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

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