﻿ 具非局部边界条件的奇异分数阶微分方程正解的存在性和唯一性

# 具非局部边界条件的奇异分数阶微分方程正解的存在性和唯一性Existence and Uniqueness of Positive Solutions for Singular Fractional Differential Equation with Nonlocal Boundary Conditions

Abstract: In this paper, we mainly study a class of singular fractional order differential equations with nonlocal boundary conditions. Firstly, the properties of the Green function are discussed. Then, under some appropriate assumptions, by using the Banach contraction mapping principle and the Krasnoselskii Fixed Point theorem, the existence and uniqueness of positive solutions for the singular boundary value problems are obtained. An example is given to illustrate the feasibility of the main results.

1. 引言

$\left\{\begin{array}{l}{}^{C}D{}_{{0}^{+}}^{\alpha }u\left(t\right)=\lambda f\left(t,u\left(t\right)\right),\text{}00,\\ {a}_{1}u\left(0\right)-{b}_{1}{u}^{\prime }\left(0\right)=0,\text{}{a}_{2}u\left(1\right)+{b}_{2}{u}^{\prime }\left(1\right)={\int }_{0}^{1}m\left(s\right)u\left(s\right)\text{d}p\left(s\right),\\ {u}^{‴}\left(0\right)=0,\text{}{u}^{″}\left(1\right)={\int }_{0}^{1}n\left(s\right){u}^{″}\left(s\right)\text{d}q\left(s\right)\end{array}$

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

$\left\{\begin{array}{l}-{D}_{t}^{\beta }\left({\phi }_{p}\left({D}_{t}^{\alpha }u\right)\right)\left(t\right)=\lambda f\left(t,u\left(t\right)\right),\text{}0

$\left\{\begin{array}{l}{D}_{{0}^{+}}^{\alpha }x\left(t\right)+\lambda f\left(t,x\left(t\right),\left(\phi x\right)\left(t\right)\right)=0,\text{\hspace{0.17em}}2<\alpha \le 3,\text{\hspace{0.17em}}t\in \left[0,1\right],\\ x\left(0\right)={x}^{\prime }\left(0\right)=0,\text{}{x}^{\prime }\left(1\right)={\int }_{0}^{1}\left(\alpha -1\right)q\left(x\left(s\right)\right)\text{d}s\end{array}$ (1.1)

2. 预备知识

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

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

(i) 当时，有

(ii) S是全连续算子，

(iii) T是压缩映射，

.

3. 正解的存在性

(3.1)

(3.2)

(1)，对

(2)，对

(2) 对，有

。在上定义两个算子，其中：

，有：

，得到。又对任意的，有：

，由上是连续的，故上是一致连续的，因此，对，当时，使得

(i) 若

(ii) 若

4. 正解的唯一性

。对，证

，则有。现在，对，有：

5. 应用举例

(5.1)

NOTES

*通讯作者。

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