Some relations between nonexpansive and order preserving mappings

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