{
  "id": "fixed-points-of-nonexpansive-maps",
  "title": "Fixed points of nonexpansive maps",
  "description": "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.",
  "units": [
    {
      "id": "minimal-invariant-sets-and-diametral-sequences",
      "title": "Minimal invariant sets and diametral sequences",
      "reader": "minimal-invariant-sets-and-diametral-sequences.html",
      "source": "src/minimal-invariant-sets-and-diametral-sequences.md",
      "source_sha256": "bef15d3495481e27d61b0981f5be522ccf35993c49d90ff8fdf8067c5c63ecde",
      "status": "author_self_checked"
    },
    {
      "id": "resolvents-and-an-anchor-point",
      "title": "Resolvents and an anchor point",
      "reader": "resolvents-and-an-anchor-point.html",
      "source": "src/resolvents-and-an-anchor-point.md",
      "source_sha256": "64145c206c84b5b6587ac9861abf00fc962fd6a044bec8056eb10f82429331b9",
      "status": "author_self_checked"
    },
    {
      "id": "a-tree-of-predicted-outputs",
      "title": "A tree of predicted outputs",
      "reader": "a-tree-of-predicted-outputs.html",
      "source": "src/a-tree-of-predicted-outputs.md",
      "source_sha256": "e24d15dc596c995b4ebcb55bc1dbb7e4df67771dd470141abe0e45d16217dada",
      "status": "author_self_checked"
    },
    {
      "id": "kirks-problem",
      "title": "Kirk's problem",
      "reader": "kirks-problem.html",
      "source": "src/kirks-problem.md",
      "source_sha256": "458362c79f6fc748a4a624464455efe9f45ca0cf1cdbce7cbde441ff49fd7653",
      "status": "author_self_checked"
    }
  ],
  "prerequisites": [
    "foundations-of-von-neumann-algebras lesson hahn-banach-baire-and-the-basic-theorems-on-banach-spaces (Theorem 1.1 Zorn; Corollary 2.3 Hahn-Banach; Theorem 4.2 uniform boundedness; Theorem 6.3 strict separation); lesson weak-topologies-tychonoff-banach-alaoglu-mazur-bipolars-krein-milman-and-eberlein-smulian (Theorem 2.3 Tychonoff; Theorem 3.1 Banach-Alaoglu; Theorem 4.1 Mazur)",
    "tingleys-problem lesson ultrapowers-and-darbos-fixed-point-theorem (Lemma 1.1, free ultrafilters on N)"
  ],
  "status": "author_self_checked",
  "proof_providers": "providers.json",
  "provenance": "provenance.json",
  "licence": "CC0-1.0"
}