Br{\o}ndsted-Rockafellar property of subdifferentials of prox-bounded functions

TL;DR

This paper offers a new proof that subdifferentials of certain convex functions are maximal monotone and demonstrates they have the Br{ }ondsted-Rockafellar property, extending known results to Banach spaces.

Contribution

It provides a novel proof technique for maximal monotonicity and extends the Br{ }ondsted-Rockafellar property to subdifferentials of prox-bounded functions in Banach spaces.

Findings

Subdifferentials are maximal monotone in Banach spaces.

Subdifferentials of prox-bounded functions have the Br{ }ondsted-Rockafellar property.

Extension of Hilbert space results to Banach space setting.

Abstract

We provide a new proof that the subdifferential of a proper lower semicontinuous convex function on a Banach space is maximal monotone by adapting the pattern commonly used in the Hilbert setting. We then extend the arguments to show more precisely that subdifferentials of proper lower semicontinuous prox-bounded functions possess the Br{\o}ndsted-Rockafellar property.