2002 Fiscal Year Final Research Report Summary
Nonlinear functional analysis and convex analysis problem by using fixed point theory
| Project/Area Number |
12640157
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Basic analysis
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| Research Institution | Tokyo Institute of Technology |
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Principal Investigator |
TAKAHASHI Wataru Tokyo Institute of Technology, Graduate School of Information Science and Engineering, Professor, 大学院・情報理工学研究科, 教授 (40016142)
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| Co-Investigator(Kenkyū-buntansha) |
KIUCHI Hirobumi Takushoku University, Department of Engineering, Associate Professor, 工学部, 助教授 (00251611)
TANIGUCHI Masahara Tokyo Institute of Technology, Graduate School of Information Science and Engineering, Associate Professor, 大学院・情報理工学研究科, 助教授 (30260623)
KOJIMA Masakazu Tokyo Institute of Technology, Graduate School of Information Science and Engineering, Professor, 大学院・情報理工学研究科, 教授 (90092551)
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| Project Period (FY) |
2000 – 2002
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| Keywords | Nonlinear Functional Analysis / Nonlinear Operators / Nonlinear Ergodic Theorem / Nonlinear Evolution Equation / Convex Analysis / Fixed Point Theorem / Mini-max Theorem / Nonlinear Variational Inequality |
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| Research Abstract |
We studied some problems concerning nonlinear functional analysis and convex analysis by using fixed point theory. We first considered iteration schemes given by an infinite family of nonexpansive mappings in Hilbert spaces or Banach spaces and then proved strong convergence theorems for the family of nonexpansive mappings. Using these results, we also considered the feasibility problem of finding a common fixed point of infinite nonexpansive mappings. Next, we introduced two proximal point algorithms suggested by the iterative schemes introduced by Solodov and Svaiter in order to find a solution of $v \in T^∧{-1}0$, where $T$ is a maximal monotone operator. Main results were established by using metric projections and generalized projections in the case of the strong convergence. We also applied these results to find a minimizer of a lower semicontinuous convex function in a Banach space. Finally, we introduced iteration schemes of finding a common element of the set of fixed points of nonexpansive mappings and the set of solutions of the variational inequality for inverse-strongly-monotone mappings. Using these results, we considered the problem of finding a common element of the set of zeros of a maximal monotone mapping and the set of zeros of an inverse-strongly-monotone mapping.
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