Abstract

We show how convex analysis can be applied to the theory of sets that are "positive" with respect to a continuous quadratic form on a Banach space. Monotone sets can be considered as a special case of positive sets, and we show how our results lead to very efficient proofs of a number of results on monotone sets. One of the key techniques that we use is a generalization of the Fitzpatrick function from monotone set theory to an analogous function for positive sets

Author information

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

Stephen Simons

Dept. of Mathematics, University of California, Santa Barbara, CA 93106-3080, U.S.A.

simons@math.ucsb.edu

References

  1. J. Bognár: Indefinite Inner Product Spaces, Ergebnisse der Mathematik und ihrer Grenzgebiete 78, Springer, Berlin (1974).https://doi.org/10.1007/978-3-642-65567-8Google Scholar
  2. R. S. Burachik, B. F. Svaiter: Maximal monotone operators, convex functions and a special family of enlargements, Set-Valued Anal. 10 (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 (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. E. Krauss: A representation of maximal monotone operators by saddle functions, Rev. Roum. Math. Pures Appl. 30 (1985) 823–837.Google Scholar
  6. J.-E. Martínez-Legaz, B. F. Svaiter: Monotone operators representable by l.s.c. convex functions, Set-Valued Anal. 13 (2005) 21–46.https://doi.org/10.1007/s11228-004-4170-4Google Scholar
  7. J.-E. Martínez-Legaz, M. Théra: A convex representation of maximal monotone operators, J. Nonlinear Convex Anal. 2 (2001) 243–247.Google Scholar
  8. J.-J. Moreau: Fonctionelles Convexes, Séminaire sur les équations aux derivées partielles, Collège de France, Paris (1966).Google Scholar
  9. J.-P. Penot: The relevance of convex analysis for the study of monotonicity, Nonlinear Anal., Theory Methods Appl. 58 (2004) 855–871.https://doi.org/10.1016/j.na.2004.05.018Google Scholar
  10. R. R. Phelps: Lectures on maximal monotone operators, Extr. Math. 12 (1997) 193–230.Google Scholar
  11. R. T. Rockafellar: Extension of Fenchel’s duality theorem for convex functions, Duke Math. J. 33 (1966) 81–89.https://doi.org/10.1215/S0012-7094-66-03312-6Google Scholar
  12. S. Simons: The range of a monotone operator, J. Math. Anal. Appl. 199 (1996) 176–201.https://doi.org/10.1006/jmaa.1996.0135Google Scholar
  13. S. Simons: Minimax and Monotonicity, Lecture Notes in Mathematics 1693, Springer, Berlin (1998).https://doi.org/10.1007/BFb0093633Google Scholar
  14. S. Simons, C. Zălinescu: A new proof for Rockafellar’s characterization of maximal monotone operators, Proc. Amer. Math. Soc., 132 (2004) 2969–2972.https://doi.org/10.1090/S0002-9939-04-07462-3Google Scholar
  15. S. Simons, C. Zălinescu: Fenchel duality, Fitzpatrick functions and maximal monotonicity, J. Nonlinear Convex Anal. 6 (2005) 1–22.Google Scholar
  16. S. Simons: The Fitzpatrick functions and nonreflexive spaces, J. Convex Analysis 13 (2006) 861–881.https://doi.org/10.68381/jca13064Google Scholar
  17. B. F. Svaiter: Personal communication (November 2004).Google Scholar
  18. C. Zălinescu: Convex Analysis in General Vector Spaces, World Scientific, Singapore (2002).https://doi.org/10.1142/5021Google Scholar

Suggested citation

S. Simons. “Positive Sets and Monotone Sets.” Journal of Convex Analysis 14 (2007), No. 2, 297–317. https://doi.org/10.68381/jca14021

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Published by Heldermann Verlag, 2007. Rights now held by Banach Press LLC.