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Convex analysis on complete geodesic spaces using the techniques of fixed point theory

Research Project

Project/Area Number 15K05007
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeMulti-year Fund
Section一般
Research Field Foundations of mathematics/Applied mathematics
Research InstitutionToho University

Principal Investigator

KIMURA Yasunori  東邦大学, 理学部, 教授 (20313447)

Co-Investigator(Kenkyū-buntansha) 高阪 史明  東海大学, 理学部, 教授 (20434003)
佐藤 健治  玉川大学, 工学部, 教授 (70307164)
Project Period (FY) 2015-04-01 – 2019-03-31
Project Status Completed (Fiscal Year 2018)
Budget Amount *help
¥4,420,000 (Direct Cost: ¥3,400,000、Indirect Cost: ¥1,020,000)
Fiscal Year 2018: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2017: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Fiscal Year 2016: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Fiscal Year 2015: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Keywords測地距離空間 / 凸解析学 / 不動点理論 / リゾルベント / 解析学 / 非線形解析 / 凸解析
Outline of Final Research Achievements

This study aims to develop the theory of convex analysis in the setting of complete geodesic spaces, which has been traditionally studied in function spaces. In particular, we define a resolvent operator for a convex function on a geodesic space, and by applying fixed point theory to this operator, we obtain approximation results for various types of nonlinear problems including convex minimization.
By choosing appropriate perturbation functions, we obtained resolvents having suitable properties for the curvature of the space. This technique enables us to apply a number of results in fixed point theory to the resolvent operators, and moreover, we succeeded to find new results about fixed point approximation by the new properties of our operators.

Academic Significance and Societal Importance of the Research Achievements

関数空間を舞台として研究されてきた従来の凸解析学はバナッハ空間における不動点理論と両輪をなし、互いに影響しあいながら発展を遂げてきた。その一方で、今世紀初頭から研究が精力的に行われている測地距離空間における不動点理論と比較して、測地距離空間上の凸解析学の研究は発展が望まれている分野であった。本研究課題はその要望にこたえるものとしての学術的意義がある。
また、この研究によって、測地距離空間上で定義される種々の非線形問題の解法に関する知見が得られている。非線形問題は現実社会における問題を数理的視点で定式化したものであり、これらの問題の解法が解明されることは社会的に大きな意義があると考えられる。

Report

(5 results)
  • 2018 Annual Research Report   Final Research Report ( PDF )
  • 2017 Research-status Report
  • 2016 Research-status Report
  • 2015 Research-status Report
  • Research Products

    (41 results)

All 2019 2018 2017 2016 2015

All Journal Article (13 results) (of which Int'l Joint Research: 1 results,  Peer Reviewed: 13 results,  Open Access: 8 results,  Acknowledgement Compliant: 4 results) Presentation (28 results) (of which Int'l Joint Research: 27 results,  Invited: 18 results)

  • [Journal Article] Two modified proximal point algorithms in geodesic spaces with curvature bounded above2019

    • Author(s)
      Yasunori Kimura and Fumiaki Kohsaka
    • Journal Title

      Rendiconti del Circolo Matematico di Palermo Series 2

      Volume: 68-1 Issue: 1 Pages: 83-104

    • DOI

      10.1007/s12215-018-0340-3

    • Related Report
      2018 Annual Research Report
    • Peer Reviewed / Open Access
  • [Journal Article] Resolvents of convex functions in complete geodesic metric spaces with negative curvature2019

    • Author(s)
      T. Kajimura and Y. Kimura
    • Journal Title

      Journal of Fixed Point Theory and Applications

      Volume: 印刷中 Issue: 1

    • DOI

      10.1007/s11784-019-0667-1

    • Related Report
      2018 Annual Research Report
    • Peer Reviewed / Open Access
  • [Journal Article] Convergence to a fixed point of a balanced mapping by the Mann algorithm in a Hadamard space2018

    • Author(s)
      T. Hasegawa and Y. Kimura
    • Journal Title

      Linear and Nonlinear Analysis

      Volume: 4 Pages: 405-412

    • Related Report
      2018 Annual Research Report
    • Peer Reviewed / Open Access
  • [Journal Article] Existence and approximation of fixed points of vicinal mappings in geodesic spaces2018

