﻿ 一类分数阶三点边值问题解的存在性

# 一类分数阶三点边值问题解的存在性Existence of Solutions for a Class of Fractional Three-Point Boundary Value Problems

Abstract: In this paper, three-point boundary value problem for a class of fractional differential equations are studied by using Leray-Schauder nonlinear alternative. The existence of the solution is obtained. Furthermore, it is proved that the function k(t), which is limited to the nonlinear term, exists at least one solution of the problem when it is controlled by a function of the form Btμ. Finally, an example is given to verify the validity of the conclusion.

1. 引言

$\left\{\begin{array}{l}{}^{c}{D}_{{0}^{+}}^{\alpha }u\left(t\right)+f\left(t,u\left(t\right)\right)=0,0

Yan Sun在文献 [9] 等利用锥拉伸与压缩不动点定理得到了四阶三点边值问题

$\left\{\begin{array}{l}{u}^{\left(4\right)}\left(t\right)+f\left(t,u\left(t\right)\right)=0,0\le t\le 1,\\ u\left(0\right)={u}^{\prime }\left(0\right)={u}^{″}\left(0\right)=0,{u}^{″}\left(1\right)-\alpha {u}^{″}\left(\eta \right)=\lambda \end{array}$

$\left\{\begin{array}{l}{D}^{\alpha }\left(D+\lambda \right)x\left(t\right)=f\left(t,x\left(t\right)\right),t\in \left[0,1\right],\\ x\left(0\right)=0,{x}^{\prime }\left(0\right)=0,x\left(1\right)=\beta x\left(\eta \right)\end{array}$ (1.1)

2. 预备知识

${D}^{\alpha }x\left(t\right)=\underset{\epsilon \to 0}{\mathrm{lim}}\frac{x\left(t+\epsilon {t}^{1-\alpha }\right)-x\left(t\right)}{\epsilon },\forall t>0$

${D}^{\alpha }x\left(t\right)=\underset{\epsilon \to 0}{\mathrm{lim}}\frac{{x}^{\left(\left[\alpha \right]-1\right)}\left(t+\epsilon {t}^{\left(\left[\alpha \right]-\alpha \right)}\right)-{x}^{\left(\left[\alpha \right]-1\right)}\left(t\right)}{\epsilon },$

${D}^{-\alpha }x\left(t\right)\equiv {I}^{\alpha }x\left(t\right)={I}^{1}\left({t}^{\alpha -1}x\right)\left(t\right)={\int }_{0}^{t}\frac{x\left(s\right)}{{s}^{1-\alpha }}ds$

${D}^{-\alpha }x\left(t\right)\equiv {I}^{\alpha }x\left(t\right)={I}^{n}\left({t}^{\alpha -\left[\alpha \right]}x\right)\left(t\right)$

1) 存在 $u\in \partial \Omega$$\theta >1$，使得 $Tu=\theta u$

2) 存在 $T$ 上的不动点 $u*\in \stackrel{¯}{\Omega }$

$\left\{\begin{array}{l}{D}^{\alpha }\left(D+\lambda \right)x\left(t\right)=\sigma \left(t\right),t\in \left[0,1\right],\\ x\left(0\right)=0,{x}^{\prime }\left(0\right)=0,x\left(1\right)=\beta x\left(\eta \right),\end{array}$

$\begin{array}{c}x\left(t\right)={\int }_{0}^{t}{\text{e}}^{-\lambda \left(t-s\right)}\left({\int }_{0}^{s}\sigma \left(u\right){u}^{\alpha -2}\left(s-u\right)du\right)ds+A\left(t\right)\left[\beta {\int }_{0}^{\eta }{\text{e}}^{-\lambda \left(\eta -s\right)}\left({\int }_{0}^{s}\sigma \left(u\right){u}^{\alpha -2}\left(s-u\right)du\right)ds\\ -{\int }_{0}^{\text{1}}{\text{e}}^{-\lambda \left(1-s\right)}\left({\int }_{0}^{s}\sigma \left(u\right){u}^{\alpha -2}\left(s-u\right)du\right)ds\right],\end{array}$

