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
. Then the map T
admits one and only one fixed point x*
(this means Tx*
). Furthermore, this fixed point can be found as follows: start with an arbitrary element x0
and define a sequence by xn
= 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
) < d(x
) for all unequal x
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
, which lacks a fixed point. However, if the space X
, 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