Remarks

Since functions can be viewed as singleton-valued correspondences, Brouwer's fixed point theorem can be viewed as a fixed point theorem for continuous singleton-valued correspondences. The assumption of singleton values can be relaxed. A fixed point of a correspondence μ is a point x satisfying x ∈ μ(x).

Kakutani [1941] proved a fixed point theorem (Corollary 15.3) for closed correspondences with nonempty convex values mapping a compact convex set into itself. His theorem can be viewed as a useful special case of von Neumann's intersection lemma (16.4). (See 21.1.) A useful generalization of Kakutani's theorem is Theorem 15.1 below. Loosely speaking, the theorem says that if a correspondence mapping a compact convex set into itself is the continuous image of a closed correspondence with nonempty convex values into a compact convex set, then it has a fixed point. This theorem is a slight variant of a theorem of Cellina [1969] and the proof is based on von Neumann's approximation lemma (13.3) and the Brouwer fixed point theorem. Another generalization of Kakutani's theorem is due to Eilenberg and Montgomery [1946]. Their theorem is discussed in Section 15.8, and relies on algebraic topological notions beyond the scope of this text. While the Eilenberg-Montgomery theorem is occasionally quoted in the mathematical economics literature (e.g. Debreu [1952], Kuhn [1956], Mas-Colell [1974]), Theorem 15.1 seems general enough for many applications. (In particular see 21.5.)

The theorems above apply to closed correspondences into a compact set.