Journal of Convex Analysis 26 (2019), No. 1, 275--298
Copyright Heldermann Verlag 2019
Fixed Points of Legendre-Fenchel Type Transforms
Alfredo N. Iusem
IMPA - Instituto Nacional de Matemática Pura e Aplicada, Estrada Dona 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
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.
Keywords: Convex conjugation, fixed point, functional equation, lower semicontinous proper convex function, Legendre-Fenchel transform, monotone operator, order reversing operator, positive definite, quadratic function.
MSC: 47H10, 26B25, 52A41, 47N10, 47H05, 47J05, 39B42
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