Schauder fixed-point theorem

Remarks

The proof reduces to the Brouwer fixed-point theorem. For $\varepsilon > 0$, compactness gives a finite $\varepsilon$-net $\set{y_1, \ldots, y_m}$ of $K$. The Schauder projection maps $K$ continuously into the convex hull of the $y_i$ and moves each point by less than $\varepsilon$. Brouwer's theorem gives a fixed point of the projected map on this finite-dimensional convex set, and it is an $\varepsilon$-fixed point of $f$. A limit as $\varepsilon \to 0$ gives a fixed point of $f$.

An equivalent form is often used in analysis. Let $C$ be a nonempty closed convex subset of a Banach space and let $f \colon C \to C$ be continuous with $f(C)$ contained in a compact set; then $f$ has a fixed point.

Example: Peano's existence theorem for $y' = F(t, y)$, $y(0) = y_0$ with continuous $F$ follows. The integral operator $y \mapsto y_0 + \int_0^t F(s, y(s)) \, ds$ maps a closed convex set of continuous functions into a compact set, by the Arzelà–Ascoli theorem. Its fixed point is a solution.

Compactness cannot be dropped: on the closed unit ball of $\ell^2$, Kakutani's map $x \mapsto \big(\sqrt{1 - \norm{x}^2}, x_1, x_2, \ldots\big)$ is continuous with no fixed point.

Going deeper

For a student

The theorem is the main tool for nonlinear problems without uniqueness, where the Banach fixed point theorem does not apply. Compactness usually comes from an Arzelà–Ascoli argument or a compact Sobolev embedding. Tychonoff extended it to compact convex subsets of locally convex spaces, and Schaefer's theorem replaces the invariant set by an a priori bound. For set-valued maps the analogue is the Kakutani fixed-point theorem and its infinite-dimensional versions.