Ergodic retractions for amenable semigroups in Banach spaces with normal structure

In this paper, we discuss the existence of nonexpansive retraction onto the set of common fixed points. Assume that φ={Ts:s∈S}φ={Ts:s∈S} is an amenable semigroup of nonexpansive mappings on a closed, convex subset CC in a reflexive Banach space EE such that the set F(φ)F(φ) of common fixed points of φφ is nonempty. Among other things, it is shown that if either CC has normal structure, or the TsTs’s are affine, then there exists a nonexpansive retraction PP from CC onto F(φ)F(φ) such that PTt=TtP=PPTt=TtP=P for each t∈St∈S and every closed convex φφ-invariant subset of CC is also PP-invariant; in the case that the mappings are affine, PP is also affine, and Px∈co¯{Ttx:t∈S} for each x∈Cx∈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].