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アトラクティブポイントを用いた不動点理論に基づく非線形関数解析・非線形問題の究明

Research Project

Project/Area Number 19K03582
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeMulti-year Fund
Section一般
Review Section Basic Section 12020:Mathematical analysis-related
Research InstitutionTokyo Woman's Christian University (2020-2023)
University of Yamanashi (2019)

Principal Investigator

厚芝 幸子  東京女子大学, 現代教養学部, 教授 (20327761)

Project Period (FY) 2019-04-01 – 2025-03-31
Project Status Granted (Fiscal Year 2023)
Budget Amount *help
¥4,290,000 (Direct Cost: ¥3,300,000、Indirect Cost: ¥990,000)
Fiscal Year 2022: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
Fiscal Year 2021: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2020: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Fiscal Year 2019: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
Keywords不動点 / 単調写像 / 単調点列 / 非拡大写像 / 不動点近似 / バナッハ空間 / 非拡大半群 / アトラクティブポイント / 凸解析学 / 最適化理論 / 不動点理論 / 非線形問題 / 非線形関数解析学
Outline of Research at the Start

不動点理論は非線形数学の中で中核をなし,非線形問題は各々何らかの非線形写像の不動点の問題と捉えられ,その非線形問題の解を求める事は不動点近似に繋がる.未解決の非線形問題・新しいタイプの非線形問題へ繋がる事から,不動点を一般化したattractive pointを考慮した不動点理論の重要性が分かってきた.そこで,不動点・attractive pointも考慮して,不動点定理・不動点近似に関連する非線形関数解析学を研究する.特に最近非線形問題に強く結びつくことがわかってきた写像・作用素に対して,基礎的性質の研究から始め,不動点近似法の適用を研究する.それを基に非線形問題の解を求める問題を研究する.

Outline of Annual Research Achievements

2023年度は,凸解析・不動点理論を介した非線形関数解析学のうち, 収束定理の構造的考察を中心にすすめ, 特に点列近似法による写像族の共通不動点および共通アトラクティブポイントへの収束定理の研究をした. 未解決事項が多い, 軌道自体の収束定理に繋がるorderedバナッハ空間における共通不動点への収束定理を中心に研究した.それを基にあらたに使いやすい不動点近似法を探究した. 最近注目されている最適化問題や均衡問題等の非線形問題の課題・問題点を的確に把握し, 数学的(関数解析学的)に再構成して問題点を洗い出すことから始めた. 非線形問題への結びつきを考慮して1つの写像だけなく, 特に任意有限個のmonotone nonexpansive mappingの族の共通不動点集合, 共通アトラクティブポイント集合を考察し, それらの相互関係および基礎性質も探究し, 以下のような研究成果を得られた. 主たる成果として, ordered一様凸バナッハ空間の有界部分集合において, 平均(average)のアイディアを用いたMann型の不動点近似法により任意有限個のmonotone onexpansive mappingからなる族の共通不動点, 共通アトラクティブポイント集合への収束定理を得た. さらにW-mappingのアイディアを用いて非可換な任意有限個のmonotone onexpansive mappingからなる族の共通不動点への収束定理も得られた. またmonotone nonexpansive mappingからなる半群に対する収束定理も得た. 得られた収束定理は他のタイプの収束定理にも用いられる考え・補題を与える研究成果となった. 軌道収束の構造に関する重要な研究成果を得られ, 最近の非線形問題への足掛かりとなる成果となり, それらは国際的論文誌に掲載または掲載決定された.

Current Status of Research Progress
Current Status of Research Progress

2: Research has progressed on the whole more than it was originally planned.

Reason

コロナ禍の影響で開催日程が延期になっていた研究集会が多かったため, 研究集会で論文発表することが遅くなったものもあったが, 以下のようにさらに進展した研究成果も併せて論文発表・議論できたこともさらに研究を進展させることにつながった. 具体的には, 研究代表者のこれまでの準備や知識もとにして, さらに多くの文献収集やその整理, ならびに, この研究に興味を持っている研究者と対面だけでなくてオンラインで多くの研究討論をしたことなどが功を奏して, 研究を進められた. さらに2023年度は対面で議論をすすめたり, 評価を得られたことも理由となって順調な進展を得た.

