Schauder fixed point theorem

From Wikipedia, the free encyclopedia

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.

[edit] 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

[edit] External links