A logical analysis of the generalized Banach contractions principle | Zendy

Alexander Kreuzer | Zendy

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A logical analysis of the generalized Banach contractions principle

Author(s) -

Alexander Kreuzer

Publication year - 2012

Publication title -

journal of logic and analysis

Language(s) - English

Resource type - Journals

SCImago Journal Rank - 0.278

H-Index - 4

ISSN - 1759-9008

DOI - 10.4115/jla.2012.4.17

Subject(s) - mathematics , banach space , pure mathematics , contraction principle , discrete mathematics , calculus (dental) , fixed point theorem , medicine , dentistry

International audienceLet (X,d) be a complete metric space, m a natural number, and w a real with 0<= w < 1. A g-contraction is a mapping T: X->X such that for all x,y in X there is an i in [1,m] with d(T^ix, T^iy) < w^i d(x,y)$. The generalized Banach contractions principle states that each g-contraction has a fixed point. We show that this principle is a consequence of Ramsey's theorem for pairs over, roughly, RCA_0 + \Sigma^0_2-IA

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