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

Associated With

Brouwer Fixed-Point Theorem

In Branch

Analysis

Sources

1. Banach Fixed-Point Theorem (Wikipedia)

Wikimedia Foundation

Statement section
Banach fixed-point theorem. Let (X, d) be a non-empty complete metric space with a contraction mapping T: X → X.
lead section, naming sentence
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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