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
- 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 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:
qn
d(x*, xn) ≤ ----- d(x1,x0)
1-q
Note that all hypotheses of the theorem are necessary: if the space X is not complete, no fixed point need exist. Also, the requirement d(Tx, Ty) < d(x, y) for all x and y is not enough to ensure the existence of a fixed point.
An earlier version of this article was posted on Planet Math. This article is open content