The Brezis-Browder Theorem revisited and properties of Fitzpatrick functions of order n

TL;DR

This paper revisits the Brezis-Browder theorem, providing a simpler proof for maximal monotonicity of linear relations, and explores Fitzpatrick functions of order n, offering explicit formulas for symmetric cases.

Contribution

It offers a new proof of a key theorem on maximal monotonicity and derives explicit formulas for Fitzpatrick functions of order n for symmetric linear relations.

Findings

Simplified proof of Brezis-Browder theorem.

Explicit formulas for Fitzpatrick functions of order n.

Characterization of maximal monotonicity via adjoint properties.

Abstract

In this note, we study maximal monotonicity of linear relations (set-valued operators with linear graphs) on reflexive Banach spaces. We provide a new and simpler proof of a result due to Brezis-Browder which states that a monotone linear relation with closed graph is maximal monotone if and only if its adjoint is monotone. We also study Fitzpatrick functions and give an explicit formula for Fitzpatrick functions of order n for monotone symmetric linear relations.