Weak subdifferentials, $r_L$-density and maximal monotonicity

TL;DR

This paper introduces the concept of $r_L$-density in Banach spaces and demonstrates its significance in establishing maximal monotonicity of certain subdifferentials, extending classical results in convex analysis.

Contribution

It defines $r_L$-density for subsets of Banach space products and proves that closed $r_L$-dense monotone sets are maximally monotone, generalizing known subdifferential properties.

Findings

Closed $r_L$-dense monotone sets are maximally monotone

$r_L$-density extends classical subdifferential results

The approach uses Ekeland's variational principle

Abstract

In this paper, we first investigate an abstract subdifferential for which (using Ekeland's variational principle) we can prove an analog of the Br{\o}ndsted-Rockafellar property. We introduce the "$r_L$-density" of a subset of the product of a Banach space with its dual. A closed $r_L$-dense monotone set is maximally monotone, but we will also consider the case of nonmonotone closed $r_L$-dense sets. As a special case of our results, we can prove Rockafellar's result that the subdifferential of a proper convex lower semicontinuous function is maximally monotone.