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 and that those which commute with translation by a constant are order preserving if and only if they are nonexpansive in <!-- MATH ${L^\infty }$ --> . Examples are presented involving partial differential equations, difference approximations and rearrangements.