Banach fixed-point theorem: Difference between revisions

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{{Short description|Theorem about metric spaces}}

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 space]]s; it guarantees the existence and uniqueness of [[fixed point (mathematics)|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]].

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 [[Convergence proof techniques#contraction mapping|tool]] in the theory of [[metric space]]s; it guarantees the existence and uniqueness of [[fixed point (mathematics)|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 [[Fixed-point iteration|Picard's method of successive approximations]].<ref>{{cite book |first1=David |last1=Kinderlehrer |author-link=David Kinderlehrer |first2=Guido |last2=Stampacchia |author-link2=Guido Stampacchia |chapter=Variational Inequalities in '''R'''^N |title=An Introduction to Variational Inequalities and Their Applications |___location=New York |publisher=Academic Press |year=1980 |isbn=0-12-407350-6 |pages=7–22 |chapter-url=https://books.google.com/books?id=eCDnoB3Np5oC&pg=PA7 }}</ref> The theorem is named after [[Stefan Banach]] (1892–1945) who first stated it in 1922.<ref>{{cite journal |last=Banach|first= Stefan|author-link=Stefan Banach| title=Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales|journal=[[Fundamenta Mathematicae]]|volume= 3|year=1922|pages= 133–181 |url=http://matwbn.icm.edu.pl/ksiazki/or/or2/or215.pdf |archive-url=https://web.archive.org/web/20110607002842/http://matwbn.icm.edu.pl/ksiazki/or/or2/or215.pdf |archive-date=2011-06-07 |url-status=live |doi=10.4064/fm-3-1-133-181}}</ref><ref>{{cite journal |first=Krzysztof |last=Ciesielski |title=On Stefan Banach and some of his results |journal=Banach J. Math. Anal. |volume=1 |year=2007 |issue=1 |pages=1–10 |url=http://www.emis.de/journals/BJMA/tex_v1_n1_a1.pdf |archive-url=https://web.archive.org/web/20090530012258/http://www.emis.de/journals/BJMA/tex_v1_n1_a1.pdf |archive-date=2009-05-30 |url-status=live |doi=10.15352/bjma/1240321550 |doi-access=free }}</ref>

== The theoremStatement==

''Definition.'' Let <math>(X, d)</math> be a [[metric space]]. Then a map <math>T : X \to X</math> is called a [[contraction mapping]] on ''X'' if there exists <math>q \in [0, 1)</math> such that

:<math>d(T(x),T(y)) \le q d(x,y)</math>

for all <math>x, y \in X.</math>

<blockquote>

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''&nbsp;<&nbsp;1 such that

'''Banach fixed-point theorem.''' Let <math>(X, d)</math> be a non-[[Empty set|empty]] [[complete metric space]] with a contraction mapping <math>T : X \to X.</math> Then ''T'' admits a unique [[Fixed point (mathematics)|fixed-point]] <math>x^*</math> in ''X'' (i.e. <math>T(x^*) = x^*</math>). Furthermore, <math>x^*</math> can be found as follows: start with an arbitrary element <math>x_0 \in X</math> and define a [[sequence]] <math>(x_n)_{n\in\mathbb N}</math> by <math>x_n = T(x_{n-1})</math> for <math>n \geq 1.</math> Then <math>\lim_{n \to \infty} x_n = x^*</math>.</blockquote>

:<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 ''x''₀ in ''X'' and define an [[iterative method|iterative]] sequence by ''x''_''n'' = ''Tx''_''n''-1 for ''n'' = 1, 2, 3, ... This sequence [[limit (mathematics)|converges]], and its limit is ''x''^*. The following inequality describes the speed of convergence:

''Remark 1.'' The following inequalities are equivalent and describe the [[Rate of convergence|speed of convergence]]:

:<math>d(x^*, x_n) \leq \frac{q^n}{1-q} d(x_1,x_0)</math>.

