动点.

接下来, 根据Schaefer不动点定理给出本文主要结果. 为了下文叙述方便, 作如下记号

$ \;Q = {{{6^{{1 \mathord{\left/ {\vphantom {1 {({P_m} - 1)}}} \right. } {({P_m} - 1)}}}}} \mathord{\left/ {\vphantom {{{6^{{1 \mathord{\left/ {\vphantom {1 {({P_m} - 1)}}} \right. } {({P_m} - 1)}}}}} {\Gamma (\alpha ){{(2\Gamma (\beta + 1))}^{{1 \mathord{\left/ {\vphantom {1 {({P_M} - 1)}}} \right. } {({P_M} - 1)}}}}}}} \right. } {\Gamma (\alpha ){{(2\Gamma (\beta + 1))}^{{1 \mathord{\left/ {\vphantom {1 {({P_M} - 1)}}} \right. } {({P_M} - 1)}}}}}}, \;{P_m}: = {\min _{t \in [0, 1]}}p(t), \;{P_M}: = {\max _{t \in [0, 1]}}p(t). $

定理 3.1    设$ f:[0, 1] \times {{{{\Bbb R}} }^2} \to {{{\Bbb R}} } $连续, 且满足条件

(H)     存在非负函数$ \zeta , \phi , \psi \in C[0, 1] $使得

$ \left| {f(t, u, v)} \right| \le \zeta (t) + \phi (t)|u{|^{r - 1}} + \psi (t)|v{|^{r - 1}}, \;(u, v) \in {{{{\Bbb R}} }^2}, \;1 < r \le {P_m}, \;t \in [0, 1], $

则当

(3.6) $ \begin{equation} \ell \left( {\frac{{3Q}}{{2\alpha }} + Q\Gamma (\alpha)}\right) + \frac{{3\lambda }}{{2\Gamma (\alpha + 1)}} + \lambda < 1 \end{equation} $

时, 边值问题(1.8) 在$ X $上至少有一个解, 其中

$ \begin{eqnarray*} \ell : = \max \left\{ {{{(||\phi |{|_\infty } + ||\psi |{|_\infty })}^{1/({p_m} - 1)}}, \;{{(||\phi {\rm{|}}{{\rm{|}}_\infty } + ||\psi |{|_\infty })}^{1/({p_M} - 1)}}} \right\}. \end{eqnarray*} $

证    关于定理的证明分为两步完成. 第一步. 证明$ T $是全连续算子. 事实上, 对任意的$ \delta > 0 $, 定义$ X $上的有界开集$ \Omega = \left\{ {x \in X:{\rm{||}}x{\rm{|}}{{\rm{|}}_X} < \delta } \right\} $. 由$ f $与$ \varphi _{p(t)}^{ - 1}( \cdot ) $的连续性易证$ T $在$ [0, 1] $上是连续的, 且存在常数$ M > 0 $使得

$ \left| {\varphi _{p(t)}^{ - 1}(I_{0 + }^\beta Nx(t) + ANx(t))} \right| \le M, \;x(t) \in \Omega, \;t \in [0, 1], $

于是有

$ \begin{eqnarray*} ||Tx|{|_\infty } & = &\mathop {\max }\limits_{t \in [0, 1]} |I_{0 + }^\alpha \varphi _{p(t)}^{ - 1}(I_{0 + }^\beta Nx(t) + ANx(t)) + BNx(t) - \lambda I_{0 + }^\alpha x(t)|\\ &\le& \mathop {\max }\limits_{t \in [0, 1]} |I_{0 + }^\alpha \varphi _{p(t)}^{ - 1}(I_{0 + }^\beta Nx(t) + ANx(t))| + \mathop {\max }\limits_{t \in [0, 1]} |BNx(t)| + \lambda\mathop {\max }\limits_{t \in [0, 1]} |I_{0 + }^\alpha x(t)|\\ &\le& \frac{{3M + 3\lambda \delta }}{{2\Gamma (\alpha + 1)}} < + \infty . \end{eqnarray*} $

注意到

$ {}^CD_{0 + }^\alpha Tx(t) = \varphi _{p(t)}^{ - 1}(I_{0 + }^\beta Nx(t) + ANx(t)) - \lambda x(t). $

