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 nonempty 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
x_{0} in
X and define a sequence by
x_{n} =
Tx_{n1} 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 PicardLindelöf 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.
An earlier version of this article was posted on
Planet Math. This article is
open content