:<math>

Equivalently,

\begin{align}

d(x^*, x_n) & \leq \frac{q^n}{1-q} d(x_1,x_0), \\[5pt]

d(x^*, x_{n+1}) & \leq \frac{q}{1-q} d(x_{n+1},x_n), \\[5pt]

d(x^*, x_{n+1}) & \leq q d(x^*,x_n).

\end{align}

</math>

Any such value of ''q'' is called a ''[[Lipschitz constant]]'' for <math>T</math>, and the smallest one is sometimes called "the best Lipschitz constant" of <math>T</math>.

:<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>.

''Remark 2.'' <math>d(T(x),T(y))<d(x,y)</math> for all <math>x \neq y</math> is in general not enough to ensure the existence of a fixed point, as is shown by the map

The smallest such value of ''q'' is sometimes called the ''[[Lipschitz constant]]''.

:<math>T : [1,\infty) \to [1,\infty), \,\, T(x)=x+\tfrac{1}{x}\,,</math>

which lacks a fixed point. However, if <math>X</math> is [[Compact space|compact]], then this weaker assumption does imply the existence and uniqueness of a fixed point, that can be easily found as a minimizer of <math>d(x,T(x))</math>, indeed, a minimizer exists by compactness, and has to be a fixed point of <math>T.</math> It then easily follows that the fixed point is the limit of any sequence of iterations of <math>T.</math>

''Remark 3.'' When using the theorem in practice, the most difficult part is typically to define <math>X</math> properly so that <math>T(X) \subseteq X.</math>

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'' : <nowiki>[1,&infin;) &rarr; [1,&infin;)</nowiki> with ''T''(''x'') = ''x'' + 1/''x'', which lacks a fixed point. However, if the space ''X'' is [[Compact space|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''.

==Proof==

<!-- The \,\! at the end of some math markup is to keep the formula rendered as PNG instead of HTML. Please don't remove it.-->

Choose anyLet <math>x_0 \in (X, d)</math>. Forbe arbitrary and define a each[[sequence]] <math>(x_n)_{n \in \{1,mathbb 2, \ldots\N}</math>, defineby setting <math>x_n = Tx_T(x_{n-1}\,\!)</math>. We claimfirst note that for all <math>n \in \{1N, 2, \dots\}</math>, thewe followinghave isthe true:inequality

::<math>d(x_{n+1}, x_n) \leqle q^n d(x_1, x_0).</math>.

This follows by [[Principle of mathematical induction|induction]] on <math>n</math>, using the fact that <math>T</math> is a contraction mapping. Then we can show that <math>(x_n)_{n\in\mathbb N}</math> is a [[Cauchy sequence]]. In particular, let <math>m, n \in \N</math> such that <math>m > n </math>:

To show this, we will proceed using induction. The above statement is true for the case <math>n = 1\,\!</math>, for

: <math>\begin{align}

::<math>d(x_{1+1}, x_1) = d(x_2, x_1) = d(Tx_1, Tx_0) \leq qd(x_1, x_0)</math>.

d(x_m, x_n) & \leq d(x_m, x_{m-1}) + d(x_{m-1}, x_{m-2}) + \cdots + d(x_{n+1}, x_n) \\[5pt]

& \leq q^{m-1}d(x_1, x_0) + q^{m-2}d(x_1, x_0) + \cdots + q^nd(x_1, x_0) \\[5pt]

& = q^n d(x_1, x_0) \sum_{k=0}^{m-n-1} q^k \\[5pt]

& \leq q^n d(x_1, x_0) \sum_{k=0}^\infty q^k \\[5pt]

& = q^n d(x_1, x_0) \left ( \frac{1}{1-q} \right ).

\end{align}</math>

SupposeLet the<math>\varepsilon above> statement0</math> holdsbe forarbitrary. someSince <math>kq \in \{[0,1)</math>, 2,we can find a large <math>N \ldotsin \}N</math>. Then weso havethat

:<math>q^N < \frac{\varepsilon(1-q)}{d(x_1, x_0)}.</math>

::{|

|-

|<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>.

|}

Therefore, by choosing <math>m</math> and <math>n</math> greater than <math>N</math> we may write:

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.

