Banach fixed point theorem
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The Banach fixed point theorem (also known as the contraction mapping theorem or contraction mapping principle) 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. The theorem is named after Stefan Banach (1892-1945), and was first stated by Banach in 1922.
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[edit] The theorem
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 nonnegative real number q < 1 such that
- <math>d(Tx,Ty) \le q\cdot d(x,y)</math>
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 an iterative 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:
- <math>d(x^*, x_n) \leq \frac{q^n}{1-q} d(x_1,x_0)</math>.
Equivalently,
- <math>d(x^*, x_{n+1}) \leq \frac{q}{1-q} d(x_{n+1},x_n)</math>
and
- <math>d(x^*, x_{n+1}) \leq q d(x_n,x^*)</math>.
The smallest such value of q is sometimes called the Lipschitz constant.
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.
[edit] Proof
Choose any <math>x_0 \in (X, d)</math>. For each <math>n \in \{1, 2, \ldots\}</math>, define <math>x_n = Tx_{n-1}\,\!</math>. We claim that for all <math>n \in \{1, 2, \dots\}</math>, the following is true:
- <math>d(x_{n+1}, x_n) \leq q^n d(x_1, x_0)</math>.
To show this, we will proceed using induction. The above statement is true for the case <math>n = 1\,\!</math>, for
- <math>d(x_{1+1}, x_1) = d(x_2, x_1) = d(Tx_1, Tx_0) \leq qd(x_1, x_0)</math>.
Suppose the above statement holds for some <math>k \in \{1, 2, \ldots\}</math>. Then we have
<math>d(x_{(k + 1) + 1}, x_{k + 1})\,\!</math> <math>= d(x_{k + 2}, x_{k + 1})\,\!</math> <math>= d(Tx_{k + 1}, Tx_k)\,\!</math> <math>\leq q d(x_{k + 1}, x_k)</math> <math>\leq q \cdot q^kd(x_1, x_0)</math> <math>= q^{k + 1}d(x_1, x_0)\,\!</math>.
The inductive assumption is used going from line three to line four. By the principle of mathematical induction, for all <math>n \in \{1, 2, \ldots\}</math>, the above claim is true.
Let <math>\epsilon > 0\,\!</math>. Since <math>0 \leq q < 1</math>, we can find a large <math>N \in \{1, 2, \ldots\}</math> so that
- <math>q^N < \frac{\epsilon(1-q)}{d(x_1, x_0)}</math>.
Using the claim above, we have that for any <math>m\,\!</math>, <math>n \in \{0, 1, \ldots\}</math> with <math>m > n \geq N</math>,
<math>d\left(x_m, x_n\right)</math> <math>\leq d(x_m, x_{m-1}) + d(x_{m-1}, x_{m-2}) + \cdots + d(x_{n+1}, x_n)</math> <math>\leq q^{m-1}d(x_1, x_0) + q^{m-2}d(x_1, x_0) + \cdots + q^nd(x_1, x_0)</math> <math>= d(x_1, x_0)q^n \cdot \sum_{k=0}^{m-n-1} q^k</math> <math>< d(x_1, x_0)q^n \cdot \sum_{k=0}^\infty q^k</math> <math>= d(x_1, x_0)q^n \frac{1}{1-q}</math> <math>= q^n \frac{d(x_1, x_0)}{1-q}</math> <math>< \frac{\epsilon(1-q)}{d(x_1, x_0)}\cdot\frac{d(x_1, x_0)}{1-q}</math> <math>= \epsilon\,\!</math>.
The inequality in line one follows from repeated applications of the triangle inequality; the series in line four is a geometric series with <math>0 \leq q < 1</math> and hence it converges. The above shows that <math>\{x_n\}_{n\geq 0}</math> is a Cauchy sequence in <math>(X, d)\,\!</math> and hence convergent by completeness. So let <math>x^* = \lim_{n\to\infty} x_n</math>. We make two claims: (1) <math>x^*\,\!</math> is a fixed point of <math>T\,\!</math>. That is, <math>Tx^* = x^*\,\!</math>; (2) <math>x^*\,\!</math> is the only fixed point of <math>T\,\!</math> in <math>(X, d)\,\!</math>.
To see (1), we note that for any <math>n \in \{0, 1, \ldots\}</math>,
- <math>0 \leq d(x_{n+1}, Tx^*) = d(Tx_n, Tx^*) \leq q d(x_n, x^*)</math>.
Since <math>qd(x_n, x^*) \to 0</math> as <math>n \to \infty</math>, the squeeze theorem shows that <math>\lim_{n\to\infty} d(x_{n+1}, Tx^*) = 0</math>. This shows that <math>x_n \to Tx^*</math> as <math>n \to \infty</math>. But <math>x_n \to x^*</math> as <math>n \to \infty</math>, and limits are unique; hence it must be the case that <math>x^* = Tx^*\,\!</math>.
To show (2), we suppose that <math>y\,\!</math> also satisfies <math>Ty = y\,\!</math>. Then
- <math>0 \leq d(x^*, y) = d(Tx^*, Ty) \leq q d(x^*, y)</math>.
Remembering that <math>0 \leq q < 1</math>, the above implies that <math>0 \leq (1-q) d(x^*, y) \leq 0</math>, which shows that <math>d(x^*, y) = 0\,\!</math>, whence by positive definiteness, <math>x^* = y\,\!</math> and the proof is complete.
[edit] 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 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.
[edit] Converses
Several converses of the Banach contraction principle exist. The following is due to Czeslaw Bessaga, from 1959:
Let <math>f:X\rightarrow X</math> be a map of an abstract set such that each iterate f n has a unique fixed point. Let q be a real number, 0 < q < 1. Then there exists a complete metric on X such that f is contractive, and q is the contraction constant.
[edit] Generalizations
See the article on fixed point theorems in infinite-dimensional spaces for generalizations.
[edit] References
- Vasile I. Istratescu, Fixed Point Theory, An Introduction, D.Reidel, the Netherlands (1981). ISBN 90-277-1224-7 See chapter 7.
- Andrzej Granas and James Dugundji, Fixed Point Theory (2003) Springer-Verlag, New York, ISBN 0-387-00173-5.
- William A. Kirk and Brailey Sims, Handbook of Metric Fixed Point Theory (2001), Kluwer Academic, London ISBN 0-7923-7073-2.
An earlier version of this article was posted on Planet Math. This article is open content.de:Fixpunktsatz von Banach fr:Application contractante it:Teorema del punto fisso di Banach he:משפט נקודת השבת של בנך pl:Twierdzenie Banacha o kontrakcji fi:Banachin kiintopistelause ru:Теорема Банаха о неподвижной точке
