Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Minimal convex functions bounded below by the duality product

Authors: J.-E. Martínez-Legaz and B. F. Svaiter
Journal: Proc. Amer. Math. Soc. 136 (2008), 873-878
MSC (2000): Primary 47H05; Secondary 52A41, 26B25
Published electronically: November 30, 2007
MathSciNet review: 2361859
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: It is well known that the Fitzpatrick function of a maximal monotone operator is minimal in the class of convex functions bounded below by the duality product. Our main result establishes that, in the setting of reflexive Banach spaces, the converse also holds; that is, every such minimal function is the Fitzpatrick function of some maximal monotone operator. Whether this converse also holds in a nonreflexive Banach space remains an open problem.

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

  • 1. R. S. Burachik and S. Fitzpatrick. On a family of convex functions associated to subdifferentials. J. Nonlinear Convex Anal., 6(1):165-171, 2005. MR 2138108 (2006b:47081)
  • 2. R. S. Burachik and B. F. Svaiter. Maximal monotone operators, convex functions and a special family of enlargements. Set-Valued Anal., 10(4):297-316, 2002. MR 1934748 (2003k:90085)
  • 3. R. S. Burachik and B. F. Svaiter. Maximal monotonicity, conjugation and the duality product. Proc. Amer. Math. Soc., 131(8):2379-2383 (electronic), 2003. MR 1974634 (2004a:49037)
  • 4. S. Fitzpatrick. Representing monotone operators by convex functions. In Workshop/Miniconference on Functional Analysis and Optimization (Canberra, 1988), volume 20 of Proc. Centre Math. Anal. Austral. Nat. Univ., pages 59-65. Austral. Nat. Univ., Canberra, 1988. MR 1009594 (90i:47054)
  • 5. J.-E. Martınez-Legaz and B. F. Svaiter. Monotone operators representable by l.s.c. convex functions. Set-Valued Anal., 13(1):21-46, 2005. MR 2128696 (2005m:47104)
  • 6. J.-E. Martınez-Legaz and M. Théra. A convex representation of maximal monotone operators. J. Nonlinear Convex Anal., 2(2):243-247, 2001. Special issue for Professor Ky Fan. MR 1848704 (2002e:49035)
  • 7. J.-P. Penot. The relevance of convex analysis for the study of monotonicity. Nonlinear Anal., 58(7-8):855-871, 2004. MR 2086060 (2005g:49026)
  • 8. S. Reich and S. Simons. Fenchel duality, Fitzpatrick functions and the Kirszbraun-Valentine extension theorem. Proc. Amer. Math. Soc., 133(9):2657-2660 (electronic), 2005. MR 2146211 (2006d:46025)
  • 9. J. P. Revalski and M. Théra. Enlargements and sums of monotone operators. Nonlinear Anal., 48(4, Ser. A: Theory Methods):505-519, 2002. MR 1871464 (2002k:47104)
  • 10. S. Simons and C. Zălinescu. A new proof for Rockafellar's characterization of maximal monotone operators. Proc. Amer. Math. Soc., 132(10):2969-2972 (electronic), 2004. MR 2063117 (2005f:47121)
  • 11. S. Simons and C. Zălinescu. Fenchel duality, Fitzpatrick functions and maximal monotonicity. J. Nonlinear Convex Anal., 6(1):1-22, 2005. MR 2138099 (2005k:49102)
  • 12. B. F. Svaiter. Fixed points in the family of convex representations of a maximal monotone operator. Proc. Amer. Math. Soc., 131(12):3851-3859 (electronic), 2003. MR 1999934 (2004h:49016)
  • 13. C. Zălinescu. A new proof of the maximal monotonicity of the sum using the Fitzpatrick function. In Variational analysis and applications, volume 79 of Nonconvex Optim. Appl., pages 1159-1172. Springer, New York, 2005.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 47H05, 52A41, 26B25

Retrieve articles in all journals with MSC (2000): 47H05, 52A41, 26B25

Additional Information

J.-E. Martínez-Legaz
Affiliation: Departament d’Economia i d’Història Econòmica, Universitat Autònoma de Barce- lona, 08193 Bellaterra, Spain

B. F. Svaiter
Affiliation: Instituto de Matemática Pura e Aplicada (IMPA), Estrada Dona Castorinha 110, Jardim Botânico, Rio de Janeiro, CEP 22460-320, Brazil

Received by editor(s): June 20, 2006
Published electronically: November 30, 2007
Additional Notes: The first author was partially supported by the Ministerio de Ciencia y Tecnología, Project MTM2005-08572-C03-03. He also thanks the support of the Barcelona Economics Program of CREA
The second author was partially suported by CNPq grant n. 300755/2005-8 and Edital Universal 476842/03-2
This work was initiated during a visit of the second author to the Universitat Autònoma de Barcelona in March 2006
Communicated by: Jonathan M. Borwein
Article copyright: © Copyright 2007 American Mathematical Society

American Mathematical Society