Banach Fixed-Point Theorem

The Banach Fixed-Point Theorem (also called the Contraction Mapping Theorem) guarantees that every contraction on a complete metric space has a unique fixed point. This powerful result is the foundation for proving existence and uniqueness of solutions to differential equations, integral equations, and iterative algorithms.

Statement

Theorem8.1Banach Fixed-Point Theorem

Let $(X, d)$ be a complete metric space and $f : X \to X$ a contraction mapping: there exists $0 \leq \lambda < 1$ such that

$d(f(x), f(y)) \leq \lambda d(x, y) \quad \text{for all } x, y \in X.$

Then:

$f$ has a unique fixed point $x^* \in X$ : $f(x^*) = x^*$ .
For any $x_0 \in X$ , the sequence $x_{n+1} = f(x_n)$ converges to $x^*$ .
The error satisfies $d(x_n, x^*) \leq \frac{\lambda^n}{1 - \lambda} d(f(x_0), x_0)$ .

Proof

Existence: Fix $x_0 \in X$ and define $x_{n+1} = f(x_n)$ . We show $(x_n)$ is Cauchy.

For $m > n$ ,

$d(x_m, x_n) \leq \sum_{k=n}^{m-1} d(x_{k+1}, x_k) = \sum_{k=n}^{m-1} d(f(x_k), f(x_{k-1})) \leq \sum_{k=n}^{m-1} \lambda^k d(x_1, x_0) \leq \frac{\lambda^n}{1 - \lambda} d(x_1, x_0).$

As $n \to \infty$ , $d(x_m, x_n) \to 0$ , so $(x_n)$ is Cauchy. By completeness, $x_n \to x^*$ for some $x^* \in X$ . By continuity of $f$ ,

$f(x^*) = f(\lim x_n) = \lim f(x_n) = \lim x_{n+1} = x^*.$

Uniqueness: If $f(y) = y$ , then $d(x^*, y) = d(f(x^*), f(y)) \leq \lambda d(x^*, y)$ . Since $\lambda < 1$ , this implies $d(x^*, y) = 0$ , so $x^* = y$ .

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Applications

ExamplePicard-Lindelöf theorem (ODEs)

The initial value problem $y' = f(t, y)$ , $y(t_0) = y_0$ has a unique solution if $f$ is Lipschitz in $y$ . Proof: rewrite as $y(t) = y_0 + \int_{t_0}^t f(s, y(s)) \, ds$ and show the integral operator is a contraction on $C([t_0, t_1])$ with appropriate norm.

ExampleComputing √2 via iteration

Define $f(x) = (x + 2/x)/2$ on $[1, 2]$ . This is a contraction (with small enough $\lambda$ ), and its fixed point satisfies $x = (x + 2/x)/2$ , i.e., $x^2 = 2$ . Starting with $x_0 = 1$ , the iteration $x_{n+1} = (x_n + 2/x_n)/2$ converges to $\sqrt{2}$ (Newton's method).

Summary

The Banach Fixed-Point Theorem is a cornerstone of analysis:

Contractions on complete spaces have unique fixed points.
Provides constructive proof via iteration.
Applications: ODEs, PDEs, integral equations, numerical methods.

See Completeness for background.