Fixed points of nonexpansive maps

OpenAI's solution (2026) of Kirk's problem: every nonexpansive self-map of a nonempty closed bounded convex subset of a real reflexive Banach space has a fixed point, for the given norm. The course reduces to a minimal invariant set where approximate fixed points are diametral (Goebel–Karlovitz), finds an anchor point through resolvents and an idempotent in a compact semigroup of maps (Ellis), builds an infinite tree of maps whose functionals are chosen against compact families of predicted outputs, and switches its leaves to a common approximate fixed point to obtain a contradiction.

Four lessons with proofs, examples and exercises with solutions. Written and self-checked by Claude Opus 5.5 (Anthropic); independent review is not complete.

1. Minimal invariant sets and diametral sequences
2. Resolvents and an anchor point
3. A tree of predicted outputs
4. Kirk's problem

Theorems and assumptions · Reader and editable sources · Course record

Prerequisites

• Hahn–Banach, Baire and the basic theorems on Banach spaces · Foundations of von Neumann algebras
• Weak topologies: Tychonoff, Banach–Alaoglu, Mazur, bipolars, Krein–Milman and Eberlein–Šmulian · Foundations of von Neumann algebras
• Ultrapowers and Darbo's fixed-point theorem (Tingley's problem)