伊犁师范大学数学与统计学院，新疆 伊宁

收稿日期：2021年10月5日；录用日期：2021年10月26日；发布日期：2021年11月8日

摘要

研究一类具 -Laplacian算子的分数阶微分方程奇异反周期边值问题，运用Krasnosel’skiis不动点定理及Banach压缩映像原理，证明了解的存在性与唯一性。

关键词

反周期边值条件，不动点定理，P-Laplacian算子，奇异

Existence and Uniqueness on Solutions for Anti-Periodic Boundary Value Problems of Singular Fractional Differential Equations with P-Laplacian Operator

Tingting Zhang, Weimin Hu

School of Mathematics and Statistic, Yili Normal University, Yining Xinjiang

Received: Oct. 5^th, 2021; accepted: Oct. 26^th, 2021; published: Nov. 8^th, 2021

ABSTRACT

The existence and uniqueness of solutions about anti-periodic boundary value problems of singular fractional differential equations with P-Laplacian operator will be studied. I will apply Krasnosel’skiis fixed point theorem and Banach compression image principle, to carry out the existence and uniqueness upon the solution of the equation.

Keywords:Anti-Periodic Boundary Value Problems, Fixed Point Theorem, P-Laplacian Operator, Singular

Copyright © 2021 by author(s) and Hans Publishers Inc.

This work is licensed under the Creative Commons Attribution International License (CC BY 4.0).

http://creativecommons.org/licenses/by/4.0/

1. 引言

分数阶微分方程作为一个重要的数学工具，具有良好的研究意义，其中反周期边值问题因在不少学科中起到了关键性作用，故引起了数学工作者浓厚的研究兴趣，并取得了一定的研究成果 [1] [2] [3] [4]。P-Laplacian算子在非牛顿流体力学、多孔介质湍流及非线性粘弹性力学等多领域起到重要作用。因此也引得大量学者对其进行研究 [5] [6] [7] [8] [9]。但很少有文献研究具P-Laplacian算子的分数阶奇异微分方程反周期边值问题，故文章是对该类问题的补充与完善。

文献 [8] 中研究了一类具P-Laplacian算子的非线性分数阶微分方程反周期边值问题

$\{\begin{array}{l}{\left({\phi }_p\left({}^{c}D{}_{0+}^{\alpha }u\left(t\right)\right)\right)}^{\prime }=f\left(t,u\left(t\right)\right),t\in \left[0,T\right];\\ u\left(0\right)=-u\left(T\right),{}^{c}D{}_{0+}^{\beta }u\left(0\right)=-{}^{c}D{}_{0+}^{\beta }u\left(T\right),\end{array}$ (1)

解的存在性与唯一性，其中 $1<\alpha \le 2,0<\beta \le 1,T>0$，${\phi }_p\left(s\right)={\left|s\right|}^{p-2}\cdot s,{\phi }_p^{-1}={\phi }_q$，其中 $\frac{1}{p}+\frac{1}{q}=1$。 ${}^{c}D{}_{0+}^{\alpha }$ 是标准的Caputo导数。

文献 [10] 研究了分数阶微分方程奇异边值问题

$\{\begin{array}{l}{D}_{0+}^{\alpha }u\left(t\right)=f\left(t,u\left(t\right)\right),0<t<1;\\ u\left(0\right)=u\prime \left(0\right)=u″\left(0\right)=u″\left(1\right)=0,\end{array}$ (2)

$D_{0+}^{\alpha }$ 是标准的Riemann-Liouville型分数阶导数， $f:\left(0,1\right]\times \left[0,+\infty \right)\to \left[0,+\infty \right)$，且f在 $t=0$ 处有奇性。

受上述文献启发，本文将研究如下具有P-Laplacian算子的奇异分数阶微分方程反周期边值问题

$\{\begin{array}{l}{}^{c}D{}_{0+}^{\beta }\left({\phi }_p\left({}^{c}D{}_{0+}^{\alpha }u\left(t\right)\right)\right)=f\left(t,u\left(t\right)\right),t\in \left[0,T\right];\\ u\left(0\right)=-u\left(T\right),{}^{c}D{}_{0+}^{\gamma }u\left(0\right)=-{}^{c}D{}_{0+}^{\gamma }u\left(T\right),\end{array}$ (3)

解的存在性与唯一性，其中 ${}^{c}D{}_{0+}^{\alpha },{}^{c}D{}_{0+}^{\beta }$ 和 ${}^{c}D{}_{0+}^{\gamma }$ 是Caputo型分数阶导数， $1<\alpha \le 2,0<\beta \le 1,0<\gamma \le 1,T>0$，非线性项 $f:\left(0,T\right]\times \left[0,+\infty \right)\to \left[0,+\infty \right)$，并且 $\underset{t\to 0+}{\lim }f\left(t,\cdot \right)=+\infty$ (即f在 $t=0$ 时是奇异的)，并且满足存在实数 $\sigma >0$，使得 $t^{\sigma }f:\left[0,1\right]\times \left[0,+\infty \right)\to \left[0,+\infty \right)$。 ${\phi }_p\left(s\right)={\left|s\right|}^{p-2}\cdot s,{\phi }_p^{-1}={\phi }_q$ 且 $\frac{1}{p}+\frac{1}{q}=1$。

