  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.

Functional Analysis