Fixed point properties for semigroups of nonlinear mappings on unbounded sets

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.

arXiv comment22 pages. arXiv admin note: text overlap with arXiv:2001.08149

References in corpus (1)

• Fixed point properties for semigroups of nonlinear mappings and amenability

Cited by in corpus (4)

• Fixed point properties and Q-nonexpansive retractions in locally convex spaces
• Algebraic and analytic properties of semigroups related to fixed point properties of non-expansive mappings
• Fixed points of asymptotically nonexpansive mappings with center 0 and applications
• Bregman nonexpansive type actions of semitopological semigroups