一类具有瞬时和非瞬时脉冲的 $\psi$-Caputo 型分数阶微分方程的多解性

数学物理学报 ›› 2025, Vol. 45 ›› Issue (3): 807-823.

一类具有瞬时和非瞬时脉冲的 $\psi$-Caputo 型分数阶微分方程的多解性

姚旺进¹(),张慧萍^2,^*()

¹莆田学院福建省金融信息处理重点实验室福建莆田 351100
²福建师范大学数学与统计学院福州 350117

收稿日期:2024-05-07 修回日期:2024-11-27 出版日期:2025-06-26 发布日期:2025-06-20
通讯作者: *张慧萍, E-mail: zhanghpmath@163.com
作者简介:姚旺进, E-mail: 13635262963@163.com
基金资助:
福建省自然科学基金(2023J01994);福建省自然科学基金(2023J01995);福建省自然科学基金(2024J01871);福建省自然科学基金(2024J01873);福建省高校创新团队培育计划(2018-39);福建省高校数学学科联盟科研项目(2024SXLMMS05);福建省中青年教师教育科研项目(JAT231093)

Multiple Solutions for a Class of $\psi$-Caputo-Type Fractional Differential Equations with Instantaneous and Non-Instantaneous Impulses

Yao Wangjin¹(),Zhang Huiping^2,^*()

¹Fujian Key Laboratory of Financial Information Processing, Putian University, Fujian Putian 351100
²School of Mathematics and Statistics, Fujian Normal University, Fuzhou 350117

Received:2024-05-07 Revised:2024-11-27 Online:2025-06-26 Published:2025-06-20
Supported by:
Natural Science Foundation of Fujian Province(2023J01994);Natural Science Foundation of Fujian Province(2023J01995);Natural Science Foundation of Fujian Province(2024J01871);Natural Science Foundation of Fujian Province(2024J01873);Program for Innovative Research Team in Science and Technology in Fujian Province University(2018-39);Fujian Alliance of Mathematics(2024SXLMMS05);Education and Research Project for Middle and Young Teachers in Fujian Province(JAT231093)

摘要/Abstract

摘要：

作为整数阶微分方程的推广, 分数阶微分方程近年来是一个十分热门的研究对象. 分数阶微分方程在反常扩散、流体流动、流行病学和粘弹性力学等许多科学与工程实际问题的建模中发挥着重要作用. 该文研究一类包含 $\psi$-Caputo 分数阶导数和具有瞬时和非瞬时脉冲的分数阶微分方程. 当参数 $\mu\in \mathbb{R}$ 时, 利用变分方法和两类三临界点定理, 获得至少三个古典解的存在性. 并且, 该文改进和推广了最近的一些结果. 最后, 给出两个例子来验证所得结果的可行性和有效性.

关键词: $\psi$-Caputo 分数阶导数, 分数阶微分方程, 变分方法, 三临界点定理

Abstract:

In recent years, as an extension of integer-order differential equations, fractional differential equations have became a popular research object. They play an important role in modeling many practical problems of science and engineering, such as anomalous diffusion, fluid flow, epidemiology, viscoelastic mechanics, etc. In this paper, a class of fractional differential equation involving $\psi$-Caputo fractional derivative with instantaneous and non-instantaneous impulses is considered. By using variational methods and two types of three critical point theorems, the existence of at least three classical solutions is obtained when $\mu\in \mathbb{R}$. Moreover, some recent results are improved and extended. In the end, two examples are given to verify the feasibility and effectiveness of the obtained results.

Key words: $\psi$-Caputo fractional derivative, fractional differential equation, variational methods, Three critical point theorems

中图分类号:

O177.91

姚旺进, 张慧萍. 一类具有瞬时和非瞬时脉冲的 $\psi$-Caputo 型分数阶微分方程的多解性[J]. 数学物理学报, 2025, 45(3): 807-823.

Yao Wangjin, Zhang Huiping. Multiple Solutions for a Class of $\psi$-Caputo-Type Fractional Differential Equations with Instantaneous and Non-Instantaneous Impulses[J]. Acta mathematica scientia,Series A, 2025, 45(3): 807-823.

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