Fixed point properties for semigroups of nonlinear mappings on unbounded sets

A well-known result of W. Ray asserts that if C is an unbounded convex subset of a Hilbert space, then there is a nonexpansive mapping T: C→C that has no fixed point. In this paper we establish some common fixed point properties for a semitopological semigroup S of nonexpansive mappings acting on a...

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Bibliographic Details
Published in:Journal of mathematical analysis and applications 2016-01, Vol.433 (2), p.1204-1219
Main Authors: Lau, Anthony To-Ming, Zhang, Yong
Format: Article
Language:English
Subjects:
Attractive point
Fixed point
Hilbert space
Invariant mean
Nonexpansive mappings
Semigroup
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Summary:A well-known result of W. Ray asserts that if C is an unbounded convex subset of a Hilbert space, then there is a nonexpansive mapping T: C→C that has no fixed point. In this paper we establish some common fixed point properties for a semitopological semigroup S of nonexpansive mappings acting on a closed convex subset C of a Hilbert space, assuming that there is a point c∈C with a bounded orbit and assuming that certain subspace of Cb(S) has a left invariant mean. Left invariant mean (or amenability) is an important notion in harmonic analysis of semigroups and groups introduced by von Neumann in 1929 [28] and formalized by Day in 1957 [5]. In our investigation we use the notion of common attractive points introduced recently by S. Atsushiba and W. Takahashi.
ISSN:0022-247X
1096-0813
DOI:10.1016/j.jmaa.2015.08.044