Existence of ergodic retractions for semigroups in Banach spaces Original Research Article

In this work, we prove among other results that if SS is a right amenable semigroup and φ={Ts:s∈S}φ={Ts:s∈S} is a (quasi-)nonexpansive semigroup on a closed, convex subset CC in a strictly convex reflexive Banach space EE such that the set F(φ)F(φ) of common fixed points of φφ is nonempty, then there exists a (quasi-)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. Moreover, if the mappings are also affine then TμTμ [G. Rode, An ergodic theorem for semigroups of nonexpansive mappings in a Hilbert space, J. Math. Anal. Appl. 85 (1982) 172–178. [12]] is a quasi-contractive affine retraction from CC onto F(φ)F(φ), such that TμTt=TtTμ=TμTμTt=TtTμ=Tμ for each t∈St∈S, and View the MathML sourceTμx∈co¯{Ttx:t∈S} for each x∈Cx∈C; and if RR is an arbitrary retraction from CC onto F(φ)F(φ) such that View the MathML sourceRx∈co¯{Ttx:t∈S} for each x∈Cx∈C, then R=TμR=Tμ. It is shown that if the TtTt’s are F(φ)F(φ)-quasi-contractive then the results hold without the strict convexity condition on EE.