从而

$ \begin{eqnarray*} ||{}^CD_{0 + }^\alpha Tx|{|_\infty } & = & \mathop {\max }\limits_{t \in [0, 1]} |\varphi _{p(t)}^{ - 1}(I_{0 + }^\beta Nx(t) + ANx(t)) - \lambda x(t)|\\ &\le& \mathop {\max }\limits_{t \in [0, 1]} |\varphi _{p(t)}^{ - 1}(I_{0 + }^\beta Nx(t) + ANx(t)){\rm{| + }}\lambda {\rm{||}}x{\rm{|}}{{\rm{|}}_\infty }\\ &\le &M + \lambda \delta < + \infty . \end{eqnarray*} $

因此, $ T $在$ \bar \Omega $上一致有界. 下证$ T $在$ \bar \Omega $上等度连续. 事实上, 对任意的$ x \in \bar \Omega $, $ 0 \le {t_1} < {t_2} \le 1 $, 则有

$ \begin{eqnarray*} \left| {Tx({t_2}) - Tx({t_1})} \right|& = & \frac{1}{{\Gamma (\alpha )}}\left| {\int_0^{{t_1}} {({{({t_2} - s)}^{\alpha - 1}} - {{({t_1} - s)}^{\alpha - 1}})\varphi _{p(s)}^{ - 1}(ANx(s) + I_{0{\rm{ + }}}^\beta Nx(s)){\rm d}s} } \right.\\ & &+ \int_{{t_1}}^{{t_2}} {{{({t_2} - s)}^{\alpha - 1}}\varphi _{p(s)}^{ - 1}(ANx(s) + I_{0{\rm{ + }}}^\beta Nx(s)){\rm d}s} \\ &&\left. { - \lambda \left[ {\int_0^{{t_1}} {({{({t_2} - s)}^{\alpha - 1}} - {{({t_1} - s)}^{\alpha - 1}})x(s){\rm d}s + } \int_{{t_1}}^{{t_2}} {{{({t_2} - s)}^{\alpha - 1}}x(s){\rm d}s} } \right]} \right|\\ & \le& \frac{1}{{\Gamma (\alpha )}}\left| {\int_0^{{t_1}} {({{({t_2} - s)}^{\alpha - 1}} - {{({t_1} - s)}^{\alpha - 1}}) \varphi _{p(s)}^{ - 1}(ANx(s) + I_{0{\rm{ + }}}^\beta Nx(s)){\rm d}s} } \right|\\ &&+ \frac{1}{{\Gamma (\alpha )}}\left| {\int_{{t_1}}^{{t_2}} {{{({t_2} - s)}^{\alpha - 1}} \varphi _{p(s)}^{ - 1}(ANx(s) + I_{0{\rm{ + }}}^\beta Nx(s)){\rm d}s} } \right|\\ &&{+} \frac{\lambda }{{\Gamma (\alpha )}}\left| {\int_0^{{t_1}} {({{({t_2} {-} s)}^{\alpha {-} 1}} {-} {{({t_1} {-} s)}^{\alpha {-} 1}})x(s){\rm d}s {+} } \int_{{t_1}}^{{t_2}} {{{({t_2} {-} s)}^{\alpha {-} 1}}x(s){\rm d}s} } \right|\\ & \le& \frac{M}{{\Gamma (\alpha )}}\left( {\int_0^{{t_1}} {({{({t_1} - s)}^{\alpha - 1}} - {{({t_2} - s)}^{\alpha - 1}}){\rm d}s} + \int_{{t_1}}^{{t_2}} {{{({t_2} - s)}^{\alpha - 1}}{\rm d}s} } \right)\\ &&+ \frac{{\delta \lambda }}{{\Gamma (\alpha )}}\left[ {\int_0^{{t_1}} {({{({t_1} - s)}^{\alpha - 1}} - {{({t_2} - s)}^{\alpha - 1}}){\rm d}s + } \int_{{t_1}}^{{t_2}} {{{({t_2} - s)}^{\alpha - 1}}{\rm d}s} } \right]\\ & = & \frac{{M + \delta \lambda }}{{\Gamma (\alpha + 1)}}(t_1^\alpha - t_2^\alpha + 2{({t_2} - {t_1})^\alpha }) \to 0, \;\rm{当}\;{t_1} \to {t_2}. \end{eqnarray*} $

