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On Contractive Families and a Fixed-Point Question of Stein

Part of: Connections with other structures, applications Spaces with richer structures

Published online by Cambridge University Press:  21 December 2009

Tim D. Austin
Tim D. Austin
Affiliation:
Trinity College, Cambridge CB2 1TQ.
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Abstract

The following conjecture generalizing the Contraction Mapping Theorem was made by Stein.

Let (X, ρ) be a complete metric space and let ℱ = {T1,…, Tn} 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.

MSC classification

Secondary: 54E40: Special maps on metric spaces 54H25: Fixed-point and coincidence theorems
Type
Research Article
Information
Mathematika , Volume 52 , Issue 1-2 , December 2005 , pp. 115 - 129
Copyright
Copyright © University College London 2005

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