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→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.