Fixed points of Legendre-Fenchel type transforms
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.
Cite
@article{arxiv.1708.00464,
title = {Fixed points of Legendre-Fenchel type transforms},
author = {Alfredo N. Iusem and Daniel Reem and Simeon Reich},
journal= {arXiv preprint arXiv:1708.00464},
year = {2019}
}
Comments
Added published data (journal, pages, etc.); correction of a typo; updated details of one reference; a slight change to the margins (to allow the last page, with the addresses, to be merged with the penultimate page)