一类带p-Laplacian算子的分数阶微分方程边值问题正解的存在性

Existence of Positive Solutions for Boundary Value Problem of Fractional Differential Equations with p-Laplacian Operator

摘要: 本文研究了一类带有p-Laplacian算子的分数阶微分方程边值问题的正解的存在性，利用Leray-Schauder非线性抉择，得出边值问题至少存在一个正解的充分条件，并给出了一个具体的例子。

Abstract: This paper is concerned with the existence of positive solution for a class of boundary value problems of fractional differential equations with p-Laplacian operator. By using  Leray-Schauder nonlinear choice, some sufficient conditions for the existence of at least one positive solution are obtained. In addition, an example is given to illustrate theoretical results.

1. 引言

分数阶微积分是在整数阶微积分的理论基础上发展而来，近年来分数阶微分方程在物理学、工程学、机械、医学、生物学等许多领域得到了广泛应用 [1] [2] [3] [4]。为了解决越来越多复杂的现象和问题，学者和专家开始研究带p-Laplacian算子的分数阶微分方程。文 [5] 研究如下带p-Laplacian算子的分数阶微分方程

$\{\begin{array}{l}{D}_{0^{+}}^{\beta }\left({\phi }_p\left(D_{0^{+}}^{\alpha }u\right)\right)\left(t\right)+f\left(t,u\left(t\right)\right)=0,0<t<1,\\ u\left(0\right)=0,u\left(1\right)+\sigma D_{0^{+}}^{\gamma }u\left(1\right)=0,D_{0^{+}}^{\alpha }u\left(0\right)=0,\end{array}$

这里 $D_{0^{+}}^{\alpha },D_{0^{+}}^{\beta }$ 和 $D_{0^{+}}^{\gamma }$ 是标准的Riemann-Liouville型分数阶导数， $1<\alpha \le 2$， $0<\beta \le 1$， $0<\gamma \le 1$， $\alpha -\gamma -1\ge 0$，常数 $\sigma$ 是一个正数， ${\phi }_p\left(s\right)={\left|s\right|}^{p-2}s,p>1$。利用锥上的不动点定理，获得了正解的一些存在性和多重性结果。

文 [6] 研究如下带有p-Laplacian算子的Riemann-Liouville型分数阶微分方程边值问题

$\{\begin{array}{l}{D}^{\alpha }\left({\phi }_p\left(D^{\beta }u\left(t\right)\right)\right)+f\left(t,u\left(t\right)\right)=0,0<t<1,\\ u\left(0\right)=0,u\left(1\right)=\lambda D^{\gamma }u\left(\xi \right),D^{\beta }u\left(0\right)=0,\end{array}$

应用凸锥上的不动点理获得了该问题正解的存在性结果。这里 $\alpha ,\beta ,\gamma \in ℝ$， $0<\alpha <1$， $1<\beta \le 2$， $\gamma =\frac{\beta -1}{2}$， $0<\xi \le \frac{1}{2}$， $\lambda \in \left[0,+\infty \right)$， $\lambda \Gamma \left(\beta \right){\xi }^{\left(\beta -1\right)/2}<\Gamma \left(\frac{\beta +1}{2}\right)$， ${\phi }_p\left(s\right)={\left|s\right|}^{p-2}s,p>1$。

基于上述文献中的研究，本文主要利用Leray-Schauder非线性抉择讨论如下带有p-Laplacian算子的Riemann-Liouville型分数阶微分方程边值问题

$\{\begin{array}{l}{\left({\varphi }_p\left(D_{0^{+}}^{\alpha }u\left(t\right)\right)\right)}^{\prime }+f\left(t,u\left(t\right),D_{0^{+}}^{\alpha }u\left(t\right)\right)=0,t\in \left[0,1\right],\\ u\left(0\right)=u\left(1\right)=D_{0^{+}}^{\alpha }u\left(0\right)=0,\end{array}$ (1)

