On contractive families and a fixed-point question of Stein

The following conjecture generalizing the Contraction Mapping Theorem was made by Stein in [9]: Let (X, ρ) be a complete metric space and let F = {T1, . . . , Tn} be a finite family of self-maps of X. Suppose there is a constant γ ∈ (0, 1) such that for any x, y ∈ X there exists T ∈ F with ρ(T (x), T (y)) ≤ γρ(x, y). Then some composition of members of F has a fixed point. In this paper we disprove this conjecture. We also show that it does hold for a (continuous) commuting F in the case n = 2. We conjecture that it holds for commuting F for any n.