2. 预备知识

定义1.1 [1] 函数 $f:\left[0,+\infty \right)\to R$ 的 $\alpha >0$ 阶分数阶积分是指

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

其中右边是在 $\left[0,+\infty \right)$ 逐点定义的。

定义1.2 [1] 函数 $f:\left[0,+\infty \right)\to R$ 的 $\alpha >0$ 阶Caputo型分数阶微分是指

${}^{c}D{}_{0+}^{\alpha }f\left(t\right)=D_{0+}^{\alpha }\left(f\left(t\right)-{\displaystyle \underset{k=0}{\overset{n-1}{\sum }}\frac{f^{\left(k\right)}\left(0\right)}{k!}{t}^k}\right)$

其中右边是在 $\left[0,+\infty \right)$ 逐点定义的 $n=\left[\alpha \right]+1$，特别的，当 $\alpha =n$ 时， ${}^{c}D{}_{0+}^{n}f\left(t\right)=f^n\left(t\right)$。

引理1.1 [1] 设 $\alpha >0$，及 $u\in \left[0,1\right]$，则 ${}^{c}D{}_{0+}^{\alpha }{I}_{0+}^{\alpha }u\left(t\right)=u\left(t\right)$。

引理1.2 [1] 设 $\alpha >0$，${}^{c}D{}_{0+}^{\alpha }u\left(t\right)\in \left[0,1\right]\cap L\prime \left[0,1\right]$，则 ${}^{c}D{}_{0+}^{\alpha }{I}_{0+}^{\alpha }u\left(t\right)=u\left(t\right)+C_0+C_{1}t+\cdots +C_{n-1}{t}^{n-1}$，其中 $C_i\in R,i=1,2,\cdots ,n,n=\left[\alpha \right]+1$。

引理1.3 [3] (Banach压缩映像原理)设E是Banach空间X的非空闭子集，如果映射T是E到其自身内的映像，它在E内满足Lipschit条件，即对任意 $x,y\in E$

$\Vert Tx-Ty\Vert \le l\Vert x-y\Vert ,\left(0\le l\le 1\right)$

则必有唯一的 $x\in E$，使得 $Tx=x$，即T在E上有唯一不动点。

引理1.4 [8] (Krasnosel’skiis不动点定理)设 $\Omega$ 为Banach空间X上的有界闭凸非空子集，其中有算子 $\Phi ,\Psi$ 满足：1) $\Phi u+\Psi v\in \Omega$，其中 $u,v\in \Omega$ ；2) 算子 $\Phi$ 是全连续的；3) 算子 $\Psi$ 是压缩印象，则存在 $z\in \Omega$，使得 $z=\Phi z+\Psi z$。

引理1.5 [6] 如果 $p>2$，并且 $\left|x\right|,\left|y\right|\le M$，则对P-Laplacian算子 ${\phi }_p$，下列不等式成立 $\left|{\phi }_p\left(x\right)-{\phi }_p\left(y\right)\right|\le \left(p-1\right)M^{P-2}\left|x-y\right|$。

引理1.6 假设 $h:(0,T]\to R^{+}$ 是连续函数， $1<\alpha \le 2,0<\beta \le 1$，则边值问题

$\{\begin{array}{l}{}^{c}D{}_{0+}^{\beta }\left({\phi }_p\left({}^{c}D{}_{0+}^{\alpha }u\left(t\right)\right)\right)=f\left(t,u\left(t\right)\right),t\in \left[0,T\right];\\ u\left(0\right)=-u\left(T\right),{}^{c}D{}_{0+}^{\gamma }u\left(0\right)=-{}^{c}D{}_{0+}^{\gamma }u\left(T\right),\end{array}$ (4)

有唯一解

$\begin{array}{c}u\left(t\right)=\frac{1}{\Gamma \left(\alpha \right)}{\displaystyle {\int }_0^t{\left(t-s\right)}^{\alpha -1}{\phi }_q\left(\frac{1}{\Gamma \left(\beta \right)}{\displaystyle {\int }_0^s{\left(s-\tau \right)}^{\beta -1}h\left(\tau \right)\text{d}\tau }\right)\text{d}s}\\ -\frac{1}{2\Gamma \left(\alpha \right)}{\displaystyle {\int }_0^T{\left(T-s\right)}^{\alpha -1}}{\phi }_q\left(\frac{1}{\Gamma \left(\beta \right)}{\displaystyle {\int }_0^s{\left(s-\tau \right)}^{\beta -1}h\left(\tau \right)\text{d}\tau }\right)\text{d}s\\ +\frac{\left(T-2t\right)T^{\gamma -1}\Gamma \left(2-\gamma \right)}{2\Gamma \left(\alpha -\gamma \right)}{\displaystyle {\int }_0^T{\left(T-s\right)}^{\alpha -\gamma -1}{\phi }_q\left(\frac{1}{\Gamma \left(\beta \right)}{\displaystyle {\int }_0^s{\left(s-\tau \right)}^{\beta -1}h\left(\tau \right)\text{d}\tau }\right)\text{d}s}\end{array}$

证明：由引理1.2，对方程两端进行 $\beta$ 阶积分，有

${\phi }_p\left({}^{c}D{}_{0+}^{\alpha }u\left(t\right)\right)=\frac{1}{\Gamma \left(\beta \right)}{\displaystyle {\int }_0^t{\left(t-s\right)}^{\beta -1}h\left(s\right)\text{d}s}-a_0$

