Abstract

We study some conditions which guarantee that a convex function represents a maximal monotone operator in non-reflexive Banach spaces.

Author information

Contact details are reproduced from the original publication and may be historical.

Maicon Marques Alves

Instituto de Matemática Pura e Aplicada, Estrada Dona Castorina 110, Jardim Botânico, Rio de Janeiro, RJ 22460-320, Brazil

maicon@impa.br

Benar Fux Svaiter

Instituto de Matemática Pura e Aplicada, Estrada Dona Castorina 110, Jardim Botânico, Rio de Janeiro, RJ 22460-320, Brazil

benar@impa.br

Keywords

  • Fitzpatrick function
  • maximal monotone operator
  • non-reflexive Banach spaces

Mathematics Subject Classification

47H05, 49J52, 47N10

References

  1. H. Attouch, H. Brezis: Duality for the sum of convex functions in general Banach spaces, in: Aspects of Mathematics and its Applications, J. A. Barroso (ed.), North-Holland Math. Library 34, North-Holland, Amsterdam (1986) 125–133.https://doi.org/10.1016/S0924-6509(09)70252-1Google Scholar
  2. R. S. Burachik, B. F. Svaiter: Maximal monotone operators, convex functions and a special family of enlargements, Set-Valued Anal. 10(4) (2002) 297–316.https://doi.org/10.1023/A:1020639314056Google Scholar
  3. R. S. Burachik, B. F. Svaiter: Maximal monotonicity, conjugation and the duality product, Proc. Amer. Math. Soc. 131(8) (2003) 2379–2383.https://doi.org/10.1090/S0002-9939-03-07053-9Google Scholar
  4. S. Fitzpatrick: Representing monotone operators by convex functions, in: Functional Analysis and Optimization, Workshop / Miniconference (Canberra, 1988), Proc. Cent. Math. Anal. Aust. Natl. Univ. 20, Australian National University, Canberra (1988) 59–65.Google Scholar
  5. M. Marques Alves, B. F. Svaiter: Brøndsted-Rockafellar property and maximality of monotone operators representable by convex functions in non-reflexive Banach spaces, J. Convex Analysis 15(4) (2008) 693–706.https://doi.org/10.68381/jca15047Google Scholar
  6. M. Marques Alves, B. F. Svaiter: A new old class of maximal monotone operators, J. Convex Analysis 16(3&4) (2009) 881–890.https://doi.org/10.68381/jca16053Google Scholar
  7. J.-E. Martínez-Legaz, B. F. Svaiter: Monotone operators representable by l.s.c. convex functions, Set-Valued Anal. 13(1) (2005) 21–46.https://doi.org/10.1007/s11228-004-4170-4Google Scholar
  8. J.-E. Martínez-Legaz, B. F. Svaiter: Minimal convex functions bounded below by the duality product, Proc. Amer. Math. Soc. 136(3) (2008) 873–878.https://doi.org/10.1090/S0002-9939-07-09176-9Google Scholar
  9. J.-E. Martínez-Legaz, M. Théra: A convex representation of maximal monotone operators, J. Nonlinear Convex Anal. 2(2) (2001) 243–247.Google Scholar
  10. S. Simons: Quadrivariate versions of the Attouch-Brezis theorem and strong representability, arXiv:0809.0325 (2008).Google Scholar

Suggested citation

M. Marques Alves, B. F. Svaiter. “Maximal Monotonicity, Conjugation and the Duality Product in Non-Reflexive Banach Spaces.” Journal of Convex Analysis 17 (2010), No. 2, 553–563. https://doi.org/10.68381/jca17037

Download BibTeXPublished by Heldermann Verlag.

Published by Heldermann Verlag, 2010. Rights now held by Banach Press LLC.