On error bounds for fixed point problems and its applications

• Project/Area Number: 20740084
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Single-year Grants
• Research Field: Basic analysis
• Research Institution: Akita Prefectural University (2009-2010); Matsue National College of Technology (2008)
• Principal Investigator: MATSUSHITA Shin-ya 秋田県立大学, システム科学技術学部, 助教 (20435449)
• Project Period (FY): 2008 – 2010
• Project Status: Completed (Fiscal Year 2010)
• Budget Amount: ¥4,160,000 (Direct Cost: ¥3,200,000、Indirect Cost: ¥960,000); Fiscal Year 2010: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2009: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000); Fiscal Year 2008: ¥1,820,000 (Direct Cost: ¥1,400,000、Indirect Cost: ¥420,000)
• Keywords: 不動点問題 / エラーバウンド / 近接点法 / 不動点 / 極大単調作用素 / リゾルベント / 変分不等式 / 関数解析 / 関数解析学

Research Abstract

This research aims to develop efficient algorithms for solving fixed point problems. Especially, we investigated notions that guarantee sequences generated by iterative methods can obtain an exact solution in finite number of steps. First, we provided a new condition for set-valued mappings based on theoretical techniques for solving the convex programming problem. Further, by using these results, we derived finite convergence results concerning nonlinear problems such as the convex minimization problem and the variational inequality.