由Caputo分数阶微分性质可知： ${}^{c}D{}_{0+}^{\beta }u\left(0\right)=0$，所以 $a_0=0$。

对上式两边作用 ${\phi }_p$ 的逆算子 ${\phi }_q$，由P-Laplacian算子的性质，有 ${}^{c}D{}_{0+}^{\alpha }u\left(t\right)={\phi }_q\left(\frac{1}{\Gamma \left(\beta \right)}{\displaystyle {\int }_0^t{\left(t-s\right)}^{\beta -1}h\left(s\right)\text{d}s}\right)$。

由引理1.1及引理1.2，对上式两边进行 $\alpha$ 阶积分，有

$u\left(t\right)=\frac{1}{\Gamma \left(\alpha \right)}{\displaystyle {\int }_0^t{\left(t-s\right)}^{\alpha -1}{\phi }_q\left(\frac{1}{\Gamma \left(\beta \right)}{\displaystyle {\int }_0^s{\left(s-\tau \right)}^{\beta -1}h\left(\tau \right)\text{d}\tau }\right)\text{d}s}-C_0-C_{1}t$

再由Caputo导数的性质，即 ${}^{c}D{}_{0+}^{\gamma }{t}^r=\frac{\Gamma \left(r+1\right)}{\Gamma \left(r+1-\gamma \right)}{t}^{r-\gamma }$，有 ${}^{c}D{}_{0+}^{\gamma }C=0$

${}^{c}D{}_{0+}^{\gamma }u\left(t\right)=\frac{1}{\Gamma \left(\alpha -\gamma \right)}{\displaystyle {\int }_0^t{\left(t-s\right)}^{\alpha -\gamma -1}{\phi }_q\left(\frac{1}{\Gamma \left(\beta \right)}{\displaystyle {\int }_0^s{\left(s-\tau \right)}^{\beta -1}h\left(\tau \right)\text{d}\tau }\right)\text{d}s}-C_1\frac{1}{\Gamma \left(2-\gamma \right)}{t}^{1-r}$

最后由边值条件 $u\left(0\right)=-u\left(T\right),{}^{c}D{}_{0+}^{\gamma }u\left(0\right)=-{}^{c}D{}_{0+}^{\gamma }u\left(T\right)$，有

$\begin{array}{l}{C}_0=\frac{1}{2\Gamma \left(\alpha \right)}{\displaystyle {\int }_0^T{\left(T-s\right)}^{\alpha -1}{\phi }_q\left(\frac{1}{\Gamma \left(\beta \right)}{\displaystyle {\int }_0^s{\left(s-\tau \right)}^{\beta -1}h\left(\tau \right)\text{d}\tau }\right)\text{d}s}\\ -\frac{T^{\gamma }\Gamma \left(2-\gamma \right)}{2\Gamma \left(\alpha -\gamma \right)}{\displaystyle {\int }_0^T{\left(T-s\right)}^{\alpha -\gamma -1}{\phi }_q\left(\frac{1}{\Gamma \left(\beta \right)}{\displaystyle {\int }_0^s{\left(s-\tau \right)}^{\beta -1}h\left(\tau \right)\text{d}\tau }\right)\text{d}s}\end{array}$

$C_1=\frac{T^{\gamma }\Gamma \left(2-\gamma \right)}{\Gamma \left(\alpha -\gamma \right)}{\displaystyle {\int }_0^T{\left(T-s\right)}^{\alpha -\gamma -1}{\phi }_q\left(\frac{1}{\Gamma \left(\beta \right)}{\displaystyle {\int }_0^s{\left(s-\tau \right)}^{\beta -1}h\left(\tau \right)\text{d}\tau }\right)\text{d}s}$

故有

$\begin{array}{c}u\left(t\right)=\frac{1}{\Gamma \left(\alpha \right)}{\displaystyle {\int }_0^t{\left(t-s\right)}^{\alpha -1}{\phi }_q\left(\frac{1}{\Gamma \left(\beta \right)}{\displaystyle {\int }_0^s{\left(s-\tau \right)}^{\beta -1}h\left(\tau \right)\text{d}\tau }\right)\text{d}s}\\ -\frac{1}{2\Gamma \left(\alpha \right)}{\displaystyle {\int }_0^T{\left(T-s\right)}^{\alpha -1}{\phi }_q\left(\frac{1}{\Gamma \left(\beta \right)}{\displaystyle {\int }_0^s{\left(s-\tau \right)}^{\beta -1}h\left(\tau \right)\text{d}\tau }\right)\text{d}s}\\ +\frac{T^{\gamma }\Gamma \left(2-\gamma \right)}{2\Gamma \left(\alpha -\gamma \right)}{\displaystyle {\int }_0^T{\left(T-s\right)}^{\alpha -\gamma -1}{\phi }_q\left(\frac{1}{\Gamma \left(\beta \right)}{\displaystyle {\int }_0^s{\left(s-\tau \right)}^{\beta -1}h\left(\tau \right)\text{d}\tau }\right)\text{d}s}\\ -\frac{t{T}^{\gamma -1}\Gamma \left(2-\gamma \right)}{\Gamma \left(\alpha -\gamma \right)}{\displaystyle {\int }_0^T{\left(T-s\right)}^{\alpha -\gamma -1}{\phi }_q\left(\frac{1}{\Gamma \left(\beta \right)}{\displaystyle {\int }_0^s{\left(s-\tau \right)}^{\beta -1}h\left(\tau \right)\text{d}\tau }\right)\text{d}s}\end{array}$

