The Leray-Schauder Fixed-Point Theorem gives conditions under which a continuous, compact map on a Banach space has a fixed point, generalizing the Brouwer Fixed-Point Theorem from finite-dimensional space to the infinite-dimensional setting by requiring the map to send bounded sets to sets with compact closure. Named for Jean Leray and Juliusz Schauder, it is a foundational existence tool used throughout the theory of nonlinear differential and integral equations.
Facts
StatementLet f be a continuous and compact mapping of a Banach space X into itself, such that the set {x in X: x = lambda f(x) for some 0 <= lambda <= 1} is bounded. Then f has a fixed point. 1 Classification
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Source Schauder fixed point theorem, Wikipedia
Sources
1. Schauder fixed point theorem, Wikipedia
History
This version is known as the Schauder-Tychonoff fixed-point theorem.
Proved By: Juliusz Schauder, Lead
The theorem was conjectured and proven for special cases, such as Banach spaces, by Juliusz Schauder in 1930.
In Group: Fixed-Point Theorems, Lead paragraph
The Schauder fixed-point theorem is an extension of the Brouwer fixed-point theorem to locally convex topological vector spaces, which may be of infinite dimension.
View the Source2. Jean Leray, Wikipedia
Section on his work with Juliusz SchauderQuote, Section on his work with Juliusz Schauder
Leray, Jean; Schauder, Juliusz (1934). Topologie et equations fonctionelles
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