:<math>d(x_m, x_n) \leq q^n d(x_1, x_0) \left ( \frac{1}{1-q} \right ) < \left (\frac{\varepsilon(1-q)}{d(x_1, x_0)} \right ) d(x_1, x_0) \left ( \frac{1}{1-q} \right ) = \varepsilon.</math>

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

This proves that the sequence <math>(x_n)_{n\in\mathbb N}</math> is Cauchy. By completeness of <math>(X, d)</math>, the sequence has a limit <math>x^* \in X.</math> Furthermore, <math>x^*</math> must be a [[Fixed point (mathematics)|fixed point]] of <math>T</math>:

::<math>q^N < \frac{\epsilon(1-q)}{d(x_1, x_0)}</math>.

:<math>x^*=\lim_{n\to\infty} x_n = \lim_{n\to\infty} T(x_{n-1}) = T\left(\lim_{n\to\infty} x_{n-1} \right) = T(x^*). </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>,

As a contraction mapping, <math>T</math> is continuous, so bringing the limit inside <math>T</math> was justified. Lastly, <math>T</math> cannot have more than one fixed point in <math>(X, d)</math>, since any pair of distinct fixed points <math>p_1</math> and <math>p_2</math> would contradict the contraction of <math>T</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>.

|}

:<math> d(T(p_1),T(p_2)) = d(p_1,p_2) > q d(p_1, p_2).</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>.

==Applications==

To see (1), we note that for any <math>n \in \{0, 1, \ldots\}</math>,

* A standard application is the proof of the [[Picard–Lindelöf theorem]] about the existence and uniqueness of solutions to certain [[ordinary differential equation]]s. 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 [[Lipschitz continuity#Definitions|bi-lipschitz]] homeomorphisms. Let Ω be an open set of a Banach space ''E''; let {{nobr|''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 {{nobr|''B''(''x'', ''r'') ⊂ Ω}} one has {{nobr|''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 {{nobr|''I'' + ''h'' : Ω → Ω′}} with ''h'' a Lipschitz map of constant ''k''/(1&nbsp;−&nbsp;''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 give a proof to the [[Nash embedding theorem]].<ref>{{cite journal |first=Matthias|last=Günther|title=Zum Einbettungssatz von J. Nash | trans-title=On the embedding theorem of J. Nash | language=de | journal=[[Mathematische Nachrichten]]|volume= 144 |year=1989|pages= 165–187|doi=10.1002/mana.19891440113 | mr=1037168}}</ref>

* It can be used to prove existence and uniqueness of solutions to value iteration, policy iteration, and policy evaluation of [[reinforcement learning]].<ref>{{cite book |first1=Frank L. |last1=Lewis |first2=Draguna |last2=Vrabie |first3=Vassilis L. |last3=Syrmos |title=Optimal Control |chapter=Reinforcement Learning and Optimal Adaptive Control |___location=New York |publisher=John Wiley & Sons |year=2012 |isbn=978-1-118-12272-3 |pages=461–517 [p. 474] |chapter-url=https://books.google.com/books?id=U3Gtlot_hYEC&pg=PA474 }}</ref>

* It can be used to prove existence and uniqueness of an equilibrium in [[Cournot competition]],<ref>{{cite journal |first1=Ngo Van |last1=Long |first2=Antoine |last2=Soubeyran |title=Existence and Uniqueness of Cournot Equilibrium: A Contraction Mapping Approach |journal=[[Economics Letters]] |volume=67 |issue=3 |year=2000 |pages=345–348 |doi=10.1016/S0165-1765(00)00211-1 |url=https://www.cirano.qc.ca/pdf/publication/99s-22.pdf |archive-url=https://web.archive.org/web/20041230225125/http://www.cirano.qc.ca/pdf/publication/99s-22.pdf |archive-date=2004-12-30 |url-status=live }}</ref> and other dynamic economic models.<ref>{{cite book |first1=Nancy L. |last1=Stokey|author1-link=Nancy Stokey |first2=Robert E. Jr. |last2=Lucas |author-link2=Robert Lucas Jr. |title=Recursive Methods in Economic Dynamics |___location=Cambridge |publisher=Harvard University Press |year=1989 |isbn=0-674-75096-9 |pages=508–516 |url=https://books.google.com/books?id=BgQ3AwAAQBAJ&pg=PA508 }}</ref>

