Abstract

Conjugation, or Legendre transformation, is a basic tool in convex analysis, rational mechanics, economics and optimization. It maps a function on a linear topological space into another one, defined in the dual of the linear space by coupling these spaces by means of the duality product. Generalized conjugation extends classical conjugation to any pair of domains, using an arbitrary coupling function between these spaces. This generalization of conjugation is now being widely used in optimal transportation problems, variational analysis and also optimization. If the coupled spaces are equal, generalized conjugations define order reversing maps of a family of functions into itself. In this case, it is natural to ask for the existence of fixed points of the conjugation, that is, functions which are equal to their (generalized) conjugated. Here we prove that any generalized symmetric conjugation has fixed points. The basic tool of the proof is a variational principle involving the order reversing feature of the conjugation. As an application of this abstract result, we will extend to real linear topological spaces a fixed-point theorem for Fitzpatrick's functions, previously proved in Banach spaces.

Author information

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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

  • Generalized conjugation
  • fixed points
  • maximal monotone operator
  • Fitzpatrick functions

Mathematics Subject Classification

49J40, 49J52; 47H05, 49J52, 47N10

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Suggested citation

M. Marques Alves, B. F. Svaiter. “Fixed Points of Generalized Conjugations.” Journal of Convex Analysis 18 (2011), No. 2, 577–588. https://doi.org/10.68381/jca18036

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