﻿ 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

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

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

$\left\{\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

$\left\{\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

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

2. 预备知识

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

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

$u\left(t\right)={c}_{1}{t}^{\alpha -1}+{c}_{2}{t}^{\alpha -2}+{c}_{n}{t}^{\alpha -n},$

${I}_{{0}^{+}}^{\alpha }{D}_{{0}^{+}}^{\alpha }u\left(t\right)=u\left(t\right)+{c}_{1}{t}^{\alpha -1}+{c}_{2}{t}^{\alpha -2}+{c}_{n}{t}^{\alpha -n},$

$\left\{\begin{array}{l}{\left({\varphi }_{p}\left({D}_{{0}^{+}}^{\alpha }u\left(t\right)\right)\right)}^{\prime }+y\left(t\right)=0,0

$u\left(t\right)={\int }_{0}^{1}G\left(t,s\right){\varphi }_{q}\left({\int }_{0}^{s}y\left(\tau \right)\text{d}\tau \right)\text{d}s,$

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

${\varphi }_{p}\left({D}_{{0}^{+}}^{\alpha }u\left(t\right)\right)=-{\int }_{0}^{t}y\left(s\right)\text{d}s+{c}_{0},$

${D}_{{0}^{+}}^{\alpha }u\left(t\right)=-{\varphi }_{q}\left({\int }_{0}^{t}y\left(s\right)\text{d}s+{c}_{0}\right).$

$u\left(t\right)=-\frac{1}{\Gamma \left(\alpha \right)}{\int }_{0}^{t}{\left(t-s\right)}^{\alpha -1}{\varphi }_{q}\left({\int }_{0}^{s}y\left(\tau \right)\text{d}\tau +{c}_{0}\right)\text{d}s+{c}_{1}{t}^{\alpha -1}+{c}_{2}{t}^{\alpha -2}.$

${D}_{{0}^{+}}^{\alpha }u\left(0\right)=0$${c}_{0}=0$，由 $u\left(0\right)=0$，得 ${c}_{2}=0$，由 $u\left(1\right)=0$，得

${c}_{1}=\frac{1}{\Gamma \left(\alpha \right)}{\int }_{0}^{1}{\left(1-s\right)}^{\alpha -1}{\varphi }_{q}\left({\int }_{0}^{s}y\left(\tau \right)\text{d}\tau \right)\text{d}s.$

$\begin{array}{c}u\left(t\right)=\frac{1}{\Gamma \left(\alpha \right)}{\int }_{0}^{1}{t}^{\alpha -1}{\left(1-s\right)}^{\alpha -1}{\varphi }_{q}\left({\int }_{0}^{s}y\left(\tau \right)\text{d}\tau \right)\text{d}s\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-\frac{1}{\Gamma \left(\alpha \right)}{\int }_{0}^{t}{\left(t-s\right)}^{\alpha -1}{\varphi }_{q}\left({\int }_{0}^{s}y\left(\tau \right)\text{d}\tau \right)\text{d}s\\ =\frac{1}{\Gamma \left(\alpha \right)}{\int }_{0}^{t}\left[{t}^{\alpha -1}{\left(1-s\right)}^{\alpha -1}-{\left(t-s\right)}^{\alpha -1}\right]{\varphi }_{q}\left({\int }_{0}^{s}y\left(\tau \right)\text{d}\tau \right)\text{d}s\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\frac{1}{\Gamma \left(\alpha \right)}{\int }_{t}^{1}{t}^{\alpha -1}{\left(1-s\right)}^{\alpha -1}{\varphi }_{q}\left({\int }_{0}^{s}y\left(\tau \right)\text{d}\tau \right)\text{d}s\\ ={\int }_{0}^{1}G\left(t,s\right){\varphi }_{q}\left({\int }_{0}^{s}y\left(\tau \right)\text{d}\tau \right)\text{d}s.\end{array}$

