TY - JOUR

T1 - On contractive families and a fixed-point question of stein

AU - Austin, Tim D.

N1 - Funding Information:
Acknowledgements. The above work was carried out under a summer research studentship funded by Trinity College, Cambridge over the long vacation period of 2004. My thanks go to Dr I. Leader (Department of Pure Mathematics and Mathematical Statistics, University of Cambridge) for the supervision of the project and the many protracted discussions this entailed, and to Trinity College for their support over this long vacation period.

PY - 2005

Y1 - 2005

N2 - 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, ...,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.

AB - 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, ...,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.

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U2 - 10.1112/S0025579300000395

DO - 10.1112/S0025579300000395

M3 - Article

AN - SCOPUS:33846177278

SN - 0025-5793

VL - 52

SP - 115

EP - 129

JO - Mathematika

JF - Mathematika

IS - 1-2

ER -