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 ℑ = { T t : t ∈ S } 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 ⋂ s ∈ S c o ¯ { T t x : t ≽ s } ∩ F ( ℑ ) consists of at most one point. We also prove that if S is reversible, then the intersection ⋂ s ∈ S c o ¯ { T t x : t ≽ s } ∩ F ( ℑ ) is nonempty for each x ∈ C if and only if there exists a nonexpansive retraction P of C onto F ( ℑ ) such that P T t = T t P = P for all t ∈ S and Px is in the closed convex hull of { T t x : t ∈ S } for each x ∈ C .