Optimization Letters

, Volume 6, Issue 8, pp 1875–1881 | Cite as

Every maximally monotone operator of Fitzpatrick–Phelps type is actually of dense type

  • Heinz H. Bauschke
  • Jonathan M. Borwein
  • Xianfu Wang
  • Liangjin YaoEmail author
Original Paper


We show that every maximally monotone operator of Fitzpatrick–Phelps type defined on a real Banach space must be of dense type. This provides an affirmative answer to a question posed by Stephen Simons in 2001 and implies that various important notions of monotonicity coincide.


Fitzpatrick function Maximally monotone operator Monotone operator Multifunction Operator of type (D) Operator of type (FP) Operator of type (NI) Set-valued operator 


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Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  • Heinz H. Bauschke
    • 1
  • Jonathan M. Borwein
    • 2
  • Xianfu Wang
    • 1
  • Liangjin Yao
    • 1
    Email author
  1. 1.Mathematics, Irving K. Barber SchoolUniversity of British ColumbiaKelownaCanada
  2. 2.CARMA, University of NewcastleNewcastleAustralia

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