Spaces not containing [formula omitted] have weak approximate fixed point property

Abstract

A nonempty closed convex bounded subset C of a Banach space is said to have the weak approximate fixed point property if for every continuous map f : C → C there is a sequence { x n } in C such that x n − f ( x n ) converge weakly to 0. We prove in particular that C has this property whenever it contains no sequence equivalent to the standard basis of ℓ 1 . As a byproduct we obtain a characterization of Banach spaces not containing ℓ 1 in terms of the weak topology.