$E=C\left[0,1\right]$，对 $x\in E$$E$ 在范数 $‖x‖=\underset{\begin{array}{l}t\in \left[0,1\right]\\ x\in \stackrel{¯}{\Omega }\end{array}}{\mathrm{max}}|x\left(t\right)|$ 下成为Banach空间。

$\begin{array}{c}\left(Tx\right)\left(t\right)={\int }_{0}^{t}{\text{e}}^{-\lambda \left(t-s\right)}\left({\int }_{0}^{s}f\left(u,x\left(u\right)\right){u}^{\alpha -2}\left(s-u\right)du\right)ds+A\left(t\right)\left[\beta {\int }_{0}^{\eta }{\text{e}}^{-\lambda \left(\eta -s\right)}\left({\int }_{0}^{s}f\left(u,x\left(u\right)\right){u}^{\alpha -2}\left(s-u\right)du\right)ds\\ -{\int }_{0}^{\text{1}}{\text{e}}^{-\lambda \left(1-s\right)}\left({\int }_{0}^{s}f\left(u,x\left(u\right)\right){u}^{\alpha -2}\left(s-u\right)du\right)ds\right]\end{array}$

$\forall x\in \stackrel{¯}{\Omega }$，由算子 $T$ 的定义可得

$\begin{array}{c}‖Tx‖=\underset{t\in \left[0,1\right]}{\mathrm{max}}|{\int }_{0}^{t}{\text{e}}^{-\lambda \left(t-s\right)}\left({\int }_{0}^{s}f\left(u,x\left(u\right)\right){u}^{\alpha -2}\left(s-u\right)du\right)ds+A\left(t\right)\left[\beta {\int }_{0}^{\eta }{\text{e}}^{-\lambda \left(\eta -s\right)}\left({\int }_{0}^{s}f\left(u,x\left(u\right)\right){u}^{\alpha -2}\left(s-u\right)du\right)ds\\ -{\int }_{0}^{\text{1}}{\text{e}}^{-\lambda \left(1-s\right)}\left({\int }_{0}^{s}f\left(u,x\left(u\right)\right){u}^{\alpha -2}\left(s-u\right)du\right)ds\right]|\\ \le {\int }_{0}^{\text{1}}{\text{e}}^{-\lambda \left(\text{1}-s\right)}\left({\int }_{0}^{s}|f\left(u,x\left(u\right)\right)|{u}^{\alpha -2}\left(s-u\right)du\right)ds+A\left(\text{1}\right)\left[\beta {\int }_{0}^{\eta }{\text{e}}^{-\lambda \left(\eta -s\right)}\left({\int }_{0}^{s}|f\left(u,x\left(u\right)\right)|{u}^{\alpha -2}\left(s-u\right)\text{d}u\right)\text{d}s\\ +{\int }_{0}^{\text{1}}{\text{e}}^{-\lambda \left(1-s\right)}\left({\int }_{0}^{s}|f\left(u,x\left(u\right)\right)|{u}^{\alpha -2}\left(s-u\right)du\right)ds\right]\end{array}$

$\begin{array}{l}\le R\left[\left(1+A\left(1\right)\right){\int }_{0}^{\text{1}}{\text{e}}^{-\lambda \left(\text{1}-s\right)}\left({\int }_{0}^{s}{u}^{\alpha -2}\left(s-u\right)du\right)ds+A\left(\text{1}\right)\beta {\int }_{0}^{\eta }{\text{e}}^{-\lambda \left(\eta -s\right)}\left({\int }_{0}^{s}{u}^{\alpha -2}\left(s-u\right)\text{d}u\right)\text{d}s\right]\\ \le \frac{R}{\alpha \left(\alpha -1\right)}\left(\left(1+A\left(1\right)\right){\int }_{0}^{\text{1}}{\text{e}}^{-\lambda \left(\text{1}-s\right)}{s}^{\alpha }ds+A\left(\text{1}\right)\beta {\int }_{0}^{\eta }{\text{e}}^{-\lambda \left(\eta -s\right)}{s}^{\alpha }\text{d}s\right)\\ \le \frac{R\left(\left(1+A\left(1\right)\right)\left(1-{\text{e}}^{-\lambda }\right)+A\left(\text{1}\right)\beta {\eta }^{\alpha }\left(1-{\text{e}}^{-\lambda \eta }\right)\right)}{\alpha \left(\alpha -1\right)}.\end{array}$