Strategy for Future Research Activity

2023年度までに得られた研究成果及び既存の不動点近似に関する成果の考えを基に以下の研究をすすめる: 1.最近注目されつつある非線形問題および他の不動点近似にも有効であるとわかってきた写像であるmonotone nonexpansive mapping, monotone generalized-nonspreading mapping, Suzuki-type nonexpansive mapping, (α,β)-nonexpansive mappingおよびそれらに関連する写像, それら写像族の(共通)不動点集合, 共通attractive point集合および共通acute point集合について基礎性質から探究し, さらに相互関係について系統立てた成果をだせるように研究をすすめる. 2. ordered狭義凸バナッハ空間のコンパクト凸集合において, またorderd一様凸バナッハ空間において, 1で記した写像族の共通不動点へ, および共通attractive pointへの収束定理について, 近接点法,マン型,ハルパーン型, CQ型を改良した不動点近似法を用いる形で研究する. 3.1.2の研究をもとに測地空間における不動点へ,attractive pointへの収束定理の研究をすすめる. 特に写像族に対する収束定理の研究は 測地空間ではまだあまり進められていないが有用性が認識されはじめたので測地空間おける写像族に関する基礎性質の研究からはじめて, 写像族に対する収束定理の研究をすすめる. 4. 不動点やattractive pointをもとめる点列近似法に関する理論の発展とその非線形問題への応用の研究のうち, 特に融合問題に関して, 1~3の研究成果を基にして, 使いやすい近似法を探求する.

Report

(5 results)
  • 2023 Research-status Report
  • 2022 Research-status Report
  • 2021 Research-status Report
  • 2020 Research-status Report
  • 2019 Research-status Report
  • Research Products

    (55 results)

All 2024 2023 2022 2021 2020 2019

All Journal Article (33 results) (of which Peer Reviewed: 10 results,  Open Access: 7 results) Presentation (22 results) (of which Int'l Joint Research: 7 results,  Invited: 3 results)

  • [Journal Article] Weak convergence theorems for a finite family of monotone nonexpansive mappings in Banach spaces2024

    • Author(s)
      Sachiko Atsushiba
    • Journal Title

      Fixed Point Theory

      Volume: -

    • Related Report
      2023 Research-status Report
    • Peer Reviewed
  • [Journal Article] Weak and strong convergence theorems for nonexpansive semigroups in ordered Banach spaces2024

    • Author(s)
      Sachiko Atsushiba
    • Journal Title

      Nonlinear Functional Analisis and Applications

      Volume: -

    • Related Report
      2023 Research-status Report
  • [Journal Article] Convergence theorems for families of monotone nonexpansive mappings in uniformly convex Banach spaces2024

    • Author(s)
      Sachiko Atsushiba
    • Journal Title

      Fixed Point Theory and Optimizations

      Volume: -

    • Related Report
      2023 Research-status Report
  • [Journal Article] Weak and strong convergence theorems for families of nonexpansive mappings in Banach spaces2023

    • Author(s)
      Sachiko Atsushiba
    • Journal Title

      Linear and Nonlinear Analysis

      Volume: 9 Pages: 175-186

    • Related Report
      2023 Research-status Report
    • Peer Reviewed
  • [Journal Article] Weak and strong convergence theorems for families of nonexpansive mappings in Banach spaces2023

    • Author(s)
      Sachiko Atsushiba
    • Journal Title

      日本大学経済学部紀要

      Volume: 98 Pages: 179-192

    • Related Report
      2023 Research-status Report
    • Open Access
  • [Journal Article] Convergence theorems for monotone generalized nonexpansive mappings2023

    • Author(s)
      Sachiko Atsushiba
    • Journal Title

      実解析学シンポジウム2023報告集

      Volume: 54 Pages: 98-103

    • Related Report
      2023 Research-status Report
    • Open Access
  • [Journal Article] Convergence theorems for families of monotone Lipschitzian mappings in ordered Banach spaces2023

    • Author(s)
      Sachiko Atsushiba
    • Journal Title

      RIMS kokyuroku, Theory of function spaces and related topics

      Volume: 2260 Pages: 68-76

    • Related Report
      2023 Research-status Report
  • [Journal Article] Convergence theorems for families of monotone nonexpansive mappings2023

    • Author(s)
      Sachiko Atsushiba
    • Journal Title

      RIMS Kokyuroku, Study on Nonlinear Analysis and Convex Analysis

      Volume: 2240

    • Related Report
      2022 Research-status Report
  • [Journal Article] Attractive points, acute points and convergence theorems for families of nonlinear mappings2023

    • Author(s)
      Sachiko Atsushiba
    • Journal Title

      RIMS Kokyuroku, Theory of function spaces and related topics

      Volume: 2250

    • Related Report
      2022 Research-status Report
  • [Journal Article] Convergence theorems for families of monotone Lipschitzian mappings in ordered Banach spaces2023