另一方面, 由于$ \varphi _{p(t)}^{ - 1}(I_{0 + }^\beta Nx(t) + ANx(t)) $与$ x(t) $在$ [0, 1] $上连续, 从而, $ {}^CD_{0{\rm{ + }}}^\alpha Tx $在$ [0, 1] $上一致连续. 进而, $ T $在$ \bar \Omega $上是等度连续的. 综上, 根据Arzelà-Ascoli定理知$ T $是全连续算子. 第二步, 证明算子$ T $在$ X $上存在不动点. 为此, 我们在$ X $上定义集合

$ S = \left\{ {x \in X|x = \mu Tx, \mu \in (0, 1)} \right\}. $

由Schaefer不动点定理知, 要证明算子$ T $在$ X $上存在不动点只需说明$ S $有界即可. 事实上, 对任意的$ x \in S $, 由(H) 可得

$ \begin{eqnarray*} |ANx(t)| &{ = }& \left| {\frac{1}{{2\Gamma (\beta )}}\int_0^1 {{{(1 - s)}^{\beta - 1}}f(s, x(s), {}^CD_{0{\rm{ + }}}^\alpha x(s))} {\rm d}s} \right|\\ &{\le}& \frac{1}{{2\Gamma (\beta )}}\int_0^1 {{{(1 - s)}^{\beta - 1}}|f(s, x(s), {}^CD_{0{\rm{ + }}}^\alpha x(s))} |{\rm d}s\\ &{\le} &\frac{1}{{2\Gamma (\beta )}}\int_0^1 {{{(1 - s)}^{\beta - 1}}(\zeta (s) + \phi (s)|x(s){|^{r - 1}} + \psi (s)|{}^CD_{0{\rm{ + }}}^\alpha x(s){|^{r - 1}})} {\rm d}s\\ &{\le} &\frac{1}{{2\Gamma (\beta + 1)}}(||\zeta |{|_\infty } + ||\phi |{|_\infty }||x||_\infty ^{r - 1} + ||\psi |{|_\infty }||{}^CD_{0{\rm{ + }}}^\alpha x||_\infty ^{r - 1})\\ &{\le} &\frac{1}{{2\Gamma (\beta + 1)}}(||\zeta |{|_\infty } + (||\phi |{|_\infty } + ||\psi |{|_\infty })||x||_X^{r - 1}). \end{eqnarray*} $

从而有

$ \begin{eqnarray*} |ANx(t) + I_{0{\rm{ + }}}^\beta Nx(t))| &{\le}& |ANx(t)| + |I_{0{\rm{ + }}}^\beta Nx(t)|\\ &{\le} &|ANx(t)| + \frac{1}{{\Gamma (\beta )}}\int_0^t {{{(t - s)}^{\beta - 1}}|} Nx(s)|{\rm d}s\\ &{\le}& \frac{3}{{2\Gamma (\beta + 1)}}(||\zeta |{|_\infty } + (||\phi |{|_\infty } + ||\psi |{|_\infty })||x||_X^{r - 1}). \end{eqnarray*} $

考虑到不等式$ {(x + y)^p} \le {2^p}({x^p} + {y^p}) $$ (x, y, p>0) $以及$ {x^k} \le x + 1(k \in [0, 1], x \ge 0) $, 可知对任意的$ t \in [0, 1] $有