解的存在性，其中 $1<\alpha \le 2$， ${\varphi }_p\left(s\right)={\left|s\right|}^{p-2}s$。 $q>1$， ${\varphi }_p^{-1}={\varphi }_q$， $\frac{1}{p}+\frac{1}{q}=1$。

2. 预备知识

定义1 [7] 函数 $y:\left(0,+\infty \right)\to R$ 的 $\alpha >0$ 阶Riemann-Liouville分数阶积分为

$I_{0^{+}}^{\alpha }y\left(t\right)=\frac{1}{\Gamma \left(\alpha \right)}{\displaystyle {\int }_0^t{\left(t-s\right)}^{\alpha -1}y\left(s\right)\text{d}s},$

等式的右端在 $\left(0,+\infty \right)$ 有定义，其中 $\Gamma (\cdot )$ 为Gamma函数。

定义2 [7] 连续函数 $y:\left(0,+\infty \right)\to R$ 的 $\alpha >0$ 阶Riemann-Liouville分数阶导数为

$D_{0^{+}}^{\alpha }y\left(t\right)=\frac{1}{\Gamma \left(n-\alpha \right)}{\left(\frac{\text{d}}{\text{d}t}\right)}^n{\displaystyle {\int }_0^t\frac{y\left(s\right)}{{\left(t-s\right)}^{\alpha -n+1}}\text{d}s},$

等式的右端在 $\left(0,+\infty \right)$ 有定义，其中 $n=\min \left\{m\in Z:m\ge \alpha \right\}$， $\Gamma (\cdot )$ 为Gamma函数。

引理1 [7] $\alpha >0$， $u\in C\left(0,1\right)\cap L\left(0,1\right)$， $D_{0^{+}}^{\alpha }u\in C\left(0,1\right)\cap L\left(0,1\right)$，则分数阶微分方程 $D_{0^{+}}^{\alpha }u\left(t\right)=0$ 有唯一解

$u\left(t\right)=c_1{t}^{\alpha -1}+c_2{t}^{\alpha -2}+c_n{t}^{\alpha -n},$

其中 $n=\min \left\{m\in Z:m\ge \alpha \right\}$， $c_i\in R,i=1,2,\cdots ,n$。

引理2 [7] $\alpha >0$， $u\in C\left(0,1\right)\cap L\left(0,1\right)$， $D_{0^{+}}^{\alpha }u\in C\left(0,1\right)\cap L\left(0,1\right)$，则

$I_{0^{+}}^{\alpha }{D}_{0^{+}}^{\alpha }u\left(t\right)=u\left(t\right)+c_1{t}^{\alpha -1}+c_2{t}^{\alpha -2}+c_n{t}^{\alpha -n},$

其中 $n=\min \left\{m\in Z:m\ge \alpha \right\}$， $c_i\in R,i=1,2,\cdots ,n$。

引理3 若 $y\left(t\right)\in C\left[0,1\right]$，且 $1<\alpha \le 2$，则分数阶微分方程

$\{\begin{array}{l}{\left({\varphi }_p\left(D_{0^{+}}^{\alpha }u\left(t\right)\right)\right)}^{\prime }+y\left(t\right)=0,0<t<1,\\ u\left(0\right)=u\left(1\right)=D_{0^{+}}^{\alpha }u\left(0\right)=0,\end{array}$

有唯一解

$u\left(t\right)={\displaystyle {\int }_0^{1}G\left(t,s\right){\varphi }_q}\left({\displaystyle {\int }_0^{s}y\left(\tau \right)\text{d}\tau }\right)\text{d}s,$

其中

$G\left(t,s\right)=\frac{1}{\Gamma \left(\alpha \right)}\{\begin{array}{l}{t}^{\alpha -1}{\left(1-s\right)}^{\alpha -1}-{\left(t-s\right)}^{\alpha -1},0\le s\le t\le 1,\\ t^{\alpha -1}{\left(1-s\right)}^{\alpha -1},0\le t\le s\le 1.\end{array}$

证明 由 ${\left({\varphi }_p\left(D_{0^{+}}^{\alpha }u\left(t\right)\right)\right)}^{\prime }+y\left(t\right)=0$ 可得

