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Two kinds of fixed point theorems and reverse mathematics
Author(s) -
Peng Weiguang,
Yamazaki Takeshi
Publication year - 2017
Publication title -
mathematical logic quarterly
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.473
H-Index - 28
eISSN - 1521-3870
pISSN - 0942-5616
DOI - 10.1002/malq.201600096
Subject(s) - fixed point theorem , mathematics , kakutani fixed point theorem , brouwer fixed point theorem , fixed point , fixed point property , schauder fixed point theorem , least fixed point , discrete mathematics , contraction principle , contraction (grammar) , reverse mathematics , picard–lindelöf theorem , coincidence point , mathematical analysis , medicine , geometry , axiom
In this paper, we investigate the logical strength of two types of fixed point theorems in the context of reverse mathematics. One is concerned with extensions of the Banach contraction principle. Among theorems in this type, we mainly show that the Caristi fixed point theorem is equivalent to ACA over RCA 0 . The other is dedicated to topological fixed point theorems such as the Brouwer fixed point theorem. We introduce some variants of the Fan‐Browder fixed point theorem and the Kakutani fixed point theorem, which we call FBFP and KFP , respectively. Then we show that FBFP is equivalent to WKL and KFP is equivalent to ACA , over RCA 0 . In addition, we also study the application of the Fan‐Browder fixed point theorem to game systems.