Journal of Convex Analysis 18 (2011), No. 2, 577--588
Copyright Heldermann Verlag 2011
Fixed Points of Generalized Conjugations
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
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.
Keywords: Generalized conjugation, fixed points, maximal monotone operator, Fitzpatrick functions.
MSC: 49J40, 49J52; 47H05, 49J52, 47N10
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