$\begin{array}{l}=\frac{1}{\Gamma \left(\alpha \right)}{\displaystyle {\int }_0^t{\left(t-s\right)}^{\alpha -1}{\phi }_q\left(\frac{1}{\Gamma \left(\beta \right)}{\displaystyle {\int }_0^s{\left(s-\tau \right)}^{\beta -1}h\left(\tau \right)\text{d}\tau }\right)\text{d}s}\\ -\frac{1}{2\Gamma \left(\alpha \right)}{\displaystyle {\int }_0^T{\left(T-s\right)}^{\alpha -1}{\phi }_q\left(\frac{1}{\Gamma \left(\beta \right)}{\displaystyle {\int }_0^s{\left(s-\tau \right)}^{\beta -1}h\left(\tau \right)\text{d}\tau }\right)\text{d}s}\\ +\frac{\left(T-2t\right)T^{\gamma -1}\Gamma \left(2-\gamma \right)}{2\Gamma \left(\alpha -\gamma \right)}{\displaystyle {\int }_0^T{\left(T-s\right)}^{\alpha -\gamma -1}{\phi }_q\left(\frac{1}{\Gamma \left(\beta \right)}{\displaystyle {\int }_0^s{\left(s-\tau \right)}^{\beta -1}h\left(\tau \right)\text{d}\tau }\right)\text{d}s}\end{array}$

3. 主要结果

定义 $E=C\left(\left(0,T\right],R^{+}\right)$，则E是以 $\Vert u\Vert =\underset{t\in J}{\sup }\left|u\left(t\right)\right|$ 为范数的Banach空间。

定义算子 $F:E\to E$ 为：

$\begin{array}{c}Fu\left(t\right)=\frac{1}{\Gamma \left(\alpha \right)}{\displaystyle {\int }_0^t{\left(t-s\right)}^{\alpha -1}{\phi }_q\left(\frac{1}{\Gamma \left(\beta \right)}{\displaystyle {\int }_0^s{\left(s-\tau \right)}^{\beta -1}f\left(\tau ,u\left(\tau \right)\right)\text{d}\tau }\right)\text{d}s}\\ -\frac{1}{2\Gamma \left(\alpha \right)}{\displaystyle {\int }_0^T{\left(T-s\right)}^{\alpha -1}}{\phi }_q\left(\frac{1}{\Gamma \left(\beta \right)}{\displaystyle {\int }_0^s{\left(s-\tau \right)}^{\beta -1}f\left(\tau ,u\left(\tau \right)\right)\text{d}\tau }\right)\text{d}s\\ +\frac{\left(T-2t\right)T^{\gamma -1}\Gamma \left(2-\gamma \right)}{2\Gamma \left(\alpha -\gamma \right)}{\displaystyle {\int }_0^T{\left(T-s\right)}^{\alpha -\gamma -1}{\phi }_q\left(\frac{1}{\Gamma \left(\beta \right)}{\displaystyle {\int }_0^s{\left(s-\tau \right)}^{\beta -1}f\left(\tau ,u\left(\tau \right)\right)\text{d}\tau }\right)\text{d}s}\end{array}$

在本节中，需附加以下条件来确保解的存在性与唯一性。

(H₁) 对于实数 $\sigma >0$，使得 $t^{\sigma }f\left(t,u\left(t\right)\right)$ 在 $\left[0,T\right]\times \left[0,+\infty \right)$ 上连续，并且存在常数 $M_1>0$，有 $\left|t^{\sigma }f\left(t,u\left(t\right)\right)\right|<M_1$ ；

(H₂) 对任意 $u\left(t\right),v\left(t\right)\in E$，存在实数 $M_2>0$，有 $\left|t^{\sigma }f\left(t,u\left(t\right)\right)-t^{\sigma }f\left(t,v\left(t\right)\right)\right|\le M_2\underset{t\in J}{\sup }\left|u-v\right|$ ；

(H₃) 记 $\eta =\frac{M_2\Gamma \left(1-\sigma \right)\left(q-1\right){\xi }^{q-2}{T}^{\alpha +\beta -\sigma }}{2\Gamma \left(\beta -\sigma +1\right)}\left[\frac{3}{\Gamma \left(\alpha +1\right)}+\frac{\Gamma \left(2-\gamma \right)T^{-\gamma }}{\Gamma \left(\alpha -\gamma +1\right)}\right]\le 1$ ；

(H₄) 记 $N=\frac{M_2\Gamma \left(2-\gamma \right)\Gamma \left(1-\sigma \right)\left(q-1\right){\xi }^{q-2}{T}^{\alpha +\beta -\sigma }}{2\Gamma \left(\alpha -\gamma +1\right)\Gamma \left(\beta -\sigma +1\right)}\le 1$。

为方便下文计算，我们 $H\left(t\right)={\phi }_q\left(\frac{1}{\Gamma \left(\beta \right)}{\displaystyle {\int }_0^t{\left(t-s\right)}^{\beta -1}f\left(s,u\left(s\right)\right)\text{d}s}\right)$。

定理2.1 假如 $1<p<2$，若条件(H₁)成立，则 $H\left(t\right)={\phi }_q\left(\frac{1}{\Gamma \left(\beta \right)}{\displaystyle {\int }_0^t{\left(t-s\right)}^{\beta -1}{s}^{-\sigma }{s}^{\sigma }f\left(s,u\left(s\right)\right)\text{d}s}\right)$ 在 $\left[0,T\right]$ 上连续，且存在实数 $L>0$，对任意 $t\in \left[0,T\right]$，有 $\left|H\left(t\right)\right|\le L$。

证明：由于 $1<p<2$ 时，根据 $\frac{1}{p}+\frac{1}{q}=1$，可得 $q>2$，并且有

$\frac{1}{\Gamma \left(\beta \right)}{\displaystyle {\int }_0^t{\left(t-s\right)}^{\beta -1}{s}^{-\sigma }{s}^{\sigma }f\left(s,u\left(s\right)\right)\text{d}s}\le \frac{W\Gamma \left(1-\sigma \right)}{\Gamma \left(\beta -\sigma +1\right)}:=\xi 1$.

