Fixed Points of Generalized Conjugations

TL;DR

This paper proves that any generalized symmetric conjugation in linear topological spaces has fixed points, extending classical results and applying a variational principle to broader contexts in convex analysis and optimization.

Contribution

It establishes the existence of fixed points for generalized symmetric conjugations in linear topological spaces, extending prior fixed-point theorems to more general settings.

Findings

Fixed points exist for generalized symmetric conjugations.

The proof uses a variational principle based on order-reversing properties.

Application extends Fitzpatrick's fixed-point theorem beyond Banach spaces.

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 space by meas 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 optima 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, is natural to ask for the existence of fixed points of the conjugation, that is, functions which are equal to their (generalized) conjugateds. 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.