Fixed point theory in weak second-order arithmetic

Naoki Shioji, Kazuyuki Tanaka

Research output: Contribution to journalArticlepeer-review

20 Citations (Scopus)

Abstract

We develop a basic part of fixed point theory in the context of weak subsystems of second-order arithmetic. RCA0 is the system of recursive comprehension and Σ01 induction. WKL0 is RCA0 plus the weak König's lemma: every infinite tree of sequences of 0's and 1's has an infinite path. A topological space X is said to possess the fixed point property if every continuous function f:X→X has a point x ε{lunate} X such that f(x) = x. Within WKL0 (indeed RCA0), we prove Brouwer's theorem asserting that every nonempty compact convex closed set C in Rn has the fixed point property, provided that C is expressed as the completion of a countable subset of Qn. We then extend Brouwer's theorem to its infinite dimensional analogue (the Tychonoff-Schauder theorem for RN) still within RCA0. As an application of this theorem, we prove the Cauchy-Peano theorem for ordinary differential equations within WKL0, which was first shown by Simpson without reference to the fixed point theorem. Within RCA0, we also prove the Markov-Kakutani theorem which asserts the existence of a common fixed point for certain families of affine mappings. Adapting Kakutani's ingenious proof for deducing the Hahn-Banach theorem from the Markov-Kakutani theorem, we also establish the Hahn-Banach theorem for seperable Banach spaces within WKL0, which was first shown by Brown and Simpson in a different way.

Original languageEnglish
Pages (from-to)167-188
Number of pages22
JournalAnnals of Pure and Applied Logic
Volume47
Issue number2
DOIs
Publication statusPublished - 1990 May 22
Externally publishedYes

ASJC Scopus subject areas

  • Logic

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