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