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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

Information

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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References

1Arvanitakis, A. D., A proof of the generalized Banach contraction conjecture, Proc. Amer. Math. Soc. 131 (2003), no.12, 3647–3656.CrossRefGoogle Scholar
2Banach, S., Sur les opérations dans les ensembles abstraits et leur applicationaux équations intégrales, Fund. Math. 3 (1922), 133–181.CrossRefGoogle Scholar
3Conway, J. B., A Course in Functional Analysis, Springer-Verlag (New York, 1990).Google Scholar
4Dahlhaus, E., Fast Parallel Recognition of Ultrametrics and Tree Metrics, SIAM J. Discrete Math. 6 (1993), 523–532.CrossRefGoogle Scholar
5Hatcher, A., Algebraic Topology, Cambridge University Press (Cambridge, 2002).Google Scholar
6Jachymski, J. R., Schroder, B. & Stein, J. D. Jr., A connection between fixed-point theorems and tiling problems, J. Combin. Theory Ser. A 87 (1999), 273–286.CrossRefGoogle Scholar
7Merryfield, J., Rothschild, B. & Stein, J. D. Jr., An application of Ramsey's theorem to the Banach contraction mapping principle, Proc. Amer. Math. Soc. 130 (2002), 927–933.CrossRefGoogle Scholar
8Merryfield, J. & Stein, J. D. Jr., A Generalization of the Banach Contraction Mapping Theorem, J. Math. Anal. Appl. 273 (2002), 112–120.CrossRefGoogle Scholar
9Stein, J. D. Jr., A systematic generalization procedure for fixed-point theorems, Rocky Mountain J. Math. 30 (2000), 735–754.CrossRefGoogle Scholar