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CONTRACTIVE FAMILIES ON COMPACT SPACES
Author(s) -
Milićević Luka
Publication year - 2014
Publication title -
mathematika
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.955
H-Index - 29
eISSN - 2041-7942
pISSN - 0025-5793
DOI - 10.1112/s0025579313000296
Subject(s) - mathematics , counterexample , conjecture , metric space , space (punctuation) , compact space , pure mathematics , metric (unit) , discrete mathematics , philosophy , linguistics , operations management , economics
Abstract A familyf 1 , … , f nof operators on a complete metric space X is called contractive if there exists λ < 1 such that for any x , y in X we have d ( f i ( x ) , f i ( y ) ) ≤ λ d ( x , y )for some i . Stein conjectured that for any contractive family there is some composition of the operators f i that has a fixed point. Austin gave a counterexample to this, and asked whether Stein's conjecture is true if we restrict to compact spaces. Our aim in this paper is to show that, even for compact spaces, Stein's conjecture is false.