Ergodic retractions for amenable semigroups in Banach spaces with normal structure

Abstract

In this paper, we discuss the existence of nonexpansive retraction onto the set of common fixed points. Assume that φ = { T s : s ∈ S } is an amenable semigroup of nonexpansive mappings on a closed, convex subset C in a reflexive Banach space E such that the set F ( φ ) of common fixed points of φ is nonempty. Among other things, it is shown that if either C has normal structure, or the T s ’s are affine, then there exists a nonexpansive retraction P from C onto F ( φ ) such that P T t = T t P = P for each t ∈ S and every closed convex φ -invariant subset of C is also P -invariant; in the case that the mappings are affine, P is also affine, and P x ∈ c o ¯ { T t x : t ∈ S } for each x ∈ C , and it is unique regarding the latter property. Our results extend corresponding results of [T. Suzuki, Some remarks on the set of common fixed points of one-parameter semigroups of nonexpansive mappings in Banach spaces with the Opial property, Nonlinear Anal. 58 (2004), 441–458] and [R. E. Bruck, A common fixed point theorem for a commuting family of nonexpansive mappings, Pacific J. Math. 53 (1974), 59-71].