Some Relations Between Nonexpansive and Order Preserving Mappings

Abstract

It is shown that nonlinear operators which preserve the integral are order preserving if and only if they are nonexpansive in ${L^1}$ and that those which commute with translation by a constant are order preserving if and only if they are nonexpansive in ${L^\infty }$. Examples are presented involving partial differential equations, difference approximations and rearrangements.