### Abstract

The following conjecture generalizing the Contraction Mapping Theorem was made by Stein. Let (X,ρ) be a complete metric space and let ℱ ={T _{1}, ...,T_{n}] be a finite family of self-maps of X. Suppose that 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 this conjecture is disproved, We also show that it does hold for a (continuous) commuting ℱ in the case n = 2. It is conjectured that it holds for commuting ℱ for any n.

Original language | English (US) |
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Pages (from-to) | 115-129 |

Number of pages | 15 |

Journal | Mathematika |

Volume | 52 |

Issue number | 1-2 |

DOIs | |

State | Published - 2005 |

### ASJC Scopus subject areas

- Mathematics(all)

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## Cite this

Austin, T. D. (2005). On contractive families and a fixed-point question of stein.

*Mathematika*,*52*(1-2), 115-129. https://doi.org/10.1112/S0025579300000395