${\varphi }_p\left(D_{0^{+}}^{\alpha }u\left(t\right)\right)=-{\displaystyle {\int }_0^{t}y\left(s\right)\text{d}s}+c_0,$

于是

$D_{0^{+}}^{\alpha }u\left(t\right)=-{\varphi }_q\left({\displaystyle {\int }_0^{t}y\left(s\right)\text{d}s}+c_0\right).$

根据引理2可得

$u\left(t\right)=-\frac{1}{\Gamma \left(\alpha \right)}{\displaystyle {\int }_0^t{\left(t-s\right)}^{\alpha -1}{\varphi }_q\left({\displaystyle {\int }_0^{s}y\left(\tau \right)\text{d}\tau }+c_0\right)\text{d}s}+c_1{t}^{\alpha -1}+c_2{t}^{\alpha -2}.$

由 $D_{0^{+}}^{\alpha }u\left(0\right)=0$ 得 $c_0=0$，由 $u\left(0\right)=0$，得 $c_2=0$，由 $u\left(1\right)=0$，得

$c_1=\frac{1}{\Gamma \left(\alpha \right)}{\displaystyle {\int }_0^1{\left(1-s\right)}^{\alpha -1}{\varphi }_q\left({\displaystyle {\int }_0^{s}y\left(\tau \right)\text{d}\tau }\right)\text{d}s}.$

所以

$\begin{array}{c}u\left(t\right)=\frac{1}{\Gamma \left(\alpha \right)}{\displaystyle {\int }_0^1{t}^{\alpha -1}{\left(1-s\right)}^{\alpha -1}{\varphi }_q}\left({\displaystyle {\int }_0^{s}y\left(\tau \right)\text{d}\tau }\right)\text{d}s\\ -\frac{1}{\Gamma \left(\alpha \right)}{\displaystyle {\int }_0^t{\left(t-s\right)}^{\alpha -1}{\varphi }_q\left({\displaystyle {\int }_0^{s}y\left(\tau \right)\text{d}\tau }\right)\text{d}s}\\ =\frac{1}{\Gamma \left(\alpha \right)}{\displaystyle {\int }_0^t\left[t^{\alpha -1}{\left(1-s\right)}^{\alpha -1}-{\left(t-s\right)}^{\alpha -1}\right]{\varphi }_q}\left({\displaystyle {\int }_0^{s}y\left(\tau \right)\text{d}\tau }\right)\text{d}s\\ +\frac{1}{\Gamma \left(\alpha \right)}{\displaystyle {\int }_t^1{t}^{\alpha -1}{\left(1-s\right)}^{\alpha -1}{\varphi }_q}\left({\displaystyle {\int }_0^{s}y\left(\tau \right)\text{d}\tau }\right)\text{d}s\\ ={\displaystyle {\int }_0^{1}G\left(t,s\right){\varphi }_q}\left({\displaystyle {\int }_0^{s}y\left(\tau \right)\text{d}\tau }\right)\text{d}s.\end{array}$

证毕。

注 由引理3的证明容易看到 $D_{0^{+}}^{\alpha }u\left(t\right)=-{\varphi }_q\left({\displaystyle {\int }_0^{t}y\left(s\right)\text{d}s}\right),t\in \left[0,1\right]$。

引理4 [8] $G\left(t,s\right)$ 有下面的性质

1) $G\left(t,s\right)\ge 0$，对 $\forall s,t\in \left(0,1\right)$ ；

2)存在正函数 $r\in C\left[0,1\right]$ 使得

$\underset{1/4\le t\le 3/4}{\min }G\left(t,s\right)\ge r\left(s\right)\underset{t\in \left[0,1\right]}{\max }G\left(t,s\right)=r\left(s\right)G\left(s,s\right),0<s<1;$

3) $\underset{t\in \left[0,1\right]}{\max }{\displaystyle {\int }_0^{1}G\left(t,s\right)\text{d}s}=\frac{1}{2^{2\left(\alpha -1\right)}\Gamma \left(\alpha \right)}$。

