A nonlinear ergodic theorem for a reversible semigroup of Lipschitzian mappings in a Hilbert space

Abstract

Let C C be a nonempty closed convex subset of a Hilbert space, S S a right reversible semitopological semigroup, S = { T t : t ∈ S } \mathcal {S} = \{ {T_t}:t \in S\} a continuous representation of S S as Lipschitzian mappings on a closed convex subset C C into C C , and F ( S ) F(\mathcal {S}) the set of common fixed points of mappings T t , t ∈ S {T_t},t \in S . Then we deal with the existence of a nonexpansive retraction P P of C C onto F ( S ) F(\mathcal {S}) such that P T t = T t P = P P{T_t} = {T_t}P = P for each t ∈ S t \in S and P x {P_x} is contained in the closure of the convex hull of { T t x : t ∈ S } \left \{ {{T_t}x:t \in S} \right \} for each x ∈ C x \in C .