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}{\mathrm{min}}G\left(t,s\right)\ge r\left(s\right)\underset{t\in \left[0,1\right]}{\mathrm{max}}G\left(t,s\right)=r\left(s\right)G\left(s,s\right),0

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

3. 主要结果

(H1) $f:\left[0,1\right]×\left[0,+\infty \right)×R\to \left[0,+\infty \right)$ 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

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

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

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

$\frac{{2}^{2\left(\alpha -1\right)}\Gamma \left(\alpha \right)\rho }{{\left[{\rho }^{p-1}\left({\int }_{0}^{1}j\left(\tau \right)\text{d}\tau +{\int }_{0}^{1}l\left(\tau \right)\text{d}\tau \right)+{\int }_{0}^{1}w\left(\tau \right)\text{d}\tau \right]}^{q-1}}\le 1,$ (2.1)

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

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

$\begin{array}{c}|Tu\left({t}_{2}\right)-Tu\left({t}_{1}\right)|=|{\int }_{0}^{1}G\left({t}_{2},s\right){\varphi }_{q}\left({\int }_{0}^{s}f\left(\tau ,u\left(\tau \right),{D}_{{0}^{+}}^{\alpha }u\left(\tau \right)\right)\text{d}\tau \right)\text{d}s\\ -{\int }_{0}^{1}G\left({t}_{1},s\right){\varphi }_{q}\left({\int }_{0}^{s}f\left(\tau ,u\left(\tau \right),{D}_{{0}^{+}}^{\alpha }u\left(\tau \right)\right)\text{d}\tau \right)\text{d}s|\\ =|{\int }_{0}^{1}\left[G\left({t}_{2},s\right)-G\left({t}_{1},s\right)\right]{\varphi }_{q}\left({\int }_{0}^{s}f\left(\tau ,u\left(\tau \right),{D}_{{0}^{+}}^{\alpha }u\left(\tau \right)\right)\text{d}\tau \right)\text{d}s|\\ \le {\left[{M}^{p-1}\left({\int }_{0}^{1}j\left(\tau \right)\text{d}\tau +{\int }_{0}^{1}l\left(\tau \right)\text{d}\tau \right)+{\int }_{0}^{1}w\left(\tau \right)\text{d}\tau \right]}^{q-1}{\int }_{0}^{1}|G\left({t}_{2},s\right)-G\left({t}_{1},s\right)|\text{d}s.\end{array}$

$\begin{array}{l}|{D}_{{0}^{+}}^{\alpha }Tu\left({t}_{2}\right)-{D}_{{0}^{+}}^{\alpha }Tu\left({t}_{1}\right)|\\ =|-\left[{\varphi }_{q}\left({\int }_{0}^{{t}_{2}}f\left(s,u\left(s\right),{D}_{{0}^{+}}^{\alpha }u\left(s\right)\right)\text{d}s\right)-{\varphi }_{q}\left({\int }_{0}^{{t}_{1}}f\left(s,u\left(s\right),{D}_{{0}^{+}}^{\alpha }u\left(s\right)\right)\text{d}s\right)\right]|\\ =|{\left({\int }_{0}^{{t}_{2}}f\left(s,u\left(s\right),{D}_{{0}^{+}}^{\alpha }u\left(s\right)\right)\text{d}s\right)}^{q-1}-{\left({\int }_{0}^{{t}_{1}}f\left(s,u\left(s\right),{D}_{{0}^{+}}^{\alpha }u\left(s\right)\right)\text{d}s\right)}^{q-1}|,\end{array}$