引理5 [8] (Leray-Schauder非线性抉择)假设 $\Omega$ 是线性赋范空间X中包含原点的开集， $F:\bar{\Omega }\to X$ 全连续，并且满足边界条件，即当 $x\in \partial \Omega$， $\text{0}<\lambda <\text{1}$ 时 $F\ne \lambda x$，则F在 $\bar{\Omega }$ 上至少有一个不动点。

3. 主要结果

下节将用到如下假设

(H1) $f:\left[0,1\right]\times \left[0,+\infty \right)\times R\to \left[0,+\infty \right)$ 为连续函数，假设存在非负连续函数 $j\left(t\right),l\left(t\right),w\left(t\right)$ 使得

$\left|f\left(t,u,v\right)\right|\le j\left(t\right){\varphi }_p\left(\left|u\right|\right)+l\left(t\right){\varphi }_p\left(\left|v\right|\right)+w\left(t\right),t\in \left[0,1\right].$

设 $X=\left\{u\left(t\right)\in C\left[0,1\right]|D_{0^{+}}^{\alpha }u\left(t\right)\in C\left[0,1\right]\right\}$，定义范数 $\Vert u\Vert =\underset{t\in \left[0,1\right]}{\max }\left|u\left(t\right)\right|+\underset{t\in \left[0,1\right]}{\max }\left|D_{0^{+}}^{\alpha }u\left(t\right)\right|$，容易证明X是Banach空间。定义算子

$Tu\left(t\right)={\displaystyle {\int }_0^{1}G\left(t,s\right){\varphi }_q\left({\displaystyle {\int }_0^{s}f\left(\tau ,u\left(\tau \right),D_{0^{+}}^{\alpha }u\left(\tau \right)\right)\text{d}\tau }\right)\text{d}s},t\in \left[0,1\right].$

定理1假设(H1)成立。若(H1)中的函数 $j,l,w$ 满足存在常数 $\rho >0$ 使得

$\frac{2^{2\left(\alpha -1\right)}\Gamma \left(\alpha \right)\rho }{{\left[{\rho }^{p-1}\left({\displaystyle {\int }_0^{1}j\left(\tau \right)\text{d}\tau }+{\displaystyle {\int }_0^{1}l\left(\tau \right)\text{d}\tau }\right)+{\displaystyle {\int }_0^{1}w\left(\tau \right)\text{d}\tau }\right]}^{q-1}}\le 1,$ (2.1)

则边值问题(1)存在解 $u=u\left(t\right)$ 且满足 $0<\Vert u\Vert <\rho$。

证明 首先证明 $T:X\to X$ 为全连续算子。由函数f的连续性容易证明T是连续算子。

以下证明T为紧的。设 $\Omega$ 是X中的有界子集，于是存在正数 $M>0$ 使得 $u\in \Omega$， $\Vert u\Vert \le M$。从而对于任意的 $u\in \Omega$ 有

