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arXiv · 2001.08158

Fixed point properties for semigroups of nonlinear mappings on unbounded sets

Anthony T. -M. Lau·Yong Zhang
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Abstract

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 $T$: $C\to C$ 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 $c\in C$ with a bounded orbit and assuming that certain subspace of $C_b(S)$ 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 \cite{Neu} and formalized by Day in 1957 \cite{Day}. In our investigation we use the notion of common attractive points introduced recently by S. Atsushiba and W. Takahashi.

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Anthony T. -M. Lau, Yong Zhang. 2020-01-22. Fixed point properties for semigroups of nonlinear mappings on unbounded sets. https://arxiv.org/abs/2001.08158

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