Abstract

A recent result characterizes the fully order reversing operators acting on the class of lower semicontinuous proper convex functions in a real Banach space as certain linear deformations of the Legendre-Fenchel transform. Motivated by the Hilbert space version of this result and by the well-known result saying that this convex conjugation transform has a unique fixed point (namely, the normalized energy function), we investigate the fixed point equation in which the involved operator is fully order reversing and acts on the above-mentioned class of functions. It turns out that this nonlinear equation is very sensitive to the involved parameters and can have no solution, a unique solution, or several (possibly infinitely many) ones. Our analysis yields a few by-products, such as results related to positive definite operators, and to functional equations and inclusions involving monotone operators.

Author information

Contact details are reproduced from the original publication and may be historical.

Alfredo N. Iusem

IMPA - Instituto Nacional de Matemática Pura e Aplicada, Estrada Doña Castorina 110, Jardim Botânico, 22460-320 Rio de Janeiro, Brazil

iusp@impa.br

Daniel Reem

Dept. of Mathematics, The Technion - Israel Institute of Technology, 3200003 Haifa, Israel

dream@technion.ac.il

Simeon Reich

Dept. of Mathematics, The Technion - Israel Institute of Technology, 3200003 Haifa, Israel

sreich@technion.ac.il

Keywords

  • Convex conjugation
  • fixed point
  • functional equation
  • lower semicontinous proper convex function
  • Legendre-Fenchel transform
  • monotone operator
  • order reversing operator
  • positive definite
  • quadratic function

Mathematics Subject Classification

47H10, 26B25, 52A41, 47N10, 47H05, 47J05, 39B42

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Suggested citation

A. N. Iusem, D. Reem, S. Reich. “Fixed Points of Legendre-Fenchel Type Transforms.” Journal of Convex Analysis 26 (2019), No. 1, 275–298. https://doi.org/10.68381/jca26016

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Published by Heldermann Verlag, 2019. Rights now held by Banach Press LLC.