Nonlinear ergodic theorems for asymptotically nonexpansive semigroups in Banach spaces

In this paper, we study nonlinear ergodic properties for an asymptotically nonexpansive semigroup in a Banach space. We prove that if S is amenable and S = {T-t : t is an element of S} is an asymptotically nonexpansive semigroup on a nonempty closed convex subset C of a uniformly convex Banach space E such that the set F(S) of common fixed points of S is nonempty, then there exists a nonexpansive retraction P of C onto F(S) such that PTt = TtP = P for every t is an element of S and Px is an element of (co) over bar {T(t)x : t is an element of S} for every x is an element of C. Also, if the norm of E is Frechet differentiable, then for each x is an element of C, Px is the unique common fixed point in boolean AND(sis an element ofS) (co) over bar {T(ts)x : t is an element of S}. Further, if {mu(alpha)} is an asymptotically invariant net of means, then for each x is an element of C, {T(mualpha)x} converges weakly to Px. Finally, we provide a necessary and sufficient condition for the existence of such a retraction P.