In mathematics, the Banach fixed-point theorem (also known as the contraction mapping theorem or contractive mapping theorem or Banach–Caccioppoli 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.
It can be understood as an abstract formulation of Picard's method of successive approximations.
[1] The theorem is named after Stefan Banach (1892–1945) who first stated it in 1922.
be a non-empty complete metric space with a contraction mapping
The following inequalities are equivalent and describe the speed of convergence: Any such value of q is called a Lipschitz constant for
, and the smallest one is sometimes called "the best Lipschitz constant" of
is in general not enough to ensure the existence of a fixed point, as is shown by the map which lacks a fixed point.
is compact, then this weaker assumption does imply the existence and uniqueness of a fixed point, that can be easily found as a minimizer of
, indeed, a minimizer exists by compactness, and has to be a fixed point of
It then easily follows that the fixed point is the limit of any sequence of iterations of
When using the theorem in practice, the most difficult part is typically to define
: Several converses of the Banach contraction principle exist.
The following is due to Czesław Bessaga, from 1959: Let f : X → X be a map of an abstract set such that each iterate fn has a unique fixed point.
Indeed, very weak assumptions suffice to obtain such a kind of converse.
is a map on a T1 topological space with a unique fixed point a, such that for each
we have fn(x) → a, then there already exists a metric on X with respect to which f satisfies the conditions of the Banach contraction principle with contraction constant 1/2.
[9] Let T : X → X be a map on a complete non-empty metric space.
Then, for example, some generalizations of the Banach fixed-point theorem are: In applications, the existence and uniqueness of a fixed point often can be shown directly with the standard Banach fixed point theorem, by a suitable choice of the metric that makes the map T a contraction.
Indeed, the above result by Bessaga strongly suggests to look for such a metric.
See also the article on fixed point theorems in infinite-dimensional spaces for generalizations.
In a non-empty compact metric space, any function
The proof is simpler than the Banach theorem, because the function
is continuous, and therefore assumes a minimum, which is easily shown to be zero.
A different class of generalizations arise from suitable generalizations of the notion of metric space, e.g. by weakening the defining axioms for the notion of metric.
[10] Some of these have applications, e.g., in the theory of programming semantics in theoretical computer science.
[11] An application of the Banach fixed-point theorem and fixed-point iteration can be used to quickly obtain an approximation of π with high accuracy.
It can be verified that π is a fixed point of f, and that f maps the interval
Therefore, by an application of the mean value theorem, f has a Lipschitz constant less than 1 (namely
Applying the Banach fixed-point theorem shows that the fixed point π is the unique fixed point on the interval, allowing for fixed-point iteration to be used.
The Banach fixed-point theorem may be used to conclude that Applying f to 3 only three times already yields an expansion of π accurate to 33 digits: This article incorporates material from Banach fixed point theorem on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.