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 $K$ is a convex subset of a topological vector space $V$ and $T$ is a continuous mapping of $K$ into itself so that $T(K)$ is contained in a compact subset of $K$, then $T$ has a fixed point. It was conjectured and proven for special cases, such as Banach spaces, by Juliusz Schauder in 1930. The full result was proven by Robert Cauty in 2001.

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 discovered earlier by Schauder and Jean Leray. The statement is as follows. Let $T$ be a continuous and compact mapping of a Banach space $X$ into itself, such that the set

$\{x\in X:x=\lambda Tx{\mbox{ for some }}0\leq \lambda \leq 1\}$

is bounded. Then $T$ has a fixed point.

References

• D. Gilbarg, N. Trudinger, Elliptic Partial Differential Equations of Second Order. ISBN 3-540-41160-7.
• Robert Cauty, Solution du problème de point fixe de Schauder, Fund. Math. 170 (2001), 231-246
• E. Zeidler, Nonlinear Functional Analysis and its Applications, I - Fixed-Point Theorems

External links

• "Schauder fixed point theorem". PlanetMath. with attached proof (for the Banach space case).