Properties of fixed-point sets of nonexpansive mappings in Banach spaces

Let C be a closed convex subset of the Banach space X. A subset F of C is called a nonexpansive retract of C if either F = ∅ F = \emptyset or there exists a retraction of C onto F which is a nonexpansive mapping. The main theorem of this paper is that if T : C → C T:C \to C is nonexpansive and satisfies a conditional fixed point property, then the fixed-point set of T is a nonexpansive retract of C. This result is used to generalize a theorem of Belluce and Kirk on the existence of a common fixed point of a finite family of commuting nonexpansive mappings.