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A nonlinear ergodic theorem for a reversible semigroup of nonexpansive mappings in a Hilbert space


Author: Wataru Takahashi
Journal: Proc. Amer. Math. Soc. 97 (1986), 55-58
MSC: Primary 47H10; Secondary 47A35
DOI: https://doi.org/10.1090/S0002-9939-1986-0831386-4
MathSciNet review: 831386
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Abstract: Let $ C$ be a nonempty closed convex subset of a Hilbert space, $ S$ a right reversible semitopological semigroup, $ \mathcal{S} = \{ {T_t}:t \in S\} $ a continuous representation of $ S$ as nonexpansive mappings on a closed convex subset $ C$ into $ C$, and $ F(\mathcal{S})$ the set of common fixed points of mappings $ {T_t},\;t \in S$. Then we deal with the existence of a nonexpansive retraction $ P$ of $ C$ onto $ F(\mathcal{S})$ such that $ P{T_t} = {T_t}P = P$ for each $ t \in S$ and $ {P_x}$ is contained in the closure of the convex hull of $ \{ {T_t}x:t \in S\} $ for each $ x \in C$.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1986-0831386-4
Keywords: Ergodic theorem, reversible semigroup, nonexpansive mapping, fixed point
Article copyright: © Copyright 1986 American Mathematical Society

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