Fixed point properties for semigroups of nonlinear mappings on unbounded sets
arXiv:2001.08158
Abstract
A well-known result of W. Ray asserts that if is an unbounded convex subset of a Hilbert space, then there is a nonexpansive mapping : that has no fixed point. In this paper we establish some common fixed point properties for a semitopological semigroup of nonexpansive mappings acting on a closed convex subset of a Hilbert space, assuming that there is a point with a bounded orbit and assuming that certain subspace of 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.
22 pages. arXiv admin note: text overlap with arXiv:2001.08149