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 <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L Superscript 1"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>1</mml:mn> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">{L^1}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and that those which commute with translation by a constant are order preserving if and only if they are nonexpansive in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L Superscript normal infinity"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>L</mml:mi> <mml:mi mathvariant="normal">∞</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">{L^\infty }</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Examples are presented involving partial differential equations, difference approximations and rearrangements.