$ \begin{eqnarray*} |BNx(t)|& { = }& \left| {\frac{1}{{2\Gamma (\alpha )}}\int_0^1 {{{(1 {-} s)}^{\alpha {-} 1}}} \varphi _{p(s)}^{ {-} 1}(ANx(s) {+} I_{0{\rm{ {+} }}}^\beta Nx(s)){\rm d}s {+} \frac{\lambda }{{2\Gamma (\alpha )}}\int_0^1 {{{(1 {-} s)}^{\alpha {-} 1}}x(s){\rm d}s} } \right|\\ & {\le}& \frac{1}{{2\Gamma (\alpha )}}\int_0^1 {{{(1 - s)}^{\alpha - 1}}} \varphi _{p(s)}^{ - 1}(|ANx(s) + I_{0{\rm{ + }}}^\beta Nx(s)|){\rm d}s + \frac{\lambda }{{2\Gamma (\alpha + 1)}}{\rm{||}}x{\rm{|}}{{\rm{|}}_\infty }\\ &{\le}& \frac{{{6^{{1 \mathord{\left/ {\vphantom {1 {({P_m} {-} 1)}}} \right. } {({P_m} {-} 1)}}}}\int_0^1 {{{(1 {-} s)}^{\alpha {-} 1}}(} {\rm{||}}\zeta {\rm{||}}_\infty ^{{1 \mathord{\left/ {\vphantom {1 {(p(s) {-} 1)}}} \right. } {(p(s) {-} 1)}}} {+} {{({\rm{||}}\phi {\rm{|}}{{\rm{|}}_\infty } {+} {\rm{||}}\psi {\rm{|}}{{\rm{|}}_\infty })}^{{1 \mathord{\left/ {\vphantom {1 {(p(s) {-} 1)}}} \right. } {(p(s) {-} 1)}}}}({\rm{||}}x{\rm{|}}{{\rm{|}}_X} {+} 1)){\rm d}s}}{{2\Gamma (\alpha ){{(2\Gamma (\beta {+} 1))}^{{1 \mathord{\left/ {\vphantom {1 {({P_M} {-} 1)}}} \right. } {({P_M} {-} 1)}}}}}}\\ & &{+} \frac{\lambda }{{2\Gamma (\alpha {+} 1)}}{\rm{||}}x{\rm{|}}{{\rm{|}}_X}. \end{eqnarray*} $

因此

(3.7) $ \begin{eqnarray} |x(t)| &{\le}& |I_{{0^ + }}^\alpha \varphi _{p(t)}^{ - 1}(I_{{0^ + }}^\beta Nx(t) + ANx(t))| + |BNx(t)| + \lambda |I_{{0^ + }}^\alpha x(t)|\\ & {\le}& Q\int_0^t {{{(t - s)}^{\alpha {-} 1}}} (||\zeta ||_\infty ^{{1 \mathord{\left/ {\vphantom {1 {(p(s) - 1)}}} \right. } {(p(s) - 1)}}} + {(||\phi |{|_\infty } + ||\psi |{|_\infty })^{{1 \mathord{\left/ {\vphantom {1 {(p(s) - 1)}}} \right. } {(p(s) - 1)}}}}(||x|{|_X} + 1)){\rm d}s\\ &&{+} \frac{Q}{2}\int_0^1 {{{(1 - s)}^{\alpha - 1}}} (||\zeta ||_\infty ^{{1 \mathord{\left/ {\vphantom {1 {(p(s) - 1)}}} \right. } {(p(s) - 1)}}} + {(||\phi |{|_\infty } + ||\psi |{|_\infty })^{{1 \mathord{\left/ {\vphantom {1 {(p(s) - 1)}}} \right. } {(p(s) - 1)}}}}(||x|{|_X} + 1)){\rm d}s\\ & &{+} \frac{{3\lambda }}{{2\Gamma (\alpha {+} 1)}}||x|{|_X}\\ & \le& \frac{{3Q}}{{2\alpha }}\left( {\hbar + \ell (||x|{|_X} + 1)} \right) + \frac{{3\lambda }}{{2\Gamma (\alpha {+} 1)}}||x|{|_X}, \end{eqnarray} $

这里$ \hbar : = \max \left\{ {||\zeta ||_\infty ^{1/({p_m} - 1)}, \;||\zeta ||_\infty ^{1/({p_M} - 1)}} \right\} $. 又因为

