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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Entity-backed identity for the statement-form enum value this theorem already carries, resolved to a mathematics concept by an explicit value-to-entity map (phase 3 bucket conversion, docs\design_entity_backed_browse_buckets_20260928.md). The statement-form fact itself stays on the theorem unchanged.
Entity-backed identity for the statement-form enum value this theorem already carries, resolved to a mathematics concept by an explicit value-to-entity map (phase 3 bucket conversion, docs\design_entity_backed_browse_buckets_20260928.md). The statement-form fact itself stays on the theorem unchanged.
Entity-backed identity for the statement-form enum value this theorem already carries, resolved to a mathematics concept by an explicit value-to-entity map (phase 3 bucket conversion, docs\design_entity_backed_browse_buckets_20260928.md). The statement-form fact itself stays on the theorem unchanged.
Entity-backed identity for the statement-form enum value this theorem already carries, resolved to a mathematics concept by an explicit value-to-entity map (phase 3 bucket conversion, docs\design_entity_backed_browse_buckets_20260928.md). The statement-form fact itself stays on the theorem unchanged.
Sources
1. Schauder fixed point theorem, Wikipedia
Section on the Leray-Schauder theoremQuote, Section on the Leray-Schauder theorem
Let 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.
2. 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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