$\begin{array}{l}|\left(Tx\right)\left({t}_{\text{2}}\right)-\left(Tx\right)\left({t}_{1}\right)|=|{\int }_{0}^{{t}_{2}}{\text{e}}^{-\lambda \left({t}_{2}-s\right)}\left({\int }_{0}^{s}f\left(u,x\left(u\right)\right){u}^{\alpha -2}\left(s-u\right)du\right)ds\\ \text{}+A\left({t}_{2}\right)\left[\beta {\int }_{0}^{\eta }{\text{e}}^{-\lambda \left(\eta -s\right)}\left({\int }_{0}^{s}f\left(u,x\left(u\right)\right){u}^{\alpha -2}\left(s-u\right)du\right)ds-{\int }_{0}^{\text{1}}{\text{e}}^{-\lambda \left(1-s\right)}\left({\int }_{0}^{s}f\left(u,x\left(u\right)\right){u}^{\alpha -2}\left(s-u\right)du\right)ds\right]\\ \text{}-{\int }_{0}^{{t}_{1}}{\text{e}}^{-\lambda \left({t}_{1}-s\right)}\left({\int }_{0}^{s}f\left(u,x\left(u\right)\right){u}^{\alpha -2}\left(s-u\right)du\right)ds\\ \text{}-A\left({t}_{1}\right)\left[\beta {\int }_{0}^{\eta }{\text{e}}^{-\lambda \left(\eta -s\right)}\left({\int }_{0}^{s}f\left(u,x\left(u\right)\right){u}^{\alpha -2}\left(s-u\right)du\right)ds-{\int }_{0}^{\text{1}}{\text{e}}^{-\lambda \left(1-s\right)}\left({\int }_{0}^{s}f\left(u,x\left(u\right)\right){u}^{\alpha -2}\left(s-u\right)du\right)ds\right]|\end{array}$

$\begin{array}{l}\le |{\int }_{0}^{{t}_{2}}{\text{e}}^{-\lambda \left({t}_{2}-s\right)}\left({\int }_{0}^{s}f\left(u,x\left(u\right)\right){u}^{\alpha -2}\left(s-u\right)du\right)ds-{\int }_{0}^{{t}_{1}}{\text{e}}^{-\lambda \left({t}_{2}-s\right)}\left({\int }_{0}^{s}f\left(u,x\left(u\right)\right){u}^{\alpha -2}\left(s-u\right)du\right)ds|\\ \text{}+|{\int }_{0}^{{t}_{\text{1}}}{\text{e}}^{-\lambda \left({t}_{2}-s\right)}\left({\int }_{0}^{s}f\left(u,x\left(u\right)\right){u}^{\alpha -2}\left(s-u\right)du\right)ds-{\int }_{0}^{{t}_{1}}{\text{e}}^{-\lambda \left({t}_{1}-s\right)}\left({\int }_{0}^{s}f\left(u,x\left(u\right)\right){u}^{\alpha -2}\left(s-u\right)du\right)ds|\\ \text{}+|A\left({t}_{2}\right)-A\left({t}_{1}\right)||\left[\beta {\int }_{0}^{\eta }{\text{e}}^{-\lambda \left(\eta -s\right)}\left({\int }_{0}^{s}f\left(u,x\left(u\right)\right){u}^{\alpha -2}\left(s-u\right)du\right)ds\\ \text{}-{\int }_{0}^{\text{1}}{\text{e}}^{-\lambda \left(1-s\right)}\left({\int }_{0}^{s}f\left(u,x\left(u\right)\right){u}^{\alpha -2}\left(s-u\right)du\right)ds\right]|\end{array}$

