Schauder fixed point theorem

The Schauder fixed point theorem is an extension of the Brouwer fixed point theorem to topological vector spaces such as Banach spaces. It asserts that if $K$ is a compact, convex subset of a topological vector space and $T$ is a continuous mapping of $K$ into itself, then $T$ has a fixed point. It was conjectured and proved for special cases (i.e. the banach case) by Juliusz Schauder. 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

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

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

* with attached proof.