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.[2][3]
Banach fixed-point theorem. Let be a non-
emptycomplete metric space with a contraction mapping Then T admits a unique
fixed-point in X (i.e. ). Furthermore, can be found as follows: start with an arbitrary element and define a
sequence by for Then .
Remark 1. 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 .
Remark 2. for all is in general not enough to ensure the existence of a fixed point, as is shown by the map
which lacks a fixed point. However, if 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
Remark 3. When using the theorem in practice, the most difficult part is typically to define properly so that
Proof
Let be arbitrary and define a
sequence by setting . We first note that for all we have the inequality
This follows by
induction on n, using the fact that T is a contraction mapping. Then we can show that is a
Cauchy sequence. In particular, let such that :
Let ε > 0 be arbitrary. Since , we can find a large so that
Therefore, by choosing and greater than we may write:
This proves that the sequence is Cauchy. By completeness of (X,d), the sequence has a limit Furthermore, must be a
fixed point of T:
As a contraction mapping, T is continuous, so bringing the limit inside T was justified. Lastly, T cannot have more than one fixed point in (X,d), since any pair of distinct fixed points p1 and p2 would contradict the contraction of T:
Applications
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 on the space of continuous functions under the
uniform norm. The Banach fixed-point theorem is then used to show that this integral operator has a unique fixed point.
One consequence of the Banach fixed-point theorem is that small Lipschitz perturbations of the identity are
bi-lipschitz homeomorphisms. Let Ω be an open set of a Banach space E; let I : Ω → E denote the identity (inclusion) map and let g : Ω → E be a Lipschitz map of constant k < 1. Then
Ω′ := (I + g)(Ω) is an open subset of E: precisely, for any x in Ω such that B(x, r) ⊂ Ω one has B((I + g)(x), r(1 − k)) ⊂ Ω′;
I + g : Ω → Ω′ is a bi-Lipschitz homeomorphism;
precisely, (I + g)−1 is still of the form I + h : Ω → Ω′ with h a Lipschitz map of constant k/(1 − k). A direct consequence of this result yields the proof of the
inverse function theorem.
It can be used to give sufficient conditions under which Newton's method of successive approximations is guaranteed to work, and similarly for Chebyshev's third-order method.
It can be used to prove existence and uniqueness of solutions to integral equations.
It can be used to prove existence and uniqueness of solutions to value iteration, policy iteration, and policy evaluation of
reinforcement learning.[5]
It can be used to prove existence and uniqueness of an equilibrium in
Cournot competition,[6] and other dynamic economic models.[7]
Converses
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
iteratefn has a unique fixed point. Let then there exists a complete metric on X such that f is contractive, and q is the contraction constant.
Indeed, very weak assumptions suffice to obtain such a kind of converse. For example if is a map on a
T1 topological space with a unique
fixed pointa, 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.[8] In this case the metric is in fact an
ultrametric.
Generalizations
There are a number of generalizations (some of which are immediate
corollaries).[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:
Assume that some iterate Tn of T is a contraction. Then T has a unique fixed point.
Assume that for each n, there exist cn such that d(Tn(x), Tn(y)) ≤ cnd(x, y) for all x and y, and that
Then T has a unique fixed point.
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.
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]
Example
Banach theorem allows for example fast and accurate calculation of the π number using the trigonometric
functions which numerically are the power
Taylor series.
Because and the π is the fixed point of for example the function
i.e.
and also the function is around π the contraction mapping from the obvious reasons because its derivative in π vanishes therefore π can be obtained from the infinite superposition for example for the argument value 3:
Already the triple superposition of this function at gives π with accuracy to 33 digits:
^Hitzler, Pascal; Seda, Anthony K. (2001). "A 'Converse' of the Banach Contraction Mapping Theorem". Journal of Electrical Engineering. 52 (10/s): 3–6.
^Seda, Anthony K.;
Hitzler, Pascal (2010). "Generalized Distance Functions in the Theory of Computation". The Computer Journal. 53 (4): 443–464.
doi:
10.1093/comjnl/bxm108.
References
Agarwal, Praveen; Jleli, Mohamed; Samet, Bessem (2018). "Banach Contraction Principle and Applications". Fixed Point Theory in Metric Spaces. Singapore: Springer. pp. 1–23.
doi:
10.1007/978-981-13-2913-5_1.
ISBN978-981-13-2912-8.
Chicone, Carmen (2006).
"Contraction". Ordinary Differential Equations with Applications (2nd ed.). New York: Springer. pp. 121–135.
ISBN0-387-30769-9.