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

Authors: Hajime Ishihara and Wataru Takahashi
Journal: Proc. Amer. Math. Soc. 104 (1988), 431-436
MSC: Primary 47H20; Secondary 47A35
MathSciNet review: 962809
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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 Lipschitzian 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 $ \left\{ {{T_t}x:t \in S} \right\}$ for each $ x \in C$.

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Keywords: Ergodic theorem, reversible semigroup, asymptotically nonexpansive mapping, fixed point
Article copyright: © Copyright 1988 American Mathematical Society