$\begin{array}{l}\le {\int }_{{t}_{1}}^{{t}_{2}}{\text{e}}^{-\lambda \left({t}_{2}-s\right)}\left({\int }_{0}^{s}|f\left(u,x\left(u\right)\right)|{u}^{\alpha -2}\left(s-u\right)du\right)ds+{\int }_{0}^{{t}_{1}}|{\text{e}}^{-\lambda \left({t}_{2}-s\right)}-{\text{e}}^{-\lambda \left({t}_{2}-s\right)}|\left({\int }_{0}^{s}|f\left(u,x\left(u\right)\right)|{u}^{\alpha -2}\left(s-u\right)du\right)ds\\ \text{}+\frac{2\lambda \left({t}_{2}-{t}_{1}\right)}{\Delta }\left[\beta {\int }_{0}^{\eta }{\text{e}}^{-\lambda \left(\eta -s\right)}\left({\int }_{0}^{s}|f\left(u,x\left(u\right)\right)|{u}^{\alpha -2}\left(s-u\right)du\right)ds+{\int }_{0}^{\text{1}}{\text{e}}^{-\lambda \left(1-s\right)}\left({\int }_{0}^{s}|f\left(u,x\left(u\right)\right)|{u}^{\alpha -2}\left(s-u\right)du\right)ds\right]\\ \le \frac{R}{\alpha \left(\alpha -1\right)}\left[{\int }_{{t}_{1}}^{{t}_{2}}{\text{e}}^{-\lambda \left({t}_{2}-s\right)}{s}^{\alpha }ds+{\int }_{0}^{{t}_{1}}|{\text{e}}^{-\lambda \left({t}_{2}-s\right)}-{\text{e}}^{-\lambda \left({t}_{2}-s\right)}|{s}^{\alpha }ds+\frac{\text{2}\lambda \left({t}_{2}-{t}_{1}\right)}{\Delta }\left(\beta {\int }_{0}^{\eta }{\text{e}}^{-\lambda \left(\eta -s\right)}{s}^{\alpha }ds+{\int }_{0}^{\text{1}}{\text{e}}^{-\lambda \left(1-s\right)}{s}^{\alpha }ds\right)\right]\\ =\frac{R}{\alpha \left(\alpha -1\right)}\left[\frac{{t}_{2}^{\alpha }-{t}_{1}^{\alpha }}{\lambda }\left(1-{\text{e}}^{-\lambda {t}_{2}}\right)+\frac{{t}_{1}^{\alpha }}{\lambda }\left({\text{e}}^{-\lambda \left({t}_{2}-s\right)}-{\text{e}}^{-\lambda \left({t}_{2}-s\right)}\right)+\frac{\text{2}\lambda \left({t}_{2}-{t}_{1}\right)}{\Delta }\frac{\beta {\eta }^{\alpha }\left(1-{\text{e}}^{-\lambda \eta }\right)+1-{\text{e}}^{-\lambda }}{\lambda }\right]\\ \le \frac{R}{\alpha \left(\alpha -1\right)}\left[\frac{{t}_{2}^{\alpha }-{t}_{1}^{\alpha }}{\lambda }+\left({t}_{2}-{t}_{1}\right)+\frac{\text{2}\left(\beta {\eta }^{\alpha }\left(1-{\text{e}}^{-\lambda \eta }\right)+1-{\text{e}}^{-\lambda }\right)}{\Delta }\left({t}_{2}-{t}_{1}\right)\right].\end{array}$

3. 主要结果及证明

(C1) $|f\left(t,x\right)|\le k\left(t\right)|x|+h\left(t\right)$$a.e.\left(t,x\right)\in \left[0,1\right]×R$

(C2) $\frac{\text{2}}{\lambda \left(1-\beta \right)}{\int }_{0}^{1}\left(1-s\right){s}^{\alpha -2}k\left(s\right)ds+\frac{\beta }{\lambda \left(1-\beta \right)}{\int }_{0}^{\eta }\left(\eta -s\right){s}^{\alpha -2}k\left(s\right)ds<1$