Research Products

• [Journal Article] Finite termination of the proximal point algorithm in Banach spaces (2012)
  • Author(s): Shin-ya Matsushita, Li Xu
  • Journal Title: Journal of Mathematical Analysis and Applications Volume: 387 Pages: 765-769
  • Peer Reviewed
• [Journal Article] On convergence of the proximal point algorithm in Banach spaces (2012)
  • Author(s): Shin-ya Matsushita, Li Xu
  • Journal Title: Proceedings of the American Mathematical Society Volume: 139 Pages: 4087-4095
  • Peer Reviewed
• [Journal Article] Strong convergence theorems obtained by a generalized projections hybrid method for families of mappings in Banach spaces (2010)
  • Author(s): Shin-ya Matsushita, Kazuhide Nakajo, Wataru Takahashi
  • Journal Title: Nonlinear Analysis: Theory, method & Applications Volume: 73 Pages: 1466-1480
  • Peer Reviewed
• [Journal Article] Strong convergence theorems obtained by a generalized projections hybrid method for families of mappings in Banach spaces (2010)
  • Author(s): S.Matsushita, K.Nakajo, W.Takahashi
  • Journal Title: Nonlinear Analysis : Theory, method & Applications Volume: 73 Pages: 1466-1480
  • Peer Reviewed
• [Journal Article] The Fan Knaster-Kuratowski-Mazurkiewicz theorem is equivalent to the Takahashi fixed point theorem (2009)
  • Author(s): Shin-ya Matsushita
  • Journal Title: Proceedings of the Asian Conference on Nonlinear Analysis and Optimization Pages: 223-226
  • Peer Reviewed
• [Journal Article] Asymptotic convergence analysis of the proximal point algorithm for metrically regular mappings (2009)
  • Author(s): Shin-ya Matsushita, Li Xu
  • Journal Title: Fifth International Workshop on Computational Intelligence & Applications IEEE SMC Hiroshima Chapter, Hiroshima University, Japan, November 10, 11 & 12 Pages: 270-273
  • NAID: 120006658683
  • Peer Reviewed
• [Journal Article] The Fan-Knaster-Kuratowski-Mazurkiewicz theorem is equivalent to the Takahashi fixed point theorem (2009)
  • Author(s): Shin-ya Matsuhita
  • Journal Title: Proceedings of the Asian Conference on Nonlinear Analysis and Optimization Pages: 223-226
  • Peer Reviewed
• [Journal Article] Asymptotic convergence analysis of the proximal point algorithm for metrically regular mappings (2009)
  • Author(s): Shin-ya Matsuhita, Li Xu
  • Journal Title: Fifth International Workshop on Computational Intelligence & Applications IEEE SMC Hiroshima Chapter, Hiroshima University, Japan, November10, 11 & 12 Pages: 270-273
  • Peer Reviewed
• [Journal Article] On the existence of zero points for pseudomonotone lower samicontinuous operators in Banach spaces (2008)
  • Author(s): Shin-ya Matsushita, Wataru Takahashi
  • Journal Title: Proceedings of the Fifth International Conference on Nonlinear Analysis and Convex Analysis Pages: 121-128
  • Peer Reviewed
• [Journal Article] Existence of zero points for pseudomonotone operators in Banach spaces (2008)
  • Author(s): Shin-ya Matsushita, Wataru Takahashi
  • Journal Title: Journal of Global Optimization Volume: 42 Pages: 519-558
  • Peer Reviewed
• [Journal Article] The sequences by the hybrid method and the existence of fixed points of nonexpansive mappings in Hilbert space (2008)
  • Author(s): Shin-ya Matsushita, Wataru Takahashi
  • Journal Title: Proceedings of the Eighth International Conference on Fixed Point Theory and its Applications Pages: 109-114
  • Peer Reviewed
• [Journal Article] Zero points of set-valued operators and fixed point theorems (2008)
  • Author(s): Shin-ya Matsushita, Wataru Takahashi
  • Journal Title: 京都大学数理解析研究所講究録 Volume: 1611 Pages: 165-171
• [Journal Article] Existence of zero points for pseudomonotone operators in Barech spaces (2008)
  • Author(s): Shin-ya Matsushita Wataru Takahashi
  • Journal Title: Journal of Global Optimization 42 Pages: 519-558
  • Peer Reviewed
• [Journal Article] The sequences by the hyhrid method and the existence of fixed points of nonexpansive mappings in III lbert space (2008)
  • Author(s): Shin-ya Matsushita. Wataru Takahashi
  • Journal Title: Proceedings of the Eighth International Conference on Fixed Point Theory and its Aplications Pages: 109-114
  • Peer Reviewed
• [Journal Article] On the existence of zero points for pseudomonotone lower samicontirnuous operators in Banech spaces (2008)
  • Author(s): Shin-ya Matsushita, Wataru Takahashi
  • Journal Title: Proceedings of the Fifth International Conference on Nonlineer Analysis and Convex Analysis Pages: 121-128
  • Peer Reviewed
• [Journal Article] Zero points of set-valued operators and fixed point theorems (2008)
  • Author(s): 松下慎也, 高橋渉
  • Journal Title: 京都大学数理解析研究所講究録 1611 Pages: 165-171
• [Journal Article] On convergence of the proximal point algorithm in Banach spaces
  • Author(s): S.Matsushita, L.Xu
  • Journal Title: Proceedings of the American Mathematical Society掲載受理2010年11月
  • Peer Reviewed
• [Presentation] Proximal point algorithm for variational inequality problems (2011)
  • Author(s): Shin-ya Matsushita
  • Organizer: 日本数学会2011 年度秋季総合分科会・実関数論分科会
