Commuting contractive families

Abstract

A family f<inf>1</inf>; : : : ; fn of operators on a complete metric space X is called contractive if there exists a positive ≤ λ 1 such that for any x; y in X we have d(fi(x); fi(y)) λ <d(x; y) for some i. Austin conjectured that any commuting contractive family of operators has a common fixed point, and he proved this for the case of two operators. We show that Austin's conjecture is true for three operators, provided that λ is sufficiently small.