$M=\frac{\text{2}}{\lambda \left(1-\beta \right)}{\int }_{0}^{1}\left(1-s\right){s}^{\alpha -2}k\left(s\right)ds+\frac{\beta }{\lambda \left(1-\beta \right)}{\int }_{0}^{\eta }\left(\eta -s\right){s}^{\alpha -2}k\left(s\right)ds$

$N=\frac{\text{2}}{\lambda \left(1-\beta \right)}{\int }_{0}^{1}\left(1-s\right){s}^{\alpha -2}h\left(s\right)ds+\frac{\beta }{\lambda \left(1-\beta \right)}{\int }_{0}^{\eta }\left(\eta -s\right){s}^{\alpha -2}h\left(s\right)ds$

$h\left(t\right)\ge |f\left(t,0\right)|$，对 $a.e.\text{}t\in \left[0,1\right]$，得到 $N>0$

$r=N{\left(1-M\right)}^{-1}$$\Omega =\left\{x\in E:‖x‖。假设 $x\in \partial \Omega$$\theta >1$，使得 $Tx=\theta x$。于是有

$\begin{array}{c}\theta r=\theta ‖x‖=‖Tx‖=\underset{t\in \left[0,1\right]}{\mathrm{max}}|{\int }_{0}^{t}{\text{e}}^{-\lambda \left(t-s\right)}\left({\int }_{0}^{s}f\left(u,x\left(u\right)\right){u}^{\alpha -2}\left(s-u\right)du\right)ds\\ +A\left(t\right)\left[\beta {\int }_{0}^{\eta }{\text{e}}^{-\lambda \left(\eta -s\right)}\left({\int }_{0}^{s}f\left(u,x\left(u\right)\right){u}^{\alpha -2}\left(s-u\right)du\right)ds-{\int }_{0}^{\text{1}}{\text{e}}^{-\lambda \left(1-s\right)}\left({\int }_{0}^{s}f\left(u,x\left(u\right)\right){u}^{\alpha -2}\left(s-u\right)du\right)ds\right]|\\ \le {\int }_{0}^{\text{1}}{\text{e}}^{-\lambda \left(\text{1}-s\right)}\left({\int }_{0}^{s}|f\left(u,x\left(u\right)\right)|{u}^{\alpha -2}\left(s-u\right)du\right)ds+A\left(\text{1}\right)\left[\beta {\int }_{0}^{\eta }{\text{e}}^{-\lambda \left(\eta -s\right)}\left({\int }_{0}^{s}|f\left(u,x\left(u\right)\right)|{u}^{\alpha -2}\left(s-u\right)du\right)\text{d}s\\ +{\int }_{0}^{\text{1}}{\text{e}}^{-\lambda \left(1-s\right)}\left({\int }_{0}^{s}|f\left(u,x\left(u\right)\right)|{u}^{\alpha -2}\left(s-u\right)du\right)ds\right]\end{array}$

$\begin{array}{c}\le \left(1+A\left(1\right)\right){\int }_{0}^{\text{1}}{\text{e}}^{-\lambda \left(\text{1}-s\right)}\left({\int }_{0}^{s}k\left(u\right)|x\left(u\right)|{u}^{\alpha -2}\left(s-u\right)du\right)ds+A\left(\text{1}\right)\beta {\int }_{0}^{\eta }{\text{e}}^{-\lambda \left(\eta -s\right)}\left({\int }_{0}^{s}k\left(u\right)|x\left(u\right)|{u}^{\alpha -2}\left(s-u\right)\text{d}u\right)\text{d}s\\ +\left(1+A\left(1\right)\right){\int }_{0}^{\text{1}}{\text{e}}^{-\lambda \left(\text{1}-s\right)}\left({\int }_{0}^{s}h\left(u\right){u}^{\alpha -2}\left(s-u\right)du\right)ds+A\left(\text{1}\right)\beta {\int }_{0}^{\eta }{\text{e}}^{-\lambda \left(\eta -s\right)}\left({\int }_{0}^{s}h\left(u\right){u}^{\alpha -2}\left(s-u\right)\text{d}u\right)\text{d}s\\ \le \frac{2}{1-\beta }{\int }_{0}^{\text{1}}\left({\int }_{u}^{1}{\text{e}}^{-\lambda \left(\text{1}-s\right)}\left(s-u\right)ds\right)k\left(u\right)|x\left(u\right)|{u}^{\alpha -2}du+\frac{\beta }{1-\beta }{\int }_{0}^{\eta }\left({\int }_{u}^{\eta }{\text{e}}^{-\lambda \left(\eta -s\right)}\left(s-u\right)ds\right)k\left(u\right)|x\left(u\right)|{u}^{\alpha -2}du\\ +\frac{2}{1-\beta }{\int }_{0}^{\text{1}}\left({\int }_{u}^{1}{\text{e}}^{-\lambda \left(\text{1}-s\right)}\left(s-u\right)ds\right)h\left(u\right){u}^{\alpha -2}du+\frac{\beta }{1-\beta }{\int }_{0}^{\eta }\left({\int }_{u}^{\eta }{\text{e}}^{-\lambda \left(\eta -s\right)}\left(s-u\right)ds\right)h\left(u\right){u}^{\alpha -2}du\end{array}$

