Existence of ergodic retractions for semigroups in Banach spaces

Abstract

In this work, we prove among other results that if S is a right amenable semigroup and φ = { T s : s ∈ S } is a (quasi-)nonexpansive semigroup on a closed, convex subset C in a strictly convex reflexive Banach space E such that the set F ( φ ) of common fixed points of φ is nonempty, then there exists a (quasi-)nonexpansive retraction P from C onto F ( φ ) such that P T t = T t P = P for each t ∈ S and every closed convex φ -invariant subset of C is also P -invariant. Moreover, if the mappings are also affine then T μ [G. Rode, An ergodic theorem for semigroups of nonexpansive mappings in a Hilbert space, J. Math. Anal. Appl. 85 (1982) 172–178. [12]] is a quasi-contractive affine retraction from C onto F ( φ ) , such that T μ T t = T t T μ = T μ for each t ∈ S , and T μ x ∈ c o ¯ { T t x : t ∈ S } for each x ∈ C ; and if R is an arbitrary retraction from C onto F ( φ ) such that R x ∈ c o ¯ { T t x : t ∈ S } for each x ∈ C , then R = T μ . It is shown that if the T t ’s are F ( φ ) -quasi-contractive then the results hold without the strict convexity condition on E .