Abstract

Consider a real-valued bifunction f which is concave in its first argument and convex in its second one. We study its subdifferential with respect to the second argument, evaluated at pairs of the form (x,x), and the subdifferential of -f with respect to its first argument, evaluated at the same pairs. The resulting operators are not always monotone, and we analyze additional conditions on f which ensure their monotonicity, and furthermore their maximal monotonicity. Our main result is that these operators are maximal monotone when f is continuous and it vanishes whenever both arguments coincide. Our results have consequences in terms of the reformulation of equilibrium problems as variational inequality ones.

Author information

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Alfredo N. Iusem

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

iusp@impa.br

Keywords

  • Equilibrium problem
  • maximal monotone operator
  • diagonal subdifferential

Mathematics Subject Classification

90C47, 49J35

References

  1. E. Asplund: Averaged norms, Israel J. Math. 5 (1967) 227–233.https://doi.org/10.1007/BF02771611Google Scholar
  2. E. Blum, W. Oettli: From optimization and variational inequalities to equilibrium problems, Math. Stud. 63 (1994) 123–145.Google Scholar
  3. H. Brezis: Analyse Fonctionnelle. Théorie et Applications, Masson, Paris (1983).Google Scholar
  4. R. S. Burachik, A. N. Iusem: Set-Valued Mappings and Enlargements of Monotone Operators, Springer, Berlin (2007).Google Scholar
  5. J. Diestel: Geometry of Banach Spaces. Selected Topics, Springer, Berlin (1975).https://doi.org/10.1007/BFb0082079Google Scholar
  6. K. Fan: A generalization of Tychonoff’s fixed point theorem, Math. Ann. 142 (1961) 305–310.https://doi.org/10.1007/BF01353421Google Scholar
  7. A. N. Iusem, G. Kassay, W. Sosa: On certain conditions for the existence of solutions of equilibrium problems, Math. Program., Ser. B 116 (2009) 259–273.https://doi.org/10.1007/s10107-007-0125-5Google Scholar
  8. A. N. Iusem, W. Sosa: New existence results for equilibrium problems, Nonlinear Anal. 52 (2003) 621–635.https://doi.org/10.1016/S0362-546X(02)00154-2Google Scholar
  9. A. N. Iusem, W. Sosa: A proximal point method for equilibrium problems in Hilbert spaces, Optimization, to appear.Google Scholar
  10. G. Köthe: Topological Vector Spaces, Springer, New York (1969).Google Scholar

Suggested citation

A. N. Iusem. “On the Maximal Monotonicity of Diagonal Subdifferential Operators.” Journal of Convex Analysis 18 (2011), No. 2, 489–503. https://doi.org/10.68381/jca18031

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Published by Heldermann Verlag, 2011. Rights now held by Banach Press.