# A generalization for a finite family of functions of the converse of Browder's fixed point theorem

## TL;DR

This paper generalizes the converse of Browder's fixed point theorem to finite families of functions, introducing a set-theoretic concept of attractors and establishing conditions for phi-contractions on metric spaces.

## Contribution

It introduces a new set-theoretic framework for families of functions with attractors and extends Browder's fixed point theorem to finite families of functions.

## Key findings

- Existence of a metric making all functions phi-contractions
- Generalization of the converse of Browder's fixed point theorem
- Extension of Bessaga's and Wong's results for finite commuting functions

## Abstract

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.