The retractions onto the common fixed points of some families and semigroups of mappings

Abstract

Assume that C is a closed convex subset of a reflexive Banach space E and φ = { T i } i ∈ I is a family of self-mappings on C of type ( γ ) such that F ( φ ) , the common fixed point set of φ , is nonempty. From our results in this paper, it can be derived that: (a) If ∪ i ∈ I F ( T i ) is contained in a 3-dimensional subspace of E then F ( φ ) is a nonexpansive retract of C ; (b) If φ is commutative, there exists a retraction R of type ( γ ) from C onto F ( φ ) , such that R T i = T i R = R ( ∀ i ) , and every closed convex φ -invariant subset of C is R -invariant; the same result holds for a non-commutative right amenable semigroup φ , under some additional assumptions. Moreover, the existence of a ( T i ) -ergodic retraction R of type ( γ ) from C ˜ = { ( x i ) ∈ l ∞ ( E ) : x i ∈ C , ∀ i ∈ I } onto F ( φ ) in l ∞ ( E ) for the family φ is discussed. We also apply some of our results to find ergodic retractions for nonexpansive affine mappings.