, Volume 18, Issue 3-4, pp 327-335
Date: 24 Aug 2010

On a Sufficient Condition for Equality of Two Maximal Monotone Operators

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Abstract

We establish minimal conditions under which two maximal monotone operators coincide. Our first result is inspired by an analogous result for subdifferentials of convex functions. In particular, we prove that two maximal monotone operators T,S which share the same convex-like domain D coincide whenever \(T(x)\cap S(x)\not=\emptyset \) for every x ∈ D. We extend our result to the setting of enlargements of maximal monotone operators. More precisely, we prove that two operators coincide as long as the enlargements have nonempty intersection at each point of their common domain, assumed to be open. We then use this to obtain new facts for convex functions: we show that the difference of two proper lower semicontinuous and convex functions whose subdifferentials have a common open domain is constant if and only if their ε-subdifferentials intersect at every point of that domain.

Dedicated to Jean-Baptiste Hiriart-Urruty on the occasion of his 60th birthday.
Juan Enrique Martínez-Legaz has been supported by MICINN of Spain, Grant MTM2008-06695-C03-03, by Generalitat de Catalunya and by the Barcelona GSE Research Network. He is affiliated to MOVE (Markets, Organizations and Votes in Economics). This research started during a visit of this author to the University of South Australia, to which he is grateful for the support received.
The contribution of Marco Rocco was initiated during a research stay at the Universitat Autònoma de Barcelona.