Existence of Nonexpansive Retractions for Amenable Semigroups of Nonexpansive Mappings and Nonlinear Ergodic Theorems in Banach Spaces

In this paper, we study nonlinear ergodic properties for an amenable semigroup of nonexpansive mappings in a Banach space. We prove that ifSis an amenable semigroup and S={Tt: t∈S} is a nonexpansive semigroup on a closed, convex subsetCin a uniformly convex Banach spaceEsuch that the setF(S) of common fixed points of S is nonempty, then there exists a nonexpansive retractionPfromContoF(S) such thatPTt=TtP=Pfor eacht∈SandPx∈co{Ttx: t∈S} for eachx∈C. In this case, there exists a net {Aα} of finite averages of S such that for eacht∈Sand for each bounded subsetBofC, limα ‖AαTtx−Aαx‖=0 and limα ‖TtAαx−Aαx‖ =0 uniformly forx∈B. Also, if the norm ofEis Fréchet differentiable, then for eachx∈C,Pxis the unique common fixed point in ∩s∈S co{Ttsx: t∈S}. Furthermore, if {μα} is an asymptotically invariant net of means, then for eachx∈C, {Tμαx} converges weakly toPx.