Banach fixed point theorem

The Banach fixed point theorem is an important tool in the theory of metric spaces; it guarantees the existence and uniqueness of fixed points of certain self maps of metric spaces, and provides a constructive method to find those fixed points.

Let (X, d) be a non-empty complete metric space. Let T : X -> X be a contraction mapping on X, i.e: there is a real number q < 1 such that

for all x, y in X. Then the map T admits one and only one fixed point x^* in X (this means Tx^* = x^*). Furthermore, this fixed point can be found as follows: start with an arbitrary element x₀ in X and define a sequence by x_n = Tx_n-1 for n = 1, 2, 3, ... This sequence converges, and its limit is x^*. The following inequality describes the speed of convergence:

Note that the requirement d(Tx, Ty) < d(x, y) for all unequal x and y is in general not enough to ensure the existence of a fixed point, as is shown by the map T : [1,∞) → [1,∞) with T(x) = x + 1/x, which lacks a fixed point. However, if the space X is compact, then this weaker assumption does imply all the statements of the theorem.

When using the theorem in practice, the most difficult part is typically to define X properly so that T actually maps elements from X to X, i.e. that Tx is always an element of X.

A standard application is the proof of the Picard-Lindelöf theorem about the existence and uniqueness of solutions to certain ordinary differential equations. The sought solution of the differential equation is expressed as a fixed point of a suitable integral operator which transforms continuous functions into continuous functions. The Banach fixed point theorem is then used to show that this integral operator has a unique fixed point.

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An earlier version of this article was posted on Planet Math. This article is open content