(3.8) $ \begin{eqnarray} {\rm{|}}{}^CD_{0{\rm{ {+} }}}^\alpha x(t){\rm{|}} &\le& \left| {\varphi _{p(t)}^{ - 1}(ANx(t) {+} I_{{0^ + }}^\beta Nx(t))} \right| {+} \left| {\lambda x(t)} \right|\\ & {\le}& \left[ {\frac{3}{{2\Gamma (\beta {+} 1)}}{(||\zeta |{|_\infty } {+} (||\phi |{|_\infty } {+} ||\psi |{|_\infty })||x||_X^{r - 1})}} \right]^{1/(p(t) {-} 1)} {+} \lambda ||x|{|_X}\\ &{\le}& Q\Gamma (\alpha)(||\zeta ||_\infty ^{1/(p(t) {-} 1)} {+} {(||\phi |{|_\infty } {+} ||\psi |{|_\infty })^{1/(p(t) {-} 1)}}(||x|{|_X} {+} 1)){+} \lambda ||x|{|_X}\\ &{\le}& Q\Gamma (\alpha) \left( {\hbar {+} \ell (||x|{|_X} {+} 1)} \right) {+} \lambda ||x|{|_X}. \end{eqnarray} $

结合(3.7) 式与(3.8) 式可得

(3.9) $ \begin{equation} ||x|{|_X} \le \frac{{3Q}}{{2\alpha }}\left( {\hbar + \ell (||x|{|_X} + 1)} \right) + Q\Gamma (\alpha ) \left( {\hbar + \ell (||x|{|_X} + 1)} \right) + \frac{{3\lambda }}{{2\Gamma (\alpha + 1)}}||x|{|_X} + \lambda ||x|{|_X}. \end{equation} $

由条件(3.6) 以及(3.9) 式可推出存在常数$ m>0 $使得$ ||x|{|_X} \le m $. 根据Schaefer不动点定理, 可知$ T $在$ S $中至少存在一个不动点, 即边值问题(1.8) 至少存在一个解.

例 4.1    考虑边值问题

(4.1) $ \begin{equation} \left\{\begin{array}{ll} {}^CD_{0{\rm{ {+} }}}^{{1 \mathord{\left/ {\vphantom {1 2}} \right. } 2}}{\varphi _{( - {t^2} {+} t {+} 2)}}(({}^CD_{0{\rm{ {+} }}}^{{1 \mathord{\left/ {\vphantom {1 2}} \right. } 2}}{\rm{ {+} (}}{1 \mathord{\left/ {\vphantom {1 {10}}} \right. } {10}}{\rm{)}})x(t)) { = } \sin t {+} \frac{3}{{200}}x {+} \frac{1}{{100}}{}^CD_{0{\rm{ {+} }}}^{{1 \mathord{\left/ {\vphantom {1 2}} \right. } 2}}x, t \in (0, 1), \\ x(0) { = } - x(1), \; {}^CD_{0{\rm{ {+} }}}^{{1 \mathord{\left/ {\vphantom {1 2}} \right. } 2}}x(0) { = } - {}^CD_{0{\rm{ {+} }}}^{{1 \mathord{\left/ {\vphantom {1 2}} \right. } 2}}x(1), \end{array}\right. \end{equation} $

对应边值问题(1.8), 这里

$ f(t, x(t), {}^CD_{0 + }^\alpha x(t)) = \sin t + \frac{3}{{200}}x(t) + \frac{1}{{100}}{}^CD_{0{\rm{ + }}}^{1/2}x(t), $

$ p(t) = - {t^2} + t + 2, \;\alpha = \beta = \frac{1}{2}, \;\lambda = \frac{1}{{10}}. $

取$ r = 2, \;\zeta (t) = 1, \;\phi (t) = \frac{3}{{200}}, \;\psi (t) = \frac{1}{{100}}. $易验证

$ {P_m} = 2, \;{P_M} = \frac{9}{4}, \;|f(t, x(t), {}^CD_{0 + }^\alpha x(t))| \le 1 + \frac{3}{{200}}|x(t)| + \frac{1}{{100}}|{}^CD_{0 + }^{1/2}x(t)|, $

$ {\left( {\frac{1}{{40}}} \right)^{4/5}}\left( {\frac{{18}}{{{{(2\Gamma (3/2))}^{9/5}}}}} + {\frac{{6}}{{{{(2\Gamma (3/2))}^{4/5}}}}}\right)+ \frac{3}{{20\Gamma (3/2)}} + \frac{1}{{10}} \approx {\rm{0}}{\rm{.8036}}< 1. $

由定理3.1可知边值问题(4.1) 至少有一个解.