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

$\begin{array}{c}0\le {\left({\int }_{0}^{{t}_{2}}f\left(s,u\left(s\right),{D}_{{0}^{+}}^{\alpha }u\left(s\right)\right)\text{d}s\right)}^{q-1}-{\left({\int }_{0}^{{t}_{1}}f\left(s,u\left(s\right),{D}_{{0}^{+}}^{\alpha }u\left(s\right)\right)\text{d}s\right)}^{q-1}\\ \le {\left({\int }_{{t}_{1}}^{{t}_{2}}f\left(s,u\left(s\right),{D}_{{0}^{+}}^{\alpha }u\left(s\right)\right)\text{d}s\right)}^{q-1}\\ \le {\left({\int }_{{t}_{1}}^{{t}_{2}}\left(j\left(s\right){\varphi }_{p}\left(|u\left(s\right)|\right)+l\left(s\right){\varphi }_{p}\left(|{D}_{{0}^{+}}^{\alpha }u\left(s\right)|\right)+w\left(s\right)\right)\text{d}s\right)}^{q-1}\\ \le {\left({\varphi }_{p}\left(‖u‖\right){\int }_{{t}_{1}}^{{t}_{2}}j\left(s\right)\text{d}s+{\varphi }_{p}\left(‖u‖\right){\int }_{{t}_{1}}^{{t}_{2}}l\left(s\right)\text{d}s+{\int }_{{t}_{1}}^{{t}_{2}}w\left(s\right)\text{d}s\right)}^{q-1}\\ \le {\left({M}^{p-1}\left({\int }_{{t}_{1}}^{{t}_{2}}j\left(s\right)\text{d}s+{\int }_{{t}_{1}}^{{t}_{2}}l\left(s\right)\text{d}s\right)+{\int }_{{t}_{1}}^{{t}_{2}}w\left(s\right)\text{d}s\right)}^{q-1},\end{array}$

${\int }_{{t}_{1}}^{{t}_{2}}j\left(s\right)\text{d}s=j\left(\mu \right)\left({t}_{2}-{t}_{1}\right),{\int }_{{t}_{1}}^{{t}_{2}}l\left(s\right)\text{d}s=l\left(\eta \right)\left({t}_{2}-{t}_{1}\right),{\int }_{{t}_{1}}^{{t}_{2}}w\left(s\right)\text{d}s=w\left(\kappa \right)\left({t}_{2}-{t}_{1}\right),$

$|{D}_{{0}^{+}}^{\alpha }Tu\left({t}_{2}\right)-{D}_{{0}^{+}}^{\alpha }Tu\left({t}_{1}\right)|\le {\left[{M}^{p-1}\left(j\left(\mu \right)+l\left(\eta \right)\right)+w\left(\kappa \right)\right]}^{q-1}{\left({t}_{2}-{t}_{1}\right)}^{q-1},$

$q-1\ge 1$ 时，由拉格朗日中值定理有

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

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

$U=\left\{u\in X|‖u‖\le \rho \right\}$，有 $U\subset X$，由上述证明可知 $T:\stackrel{¯}{U}\to X$ 是全连续的。我们断言当 $u\in \partial U$$\lambda \in \left(\text{0,1}\right)$$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

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

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

4. 举例

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

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

$|f\left(t,u,v\right)|\le {\text{e}}^{t}{\varphi }_{3}\left(|u|\right)+{\text{e}}^{-t}{\varphi }_{3}\left(|v|\right)+\sqrt{t},0\le t\le 1,$

$\rho =1$，有

$\frac{{2}^{2\left(\alpha -1\right)}\Gamma \left(\alpha \right)\rho }{{\left[{\rho }^{p-1}\left({\int }_{0}^{1}j\left(\tau \right)\text{d}\tau +{\int }_{0}^{1}l\left(\tau \right)\text{d}\tau \right)+{\int }_{0}^{1}w\left(\tau \right)\text{d}\tau \right]}^{q-1}}>1,$

NOTES

*通讯作者。

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https://doi.org/10.1186/s13661-016-0548-0

[3] Lu, H.L., Han, Z.L., Sun, S.R. and Liu, J. (2013) Existence on Positive Solutions for Boundary Value Problems of Nonlinear Fractional Differential Equations with p-Laplacian. Advances in Difference Equations, 30, 1-16.

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[8] 李小平, 李辉来. 带p-Laplace算子分数阶微分方程边值问题正解的存在性[J]. 吉林大学学报(理学版), 2017, 55(3): 481-489.

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