    • Author(s)
      Fumiaki Kohsaka
    • Journal Title

      Pure and Applied Functional Analysis

      Volume: 3 Pages: 91-106

    • Related Report
      2017 Research-status Report
    • Peer Reviewed / Open Access
  • [Journal Article] Equilibrium problems and their resolvents in Hadamard spaces2018

    • Author(s)
      Yasunori Kimura and Yusuke Kishi
    • Journal Title

      Proceedings of the Fifth Asian Conference on Nonlinear Analysis and Optimization (印刷中)

      Volume: -

    • Related Report
      2017 Research-status Report
    • Peer Reviewed
  • [Journal Article] An approximation method with nonsummable errors for convex minimization problems2017

    • Author(s)
      Yasunori Kimura
    • Journal Title

      Linear and Nonlinear Analysis

      Volume: 3 Pages: 401-407

    • Related Report
      2017 Research-status Report
    • Peer Reviewed / Open Access
  • [Journal Article] The existence of nonexpansive extension on a geodesic space with constant curvature2017

    • Author(s)
      Y. Kimura and K. Sato
    • Journal Title

      Journal of Nonlinear and Convex Analysis

      Volume: 18 Pages: 123-136

    • Related Report
      2016 Research-status Report
    • Peer Reviewed / Open Access / Acknowledgement Compliant
  • [Journal Article] Two modified proximal point algorithms for convex functions in Hadamard spaces2016

    • Author(s)
      Y. Kimura and F. Kohsaka
    • Journal Title

      Linear and Nonlinear Analysis

      Volume: 2 Pages: 69-86

    • Related Report
      2016 Research-status Report
    • Peer Reviewed / Open Access / Acknowledgement Compliant
  • [Journal Article] Iterative methods for nonexpansive mappings on Hadamard spaces and their coefficient conditions2016

    • Author(s)
      Y. Kimura and H. Wada
    • Journal Title

      Linear and Nonlinear Analysis

      Volume: 2 Pages: 253-261

    • Related Report
      2016 Research-status Report
    • Peer Reviewed / Open Access / Acknowledgement Compliant
  • [Journal Article] Spherically nonspreading mappings in geodesic spaces with curvature bounded above by one2016

    • Author(s)
      Y. Kimura and F. Kohsaka
    • Journal Title

      Journal of Fixed Point Theory and Applications

      Volume: 18 Issue: 1 Pages: 93-115

    • DOI

      10.1007/s11784-015-0267-7

    • Related Report
      2015 Research-status Report
    • Peer Reviewed
  • [Journal Article] A projection method for approximating fixed points of quasinonexpansive mappings in Hadamard spaces2016

    • Author(s)
      S. Huang and Y. Kimura
    • Journal Title

      Fixed Point Theory and Applications

      Volume: 2016:36 Issue: 1 Pages: 1-13

    • DOI

      10.1186/s13663-016-0523-6

    • Related Report
      2015 Research-status Report
    • Peer Reviewed / Int'l Joint Research / Acknowledgement Compliant
  • [Journal Article] Halpern type iteration with multiple anchor points in a Hadamard space2015

    • Author(s)
      Y. Kimura and H. Wada
    • Journal Title

      Journal of Inequalities and Applications

      Volume: 182 Issue: 1 Pages: 1-11

    • DOI

      10.1186/s13660-015-0693-5

    • Related Report
      2015 Research-status Report
    • Peer Reviewed
  • [Journal Article] Approximation of a common fixed point in a geodesic space with curvature bounded above2015

    • Author(s)
      Y. Kimura and K. Sato
    • Journal Title

      Journal of Nonlinear and Convex Analysis

      Volume: 16 Pages: 2227-2234

    • Related Report
      2015 Research-status Report
    • Peer Reviewed
  • [Presentation] Proximal point argorithm and set-valued analysis2019

    • Author(s)
      Y. Kimura
    • Organizer
      International Workshop on Fixed Point Theory and its Applications (IWFPTA-2019)
    • Related Report
      2018 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Resolvent of convex functions and the curvature of geodesic spaces2019

    • Author(s)
      Y. Kimura
    • Organizer
      International Workshop on Fixed Point Theory and its Applications (IWFPTA-2019)
    • Related Report
      2018 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Asymptotic behavior of resolvent at infinity on CAT(1) spaces2019

