Nonlinear ergodic theorems of nonexpansive type mappings

Let S be a semitopological semigroup. Let C be a closed convex subset of a uniformly convex Banach space E whose norm is Fréchet differentiable and be a continuous representation of S as almost asymptotically nonexpansive type mapping of C into C such that the common fixed point set F(ℑ) of ℑ in C is nonempty. In this paper, we prove that if S is right reversible then for each x∈C, the closed convex set consists of at most one point. We also prove that if S is reversible, then the intersection is nonempty for each x∈C if and only if there exists a nonexpansive retraction P of C onto F(ℑ) such that PTt=TtP=P for all t∈S and Px is in the closed convex hull of for each x∈C.