Weak compactness is not equivalent to the fixed point property in c

We show that there exists a non-weakly compact, closed, bounded, convex subset W   of the Banach space of convergent sequences (c,‖⋅‖∞)(c,‖⋅‖∞), such that every nonexpansive mapping T:W⟶WT:W⟶W has a fixed point. This answers a question left open in the 2003 and 2004 papers of Dowling, Lennard and Turett. This is also the first example of a non-weakly compact, closed, bounded, convex subset W of a Banach space X   isomorphic to c0c0, for which W has the fixed point property for nonexpansive mappings. We also prove that the sets W   may be perturbed to a large family of non-weakly compact, closed, bounded, convex subsets WqWq of (c,‖⋅‖∞)(c,‖⋅‖∞) with the fixed point property for nonexpansive mappings; and we discuss similarities and differences with work of Goebel and Kuczumow concerning analogous subsets of ℓ1ℓ1.