$\begin{array}{c}\left|Tu\left(t\right)\right|=\left|{\displaystyle {\int }_0^{1}G\left(t,s\right){\varphi }_q\left({\displaystyle {\int }_0^{s}f\left(\tau ,u\left(\tau \right),D_{0^{+}}^{\alpha }u\left(\tau \right)\right)\text{d}\tau }\right)\text{d}s}\right|\\ \le {\displaystyle {\int }_0^{1}G\left(t,s\right){\varphi }_q\left(\left|{\displaystyle {\int }_0^{s}f\left(\tau ,u\left(\tau \right),D_{0^{+}}^{\alpha }u\left(\tau \right)\right)\text{d}\tau }\right|\right)\text{d}s}\\ \le {\displaystyle {\int }_0^{1}G\left(t,s\right){\varphi }_q}\left({\displaystyle {\int }_0^s\left|f\left(\tau ,u\left(\tau \right),D_{0^{+}}^{\alpha }u\left(\tau \right)\right)\right|\text{d}\tau }\right)\text{d}s\\ \le {\displaystyle {\int }_0^{1}G\left(t,s\right){\varphi }_q}\left({\displaystyle {\int }_0^1\left(j\left(\tau \right){\varphi }_p\left(\left|u\left(\tau \right)\right|\right)+l\left(\tau \right){\varphi }_p\left(\left|D_{0^{+}}^{\alpha }u\left(\tau \right)\right|\right)+w\left(\tau \right)\right)\text{d}\tau }\right)\text{d}s\\ \le {\displaystyle {\int }_0^{1}G\left(t,s\right){\varphi }_q}\left({\displaystyle {\int }_0^1\left(j\left(\tau \right){\varphi }_p\left(\Vert u\Vert \right)+l\left(\tau \right){\varphi }_p\left(\Vert u\Vert \right)+w\left(\tau \right)\right)\text{d}\tau }\right)\text{d}s\\ \le \frac{1}{2^{2\left(\alpha -1\right)}\Gamma \left(\alpha \right)}{\left[M^{p-1}\left({\displaystyle {\int }_0^{1}j\left(\tau \right)\text{d}\tau }+{\displaystyle {\int }_0^{1}l\left(\tau \right)\text{d}\tau }\right)+{\displaystyle {\int }_0^{1}w\left(\tau \right)\text{d}\tau }\right]}^{q-1},t\in \left[0,1\right],\end{array}$

$\begin{array}{c}\left|D_{0^{+}}^{\alpha }Tu(t)\right|=\left|-{\varphi }_q\left({\displaystyle {\int }_0^{t}f\left(s,u\left(s\right),D_{0^{+}}^{\alpha }u\left(s\right)\right)\text{d}s}\right)\right|\\ \le {\varphi }_q\left({\displaystyle {\int }_0^t\left|f\left(s,u\left(s\right),D_{0^{+}}^{\alpha }u\left(s\right)\right)\right|\text{d}s}\right)\\ \le {\varphi }_q\left({\displaystyle {\int }_0^1\left(j\left(s\right){\varphi }_p\left(\left|u\left(s\right)\right|\right)+l\left(s\right){\varphi }_p\left(\left|D_{0^{+}}^{\alpha }u\left(s\right)\right|\right)+w\left(s\right)\right)\text{d}s}\right)\\ \le \left[{\varphi }_q\left({\varphi }_p\left(\Vert u\Vert \right){\displaystyle {\int }_0^{1}j\left(s\right)\text{d}s}+{\varphi }_p\left(\Vert u\Vert \right){\displaystyle {\int }_0^{1}l\left(s\right)\text{d}s}+{\displaystyle {\int }_0^{1}w\left(s\right)\text{d}s}\right)\right]\\ \le {\left[M^{p-1}\left({\displaystyle {\int }_0^{1}j\left(s\right)\text{d}s}+{\displaystyle {\int }_0^{1}l\left(s\right)\text{d}s}\right)+{\displaystyle {\int }_0^{1}w\left(s\right)\text{d}s}\right]}^{q-1},t\in \left[0,1\right].\end{array}$

因此 $T\left(\Omega \right)$ 和 $\left\{D_{0^{+}}^{\alpha }Tu|u\in \Omega \right\}$ 皆为一致有界的子集合。

再证 $T\left(\Omega \right)$ 和 $\left\{D_{0^{+}}^{\alpha }Tu|u\in \Omega \right\}$ 皆是等度连续的。对任意的 $0\le t_1<t_2\le 1$， $u\in \Omega$ 有