$\begin{array}{c}\le \frac{2}{\lambda \left(1-\beta \right)}{\int }_{0}^{\text{1}}\left(1-s\right){s}^{\alpha -2}k\left(s\right)|x\left(s\right)|ds+\frac{\beta }{1-\beta }{\int }_{0}^{\eta }\left(1-s\right){s}^{\alpha -2}k\left(s\right)|x\left(s\right)|ds\\ +\frac{\text{2}}{\lambda \left(1-\beta \right)}{\int }_{0}^{1}\left(1-s\right){s}^{\alpha -2}h\left(s\right)ds+\frac{\beta }{\lambda \left(1-\beta \right)}{\int }_{0}^{\eta }\left(\eta -s\right){s}^{\alpha -2}h\left(s\right)ds\\ =M‖x‖+N.\end{array}$

$\theta \le M+\frac{N}{r}=M+\frac{N}{N{\left(1-M\right)}^{-1}}=M+\left(1-M\right)=1$

(C3)存在一个常数 $\mu >0$，使得

$k\left(s\right)\le B{s}^{\mu },\text{}a.e.\text{}s\in \left[0,1\right],$

$B=\frac{\lambda \left(1-\beta \right)\left(\alpha -1+\mu \right)\left(\alpha +\mu \right)}{2+\beta {\eta }^{\alpha +\mu }}$

$\begin{array}{c}M=\frac{\text{2}}{\lambda \left(1-\beta \right)}{\int }_{0}^{1}\left(1-s\right){s}^{\alpha -2}k\left(s\right)ds+\frac{\beta }{\lambda \left(1-\beta \right)}{\int }_{0}^{\eta }\left(\eta -s\right){s}^{\alpha -2}k\left(s\right)ds\\ \le B\left(\frac{\text{2}}{\lambda \left(1-\beta \right)}{\int }_{0}^{1}\left(1-s\right){s}^{\alpha -2+\mu }ds+\frac{\beta }{\lambda \left(1-\beta \right)}{\int }_{0}^{\eta }\left(\eta -s\right){s}^{\alpha -2+\mu }ds\right)\\ \le B\left(\frac{\text{2}}{\lambda \left(1-\beta \right)}\frac{\text{1}}{\left(\alpha -1+\mu \right)\left(\alpha +\mu \right)}+\frac{\beta }{\lambda \left(1-\beta \right)}\frac{{\eta }^{\alpha +\mu }}{\left(\alpha -1+\mu \right)\left(\alpha +\mu \right)}\right)\\ \le B\frac{\text{2}+\beta {\eta }^{\alpha +\mu }}{\lambda \left(1-\beta \right)\left(\alpha -1+\mu \right)\left(\alpha +\mu \right)}\\ =1.\end{array}$

