Maximal Monotone Multifunctions of Brøndsted–Rockafellar Type

Abstract

We consider whether the “inequality-splitting” property established in the Brondsted–Rockafellar theorem for the subdifferential of a proper convex lower semicontinuous function on a Banach space has an analog for arbitrary maximal monotone multifunctions. We introduce the maximal monotone multifunctions of type (ED), for which an “inequality-splitting” property does hold. These multifunctions form a subclass of Gossez"s maximal monotone multifunctions of type (D); however, in every case where it has been proved that a multifunction is maximal monotone of type (D) then it is also of type (ED). Specifically, the following maximal monotone multifunctions are of type (ED): • ultramaximal monotone multifunctions, which occur in the study of certain nonlinear elliptic functional equations; • single-valued linear operators that are maximal monotone of type (D); • subdifferentials of proper convex lower semicontinuous functions; • “subdifferentials” of certain saddle-functions. We discuss the negative alignment set of a maximal monotone multifunction of type (ED) with respect to a point not in its graph – a mysterious continuous curve without end-points lying in the interior of the first quadrant of the plane. We deduce new inequality-splitting properties of subdifferentials, almost giving a substantial generalization of the original Brondsted–Rockafellar theorem. We develop some mathematical infrastructure, some specific to multifunctions, some with possible applications to other areas of nonlinear analysis: • the formula for the biconjugate of the pointwise maximum of a finite set of convex functions – in a situation where the “obvious” formula for the conjugate fails; • a new topology on the bidual of a Banach space – in some respects, quite well behaved, but in other respects, quite pathological; • an existence theorem for bounded linear functionals – unusual in that it does not assume the existence of any a priori bound; • the 'big convexification" of a multifunction.