$\begin{array}{c}\left|Tu\left(t_2\right)-Tu\left(t_1\right)\right|=|{\displaystyle {\int }_0^{1}G\left(t_2,s\right){\varphi }_q\left({\displaystyle {\int }_0^{s}f\left(\tau ,u\left(\tau \right),D_{0^{+}}^{\alpha }u\left(\tau \right)\right)\text{d}\tau }\right)\text{d}s}\\ -{\displaystyle {\int }_0^{1}G\left(t_1,s\right){\varphi }_q\left({\displaystyle {\int }_0^{s}f\left(\tau ,u\left(\tau \right),D_{0^{+}}^{\alpha }u\left(\tau \right)\right)\text{d}\tau }\right)\text{d}s}|\\ =\left|{\displaystyle {\int }_0^1\left[G\left(t_2,s\right)-G\left(t_1,s\right)\right]{\varphi }_q}\left({\displaystyle {\int }_0^{s}f\left(\tau ,u\left(\tau \right),D_{0^{+}}^{\alpha }u\left(\tau \right)\right)\text{d}\tau }\right)\text{d}s\right|\\ \le {\left[M^{p-1}\left({\displaystyle {\int }_0^{1}j\left(\tau \right)\text{d}\tau }+{\displaystyle {\int }_0^{1}l}\left(\tau \right)\text{d}\tau \right)+{\displaystyle {\int }_0^{1}w}\left(\tau \right)\text{d}\tau \right]}^{q-1}{\displaystyle {\int }_0^1\left|G\left(t_2,s\right)-G\left(t_1,s\right)\right|\text{d}s}.\end{array}$

注意到 $G\left(t,s\right)$ 在 $\left[0,1\right]\times \left[0,1\right]$ 上的一致连续性可知， $T\left(\Omega \right)$ 是等度连续的。此外

$\begin{array}{l}\left|D_{0^{+}}^{\alpha }Tu\left(t_2\right)-D_{0^{+}}^{\alpha }Tu\left(t_1\right)\right|\\ =|-[{\varphi }_q\left({\displaystyle {\int }_0^{t_2}f\left(s,u\left(s\right),D_{0^{+}}^{\alpha }u\left(s\right)\right)\text{d}s}\right)-{\varphi }_q\left({\displaystyle {\int }_0^{t_1}f\left(s,u\left(s\right),D_{0^{+}}^{\alpha }u\left(s\right)\right)\text{d}s}\right)]|\\ =\left|{\left({\displaystyle {\int }_0^{t_2}f\left(s,u\left(s\right),D_{0^{+}}^{\alpha }u\left(s\right)\right)\text{d}s}\right)}^{q-1}-{\left({\displaystyle {\int }_0^{t_1}f\left(s,u\left(s\right),D_{0^{+}}^{\alpha }u\left(s\right)\right)\text{d}s}\right)}^{q-1}\right|,\end{array}$

当 $0<q-1<1$ 时，根据不等式 $b^m+c^m\ge {\left(b+c\right)}^m,b\ge 0,c\ge 0,0<m<1$ 可得

$\begin{array}{c}0\le {\left({\displaystyle {\int }_0^{t_2}f\left(s,u\left(s\right),D_{0^{+}}^{\alpha }u\left(s\right)\right)\text{d}s}\right)}^{q-1}-{\left({\displaystyle {\int }_0^{t_1}f\left(s,u\left(s\right),D_{0^{+}}^{\alpha }u\left(s\right)\right)\text{d}s}\right)}^{q-1}\\ \le {\left({\displaystyle {\int }_{t_1}^{t_2}f\left(s,u\left(s\right),D_{0^{+}}^{\alpha }u\left(s\right)\right)\text{d}s}\right)}^{q-1}\\ \le {\left({\displaystyle {\int }_{t_1}^{t_2}\left(j\left(s\right){\varphi }_p\left(\left|u\left(s\right)\right|\right)+l\left(s\right){\varphi }_p\left(\left|D_{0^{+}}^{\alpha }u\left(s\right)\right|\right)+w\left(s\right)\right)\text{d}s}\right)}^{q-1}\\ \le {\left({\varphi }_p\left(\Vert u\Vert \right){\displaystyle {\int }_{t_1}^{t_2}j\left(s\right)\text{d}s}+{\varphi }_p\left(\Vert u\Vert \right){\displaystyle {\int }_{t_1}^{t_2}l\left(s\right)\text{d}s}+{\displaystyle {\int }_{t_1}^{t_2}w\left(s\right)\text{d}s}\right)}^{q-1}\\ \le {\left(M^{p-1}\left({\displaystyle {\int }_{t_1}^{t_2}j\left(s\right)\text{d}s}+{\displaystyle {\int }_{t_1}^{t_2}l\left(s\right)\text{d}s}\right)+{\displaystyle {\int }_{t_1}^{t_2}w\left(s\right)\text{d}s}\right)}^{q-1},\end{array}$