    • Author(s)
      K. Shindo and Y. Kimura
    • Organizer
      International Workshop on Fixed Point Theory and its Applications (IWFPTA-2019)
    • Related Report
      2018 Annual Research Report
    • Int'l Joint Research
  • [Presentation] Resolvents for convex functions defined on geodesic spaces and its applications,2018

    • Author(s)
      Y. Kimura
    • Organizer
      International Workshop on Nonlinear Analysis and its Applications
    • Related Report
      2018 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Resolvents on complete geodesic spaces and iterative schemes2018

    • Author(s)
      Y. Kimura
    • Organizer
      The 10th Asian Conference on Fixed Point Theory and Optimizations (ACFPTO2018)
    • Related Report
      2018 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Convex minimization problems on complete geodesic spaces2018

    • Author(s)
      Y. Kimura
    • Organizer
      The 6th Asian Conference on Nonlinear Analysis and Optimization (NAO-Asia2018)
    • Related Report
      2018 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] The Halpern iteration procedure with two kinds of mappings in CAT(1) spaces2018

    • Author(s)
      K. Nakagawa and Y. Kimura
    • Organizer
      The 10th Asian Conference on Fixed Point Theory and Optimizations (ACFPTO2018)
    • Related Report
      2018 Annual Research Report
    • Int'l Joint Research
  • [Presentation] The CQ projection method by a finite number of mappings in CAT(1) spaces2018

    • Author(s)
      Y. Kimura and S. Naiki
    • Organizer
      The 10th Asian Conference on Fixed Point Theory and Optimizations (ACFPTO2018)
    • Related Report
      2018 Annual Research Report
    • Int'l Joint Research
  • [Presentation] The proximal point algorithm in complete geodesic spaces with negative curvature2018

    • Author(s)
      T. Kajimura and Y. Kimura
    • Organizer
      The 10th Asian Conference on Fixed Point Theory and Optimizations (ACFPTO2018)
    • Related Report
      2018 Annual Research Report
    • Int'l Joint Research
  • [Presentation] terative sequences for a finite number of resolvent operators on a complete geodesic space2018

    • Author(s)
      K. Kasahara and Y. Kimura
    • Organizer
      The 10th Asian Conference on Fixed Point Theory and Optimizations (ACFPTO2018)
    • Related Report
      2018 Annual Research Report
    • Int'l Joint Research
  • [Presentation] Iterative methods for balanced mappings on Hadamard spaces2018

    • Author(s)
      T. Hasegawa and Y. Kimura
    • Organizer
      The 10th Asian Conference on Fixed Point Theory and Optimizations (ACFPTO2018)
    • Related Report
      2018 Annual Research Report
    • Int'l Joint Research
  • [Presentation] Convex minimization problems and resonvents on geodesic spaces2017

    • Author(s)
      Y. Kimura
    • Organizer
      2017 Symposium on Mathematical Programming and Game Theory
    • Place of Presentation
      Indian Statistical Institute, Delhi, India
    • Year and Date
      2017-01-09
    • Related Report
      2016 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Convex functions defined on geodesic spaces and approximation of their minimizers2017

    • Author(s)
      Yasunori Kimura
    • Organizer
      The 10th Anniversary International Conference on Nonlinear Analysis and Convex Analysis
    • Related Report
      2017 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Approximation of minimizers of convex functions defined on a complete geodesic spaces2017

    • Author(s)
      Yasunori Kimura
    • Organizer
      The 11th ISAAC Congress
    • Related Report
      2017 Research-status Report
    • Int'l Joint Research
  • [Presentation] The proximal point algorithm and its variants for convex functions in geodesic metric spaces2017

    • Author(s)
      Fumiaki Kohsaka
    • Organizer
      The 10th Anniversary International Conference on Nonlinear Analysis and Convex Analysis
    • Related Report
      2017 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Approximating minimizers of convex functions in geodesic metric spaces with curvature bounded above2017

    • Author(s)
      Fumiaki Kohsaka
    • Organizer
      XII International Symposium on Generalized Convexity and Monotonicity
    • Related Report
      2017 Research-status Report
    • Int'l Joint Research
  • [Presentation] The resolvents of convex functions in geodesic metric spaces2017

