A note on “Approximation of fixed points for amenable semigroups of nonexpansive mappings in Banach spaces” [Nonlinear Anal. 67 (2007) 1211–1225]

In this note, we deal with an iterative scheme of Halpern type for a semigroup of nonexpansive mappings on a compact convex subset of a strictly convex and smooth Banach space with respect to an asymptotically left invariant sequence of means defined on an appropriate space of bounded real valued functions of the semigroup. We improve the corresponding result of [A.T. Lau, H. Miyake, W. Takahashi, Approximation of fixed points for amenable semigroups of nonexpansive mappings in Banach spaces, Nonlinear Anal. 67 (2007) 1211–1225].