Nonexpansive mappings and fixed-points in Banach spaces

A subset K of a Banach space B has norma structure [2] if for each bounded convex subset H of K which contains more than one point there is a pointWe proved in an earlier paper [1] that if K is a bounded, nonempty, weakly compact, convex subset of a Banach space B, and if K has normal structure then every nte family of commuting nonexpansive mappings of K into itself has a common fixed-point.(A mapping/ on K is nonexpansive if f(x) f(y) =< x y for each x, y K.)If the norm of B is strictly convex then this theorem holds for infinite families.(For if the norm is strictly convex then the fixed-point set for each f e is nonempty, bounded,