Theorem

Banach Fixed-Point Theorem

Analysis

Every contraction mapping on a complete metric space has exactly one fixed point, reached as the limit of repeated application of the map from any starting point. Also called the contraction mapping principle, it underlies existence and uniqueness proofs across analysis, including for ordinary differential equations.

Facts

Statement

Let (X, d) be a non-empty complete metric space with a contraction mapping T: X → X. Then T admits a unique fixed point x* ∈ X. 1

Proof Year

1922 1

Classification

Statement Form

Existence Theorem 1

Statement Form

Uniqueness Theorem 1

Connections

Source Banach Fixed-Point Theorem (Wikipedia)

Sources

1. Banach Fixed-Point Theorem (Wikipedia)

Wikimedia Foundation

Applications
The Banach fixed-point theorem is then used to show that this integral operator has a unique fixed point.
lead section, naming sentence
The theorem is named after Stefan Banach (1892-1945) who first stated it in 1922.
Proved By: Stefan Banach, Lead
The theorem is named after Stefan Banach (1892-1945) who first stated it in 1922.

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Banach Fixed-Point Theorem

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