则对 $\forall t,t_0\in \left[0,T\right]$，$\exists \left|t-t_0\right|<\delta$，有

$\begin{array}{c}\left|H\left(t\right)-H\left(t_0\right)\right|\le |{\phi }_q\left(\frac{1}{\Gamma \left(\beta \right)}{\displaystyle {\int }_0^t{\left(t-s\right)}^{\beta -1}{s}^{-\sigma }{s}^{\sigma }f\left(s,u\left(s\right)\right)\text{d}s}\right)\\ -|{\phi }_q\left(\frac{1}{\Gamma \left(\beta \right)}{\displaystyle {\int }_0^{t_0}{\left(t_0-s\right)}^{\beta -1}{s}^{-\sigma }{s}^{\sigma }f\left(s,u\left(s\right)\right)\text{d}s}\right)|\\ \le \frac{M_1}{\Gamma \left(\beta \right)}\left(q-1\right){\xi }^{q-2}|{\displaystyle {\int }_0^{t_0}{\left(t_0-s\right)}^{\beta -1}{s}^{-\sigma }\text{d}s}-{\displaystyle {\int }_0^{t_0}{\left(t_0-s\right)}^{\beta -1}{s}^{-\sigma }\text{d}s}|\\ \le \frac{M_1}{\Gamma \left(\beta \right)}\left(q-1\right){\xi }^{q-2}B\left(1-\sigma ,\beta \right)\left|t^{\beta -\sigma }-t_0^{\beta -\sigma }\right|\to 0,t\to t_0\end{array}$

即 $\left|H\left(t\right)\right|$ 在 $\left[0,1\right]$ 上连续，故 $\left|H\left(t\right)\right|\le L$。

定理2.2 若条件(H₁)成立，根据定理2.1及引理1.3，可知边值问题(4)有唯一解。

证明：定义集合 $\Omega =\left\{u\in E|\Vert u\Vert <r\right\}$，这里 $r=\frac{L{T}^{\alpha }}{2}\left(\frac{3\Gamma \left(\alpha -\gamma +1+\Gamma \left(2-\gamma \right)\Gamma \left(\alpha +1\right)\right)}{\Gamma \left(\alpha +1\right)\Gamma \left(\alpha -\gamma +1\right)}\right)$。

首先 $F:\Omega \to \Omega$ 证明，对任意 $u\in \Omega$，有

$\begin{array}{c}Fu\left(t\right)\le \frac{1}{\Gamma \left(\alpha \right)}{\displaystyle {\int }_0^t{\left(t-s\right)}^{\alpha -1}\left|H\left(S\right)\right|\text{d}s}+\frac{1}{2\Gamma \left(\alpha \right)}{\displaystyle {\int }_0^T{\left(T-s\right)}^{\alpha -1}\left|H\left(S\right)\right|\text{d}s}\\ +\frac{\left(T-2t\right)T^{\gamma -1}\Gamma \left(2-\gamma \right)}{2\Gamma \left(\alpha -\gamma \right)}{\displaystyle {\int }_0^T{\left(T-s\right)}^{\alpha -\gamma -1}\left|H\left(s\right)\right|\text{d}s}\\ \le \frac{L}{\Gamma \left(\alpha \right)}{\displaystyle {\int }_0^t{\left(t-s\right)}^{\alpha -1}ds}+\frac{L}{2\Gamma \left(\alpha \right)}{\displaystyle {\int }_0^T{\left(T-s\right)}^{\alpha -1}\text{d}s}+\frac{T^{\gamma }\Gamma \left(2-\gamma \right)L}{2\Gamma \left(\alpha -\gamma \right)}{\displaystyle {\int }_0^T{\left(T-s\right)}^{\alpha -\gamma -1}\text{d}s}\\ \le \frac{L}{2}{T}^{\alpha }\left[\frac{3\Gamma \left(\alpha -\gamma +1\right)+\Gamma \left(2-\gamma \right)\Gamma \left(\alpha +1\right)}{\Gamma \left(\alpha +1\right)\Gamma \left(\alpha -\gamma +1\right)}\right]\le r\end{array}$

即 $F:\Omega \to \Omega$，故 $F:\Omega \to \Omega$ 成立。

其次，对 $\forall u\left(t\right),v\left(t\right)\in \Omega$，当 $t\in \left[0,T\right]$ 时，

