In mathematics, the Banach fixed-point theorem (also known as the contraction mapping theorem or contractive mapping theorem or Banach–Caccioppoli 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. It can be understood as an abstract formulation of Picard's method of successive approximations. The theorem is named after Stefan Banach (1892–1945) who first stated it in 1922.

Statement

Definition. be a metric space is called a contraction mapping such that

Banach fixed-point theorem. be a non-empty complete metric space admits a unique fixed point.

Remark 1. The following inequalities are equivalent and describe the speed of convergence:

is called a Lipschitz constant.

Remark 2. is compact.

Remark 3..

Proof

be arbitrary and define a sequence we have the inequality

This follows by induction is a Cauchy sequence:

so that

we may write:

must be a fixed point:

:

.

Applications

• 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 on the space of continuous functions under the uniform norm. The Banach fixed-point theorem is then used to show that this integral operator has a unique fixed point.

• One consequence of the Banach fixed-point theorem is that small Lipschitz perturbations of the identity are bi-lipschitz. Then

• ;

• is a bi-Lipschitz homeomorphism;

• . A direct consequence of this result yields the proof of the inverse function theorem.

• It can be used to give sufficient conditions under which Newton's method of successive approximations is guaranteed to work, and similarly for Chebyshev's third-order method.

• It can be used to prove existence and uniqueness of solutions to integral equations.

• It can be used to give a proof to the Nash embedding theorem.

• It can be used to prove existence and uniqueness of solutions to value iteration, policy iteration, and policy evaluation of reinforcement learning.

• It can be used to prove existence and uniqueness of an equilibrium in Cournot competition, and other dynamic economic models.

• If X is a vector space, an alternative method for iteratively approximating the fixed point is to apply Newton's method.

• One real world illustration of the theorem is the following demonstration that can often be understood with little formal mathematics background. Suppose a person holds a sheet of paper on which is printed a map of the city they are in (or a map of their country, or a blueprint of the building they are in, etc.). The person then places the map on the floor. This can be considered to represent a continuous function from the city to itself and is clearly a contraction. Therefore, there is exactly one point in the map that lies directly above the point that it represents. The accuracy of the map and projection method do not matter as long as the map is complete and all points represented are moved closer together in the projection. What's more, should the person scrunch the map up into a ball and flatten it on the floor, as long as the paper does not tear, this will still be a contraction and the same conclusion can be reached.

Converses

Several converses of the Banach contraction principle exist. The following is due to Czesław Bessaga, from 1959:

be a map of an abstract set such that each iterate is the contraction constant.

is a map on a T1 topological space with a unique fixed point. In this case the metric is in fact an ultrametric.

Generalizations

There are a number of generalizations (some of which are immediate corollaries).

be a map on a complete non-empty metric space. Then, for example, some generalizations of the Banach fixed-point theorem are:

• has a unique fixed point.

• , and that

• has a unique fixed point.

a contraction. Indeed, the above result by Bessaga strongly suggests to look for such a metric. See also the article on fixed point theorems in infinite-dimensional spaces for generalizations.

In a non-empty compact metric space is continuous, and therefore assumes a minimum, which is easily shown to be zero.

A different class of generalizations arise from suitable generalizations of the notion of metric space, e.g. by weakening the defining axioms for the notion of metric. Some of these have applications, e.g., in the theory of programming semantics in theoretical computer science.

Example

, and it can be verified that

on this interval. Therefore, by an application of the mean value theorem is the unique fixed point on the interval, allowing for fixed-point iteration to be used.

. The Banach fixed-point theorem may be used to conclude that

digits:

See also

• Brouwer fixed-point theorem

• Caristi fixed-point theorem

• Contraction mapping

• Fichera's existence principle

• Fixed-point iteration

• Fixed-point theorems

• Infinite compositions of analytic functions

• Kantorovich theorem

Notes

References

• See chapter 7.