Nonexpansive mappings on Banach lattices
Jonathan Michael Borwein, Brailey Sims · 1983
A mapping T defined on a weakly compact convex subset C of a Banach space X is said to be nonexpansive if ||T(x)-T(y)||≼||x-y|| for all x and y in C; and X is said to have the (weak) fixed point properly (FPP) if every such mapping has a fixed point. Classical results show that every uniformly convex Banach space and those with normal structure have the fixed point property. Until recently other positive results remained fragmentary. Moreover, it was only in 1981 that Alspach showed that L₁ does not have the FPP.