Uniformly nonsquare Banach spaces have the fixed point property for nonexpansive mappings

It is shown that if the modulus Gamma(X) of nearly uniform smoothness of a reflexive Banach space satisfies Gamma'(X) (0) < 1, then every bounded closed convex subset of X has the fixed point property for nonexpansive mappings. In particular, uniformly nonsquare Banach spaces have this property since they are properly included in this class of spaces. This answers a long-standing question in the theory. (c) 2005 Elsevier Inc. All rights reserved.