The Brézis-Browder Theorem Revisited and Properties of Fitzpatrick Functions of Order n

Chapter
Part of the Springer Optimization and Its Applications book series (SOIA, volume 49)

Abstract

In this paper, 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 Brézis–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.

Keywords

Adjoint Convex function Convex set Fenchel conjugate Fitzpatrick function Linear relation Maximal monotone operator Multifunction Monotone operator Set-valued operator Symmetric operator. 

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Notes

Acknowledgements

The author thanks Dr. Heinz Bauschke and Dr. Xianfu Wang for valuable discussions. The author also thanks the two anonymous referees for their careful reading and their pertinent comments.

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© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Department of Mathematics, Irving K. Barber SchoolUniversity of British ColumbiaKelownaCanada

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