一类由变分不等式驱动的模糊分数阶微分包含系统解的存在性

2024-05-15 19:23李慧敏顾海波
吉林大学学报(理学版) 2024年2期
关键词:变分微积分微分

李慧敏 顾海波

摘要: 考虑一类动态模糊系统, 该系统由模糊Atangana-Baleanu分数阶微分包含和变分不等式组成, 称为模糊分数阶微分变分不等式(FFDVI), 它包括了模糊分数阶微分包含和变分不等式两个领域的研究, 拓宽了模糊环境下的可研究问题, 该模型在同一框架内捕获了模糊分数微分包含和分数微分变分不等式的期望特征. 利用Krasnoselskii不动点定理, 得到了FFDVI在某些温和条件下解的存在性.

关键词: Atangana-Baleanu分数阶导数; 分数阶模糊微分变分不等式; Krasnoselskii不动点定理; 解的存在性

中图分类号: O175.14文献标志码: A文章编号: 1671-5489(2024)02-0222-15

Existence of Solutions for a Class of Fuzzy Fractional DifferentialInclusion Systems Driven by Variational Inequalities

LI Huimin, GU Haibo

(School of Mathematical Sciences, Xinjiang Normal University, Urumqi 830017, China)

Abstract: We considered a class of dynamic fuzzy systems, which consisted of fuzzy Atangana-Baleanu fractional differential inclusion and variational inequalities, called fuzzy fractional differential variational inequalities (FFDVI). It included the two fields of fuzzy fractional differential inclusion and variational inequalities, expanding the researchable problems in fuzzy environments. The model captured the desired features of the fuzzy fractional differential inclusion and fractional differential variational inequalities within the same framework. By using Krasnoselskii fixed point theorem, the existence of solutions of FFDVI under some mild conditions was obtained.

Keywords: Atangana-Baleanu fractional derivative; fractional fuzzy differential variational inequality; Krasnoselskii fixed point theorem; existence of solution

0 引 言

分數阶微积分作为经典整数阶微积分的一种自然推广, 近年来备受关注. 由于整数阶微分方程具有一定的局限性, 而分数阶模糊微分方程在应用中比整数阶模糊微分方程更贴合实际, 因此对分数阶模糊变分不等式系统的研究很有必要, 已广泛应用于电动力学、 生物技术、 空气动力学、 分布式螺旋桨设计和控制动力系统中[1-8]. 最常用的分数式积分和微分算子是Riemann-Liouville,Hadamard,Grunwald-Letnikov,Caputo和Riesz-Caputo. Atangana和Baleanu[9]引入了一种新的分数阶导数, 它包含由Mittag-Leffler函数描述的非局部和非奇异核, Atangana-Baleanu分数阶微积分[9-10]是一种强大的数学工具, 可以解释和描述各种复杂的物理现象、 化学反应过程、 种群动力学行为等问题[11-15].

近年来, Caputo和Fabrizio[16]基于实值函数空间中的幂函数核, 引入了Caputo意义上的分数阶导数新定义如下:CFDαa+x(t)=M(α)/1-α∫taexp-α/1-α(t-s)d/dsx(s)ds,其中α∈(0,1), M(α)是满足M(0)=M(1)=1的归一化常数. 目前, Caputo-Fabrizio(CF)[17-18]分数阶导数已经在扩散建模和质量弹簧-阻尼器系统领域得到广泛应用.

最近, 研究者提出了一个新的分数阶导数概念, 基于将CF分数阶导数公式中的函数核exp{z}替换为函数(一个参数)Eα,1(z), 这个概念被称为定义在Caputo意义上的Atangana-Baleanu分数阶导数(ABC)[9,19]:ABCDαa+x(t)=M(α)/1-α∫taEα,1-α/1-α(t-s)αd/dsx(s)ds,ABC分数阶导数在传热、 变分问题、 混沌理论和经济模型等领域应用广泛.

模糊微分包含在人工智能、 人口动力学、 石油工程、 力学、 医学等领域[20-21]的不确定现象建模方面有许多重要应用. 最早, Hüllermeier[22]基于在知识系统中的应用, 介绍并研究了以下一类模糊微分包含:x′(t)∈[F(t,x(t))]α, α∈[0,1],

x(0)∈[x0]α.Guo等[23]建立了模糊脉冲泛函微分包含的一些存在性结果, 并在模糊总体模型中提供了一个应用; Min等[24]研究了一类隐式模糊微分包含, 并给出了其在钻井石油工程动力学中的应用; Majumdar等[25]讨论了模糊微分包含问题在大气和医学控制论中的应用; Liu等[26]进一步讨论了模糊延迟微分包含; Wu等[27]建立了半线性模糊微分包含的一些存在性结果; Dai等[28]研究了一类通用振子模糊微分方程; Liu等[26]给出了模糊过程、 混合过程和不确定过程的一些基本概念, 并发展了一种模糊微积分, 提出了一类新的模糊微分方程; Hung等[29]研究了Banach空间中一类具有可解算子的模糊微分包含体.

變分不等式理论是优化理论的重要组成部分, 它作为一种数学规划工具广泛应用于建模等许多优化和决策问题中[30-31]. 但在数学优化、 控制理论、 运筹学和博弈论等领域中出现许多决策问题, 其面临着不确定性. 针对这些不确定性, Zadeh[32]首先提出并研究了模糊集的概念. 模糊集理论由于可作为建模不确定问题的有力工具, 也获得了广泛关注. Chang等[33]提出了模糊映射的概念和简单的模糊变分不等式, 而这种模糊变分不等式之所以能引起研究者的广泛关注, 是因为它可以解决如图像处理、 接触力学和动态交通网络等问题, 包括Chang等[34]引入并研究了由模糊映射驱动的向量拟变分不等式; Chang[35]证明了模糊向量拟变分类不等式解的存在性; 在适当的条件下, Huang等[36]应用F-KKM[KG*8]定理研究了一类f互补问题; Tang等[37]研究了有限维空间中具有模糊映射的摄动变分不等式的存在定理.

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(責任编辑: 赵立芹)

收稿日期: 2023-06-02. 网络首发日期: 2024-03-02.

第一作者简介: 李慧敏(1997—), 女, 回族, 硕士研究生, 从事微分方程理论及其应用的研究, E-mail: 1275013458@qq.com.

通信作者简介: 顾海波(1982—), 男, 汉族, 博士, 教授, 从事微分方程理论及其应用的研究, E-mail: hbgu_math@163.com.

基金项目: 国家自然科学基金(批准号: 11961069)、 新疆优秀青年科技人才培训计划项目(批准号: 2019Q022)、 新疆维吾尔自治区自然科学基金(批准号: 2019D01A71)和新疆师范大学青年拔尖人才计划项目(批准号: XJNUQB2022-14).

网络首发地址: https://link.cnki.net/urlid/22.1340.o.20240228.1502.003.

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