Ergodic theorems for non-Lipschitzian semigroups without convexity

Let G be a semitopological semigroup. Let C be a nonempty subset of a Hilbert space and F = {T-t : t is an element of G} be a representation of G as asymptotically nonexpansive type mappings of C into itself such that the common fixed point set F(F) of F in C is nonempty. It is proved that [GRAPHICS] <(co)over bar> {T(ts)x : t is an element of G} boolean AND F(T) is nonempty for each x is an element of C if and only if there exists a nonexpansive retraction P of C onto F(F) such that PTs = TsP = P for all s is an element of G and P(x) is in the closed convex hull of {T(s)x : s is an element of G}, x is an element of C. This result shows that many key conditions in [1-4, 9, 12-15] are not necessary.