再根据积分第一中值定理，因 $j\left(s\right),l\left(s\right),w\left(s\right)$ 在 $\left[t_1,t_2\right]$ 上连续，则至少存在三点 $\mu ,\eta ,\kappa \in \left[t_1,t_2\right]$，使得

${\displaystyle {\int }_{t_1}^{t_2}j\left(s\right)\text{d}s}=j\left(\mu \right)\left(t_2-t_1\right),{\displaystyle {\int }_{t_1}^{t_2}l\left(s\right)\text{d}s}=l\left(\eta \right)\left(t_2-t_1\right),{\displaystyle {\int }_{t_1}^{t_2}w\left(s\right)\text{d}s}=w\left(\kappa \right)\left(t_2-t_1\right),$

于是

$\left|D_{0^{+}}^{\alpha }Tu\left(t_2\right)-D_{0^{+}}^{\alpha }Tu\left(t_1\right)\right|\le {\left[M^{p-1}\left(j\left(\mu \right)+l\left(\eta \right)\right)+w\left(\kappa \right)\right]}^{q-1}{\left(t_2-t_1\right)}^{q-1},$

当 $q-1\ge 1$ 时，由拉格朗日中值定理有

$\begin{array}{c}0\le {\left({\displaystyle {\int }_0^{t_2}f\left(s,u\left(s\right),D_{0^{+}}^{\alpha }u\left(s\right)\right)\text{d}s}\right)}^{q-1}-{\left({\displaystyle {\int }_0^{t_1}f\left(s,u\left(s\right),D_{0^{+}}^{\alpha }u\left(s\right)\right)\text{d}s}\right)}^{q-1}\\ =\left(q-1\right){\left({\displaystyle {\int }_0^{\xi }f\left(s,u\left(s\right),D_{0^{+}}^{\alpha }u\left(s\right)\right)\text{d}s}\right)}^{q-2}f\left(\xi ,u\left(\xi \right),D_{0^{+}}^{\alpha }u\left(\xi \right)\right)\left(t_2-t_1\right)\\ \le \left(q-1\right){\left[M^{p-1}\left({\displaystyle {\int }_0^{\xi }j\left(s\right)\text{d}s}+{\displaystyle {\int }_0^{\xi }l\left(s\right)\text{d}s}\right)+{\displaystyle {\int }_0^{\xi }w\left(s\right)\text{d}s}\right]}^{q-2}\left(M^{p-1}\left(j\left(\xi \right)+l\left(\xi \right)\right)+w\left(\xi \right)\right)\left(t_2-t_1\right),\end{array}$

其中 $\xi$ 是 $\left(t_1,t_2\right)$ 之间某个确定的值。故

$\begin{array}{l}\left|D_{0^{+}}^{\alpha }Tu\left(t_2\right)-D_{0^{+}}^{\alpha }Tu\left(t_1\right)\right|\\ \le \left(q-1\right){\left[M^{p-1}\left({\displaystyle {\int }_0^{\xi }j\left(s\right)\text{d}s}+{\displaystyle {\int }_0^{\xi }l\left(s\right)\text{d}s}\right)+{\displaystyle {\int }_0^{\xi }w\left(s\right)\text{d}s}\right]}^{q-2}\left(M^{p-1}\left(j\left(\xi \right)+l\left(\xi \right)\right)+w\left(\xi \right)\right)\left(t_2-t_1\right).\end{array}$

由此可得 $\left\{D_{0^{+}}^{\alpha }Tu|u\in \Omega \right\}$ 是等度连续的。

既然集合 $T\left(\Omega \right)$ 和 $\left\{D_{0^{+}}^{\alpha }Tu|u\in \Omega \right\}$ 都是一致有界且等度连续的，根据Arzela-Ascoli定理可知 $T\left(\Omega \right)$ 和 $\left\{D_{0^{+}}^{\alpha }Tu|u\in \Omega \right\}$ 皆为 $C\left[0,1\right]$ 中的相对紧集，进而可知T是全连续算子。

