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 -