Set-Valued and Variational Analysis

, Volume 18, Issue 3, pp 327–335

On a Sufficient Condition for Equality of Two Maximal Monotone Operators

  • Regina S. Burachik
  • Juan Enrique Martínez-Legaz
  • Marco Rocco

DOI: 10.1007/s11228-010-0148-6

Cite this article as:
Burachik, R.S., Martínez-Legaz, J.E. & Rocco, M. Set-Valued Anal (2010) 18: 327. doi:10.1007/s11228-010-0148-6


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.


Maximal monotone operatorsSubdifferentialConvex functionsEnlargementsε-subdifferential

Copyright information

© Springer Science+Business Media B.V. 2010

Authors and Affiliations

  • Regina S. Burachik
    • 1
  • Juan Enrique Martínez-Legaz
    • 2
  • Marco Rocco
    • 3
  1. 1.School of Mathematics and StatisticsUniversity of South AustraliaAdelaideAustralia
  2. 2.Departament d’Economia i d’Història EconòmicaUniversitat Autònoma de BarcelonaBellaterraSpain
  3. 3.Dipartimento di Matematica, Statistica, Informatica e ApplicazioniUniversità degli Studi di BergamoBergamoItaly