## Abstract

We develop a basic part of fixed point theory in the context of weak subsystems of second-order arithmetic. RCA_{0} is the system of recursive comprehension and Σ^{0}_{1} induction. WKL_{0} is RCA_{0} 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 WKL_{0} (indeed RCA_{0}), we prove Brouwer's theorem asserting that every nonempty compact convex closed set C in R^{n} has the fixed point property, provided that C is expressed as the completion of a countable subset of Q^{n}. We then extend Brouwer's theorem to its infinite dimensional analogue (the Tychonoff-Schauder theorem for R^{N}) still within RCA_{0}. As an application of this theorem, we prove the Cauchy-Peano theorem for ordinary differential equations within WKL_{0}, which was first shown by Simpson without reference to the fixed point theorem. Within RCA_{0}, 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 WKL_{0}, which was first shown by Brown and Simpson in a different way.

Original language | English |
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Pages (from-to) | 167-188 |

Number of pages | 22 |

Journal | Annals of Pure and Applied Logic |

Volume | 47 |

Issue number | 2 |

DOIs | |

Publication status | Published - 1990 May 22 |

Externally published | Yes |

## ASJC Scopus subject areas

- Logic