The Schauder fixed point theorem can be proved using the Brouwer fixed point theorem. It says that if $K$ is a convex subset of a Banach space (or more generally: topological vector space) $V$ and $T$ is a continuous map of $K$ into itself such that $T(K)$ is contained in a compact subset of $K$, then $T$ has a fixed point.

The Schauder fixed point theorem in its turn is an important tool for existence proofs in differential equations. One easy application is the Peano existence theorem, but there are also more sophisticated examples.