﻿ 一类无穷区间上的分数阶微分方程边值问题解的存在性 The Existence of Solutions for a Class of Boundary Value Problem of Fractional Differential Equations on an Infinite Interval

Pure Mathematics
Vol.07 No.04(2017), Article ID:21066,12 pages
10.12677/PM.2017.74027

The Existence of Solutions for a Class of Boundary Value Problem of Fractional Differential Equations on an Infinite Interval

Jiaxin Yao, Wenxia Wang, Jianmei Jia

Department of Mathematics, Taiyuan Normal University, Jinzhong Shanxi

Received: Jun. 3rd, 2017; accepted: Jun. 19th, 2017; published: Jun. 22nd, 2017

ABSTRACT

By using the Leray-Schauder nonlinear alternative theorem and the Leggett-Williams fixed point theorem, a class of boundary value problem for fractional differential equations with integral conditions on an infinite interval is investigated. Some sufficient conditions on the existence of at least one unbounded solution and three positive solutions are established. At last, two examples are given to illustrate the results.

Keywords:Fractional Differential Equations, Integral Boundary Conditions, Infinite Interval, Positive Solutions

1. 引言

(1.1)

(1.2)

(1.3)

2. 相关定义和引理

(C1)且对于任意的

(C2) 对任意的，有

(C3) 对任意的，有

3. 无界解的存在性

(3.1)

(3.2)

1) 对任意的关于是严格递增的；

2) 设实数，存在正实数，使得对

(H1)

(H2)是连续函数，

(H3) 存在非减，使得

1)上局部等度连续；

2)处等度收敛，即对任意的，存在使得对任意的

。从而对于任意的

(H4)使得

(3.3)

(3.4)

4. 多解的存在性

(H5)

(H6)

(H7)

，我们有

，说明定理2.2中的条件(C2)成立。

(H6)我们可以得到，由此知定理2.2中的条件(C3)成立。故至少存在三个不动点，即边值问题(1.3)至少存在三个正解使得

5. 例子

(5.1)

，有连续；

；令，有

(5.2)

，通过计算可得

The Existence of Solutions for a Class of Boundary Value Problem of Fractional Differential Equations on an Infinite Interval[J]. 理论数学, 2017, 07(04): 213-224. http://dx.doi.org/10.12677/PM.2017.74027

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