Schauder fixed-point theorem

The Schauder fixed-point theorem is an extension of the Brouwer fixed-point theorem to topological vector spaces, which may be of infinite dimension. It asserts that if ${\displaystyle K}$ is a nonempty convex closed subset of a Hausdorff topological vector space ${\displaystyle V}$ and ${\displaystyle f}$ is a continuous mapping of ${\displaystyle K}$ into itself such that ${\displaystyle f(K)}$ is contained in a compact subset of ${\displaystyle K}$, then ${\displaystyle f}$ has a fixed point.

A consequence, called Schaefer's fixed-point theorem, is particularly useful for proving existence of solutions to nonlinear partial differential equations. Schaefer's theorem is in fact a special case of the far reaching Leray–Schauder theorem which was proved earlier by Juliusz Schauder and Jean Leray. The statement is as follows:

Let ${\displaystyle f}$ be a continuous and compact mapping of a Banach space ${\displaystyle X}$ into itself, such that the set

${\displaystyle \{x\in X:x=\lambda f(x){\mbox{ for some }}0\leq \lambda \leq 1\}}$

is bounded. Then ${\displaystyle f}$ has a fixed point. (A compact mapping in this context is one for which the image of every bounded set is relatively compact.)