$M<1$ 时，有 $\theta \le 1$。则结论得证。

4. 示例

$\left\{\begin{array}{l}{D}^{\frac{\text{4}}{\text{3}}}\left(D+\text{5}\right)x\left(t\right)=\frac{t}{5}|x|\mathrm{cos}\sqrt[3]{x}+2t+1,t\in \left[0,1\right],\\ x\left(0\right)=0,{x}^{\prime }\left(0\right)=0,x\left(1\right)=\frac{1}{2}x\left( 1 3 \right)\end{array}$

$f\left(t,x\right)=\frac{t}{5}|x|\mathrm{cos}\sqrt[3]{x}+2t+1,$

$k\left(t\right)=\frac{t}{2},\text{}h\left(t\right)=1$

$|f\left(t,x\right)|\le k\left(t\right)|x|+h\left(t\right),\text{}a.e.\text{}\left(t,x\right)\in \left[0,1\right]×R,$

$\begin{array}{c}M=\frac{\text{2}}{\text{5}×\left(1-\frac{\text{1}}{\text{2}}\right)}{\int }_{0}^{1}\left(1-s\right){s}^{-\frac{2}{3}}\cdot \frac{s}{2}ds+\frac{\frac{1}{2}}{5×\left(1-\frac{1}{2}\right)}{\int }_{0}^{\frac{1}{3}}\left(\frac{1}{3}-s\right){s}^{-\frac{2}{3}}\cdot \frac{s}{2}ds\\ \approx \text{0}\text{.13105}<1.\end{array}$

[1] Bendouma, B., Cabada, A. and Hammoudi, A. (2019) Three-Point Boundary Value Problems for Conformable Frac-tional Differential Equations. Archivum Mathematicum, 55, 69-82.
https://doi.org/10.5817/AM2019-2-69

[2] Bai, Z., Cheng, Y. and Sun, S. (2020) On Solutions of a Class of Three-Point Fractional Boundary Value Problems. Boundary Value Problems, 11, 1-12.
https://doi.org/10.1186/s13661-019-01319-x

[3] 董晓玉, 白占兵, 张伟. 具有适型分数阶导数的非线性特征值问题的正解[J]. 山东科技大学学报(自然科学版), 2016, 35(3): 85-91.

[4] Li, Y. and Jiang, W. (2019) Existence and Nonexistence of Positive Solutions for Fractional Three-Point Boundary Value Problems with a Parameter. Journal of Function Spaces, 34, 1-10.
https://doi.org/10.1155/2019/9237856

[5] 郑春华, 马睿, 傅霞. 一类带积分边值条件的高阶时滞分数阶微分方程解的存在性[J]. 西南民族大学学报(自然科学版), 2020, 46(3): 303-309.

[6] Bekri, Z. and Benaicha, S. (2017) Existence of Solution for a Nonlinear Three-Point Boundary Value Problem. Boletim Sociedad Paranaense de Matematica, 14, 1120-1134.
https://doi.org/10.5269/bspm.v38i1.34767

[7] Zouaoui, B. and Slimane, B. (2020) Existence of Solution for Nonlinear Fourth-Order Three-Point Boundary Value Problem. Boletim Sociedad Paranaense de Matematica, 38, 67-82.
https://doi.org/10.5269/bspm.v38i1.34767

[8] 庞杨, 韦煜明. Caputo分数阶微分方程三点边值问题解的存在性[J]. 应用泛函分析学报, 2018, 20(1): 63-76.

[9] Sun, Y. and Zhu, C. (2013) Existence of Positive Solutions for Singular Fourth-Order Three-Point Boundary Value Problems. Advances in Difference Equations, 51, 1-13.
https://doi.org/10.1186/1687-1847-2013-51

[10] Batarfi, H. and Losada, J. (2015) Three-Point Boundary Value Problems for Conformable Fractional Differential Equations. Journal of Function Spaces, 34, 1-6.
https://doi.org/10.1155/2015/706383

[11] Khalil, R. (2014) A New Definition of Fractional Derivative. Journal Computational and Applied Mathematics, 264, 65-70.
https://doi.org/10.1016/j.cam.2014.01.002

[12] Deimling, K. (1985) Nonlinear Functional Analysis. Springer, Berlin.
https://doi.org/10.1007/978-3-662-00547-7

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