Paul Shafer

Exploring the strength of Caristi's fixed point theorem and Ekeland's variational principle

Caristi's fixed point theorem is a fixed point theorem for functions that may not themselves be continuous, but are nonetheless controlled by other continuous functions. 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 strength of Caristi's fixed point theorem and related statements, such as Ekeland's variational principle, which vary from WKL_0 in certain special cases to well beyond Pi^1_1-CA_0.