$\begin{array}{l}\left|Fu\left(t\right)-Fv\left(t\right)\right|\\ \le \frac{1}{\Gamma \left(\alpha \right)}{\displaystyle {\int }_0^t{\left(t-s\right)}^{\alpha -1}|{\phi }_q\left(\frac{1}{\Gamma \left(\beta \right)}{\displaystyle {\int }_0^s{\left(s-\tau \right)}^{\beta -1}f\left(\tau ,u\left(\tau \right)\right)\text{d}\tau }\right)}\\ -{\phi }_q\left(\frac{1}{\Gamma \left(\beta \right)}{\displaystyle {\int }_0^s{\left(s-\tau \right)}^{\beta -1}f\left(\tau ,v\left(\tau \right)\right)\text{d}\tau }\right)|\text{d}s\\ +\frac{1}{2\Gamma \left(\alpha \right)}{\displaystyle {\int }_0^T{\left(T-s\right)}^{\alpha -1}}|{\phi }_q\left(\frac{1}{\Gamma \left(\beta \right)}{\displaystyle {\int }_0^s{\left(s-\tau \right)}^{\beta -1}f\left(\tau ,u\left(\tau \right)\right)\text{d}\tau }\right)\\ -{\phi }_q\left(\frac{1}{\Gamma \left(\beta \right)}{\displaystyle {\int }_0^s{\left(s-\tau \right)}^{\beta -1}f\left(\tau ,v\left(\tau \right)\right)\text{d}\tau }\right)|\text{d}s\end{array}$

$\begin{array}{l}+\frac{T^{\gamma }\Gamma \left(2-\gamma \right)}{2\Gamma \left(\alpha -\gamma \right)}{\displaystyle {\int }_0^T{\left(T-s\right)}^{\alpha -\gamma -1}|{\phi }_q\left(\frac{1}{\Gamma \left(\beta \right)}{\displaystyle {\int }_0^s{\left(s-\tau \right)}^{\beta -1}f\left(\tau ,u\left(\tau \right)\right)\text{d}\tau }\right)}\\ -{\phi }_q\left(\frac{1}{\Gamma \left(\beta \right)}{\displaystyle {\int }_0^s{\left(s-\tau \right)}^{\beta -1}f\left(\tau ,v\left(\tau \right)\right)\text{d}\tau }\right)|\text{d}s\\ \le \frac{\left(q-1\right){\xi }^{q-2}}{\Gamma \left(\alpha \right)\Gamma \left(\beta \right)}{\displaystyle {\int }_0^t{\left(t-s\right)}^{\alpha -1}}|\left({\displaystyle {\int }_0^s{\left(s-\tau \right)}^{\beta -1}}{\tau }^{-\sigma }{\tau }^{\sigma }f\left(\tau ,u\left(\tau \right)\right)\text{d}\tau \right)\\ -\left({\displaystyle {\int }_0^s{\left(s-\tau \right)}^{\beta -1}}{\tau }^{-\sigma }{\tau }^{\sigma }f\left(\tau ,v\left(\tau \right)\right)\text{d}\tau \right)|\text{d}s\\ +\frac{\left(q-1\right){\xi }^{q-2}}{2\Gamma \left(\alpha \right)\Gamma \left(\beta \right)}{\displaystyle {\int }_0^T{\left(T-s\right)}^{\alpha -1}}|\left({\displaystyle {\int }_0^s{\left(s-\tau \right)}^{\beta -1}}{\tau }^{-\sigma }{\tau }^{\sigma }f\left(\tau ,u\left(\tau \right)\right)\text{d}\tau \right)\end{array}$

$\begin{array}{l}-\left({\displaystyle {\int }_0^s{\left(s-\tau \right)}^{\beta -1}}{\tau }^{-\sigma }{\tau }^{\sigma }f\left(\tau ,v\left(\tau \right)\right)\text{d}\tau \right)|\text{d}s\\ +\frac{\left(q-1\right){\xi }^{q-2}{T}^{\gamma }\Gamma \left(2-\gamma \right)}{2\Gamma \left(\alpha -\gamma \right)\Gamma \left(\beta \right)}{\displaystyle {\int }_0^T{\left(T-s\right)}^{\alpha -\gamma -1}}|\left({\displaystyle {\int }_0^s{\left(s-\tau \right)}^{\beta -1}}{\tau }^{-\sigma }{\tau }^{\sigma }f\left(\tau ,u\left(\tau \right)\right)\text{d}\tau \right)\\ -\left({\displaystyle {\int }_0^s{\left(s-\tau \right)}^{\beta -1}}{\tau }^{-\sigma }{\tau }^{\sigma }f\left(\tau ,v\left(\tau \right)\right)\text{d}\tau \right)\text{d}s\\ \le \frac{M_2\Gamma \left(1-\sigma \right)\left(q-1\right){\xi }^{q-2}{T}^{\alpha +\beta -\sigma }}{2\Gamma \left(\beta -\sigma +1\right)}\left[\frac{3}{\Gamma \left(\alpha +1\right)}+\frac{\Gamma \left(2-\gamma \right)T^{-\gamma }}{\Gamma \left(\alpha -\gamma +1\right)}\right]\underset{t\in J}{\sup }\left|u-v\right|\\ \le \eta \left|u-v\right|\end{array}$

由假设条件(H₃)可知 $\eta \le 1$ 故有 $\Vert Fu-Fv\Vert \le \eta \Vert u-v\Vert$，因此由引理1.3可知算子有唯一不动点。

定理2.3 若条件(H₁)~(H₄)满足，则根据定理2.1、2.2及引理1.4可知边值问题(4)至少有一个不动点。

证明：定义集合 ${\Omega }_R=\left\{u\in E|\Vert u\Vert <R\right\}$，这里 $R\ge \frac{L{T}^{\alpha }\left[\Gamma \left(\alpha -\gamma +1\right)+\Gamma \left(\alpha +1\right)\Gamma \left(2-\gamma \right)\right]}{2\Gamma \left(\alpha +1\right)\Gamma \left(\alpha -\gamma +1\right)}$，则 ${\Omega }_R$ 为E的有界闭子集。定义 ${\Omega }_R$ 上的算子 $P,Q$，其中

