On contractive families and a fixed-point question of stein

Tim D. Austin

Research output: Contribution to journalArticle

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, ...,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.

Original languageEnglish (US)
Pages (from-to)115-129
Number of pages15
JournalMathematika
Volume52
Issue number1-2
DOIs
StatePublished - 2005

ASJC Scopus subject areas

  • Mathematics(all)

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