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Journal of Convex Analysis 23 (2016), No. 4, 1017--1050
Copyright Heldermann Verlag 2016

"Densities" and Maximal Monotonicity

Stephen Simons
Dept. of Mathematics, University of California, Santa Barbara, CA 93106-3080, U.S.A.

We discuss "Banach SN spaces", which include Hilbert spaces, negative Hilbert spaces, and the product of any real Banach space with its dual. We introduce "L-positive" sets, which generalize monotone multifunctions from a Banach space into its dual. We introduce the concepts of "rL-density" and its specialization "quasidensity": the closed quasidense monotone multifunctions from a Banach space into its dual form a (generally) strict subset of the maximally monotone ones, though all surjective maximally monotone and all maximally monotone multifunctions on a reflexive space are quasidense. We give a sum theorem and a parallel sum theorem for closed monotone quasidense multifunctions under very general constraint conditions. That is to say, quasidensity obeys very nice calculus rules. We give a short proof that the subdifferential of a proper convex lower semicontinuous function on a Banach space is quasidense, and deduce generalizations of the Brezis-Browder theorem on linear relations to non reflexive Banach spaces. We also prove that any closed monotone quasidense multifunction has a number of other very desirable properties.

Keywords: Banach SN space, L-positive set, r-L-density, quasidensity, multifunction, maximal monotonicity, sum theorem, subdifferential, negative alignment, monotone linear relation, Brezis-Browder theorem.

MSC: 47H05; 47N10, 52A41, 46A20

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