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一類由變分不等式驅動的模糊分數階微分包含系統解的存在性

2024-05-15 19:23:13李慧敏顧海波
吉林大學學報(理學版) 2024年2期
關鍵詞:理論研究

李慧敏 顧海波

摘要: 考慮一類動態模糊系統, 該系統由模糊Atangana-Baleanu分數階微分包含和變分不等式組成, 稱為模糊分數階微分變分不等式(FFDVI), 它包括了模糊分數階微分包含和變分不等式兩個領域的研究, 拓寬了模糊環境下的可研究問題, 該模型在同一框架內捕獲了模糊分數微分包含和分數微分變分不等式的期望特征. 利用Krasnoselskii不動點定理, 得到了FFDVI在某些溫和條件下解的存在性.

關鍵詞: Atangana-Baleanu分數階導數; 分數階模糊微分變分不等式; Krasnoselskii不動點定理; 解的存在性

中圖分類號: O175.14文獻標志碼: A文章編號: 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

分數階微積分作為經典整數階微積分的一種自然推廣, 近年來備受關注. 由于整數階微分方程具有一定的局限性, 而分數階模糊微分方程在應用中比整數階模糊微分方程更貼合實際, 因此對分數階模糊變分不等式系統的研究很有必要, 已廣泛應用于電動力學、 生物技術、 空氣動力學、 分布式螺旋槳設計和控制動力系統中[1-8]. 最常用的分數式積分和微分算子是Riemann-Liouville,Hadamard,Grunwald-Letnikov,Caputo和Riesz-Caputo. Atangana和Baleanu[9]引入了一種新的分數階導數, 它包含由Mittag-Leffler函數描述的非局部和非奇異核, Atangana-Baleanu分數階微積分[9-10]是一種強大的數學工具, 可以解釋和描述各種復雜的物理現象、 化學反應過程、 種群動力學行為等問題[11-15].

近年來, Caputo和Fabrizio[16]基于實值函數空間中的冪函數核, 引入了Caputo意義上的分數階導數新定義如下:CFDαa+x(t)=M(α)/1-α∫taexp-α/1-α(t-s)d/dsx(s)ds,其中α∈(0,1), M(α)是滿足M(0)=M(1)=1的歸一化常數. 目前, Caputo-Fabrizio(CF)[17-18]分數階導數已經在擴散建模和質量彈簧-阻尼器系統領域得到廣泛應用.

最近, 研究者提出了一個新的分數階導數概念, 基于將CF分數階導數公式中的函數核exp{z}替換為函數(一個參數)Eα,1(z), 這個概念被稱為定義在Caputo意義上的Atangana-Baleanu分數階導數(ABC)[9,19]:ABCDαa+x(t)=M(α)/1-α∫taEα,1-α/1-α(t-s)αd/dsx(s)ds,ABC分數階導數在傳熱、 變分問題、 混沌理論和經濟模型等領域應用廣泛.

模糊微分包含在人工智能、 人口動力學、 石油工程、 力學、 醫學等領域[20-21]的不確定現象建模方面有許多重要應用. 最早, Hüllermeier[22]基于在知識系統中的應用, 介紹并研究了以下一類模糊微分包含:x′(t)∈[F(t,x(t))]α, α∈[0,1],

x(0)∈[x0]α.Guo等[23]建立了模糊脈沖泛函微分包含的一些存在性結果, 并在模糊總體模型中提供了一個應用; Min等[24]研究了一類隱式模糊微分包含, 并給出了其在鉆井石油工程動力學中的應用; Majumdar等[25]討論了模糊微分包含問題在大氣和醫學控制論中的應用; Liu等[26]進一步討論了模糊延遲微分包含; Wu等[27]建立了半線性模糊微分包含的一些存在性結果; Dai等[28]研究了一類通用振子模糊微分方程; Liu等[26]給出了模糊過程、 混合過程和不確定過程的一些基本概念, 并發展了一種模糊微積分, 提出了一類新的模糊微分方程; Hung等[29]研究了Banach空間中一類具有可解算子的模糊微分包含體.

變分不等式理論是優化理論的重要組成部分, 它作為一種數學規劃工具廣泛應用于建模等許多優化和決策問題中[30-31]. 但在數學優化、 控制理論、 運籌學和博弈論等領域中出現許多決策問題, 其面臨著不確定性. 針對這些不確定性, Zadeh[32]首先提出并研究了模糊集的概念. 模糊集理論由于可作為建模不確定問題的有力工具, 也獲得了廣泛關注. Chang等[33]提出了模糊映射的概念和簡單的模糊變分不等式, 而這種模糊變分不等式之所以能引起研究者的廣泛關注, 是因為它可以解決如圖像處理、 接觸力學和動態交通網絡等問題, 包括Chang等[34]引入并研究了由模糊映射驅動的向量擬變分不等式; Chang[35]證明了模糊向量擬變分類不等式解的存在性; 在適當的條件下, Huang等[36]應用F-KKM[KG*8]定理研究了一類f互補問題; Tang等[37]研究了有限維空間中具有模糊映射的攝動變分不等式的存在定理.

參考文獻

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(責任編輯: 趙立芹)

收稿日期: 2023-06-02. 網絡首發日期: 2024-03-02.

第一作者簡介: 李慧敏(1997—), 女, 回族, 碩士研究生, 從事微分方程理論及其應用的研究, E-mail: 1275013458@qq.com.

通信作者簡介: 顧海波(1982—), 男, 漢族, 博士, 教授, 從事微分方程理論及其應用的研究, E-mail: hbgu_math@163.com.

基金項目: 國家自然科學基金(批準號: 11961069)、 新疆優秀青年科技人才培訓計劃項目(批準號: 2019Q022)、 新疆維吾爾自治區自然科學基金(批準號: 2019D01A71)和新疆師范大學青年拔尖人才計劃項目(批準號: XJNUQB2022-14).

網絡首發地址: https://link.cnki.net/urlid/22.1340.o.20240228.1502.003.

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