$\begin{array}{l}Pu\left(t\right)=\frac{1}{\Gamma \left(\alpha \right)}{\displaystyle {\int }_0^t{\left(t-s\right)}^{\alpha -1}}{\phi }_q\left(\frac{1}{\Gamma \left(\beta \right)}{\displaystyle {\int }_0^s{\left(s-\tau \right)}^{\beta -1}}f\left(\tau ,u\left(\tau \right)\right)\text{d}\tau \right)\text{d}s\\ -\frac{1}{2\Gamma \left(\alpha \right)}{\displaystyle {\int }_0^T{\left(T-s\right)}^{\alpha -1}}{\phi }_q\left(\frac{1}{\Gamma \left(\beta \right)}{\displaystyle {\int }_0^s{\left(s-\tau \right)}^{\beta -1}}f\left(\tau ,u\left(\tau \right)\right)\text{d}\tau \right)\text{d}s\end{array}$

$Qu\left(t\right)=\frac{\left(T-2t\right)T^{\gamma -1}\Gamma \left(2-\gamma \right)}{2\Gamma \left(\alpha -\gamma \right)}{\displaystyle {\int }_0^T{\left(T-s\right)}^{\alpha -\gamma -1}}{\phi }_q\left(\frac{1}{\Gamma \left(\beta \right)}{\displaystyle {\int }_0^s{\left(s-\tau \right)}^{\beta -1}}f\left(\tau ,u\left(\tau \right)\right)\text{d}\tau \right)\text{d}s$

对 $\forall u,v\in {\Omega }_R,t\in \left[0,T\right]$ 时，有

$\begin{array}{l}\left|Pu\left(t\right)+Qv\left(t\right)\right|\\ \le \frac{1}{\Gamma \left(\alpha \right)}{\displaystyle {\int }_0^t{\left(t-s\right)}^{\alpha -1}}\left|{\phi }_q\left(\frac{1}{\Gamma \left(\beta \right)}{\displaystyle {\int }_0^s{\left(s-\tau \right)}^{\beta -1}}{\tau }^{-\sigma }{\tau }^{\sigma }f\left(\tau ,u\left(\tau \right)\right)\text{d}\tau \right)\right|\text{d}s\\ +\frac{1}{2\Gamma \left(\alpha \right)}{\displaystyle {\int }_0^T{\left(T-s\right)}^{\alpha -1}}\left|{\phi }_q\left(\frac{1}{\Gamma \left(\beta \right)}{\displaystyle {\int }_0^s{\left(s-\tau \right)}^{\beta -1}}{\tau }^{-\sigma }{\tau }^{\sigma }f\left(\tau ,u\left(\tau \right)\right)\text{d}\tau \right)\right|\text{d}s\\ +\frac{\left(T-2t\right)T^{\gamma -1}\Gamma \left(2-\gamma \right)}{2\Gamma \left(\alpha -\gamma \right)}{\displaystyle {\int }_0^T{\left(T-s\right)}^{\alpha -\gamma -1}}\left|{\phi }_q\left(\frac{1}{\Gamma \left(\beta \right)}{\displaystyle {\int }_0^s{\left(s-\tau \right)}^{\beta -1}}{\tau }^{-\sigma }{\tau }^{\sigma }f\left(\tau ,u\left(\tau \right)\right)\text{d}\tau \right)\right|\text{d}s\\ \le \frac{T^{\alpha }L}{\Gamma \left(\alpha +1\right)}+\frac{T^{\alpha }L}{2\Gamma \left(\alpha +1\right)}+\frac{T^{\alpha }L\Gamma \left(2-\gamma \right)}{2\Gamma \left(\alpha -\gamma +1\right)}\\ \le \frac{L{T}^{\alpha }\left[3\Gamma \left(\alpha -\gamma +1\right)+\Gamma \left(\alpha +1\right)\Gamma \left(2-\gamma \right)\right]}{2\Gamma \left(\alpha +1\right)\Gamma \left(\alpha -\gamma +1\right)}\le R\end{array}$

因此 $\Vert Pu\left(t\right)+Qv\left(t\right)\Vert \le R$，即 $Pu\left(t\right)+Qv\left(t\right)\in {\Omega }_R$。

