  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.