    • Author(s)
      Sachiko Atsushiba
    • Journal Title

      RIMS Kokyuroku, Theory of function spaces and related topics

      Volume: - Pages: 1-8

    • Related Report
      2022 Research-status Report
  • [Journal Article] Weak convergence theorems for monotone nonexpansive semigroups2023

    • Author(s)
      Sachiko Atsushiba
    • Journal Title

      Fixed Point Theory

      Volume: -

    • Related Report
      2022 Research-status Report
    • Peer Reviewed
  • [Journal Article] Convergence theorems for finite families of monotone nonexpansive mappings2023

    • Author(s)
      Sachiko Atsushiba
    • Journal Title

      Nonlinear Functional Analysis

      Volume: -

    • Related Report
      2022 Research-status Report
  • [Journal Article] Weak convergence theorems for finite families of monotone nonexpansive mappings2023

    • Author(s)
      Sachiko Atsushiba
    • Journal Title

      Fixed Point Theory and Applications

      Volume: -

    • Related Report
      2022 Research-status Report
  • [Journal Article] Convergence theorems for monotone nonexpansive mappings in Banach spaces2022

    • Author(s)
      Sachiko Atsushiba
    • Journal Title

      Linear and Nonlinear Analysis

      Volume: 8 Pages: 155-162

    • Related Report
      2022 Research-status Report 2021 Research-status Report
    • Peer Reviewed
  • [Journal Article] Weak and strong convergence theorems for families of monotone nonexpansive mappings2022

    • Author(s)
      Sachiko Atsushiba
    • Journal Title

      実解析シンポジウム2022報告集

      Volume: 53 Pages: 24-29

    • Related Report
      2022 Research-status Report
  • [Journal Article] Weak and strong convergence theorems for monotone nonexpansive mappings2022

    • Author(s)
      Sachiko Atsushiba
    • Journal Title

      RIMS kokyuroku, Study on Nonlinear Analysis and Convex Analysis

      Volume: 2214

    • Related Report
      2021 Research-status Report
  • [Journal Article] Convergence theorems for families of monotone nonexpansive mappings in Banach spaces2022

    • Author(s)
      Sachiko Atsushiba
    • Journal Title

      FIxed Point Theory

      Volume: -

    • Related Report
      2021 Research-status Report
    • Peer Reviewed
  • [Journal Article] On the asymptotic behavior of families of nonlinear mappings and some weak convergence theorems2021

    • Author(s)
      Sachiko Atsushiba
    • Journal Title

      RIMS Kokyuroku, Study on Nonlinear Analysis and Convex Analysis

      Volume: 2194 Pages: 152-163

    • NAID

      120007165873

    • Related Report
      2021 Research-status Report
  • [Journal Article] Fixed point property and convergence theorems for hybrid-type mappings2021

    • Author(s)
      Sachiko Atsushiba
    • Journal Title

      RIMS kokyuroku, Study on Nonlinear Analysis and Convex Analysis

      Volume: 2190

    • NAID

      120007165937

    • Related Report
      2021 Research-status Report
  • [Journal Article] Weak convergence theorems for 2-generalized nonspreading mappings2021

    • Author(s)
      Sachiko Atsushiba
    • Journal Title

      Journal of Nonlinear and Convex Analysis

      Volume: -

    • Related Report
      2020 Research-status Report
    • Peer Reviewed
  • [Journal Article] Fixed point property and convergence theorems for hybrid-type mappings2021

    • Author(s)
      Sachiko Atsushiba
    • Journal Title

      非線形解析学と凸解析学の研究(Nonlinear analysis and convex analysis), RIMS Kokyuroku

      Volume: -

    • NAID

      120007165937

    • Related Report
      2020 Research-status Report
    • Open Access
  • [Journal Article] Attractive points, acute points and convergence theorems for families of nonlinear mappings2021

    • Author(s)
      Sachiko Atsushiba
    • Journal Title

      関数空間論とその周辺(Theory of function spaces and related topics ), RIMS Kokyuroku

      Volume: -

    • Related Report
      2020 Research-status Report
    • Open Access
  • [Journal Article] Weak convergence theorems for generic generalized-hybrid-type mappings2021

    • Author(s)
      Sachiko Atsushiba
    • Journal Title

      非線形解析学と凸解析学の研究(Nonlinear analysis and convex analysis), RIMS Kokyuroku

      Volume: -

    • Related Report
      2020 Research-status Report
  • [Journal Article] Convergence of orbits of nonexpansive semigroups in Banach spaces2020

    • Author(s)
      Sachiko Atsushiba
    • Journal Title

      Journal of Nonlinear and Convex Analysis

      Volume: 21

    • Related Report
      2020 Research-status Report
    • Peer Reviewed
  • [Journal Article] Convergence of orbits of nonexpansive semigroups in Banach spaces2020

