Banach fixed point theorem Banach fixed point theorem Battle of Tours Category="Mathematical analysis"Category="Mathematical theorems"Category="Topology"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. The theorem is named after Stefan Banach (1892-1945), and was first stated by Banach in 1922.

The theorem

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
d (Tx,Ty) ≤ q • d (x,y)
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 an iterative sequence by xn = Txn-1 for n = 1, 2, 3, ... This sequence converges, and its limit is x *. The following inequality describes the speed of convergence:
d (x * , x n ) ≤  q n
–––––
1-q		  d (x 1 ,x 0 )
.

Equivalently,

d (x * , x n + 1 ) ≤  q
–––––
1-q		  d (x n + 1 ,x n )
and
d (x * , x n + 1 ) ≤ q d (x n ,x * ).

The smallest such value of q is sometimes called the Lipschitz constant.

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.

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 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.

Converses

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

Let f : X → X be a map of an abstract set such that each iterate f  n has a unique fixed point. Let q be a real number, 0 < q < 1. Then there exists a complete metric on X such that f is contractive, and q is the contraction constant.

Generalizations

See the article on Fixed point theorems in infinite-dimensional spaces for generalizations.

References

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