Abstract

We give a new proof based on the recent very elegant argument of M. Marques Alves and B. F. Svaiter [J. Convex Analysis 15 (2008) 345–348] that the subdifferential of a proper, convex lower semicontinuous function on a real Banach space is maximally monotone. We also show how the argument can be simplified in the reflexive case

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. A. Brøndsted, R. T. Rockafellar: On the subdifferentiability of convex functions, Proc. Amer. Math. Soc. 16 (1965) 605–611.https://doi.org/10.1090/S0002-9939-1965-0178103-8Google Scholar
  2. M. Marques Alves, B. F. Svaiter: A new proof for maximal monotonicity of subdifferential operators, J. Convex Analysis 15 (2008) 345–348.https://doi.org/10.68381/jca15024Google Scholar
  3. J.-J. Moreau: Fonctionelles convexes, Séminaire sur les équations aux derivées partielles, Lecture Notes, Collège de France, Paris (1966–67).Google Scholar
  4. 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
  5. R. T. Rockafellar: On the maximal monotonicity of subdifferential mappings, Pac. J. Math. 33 (1970) 209–216.https://doi.org/10.2140/pjm.1970.33.209Google Scholar
  6. S. Simons: Minimax and monotonicity, Lecture Notes in Mathematics 1693, Springer, Berlin (1998).https://doi.org/10.1007/BFb0093633Google Scholar
  7. S. Simons: Five kinds of maximal monotonicity, Set-Valued Anal. 9 (2001) 391–409.https://doi.org/10.1023/A:1012664003952Google Scholar
  8. L. Thibault: Limiting convex subdifferential calculus with applications to integration and maximal monotonicity of subdifferential, in: Constructive, Experimental, and Nonlinear Analysis (Limoges, 1999), M. Théra (ed.), CMS Conf. Proc. 27, AMS, Providence (2000) 279–289.Google Scholar
  9. D. Zagrodny: Approximate mean value theorem for upper subderivatives, Nonlinear Anal., Theory Methods Appl. 12 (1988) 1413–1428.https://doi.org/10.1016/0362-546X(88)90088-0Google Scholar

Suggested citation

S. Simons. “A New Proof of the Maximal Monotonicity of Subdifferentials.” Journal of Convex Analysis 16 (2009), No. 1, 165–168. https://doi.org/10.68381/jca16008

Download BibTeXPublished by Heldermann Verlag.

Published by Heldermann Verlag, 2009. Rights now held by Banach Press LLC.