    • Author(s)
      Sachiko Atsushiba
    • Journal Title

      Journal of Nonlinear and Convex Analaysis

      Volume: 21

    • Related Report
      2019 Research-status Report
    • Peer Reviewed
  • [Journal Article] Fixed point property and convergence theorems for hybird-type mappings2020

    • Author(s)
      Sachiko Atsushiba
    • Journal Title

      非線形解析学と凸解析学の研究(Nonlinear analysis and convex analysis), RIMS Kokyuroku,

      Volume: -

    • Related Report
      2019 Research-status Report
  • [Journal Article] Attractive points, acute points and convergence theorems for families of nonlinear mappings2020

    • Author(s)
      Sachiko Atsushiba
    • Journal Title

      RIMS Kokyuroku 関数空間とその周辺

      Volume: -

    • Related Report
      2019 Research-status Report
  • [Journal Article] Fixed point property for normally 2-generalized hybrid mappings2019

    • Author(s)
      Sachiko Atsushiba
    • Journal Title

      Journal of Nonlinear and Variational Analysis,

      Volume: 3 Pages: 149-158

    • Related Report
      2019 Research-status Report
    • Peer Reviewed
  • [Journal Article] Nonlinear ergodic theorems for normally 2-generalized hybrid sequences2019

    • Author(s)
      Sachiko Atsushiba
    • Journal Title

      Journal of Nonlinear and Convex Analaysis

      Volume: 20 Pages: 251-264

    • Related Report
      2019 Research-status Report
    • Peer Reviewed
  • [Journal Article] Attractive point and convergence theorems for hybrid-type sequences2019

    • Author(s)
      Sachiko Atsushiba
    • Journal Title

      非線形解析学と凸解析学の研究(Nonlinear analysis and convex analysis), RIMS Kokyuroku,

      Volume: 2112 Pages: 46-53

    • Related Report
      2019 Research-status Report
    • Open Access
  • [Journal Article] Approximation of attractive points of nonlinear mappings, --Common attractive points common acute points, common fixed points and convergence theorema--,2019

    • Author(s)
      Sachiko Atsushiba
    • Journal Title

      非線形解析学と凸解析学の研究(Nonlinear analysis and convex analysis), RIMS Kokyuroku,

      Volume: 2114 Pages: 103-110

    • Related Report
      2019 Research-status Report
    • Open Access
  • [Journal Article] Convergence theorems for generalized hybrid-type sequences and some nonlinear mappings,2019

    • Author(s)
      Sachiko Atsushiba
    • Journal Title

      関数空間の一般化とその周辺, RIMS Kokyuroku,

      Volume: 2143

    • NAID

      40022188905

    • Related Report
      2019 Research-status Report
    • Open Access
  • [Journal Article] Fixed points, absolute fixed points and convergence theorems for nonlinear mappings2019

    • Author(s)
      Sachiko Atsushiba
    • Journal Title

      実解析シンポジウム2019 報告集

      Volume: 2019 Pages: 61-65

    • Related Report
      2019 Research-status Report
  • [Presentation] Approximation of fixed points of monotone nonexpansive-type mappings2024

    • Author(s)
      Sachiko Atsushiba
    • Organizer
      日本数学会年会2024
    • Related Report
      2023 Research-status Report
  • [Presentation] Attractive points, fixed points, and convergence theorems for families of monotone nonexpansive mappings2023

    • Author(s)
      Sachiko Atsushiba
    • Organizer
      The 14th International Conference on Fixed Point Theory and its Applications (ICFPTA2023)
    • Related Report
      2023 Research-status Report
    • Int'l Joint Research
  • [Presentation] Attractive points and weak convergence theorems for monotone nonexpansive mappings2023

    • Author(s)
      Sachiko Atsushiba
    • Organizer
      The 11th Asian Conference on Fixed Point Theory and Optimization 2023 (ACFPTO2023)
    • Related Report
      2023 Research-status Report
    • Int'l Joint Research
  • [Presentation] Convergence theorems for monotone generalized nonexpansive mappings2023

    • Author(s)
      Sachiko Atsushiba
    • Organizer
      実解析学シンポジウム2023
    • Related Report
      2023 Research-status Report
  • [Presentation] Attractive points, fixed points and convergence theorems for monotone Lipschitzian mappings2023

    • Author(s)
      Sachiko Atsushiba
    • Organizer
      日本数学会秋季総合分科会2023
    • Related Report
      2023 Research-status Report
  • [Presentation] Convergence theorems for families of monotone Lipschitzian mappings in ordered Banach spaces2023

