In this paper we consider a mapping S of the form S = α 0 I + α 1 T + α 2 T 2 + … + α K T K , where α i ≥ 0 . 0$"> α 1 > 0 with ∑ i = 0 k α i = 1 , and show that in a uniformly convex Banach space the Picard iterates of S converge to a fixed point of T when T is nonexpansive or generalized nonexpansive or even quasinonexpansive.