The reverse mathematics of Caristi's fixed point theorem

Paul Shafer (Leeds)

Franks Adams 1,

Caristi's fixed point theorem is a fixed point theorem for functions that are controlled by continuous functions but are not necessarily continuous themselves.  Let a 'Caristi system' be a tuple \( (X,V,f) \), where  \( X \) is a complete separable metric space, \( V \) is a continuous function from \( X\) to the non-negative reals, and \(f\) is an arbitrary function from \(X\) to \(X\) such that for all \(x\) in \(X, d(x,f(x)) ≤ V(x) - V(f(x)) \).  Caristi's fixed point theorem states that if \( (X,V,f)\) is a Caristi system, then \(f\) has a fixed point.  In fact, Caristi's fixed point theorem also holds if \(V\) is only lower semi-continuous.  In this talk, we explore the strengths of Caristi's fixed point theorem and related statements, which vary from \( WKL_0 \) in certain special cases to well beyond \(Pi^1_1-CA_0\).

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