    • Author(s)
      Sachiko Atsushiba
    • Organizer
      RIMS, Theory of function spaces and related topics
    • Related Report
      2022 Research-status Report
  • [Presentation] Convergence theorems for families of monotone nonexpansive mappings in ordered Banach spaces2023

    • Author(s)
      Sachiko Atsushiba
    • Organizer
      2023 日本数学会年会実函数論分科会
    • Related Report
      2022 Research-status Report
  • [Presentation] Weak and strong convergence theorems for families of monotone nonexpansive mappings2022

    • Author(s)
      Sachiko Atsushiba
    • Organizer
      実解析シンポジウム2022
    • Related Report
      2022 Research-status Report
  • [Presentation] Convergence theorems for families of monotone nonexpansive mappings2022

    • Author(s)
      Sachiko Atsushiba
    • Organizer
      RIMS, Study on Nonlinear Analysis and Convex Analysis
    • Related Report
      2022 Research-status Report
  • [Presentation] Convergence theorems for monotone nonexpansive mappings in ordered uniformly convex Banach spaces2022

    • Author(s)
      Sachiko Atsushiba
    • Organizer
      The 3rd International Workshop on Applied Nonlinear Analysisonal (IWANA2022, Thailand)
    • Related Report
      2022 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Convergence theorems for monotone nonexpansive mappings2022

    • Author(s)
      Sachiko Atsushiba
    • Organizer
      2022年日本数学会年会実函数論分科会
    • Related Report
      2021 Research-status Report
  • [Presentation] Weak and strong convergence theorems for monotone nonexpansive mappings2021

    • Author(s)
      Sachiko Atsushiba
    • Organizer
      京都大学数理解析研究所(RIMS) 共同研究 (公開型) 「非線形解析学と凸解析学の研究」
    • Related Report
      2021 Research-status Report
  • [Presentation] Attractive points and convergence theorems for generic 2-generalized hybrid mappings2021

    • Author(s)
      Sachiko Atsushiba
    • Organizer
      2021年日本数学会秋季総合分科会実函数論分科会
    • Related Report
      2021 Research-status Report
  • [Presentation] Convergence theorems for quasi-nonexpansive-type mappings2021

    • Author(s)
      Sachiko Atsushiba
    • Organizer
      京都大学数理解析研究所(RIMS) 共同研究 (公開型) 「非線形解析学と凸解析学の研究」
    • Related Report
      2020 Research-status Report
  • [Presentation] Fixed points and convergence of orbits of nonexpansive semigroups2020

    • Author(s)
      Sachiko Atsushiba
    • Organizer
      2020年日本数学会秋季総合分科会実函数論分科会
    • Related Report
      2020 Research-status Report
  • [Presentation] Fixed point theorems and weak convergence theorems for normally 2-generalized hybrid mappings2019

    • Author(s)
      Sachiko Atsushiba
    • Organizer
      The 13th International Conference on Fixed Point Theory and Its Applications (ICFPTA2019)
    • Related Report
      2019 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Fixed point theorems and convergence theorems for hybrid type sequences2019

    • Author(s)
      Sachiko Atsushiba
    • Organizer
      NACA-ICOTA2019(
    • Related Report
      2019 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Fixed point theorems for hybrid-type mappings and convergence theorems for some sequences2019

    • Author(s)
      Sachiko Atsushiba
    • Organizer
      The 2nd International Workshop on Applied Nonlinear Analysis(Iwana2019)
    • Related Report
      2019 Research-status Report
    • Int'l Joint Research
  • [Presentation] Fixed point property and convergence theorems for hybird-type mappings2019

    • Author(s)
      Sachiko Atsushiba
    • Organizer
      RIMS workshop on Nonlinear Analysis and Convex Analyais
    • Related Report
      2019 Research-status Report
    • Int'l Joint Research
  • [Presentation] Fixed point property and convergence theorems for iterative sequences2019

    • Author(s)
      Sachiko Atsushiba
    • Organizer
      2019年度日本数学会秋季総合分科会実函数論分科会
    • Related Report
      2019 Research-status Report
  • [Presentation] Fixed points, absolute fixed points and convergence theorems for nonlinear mappings,2019

    • Author(s)
      Sachiko Atsushiba
    • Organizer
      実解析シンポジウム2019
    • Related Report
      2019 Research-status Report
  • [Presentation] Attractive points, acute points and convergence theorems for families of nonlinear mappings2019

    • Author(s)
      Sachiko Atsushiba
    • Organizer
      RIMS 共同研究(公開型)  関数空間論とその周辺
    • Related Report
      2019 Research-status Report

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Published: 2019-04-18   Modified: 2024-12-25  

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