  • Place of Presentation: 信州大学
  • Year and Date: 2011-09-28
• [Presentation] Proximal point algorithm for variational inequality problems (2011)
  • Author(s): 松下慎也
  • Organizer: 日本数学会2011年度秋季総合分科会・実関数論分科会
  • Place of Presentation: 信州大学
  • Year and Date: 2011-09-28
• [Presentation] Proximal point algorithm for variational inequality problems (2011)
  • Author(s): Shin-ya Matsushita
  • Organizer: 日本数学会2011 年度年会・実関数論分科会
  • Place of Presentation: 早稲田大学
  • Year and Date: 2011-03-20
• [Presentation] Proximal point algorithm for variational inequality problems (2011)
  • Author(s): 松下慎也
  • Organizer: 日本数学会2011年度年会・実関数論分科会
  • Place of Presentation: 早稲田大学
  • Year and Date: 2011-03-20
• [Presentation] On finite termination of algorithms (2010)
  • Author(s): Shin-ya Matsushita
  • Organizer: 日本数学会2010 年度秋季総合分科会・実関数論分科会
  • Place of Presentation: 名古屋大学
  • Year and Date: 2010-09-24
• [Presentation] On finite termination of algorithms (2010)
  • Author(s): 松下慎也
  • Organizer: 日本数学会2010年度秋季総合分科会・実関数論分科会
  • Place of Presentation: 名古屋大学
  • Year and Date: 2010-09-24
• [Presentation] On finite termination of iterative methods (2010)
  • Author(s): Shin-ya Matsushita
  • Organizer: 京都大学数理解析研究所研究集会「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研究
  • Year and Date: 2010-08-31
• [Presentation] On finite termination of iterative methods (2010)
  • Author(s): 松下慎也
  • Organizer: 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研究
  • Year and Date: 2010-08-31
• [Presentation] On the Proximal Point Method for Metrically Regular Mappings (2009)
  • Author(s): Shin-ya Matsushita
  • Organizer: The 20th International Symposium of Mathematical Programming
  • Place of Presentation: Gleacher Center, Chicago, Illinois, アメリカ
  • Year and Date: 2009-08-24
• [Presentation] On the Proximal Point Method for Metrically Regular Mappings (2009)
  • Author(s): Shin-ya Matsushita
  • Organizer: The 20th International Symposium of Mathematical Programming
  • Place of Presentation: Gleacher Center, Chicago, IIIinois, (USA)
  • Year and Date: 2009-08-24
• [Presentation] A proximal type algorithm for optimization problems (2009)
  • Author(s): Shin-ya Matsushita
  • Organizer: The 2nd International Workshop on Institutional Supply Chain Management
  • Place of Presentation: Xi'an Jiaotong University, 中国
  • Year and Date: 2009-08-09
• [Presentation] A proximal type algorithm for optimization problems (2009)
  • Author(s): Shin-ya Matsushita
  • Organizer: The 2nd International Workshop on Institutional Supply Chain Managemetn
  • Place of Presentation: Xi'an Jiaotong University, (China)
  • Year and Date: 2009-08-09
• [Presentation] Fixed point theorems for set-valued operators (2009)
  • Author(s): Shin-ya Matsushita
  • Organizer: The Sixth International Conference on Nonlinear Analysis and Convex Analysis (NACA 2009)
  • Place of Presentation: 東京工業大学
  • Year and Date: 2009-03-30
• [Presentation] On boundarv conditions for monotone operators in Banach spaces (2008)
  • Author(s): Shin-ya Matsushita
  • Organizer: 日本数学会秋季総合分科会
  • Place of Presentation: 東京工業大学
  • Year and Date: 2008-09-26
• [Presentation] On bourdarv conditions for montotone operators in Banach spaces (2008)
  • Author(s): 松下慎也
  • Organizer: 2008日本数学会秋季総合分科会
  • Place of Presentation: 東京工業大学
  • Year and Date: 2008-09-26
• [Presentation] On boundary conditions for nonlinear operators in Banach spaces (2008)
  • Author(s): Shin-ya Matsushita
  • Organizer: The Asian Conference on Nonlinear Analysis and Optimization(NAD2008)
  • Place of Presentation: くにびきメッセ, 松江市, 島根県
  • Year and Date: 2008-09-17
• [Presentation] On boundary conditions for nonlinear operators in Banach spaces (2008)
  • Author(s): 松下慎也
  • Organizer: The Asian Conference on Nonlinear Analysis and Optimization(NAD 2008)
  • Place of Presentation: くにびきメッセ、松江市, 島根県
  • Year and Date: 2008-09-17
• [Presentation] バナッハ空間における集合値写像の境界条件について (2008)
  • Author(s): Shin-ya Matsushita
  • Organizer: 京都大学数理解析研究所研究集会「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
  • Year and Date: 2008-09-03
• [Presentation] On the Existence of Zero Points of Set-valued Operators in Banach Spaces and its Applications (2008)
  • Author(s): Shin-ya Matsushita
  • Organizer: The inaugural Sino-Japanese Optimization Meeting (SJOM 2008)
  • Place of Presentation: National Cheng-Kung University, 台湾
  • Year and Date: 2008-08-28
• [Presentation] On the Existence of Zero Points of Set-valued Operators in Banach Spaces and its Applications (2008)
  • Author(s): 松下慎也
  • Organizer: The inaugural Sino-Japanese Optimization Meeting (SJOM 2008)
  • Place of Presentation: National Cheng-Kung University(台湾)
  • Year and Date: 2008-08-28
• [Presentation] On the existence of zero points for set-valued operators under same boundary conditions (2008)
  • Author(s): Shin-ya Matsushita
  • Organizer: World Congress of Nonlinear Analysts (WCNA 2008)
  • Place of Presentation: Hyatt Grand Cypress Resort Orlando, アメリカ
  • Year and Date: 2008-07-05