The Schauder Fixed-Point Theorem

Abstract

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 {\displaystyle K} is a nonempty convex subset of a Hausdorff topological vector space V {\displaystyle V} and T {\displaystyle T} is a continuous mapping of K {\displaystyle K} into itself such that T ( K ) {\displaystyle T(K)} is contained in a compact subset of K {\displaystyle K} , then T {\displaystyle T} 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 discovered earlier by Juliusz Schauder and Jean Leray. The statement is as follows: Let T {\displaystyle T} be a continuous and compact mapping of a Banach space X {\displaystyle X} into itself, such that the set { x ∈ X : x = λ T x for some 0 ≤ λ ≤ 1 } {\displaystyle \{x\in X:x=\lambda Tx{\mbox{ for some }}0\leq \lambda \leq 1\}} is bounded. Then T {\displaystyle T} has a fixed point.