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Maximal monotonicity, conjugation and the duality product

Authors: Regina Sandra Burachik and B. F. Svaiter
Journal: Proc. Amer. Math. Soc. 131 (2003), 2379-2383
MSC (2000): Primary 47H05
Published electronically: March 18, 2003
MathSciNet review: 1974634
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Abstract: Recently, the authors studied the connection between each maximal monotone operator $T$ and a family $\mathcal{H}(T)$ of convex functions. Each member of this family characterizes the operator and satisfies two particular inequalities.

The aim of this paper is to establish the converse of the latter fact. Namely, that every convex function satisfying those two particular inequalities is associated to a unique maximal monotone operator.

References [Enhancements On Off] (What's this?)

  • 1. Asplund, E.:
    Averaged norms,
    Israel Journal of Mathematics 5 (1967), 227-233. MR 36:5660
  • 2. Burachik, R.S. and Svaiter, B.F.:
    Maximal monotone operators, convex functions and a special family of enlargements,
    Set Valued Analysis (to appear).
  • 3. Moreau, J.-J.:
    Fonctionelles convexes,
    mimeographed lecture notes,
    Collège de France, 1967.
  • 4. Fitzpatrick, S.:
    Representing monotone operators by convex functions,
    Workshop/Miniconference on Functional Analysis and Optimization (Canberra, 1988) 59-65,
    Proc. Centre Math. Anal. Austral. Nat. Univ., 20
    Austral. Nat. Univ., Canberra, 1988. MR 90i:47054
  • 5. Krauss, Eckehard:
    A representation of maximal monotone operators by saddle functions,
    Rev. Roumaine Math. Pures Appl. 30 (1985), 823-837. MR 87e:47074
  • 6. Martinez-Legaz, J.-E. and Thera, M.:
    A convex representation of maximal monotone operators,
    Journal of Nonlinear and Convex Analysis 2 (2001), 243-247. MR 2002e:49035
  • 7. Rockafellar, R. T.:
    On the maximal monotonicity of subdifferential mappings,
    Pacific Journal of Mathematics 33 (1970), 209-216. MR 41:7432
  • 8. -,
    On the maximality of sums of nonlinear monotone operators,
    Transactions of the American Mathematical Society 149 (1970), 75-88. MR 43:7984
  • 9. -,
    An extension of Fenchel's duality theorem,
    Duke Math. J. 33 (1966), 81-90. MR 32:4517

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Additional Information

Regina Sandra Burachik
Affiliation: Engenharia de Sistemas e Computação, COPPE–UFRJ CP 68511, Rio de Janeiro–RJ, CEP 21945–970 Brazil

B. F. Svaiter
Affiliation: IMPA Instituto de Matemática Pura e Aplicada, Estrada Dona Castorina 110, Rio de Janeiro–RJ, CEP 22460-320 Brazil

Keywords: Convex functions, maximal monotone operators, duality product, conjugation
Received by editor(s): February 28, 2002
Published electronically: March 18, 2003
Additional Notes: The first author was partially supported by CNPq and by PRONEX–Optimization
The second author was partially supported by CNPq Grant 301200/93-9(RN) and by PRONEX–Optimization.
Communicated by: Jonathan M. Borwein
Article copyright: © Copyright 2003 American Mathematical Society

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