Vol. 112, No. 2, 1984

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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

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
Received: 21 December 1981
Revised: 2 March 1983
Published: 1 June 1984
Norimichi Hirano
Wataru Takahashi