Fixed-point theorems in infinite-dimensional spaces

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Fixed-point theorems in infinite-dimensional spaces extend the idea of Brouwer’s theorem and help prove that solutions exist in problems such as partial differential equations. The first major result was the Schauder fixed-point theorem in 1930 by Juliusz Schauder; this followed the Banach fixed-point theorem for contractions in complete spaces (1922). Since then many more results have been found. One big influence is that ideas from algebraic topology have been attempted in infinite-dimensional spaces, with Leray’s work on sheaf theory growing from extensions of Schauder.

Key fixed-point theorems (short, easy statements)
- Schauder fixed-point theorem: If C is a nonempty closed convex subset of a Banach space and f maps C into itself in a continuous way with a compact image, then f has a fixed point.
- Tikhonov (Tychonoff) fixed-point theorem: In a locally convex space, any continuous map from a nonempty compact convex set into itself has a fixed point.
- Browder fixed-point theorem: In a uniformly convex Banach space, any nonexpansive map on a nonempty closed bounded convex set has a fixed point. (Nonexpansive means it does not increase distances.)
- Markov–Kakutani fixed-point theorem: For a commuting family of affine self-maps of a nonempty compact convex set, there is a common fixed point.
- Ryll-Nardzewski fixed-point theorem: For continuous affine self-maps of a compact convex set, there is a fixed point.
- Earle–Hamilton fixed-point theorem: For holomorphic self-maps of certain open domains, fixed points exist.
- Kakutani fixed-point theorem: If a correspondence maps a compact convex set into itself with a closed graph and convex nonempty values, there is a fixed point.
- Aniki & Rauf (2019): results on the stability of partially ordered metric spaces for certain fixed-point iterations with mixed monotone mappings.

These theorems are useful because they provide guarantees that a problem has a solution, even in very large or complex spaces.