The fixed point property for nonexpansive maps in reflexive Banach spaces

Theorem 1.1: if $X$ is a real reflexive Banach space and $C\subseteq X$ is nonempty, closed, bounded and convex, every nonexpansive $F:C\to C$ has a fixed point, with no uniform convexity, normal structure or separability assumed and the norm unchanged. Corollary: for any commuting family of nonexpansive self-maps of such $C$, the common fixed set is nonempty and a nonexpansive retract (via Bruck 1974). The proof argues by contradiction from a minimal invariant set, using the Goebel-Karlovitz lemma, an adaptive anchor, a tree of approximate fixed point images and a weighted switching estimate. It says nothing about nonreflexive spaces with the weak fixed point property, and gives no constructive iteration.

Theorem 1.1: if $X$ is a real reflexive Banach space and $C\subseteq X$ is nonempty, closed, bounded and convex, every nonexpansive $F:C\to C$ has a fixed point, with no uniform convexity, normal structure or separability assumed and the norm unchanged. Corollary: for any commuting family of nonexpansive self-maps of such $C$, the common fixed set is nonempty and a nonexpansive retract (via Bruck 1974). The proof argues by contradiction from a minimal invariant set, using the Goebel-Karlovitz lemma, an adaptive anchor, a tree of approximate fixed point images and a weighted switching estimate. It says nothing about nonreflexive spaces with the weak fixed point property, and gives no constructive iteration.