2018 Fiscal Year Final Research Report
The Study of Nonlinear Functional Analysis and Nonlinear Problems Based on New Fixed Point Theory and Convex Analysis
| Project/Area Number |
15K04906
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Multi-year Fund |
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| Section | 一般 |
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| Research Field |
Basic analysis
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| Research Institution | Keio University |
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Principal Investigator |
TAKAHASHI Wataru 慶應義塾大学, 自然科学研究教育センター(日吉), 訪問教授 (40016142)
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| Co-Investigator(Kenkyū-buntansha) |
小宮 英敏 慶應義塾大学, 商学部(日吉), 教授 (90153676)
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| Project Period (FY) |
2015-04-01 – 2019-03-31
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| Keywords | 非線形関数解析学 / 凸解析学 / 不動点理論 / 最適化理論 / 非線形作用素 / 均衡点問題 / 不動点アルゴリズム / バナッハ空間 |
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| Outline of Final Research Achievements |
In this research, we studied nonlinear functional analysis and nonlinear problems by using new fixed point theory and convex analysos. We at first introduced the concept of attractive points of nonlinear mappings in Hilbert spaces and Banach spaces and then proved the existence of attractive points and mean convergence theorems. In the study of inverse problem which is important in medical science, engineering, economics and so on, we proved weak convergence theorems of Mann's type iteration and strong convergence theorems of Halpern's type iteration in Hilbert spaces. We also obtained strong convergence theorems by the hybrid method in Banach spaces. Furthermore, we proved weak and strong convergence theorems for semigroups of not necessarily continuous mappings in Hilbert spaces and Banach spaces. Using these theorems, we solved nonlinear problems which are important in many areas of applied mathematics.
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| Free Research Field |
非線形関数解析学とその応用
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| Academic Significance and Societal Importance of the Research Achievements |
本研究の学術的独自性と創造性は、非線形関数解析学と非線形問題、特に逆問題、非線形最適化や均衡問題、平均収束の問題、微分方程式の問題を、新しくつくられた不動点理論と凸解析学の立場から捉え、それを通してこれまでの理論よりも優れた非線形関数解析学の理論を構築するとともに、それらの非線形問題への直接的解明にあたったものである。凸解析学でのアイデアや、種々の不動点定理を駆使して、数学、医学、工学、経済学等で重要な逆問題、非線形最適化や均衡問題、平均収束定理の問題、微分方程式等の問題が解明でき、さらには像再生の問題や制約問題などにも応用できた。この研究による結果とその意義は大いにあるとおもう。
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