2025年04月23日 星期三

广西师范大学学报（自然科学版） ›› 2024, Vol. 42 ›› Issue (6): 177-185.doi: 10.16088/j.issn.1001-6600.2023112205

• “污水处理”专栏 • 上一篇    下一篇

含参数分数阶微分方程边值问题正解的存在性

罗茜, 许勇强*   

  1. 闽南师范大学 数学与统计学院,福建 漳州 363000
  • 收稿日期:2023-11-22 修回日期:2024-01-11 出版日期:2024-12-30 发布日期:2024-12-30
  • 通讯作者: 许勇强(1976—), 男,福建漳浦人,闽南师范大学教授,博士。E-mail: yqx458@163.com
  • 基金资助:
    国家自然科学基金(11571159);福建省自然科学基金(2017J01562)

Existence of Positive Solutions for Boundary Value Problems of Fractional Differential Equations with Parameters

LUO Xi, XU Yongqiang*   

  1. School of Mathematics and Statistics, Minnan Normal University, Zhangzhou Fujian 363000, China
  • Received:2023-11-22 Revised:2024-01-11 Online:2024-12-30 Published:2024-12-30

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摘要： 本文考虑一类含参数且具有两项分数阶导数的Caputo型非零边值的分数阶微分方程问题。首先,借助拉普拉斯变换构造Green函数,将边值问题转化为等价的第二类Fredholm积分方程;然后,利用Green函数的性质、Guo-Krasnoselskii不动点定理和Leggett-Williams不动点定理,得到边值问题正解的存在性、不存在性以及多重性的充分条件;接着,将一般分数阶微分方程边值问题正解存在性的结果推广到含有两项分数阶导数的边值问题,得到更丰富的结论;最后,通过实例论证所得结论的正确性。

关键词: 两项分数阶导数, 边值问题, Guo-Krasnoselskii不动点定理, Leggett-Williams不动点定理, 正解的存在性

Abstract: A class of parametric boundary value problems with two-term fractional derivatives and non-zero boundary values is investigated in this paper. Firstly, Green’s function is constructed by Laplace transform, and the boundary value problem is transformed into the equivalent second kind of Fredholm integral equation. Secondly, by using the properties of Green’s function, Guo-Krasnoselskii fixed point theorem and Leggett-Williams fixed point theorem, sufficient conditions for the existence, nonexistence and multiplicity of positive solutions for boundary value problems of fractional differential equations are obtained. Thirdly, the existence of positive solutions for boundary value problems of usual fractional differential equations is extended to boundary value problems with two fractional derivatives. Finally, an example is given to illustrate the feasibility of the obtained results.

Key words: two-term fractional derivatives, boundary value problem, Guo-Krasnoselskii fixed point theorem, Leggett-Williams fixed point theorem, existence of positive solutions

中图分类号:  O175.25

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