Copyright © 1999 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Let G be a semitopological semigroup, C a nonempty subset of a real Hilbert space H , and ℑ = { T t : t ∈ G } a representation of G as asymptotically nonexpansive type mappings of C into itself. Let L ( x ) = { z ∈ H : inf s ∈ G sup t ∈ G ‖ T t s   x − z ‖ = inf t ∈ G ‖ T t   x − z ‖ } for each x ∈ C and L ( ℑ ) = ∩ x ∈ C   L ( x ) . In this paper, we prove that ∩ s ∈ G conv ¯ { T t s   x : t ∈ G } ∩ L ( ℑ ) is nonempty for each x ∈ C if and only if there exists a unique nonexpansive retraction P of C into L ( ℑ ) such that P T s = P for all s ∈ G and P ( x ) ∈ conv ¯ { T s   x : s ∈ G } for every x ∈ C . Moreover, we prove the ergodic convergence theorem for a semitopological semigroup of non-Lipschitzian mappings without convexity.