令 $U=\left\{u\in X|\Vert u\Vert \le \rho \right\}$，有 $U\subset X$，由上述证明可知 $T:\bar{U}\to X$ 是全连续的。我们断言当 $u\in \partial U$， $\lambda \in \left(\text{0,1}\right)$ 时 $u\ne \lambda Tu$。如若不然存在 $u_0\in \partial U$， ${\lambda }_{\text{0}}\in \left(\text{0,1}\right)$ 使 $u_{\text{0}}={\lambda }_{\text{0}}T{u}_{\text{0}}$。于是有

$\rho =\Vert u_0\Vert =\Vert {\lambda }_{0}T{u}_0\Vert \le \frac{1}{2^{2\left(\alpha -1\right)}\Gamma \left(\alpha \right)}{\left[{\rho }^{p-1}\left({\displaystyle {\int }_0^{1}j\left(\tau \right)\text{d}\tau }+{\displaystyle {\int }_0^{1}l\left(\tau \right)\text{d}\tau }\right)+{\displaystyle {\int }_0^{1}w\left(\tau \right)\text{d}\tau }\right]}^{q-1},$

进而有

$\frac{2^{2\left(\alpha -1\right)}\Gamma \left(\alpha \right)\rho }{{\left[{\rho }^{p-1}\left({\displaystyle {\int }_0^{1}j\left(\tau \right)\text{d}\tau }+{\displaystyle {\int }_0^{1}l\left(\tau \right)\text{d}\tau }\right)+{\displaystyle {\int }_0^{1}w\left(\tau \right)\text{d}\tau }\right]}^{q-1}}\le 1,$

此与(2.1)式矛盾。由引理5可知边值问题(1)存在解 $u\left(t\right)$ 使得 $0\le \Vert u\Vert \le \rho$。证毕。

4. 举例

考虑下列具有p-laplacian算子分数阶微分方程边值问题

$\{\begin{array}{l}{\left({\varphi }_3\left(D_{0^{+}}^{3/2}u\left(t\right)\right)\right)}^{\prime }+\left(\frac{\sqrt{t}}{2}+\frac{u^2\left(t\right)}{1+t^2}+{\text{e}}^{-t}{\left|D_{0^{+}}^{3/2}u\left(t\right)\right|}^2\right)=0,0\le t\le 1,\\ u\left(0\right)=u\left(1\right)=D_{0^{+}}^{3/2}u\left(0\right)=0,\end{array}$

其中 $p=3,\alpha =3/2$，

$f\left(t,u,v\right)=\left(\frac{\sqrt{t}}{2}+\frac{u^2}{1+t^2}+{\text{e}}^{-t}{\left|v\right|}^2\right),0\le t\le 1,$

选取 $j\left(t\right)={\text{e}}^t,l\left(t\right)={\text{e}}^{-t},w\left(t\right)=\sqrt{t}$，有

$\left|f\left(t,u,v\right)\right|\le {\text{e}}^t{\varphi }_3\left(\left|u\right|\right)+{\text{e}}^{-t}{\varphi }_3\left(\left|v\right|\right)+\sqrt{t},0\le t\le 1,$

令 $\rho =1$，有

$\frac{2^{2\left(\alpha -1\right)}\Gamma \left(\alpha \right)\rho }{{\left[{\rho }^{p-1}\left({\displaystyle {\int }_0^{1}j\left(\tau \right)\text{d}\tau }+{\displaystyle {\int }_0^{1}l\left(\tau \right)\text{d}\tau }\right)+{\displaystyle {\int }_0^{1}w\left(\tau \right)\text{d}\tau }\right]}^{q-1}}>1,$

定理1的条件皆满足，所以该边值问题至少存在一个解。

基金项目

国家自然科学基金(11361047)。

NOTES

^*通讯作者。