    • Author(s)
      Fumiaki Kohsaka
    • Organizer
      The Workshop Convex Analysis in CAT(k) Spaces
    • Related Report
      2017 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Fixed point problems for spherically nonspreading mappings and vicinal mappings in CAT(1) spaces2017

    • Author(s)
      Fumiaki Kohsaka
    • Organizer
      The Workshop Convex Analysis in CAT(k) Spaces
    • Related Report
      2017 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Minimizing convex functions in geodesic metric spaces with curvature bounded above2016

    • Author(s)
      F. Kohsaka
    • Organizer
      The International Conference on New Trends in Optimization and Variational Analysis for Applications
    • Place of Presentation
      Quy Nhon University, Quy Nhon, Vietnam
    • Year and Date
      2016-12-08
    • Related Report
      2016 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Resolvent of convex functions defined on a complete geodesic space2016

    • Author(s)
      Y. Kimura
    • Organizer
      International Conference on Differential Geometry, Algebra and Analysis (ICDGAA-16)
    • Place of Presentation
      Jamia Millia Islamia, Delhi, India
    • Year and Date
      2016-11-15
    • Related Report
      2016 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Spherically nonspreading mappings in geodesic metric spaces with curvature bounded above2016

    • Author(s)
      F. Kohsaka
    • Organizer
      The 5th Asian Conference on Nonlinear Analysis and Optimization (NAO-Asia2016)
    • Place of Presentation
      朱鷺メッセ(新潟県新潟市)
    • Year and Date
      2016-08-05
    • Related Report
      2016 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Resolvents of convex functions and their properties in geodesic spaces2016

    • Author(s)
      Y. Kimura
    • Organizer
      The 5th Asian Conference on Nonlinear Analysis and Optimization (NAO-Asia2016)
    • Place of Presentation
      朱鷺メッセ(新潟県新潟市)
    • Year and Date
      2016-08-02
    • Related Report
      2016 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Resolvents of convex functions in a complete geodesic space with curvature bounded above2016

    • Author(s)
      Y. Kimura
    • Organizer
      The 9th Asian Conference on Fixed Point Theory and Optimizations (ACFPTO2016)
    • Place of Presentation
      King Mongkut's University of Technology Thonburi, Bangkok, Thailand
    • Year and Date
      2016-05-19
    • Related Report
      2016 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Existence and approximation of minimizers of convex functions in geodesic metric spaces2016

    • Author(s)
      F. Kohsaka
    • Organizer
      The 9th Asian Conference on Fixed Point Theory and Optimizations (ACFPTO2016)
    • Place of Presentation
      King Mongkut's University of Technology Thonburi, Bangkok, Thailand
    • Year and Date
      2016-05-19
    • Related Report
      2016 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Iterative methods for common fixed point problems on a geodesic space2015

    • Author(s)
      Y, Kimura
    • Organizer
      VII Symposium on Nonlinear Analysis
    • Place of Presentation
      icolaus Copernicus University, Torun (Poland)
    • Year and Date
      2015-09-15
    • Related Report
      2015 Research-status Report
    • Int'l Joint Research
  • [Presentation] 測地距離空間における近似点列の計算誤差2015

    • Author(s)
      木村泰紀
    • Organizer
      RIMS 研究集会「非線形解析学と凸解析学の研究」
    • Place of Presentation
      京都大学数理解析研究所(京都府京都市)
    • Year and Date
      2015-09-08
    • Related Report
      2015 Research-status Report
  • [Presentation] Iterative schemes with nonvanishing errors for common fixed point problems2015

    • Author(s)
      Y. Kimura
    • Organizer
      The Fifth International Symposium on Banach and Function Spaces 2015
    • Place of Presentation
      九州工業大学戸畑キャンパス(福岡県北九州市)
    • Year and Date
      2015-09-02
    • Related Report
      2015 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Approximation of a common fixed point for mappings defined on a geodesic space2015

    • Author(s)
      Y. Kimura
    • Organizer
      The 11th International Conference on Fixed Point Theory and Its Applications
    • Place of Presentation
      Galatasaray University, Istanbul (Turkey)
    • Year and Date
      2015-07-24
    • Related Report
      2015 Research-status Report
    • Int'l Joint Research / Invited

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Published: 2015-04-16   Modified: 2020-03-30  

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