Nonlinear ergodic theorems for an amenable semigroup of nonexpansive mappings in a Banach space

Norimichi Hirano and Wataru Takahashi

Vol. 112 (1984), No. 2, 333–346

DOI: 10.2140/pjm.1984.112.333

Abstract

Let C be a nonempty closed convex subset of a Banach space, S a semigroup of nonexpansive mappings t of C into itself, and F(S) the set of common fixed points of mappings t. Then we deal with the existence of a nonexpansive retraction P of C onto F(S) such that Pt = tP = P for each t ∈ S and Px is contained in the closure of the convex hull of {tx : t ∈ S} for each x ∈ C. That is, we prove nonlinear ergodic theorems for a semigroup of nonexpansive mappings in a Banach space.

Mathematical Subject Classification 2000

Primary: 47H09

Secondary: 47H20

Milestones

Received: 21 December 1981

Revised: 2 March 1983

Published: 1 June 1984