  Taking as model the attractor of an iterated function system consisting of
phi-contractions on a complete and bounded metric space, we introduce the
set-theoretic concept of family of functions having attractor. We prove that,
given such a family, there exist a metric on the set on which the functions are
defined and take values and a comparison function phi such that all the
family's functions are phi-contractions. In this way we obtain a generalization
for a finite family of functions of the converse of Browder's fixed point
theorem. As byproducts we get a particular case of Bessaga's theorem concerning
the converse of the contraction principle and a companion of Wong's result
which extends the above mentioned Bessaga's result for a finite family of
commuting functions with common fixed point.
