REFLEXIVITY AND THE FIXED-POINT PROPERTY FOR NONEXPANSIVE MAPS

Connections between reflexivity and the fixed-point property for nonex pansive self-mappings of nonempty, closed, bounded, convex subsets of a Banach space are investigated. In particular, it is shown that l(1)( Gamma) for uncountable sets Gamma and l(infinity) cannot even be renor med to have the fixed-point property. As a consequence, if an Orlicz s pace on a finite measure space that is not purely atomic is endowed wi th the Orlicz norm, the Orlicz space has the fixed-point property exac tly when it is reflexive. (C) 1996 Academic Press, Inc.