其次当 $t\in \left[0,T\right]$ 时，有

$\begin{array}{l}\left|Qu\left(t\right)-Qv\left(t\right)\right|\\ \le \frac{\left(T-2t\right)T^{\gamma -1}\Gamma \left(2-\gamma \right)}{2\Gamma \left(\alpha -\gamma \right)}{\displaystyle {\int }_0^T{\left(T-s\right)}^{\alpha -\gamma -1}}\left|{\phi }_q\left(\frac{1}{\Gamma \left(\beta \right)}{\displaystyle {\int }_0^s{\left(s-\tau \right)}^{\beta -1}}{\tau }^{-\sigma }{\tau }^{\sigma }f\left(\tau ,u\left(\tau \right)\right)\text{d}\tau \right)\right|\text{d}s\\ -\frac{\left(T-2t\right)T^{\gamma -1}\Gamma \left(2-\gamma \right)}{2\Gamma \left(\alpha -\gamma \right)}{\displaystyle {\int }_0^T{\left(T-s\right)}^{\alpha -\gamma -1}}\left|{\phi }_q\left(\frac{1}{\Gamma \left(\beta \right)}{\displaystyle {\int }_0^s{\left(s-\tau \right)}^{\beta -1}}{\tau }^{-\sigma }{\tau }^{\sigma }f\left(\tau ,v\left(\tau \right)\right)\text{d}\tau \right)\right|\text{d}s\\ \le \frac{T^{\gamma }\left(q-1\right){\xi }^{q-2}\Gamma \left(2-\gamma \right)}{2\Gamma \left(\alpha -\gamma \right)\Gamma \left(\beta \right)}{\displaystyle {\int }_0^T{\left(T-s\right)}^{\alpha -\gamma -1}}\left({\displaystyle {\int }_0^s{\left(s-\tau \right)}^{\beta -1}}{\tau }^{-\sigma }\left|{\tau }^{\sigma }f\left(\tau ,u\left(\tau \right)\right)-{\tau }^{\sigma }f\left(\tau ,v\left(\tau \right)\right)\right|\text{d}\tau \right)\text{d}s\\ \le \frac{M_2\left(q-1\right){\xi }^{q-2}\Gamma \left(2-\gamma \right)\Gamma \left(1-\sigma \right)T^{\alpha +\beta -\sigma }}{2\Gamma \left(\alpha -\gamma +1\right)\Gamma \left(\beta -\sigma +1\right)}\underset{t\in J}{\sup }\left|u-v\right|=N\left|u-v\right|\end{array}$

由条件(H₄)及引理1.4可知算子Q在 ${\Omega }_R$ 中为压缩映射。

最后证明算子P在 ${\Omega }_R$ 上是全连续的，对任意 $u\in {\Omega }_R$，有

$\begin{array}{c}\left|Pu\left(t\right)\right|=|\frac{1}{\Gamma \left(\alpha \right)}{\displaystyle {\int }_0^t{\left(t-s\right)}^{\alpha -1}}{\phi }_q\left(\frac{1}{\Gamma \left(\beta \right)}{\displaystyle {\int }_0^s{\left(s-\tau \right)}^{\beta -1}}{\tau }^{-\sigma }{\tau }^{\sigma }f\left(\tau ,u\left(\tau \right)\right)\text{d}\tau \right)\text{d}s\\ -\frac{1}{2\Gamma \left(\alpha \right)}{\displaystyle {\int }_0^T{\left(T-s\right)}^{\alpha -1}}{\phi }_q\left(\frac{1}{\Gamma \left(\beta \right)}{\displaystyle {\int }_0^s{\left(s-\tau \right)}^{\beta -1}}{\tau }^{-\sigma }{\tau }^{\sigma }f\left(\tau ,u\left(\tau \right)\right)\text{d}\tau \right)\text{d}s|\\ \le \frac{3L{T}^{\alpha }}{2\Gamma \left(\alpha +1\right)}\end{array}$

可知算子P在 ${\Omega }_R$ 上一致有界。

其次对 $\forall t_1,t_2\in \left[0,T\right]$ 当 $t_1<t_2$ 时，有

$\begin{array}{l}\left|Pu\left(t_2\right)-Pu\left(t_1\right)\right|\\ \le |\frac{1}{\Gamma \left(\alpha \right)}{\displaystyle {\int }_0^{t_2}{\left(t_2-s\right)}^{\alpha -1}}{\phi }_q\left(\frac{1}{\Gamma \left(\beta \right)}{\displaystyle {\int }_0^s{\left(s-\tau \right)}^{\beta -1}}{\tau }^{-\sigma }{\tau }^{\sigma }f\left(\tau ,u\left(\tau \right)\right)\text{d}\tau \right)\text{d}s\\ -\frac{1}{\Gamma \left(\alpha \right)}{\displaystyle {\int }_0^{t_1}{\left(t_1-s\right)}^{\alpha -1}}{\phi }_q\left(\frac{1}{\Gamma \left(\beta \right)}{\displaystyle {\int }_0^s{\left(s-\tau \right)}^{\beta -1}}{\tau }^{-\sigma }{\tau }^{\sigma }f\left(\tau ,u\left(\tau \right)\right)\text{d}\tau \right)\text{d}s|\\ \le \frac{L}{\Gamma \left(\alpha +1\right)}\left(t_2^{\alpha }-t_1^{\alpha }\right)\to 0,t_2\to t_1\end{array}$

即算子P在 ${\Omega }_R$ 上等度连续，由Arzela-Ascoli定理可知算子P为全连续算子，故由引理1.4可知边值问题(4)至少有一个解。

致谢

谨向审稿人提出的宝贵意见和建议表示诚挚的感谢！同时也要向我的指导老师胡卫敏教授表示我诚挚的谢意！

基金项目

新疆维吾尔自治区自然科学基金资助项目(2019D01C331)。

文章引用

张婷婷,胡卫敏. 一类具P-Laplacian算子的分数阶奇异微分方程反周期边值问题解的存在性与唯一性
Existence and Uniqueness on Solutions for Anti-Periodic Boundary Value Problems of Singular Fractional Differential Equations with P-Laplacian Operator[J]. 应用数学进展, 2021, 10(11): 3650-3658. https://doi.org/10.12677/AAM.2021.1011386

参考文献