==Converses==

::<math>0 \leq d(x_{n+1}, Tx^*) = d(Tx_n, Tx^*) \leq q d(x_n, x^*)</math>.

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 (mathematics)|set]] such that each [[iterated function|iterate]] ''fⁿ'' has a unique fixed point. Let <math>q \in (0, 1),</math> then there exists a complete metric on ''X'' such that ''f'' is contractive, and ''q'' is the contraction constant.

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>.

Indeed, very weak assumptions suffice to obtain such a kind of converse. For example if <math>f : X \to X</math> is a map on a [[T1 space|''T''₁ topological space]] with a unique [[fixed point (mathematics)|fixed point]] ''a'', such that for each <math>x \in X</math> 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.<ref>{{cite journal |first1=Pascal |last1=Hitzler | author-link1=Pascal Hitzler|first2=Anthony K. |last2=Seda |title=A 'Converse' of the Banach Contraction Mapping Theorem |journal=Journal of Electrical Engineering |volume=52 |issue=10/s |year=2001 |pages=3–6 }}</ref> In this case the metric is in fact an [[ultrametric]].

To show (2), we suppose that <math>y\,\!</math> also satisfies <math>Ty = y\,\!</math>. Then

==Generalizations==

::<math>0 \leq d(x^*, y) = d(Tx^*, Ty) \leq q d(x^*, y)</math>.

There are a number of generalizations (some of which are immediate [[Corollary|corollaries]]).<ref name=Latif2014>{{cite book |first=Abdul |last=Latif |title=Topics in Fixed Point Theory |pages=33–64 |chapter=Banach Contraction Principle and its Generalizations |publisher=Springer |year=2014 |doi=10.1007/978-3-319-01586-6_2 |isbn=978-3-319-01585-9 }}</ref>

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:

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.

*Assume that some iterate ''Tⁿ'' of ''T'' is a contraction. Then ''T'' has a unique fixed point.

*Assume that for each ''n'', there exist ''c_n'' such that ''d''(''T''^''n''(''x''), ''T''^''n''(''y'')) ≤ ''c''_''n''''d''(''x'', ''y'') for all ''x'' and ''y'', and that

::<math>\sum\nolimits_n c_n <\infty.</math>

: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.

In a non-empty [[compact metric space]], any function <math>T</math> satisfying <math>d(T(x),T(y))<d(x,y)</math> for all distinct <math>x,y</math>, has a unique fixed point. The proof is simpler than the Banach theorem, because the function <math>d(T(x),x)</math> is continuous, and therefore assumes a minimum, which is easily shown to be zero.

==Applications==

A standard application is the proof of the [[Picard-Lindelöf theorem]] about the existence and uniqueness of solutions to certain [[ordinary differential equation]]s. 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.

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.<ref>{{cite book |first1=Pascal |last1=Hitzler | author-link1=Pascal Hitzler|first2=Anthony |last2=Seda |title=Mathematical Aspects of Logic Programming Semantics |publisher=Chapman and Hall/CRC |year=2010 |isbn=978-1-4398-2961-5 }}</ref> Some of these have applications, e.g., in the theory of programming semantics in theoretical computer science.<ref>{{cite journal |first1=Anthony K. |last1=Seda |first2=Pascal |last2=Hitzler | author-link2=Pascal Hitzler|title=Generalized Distance Functions in the Theory of Computation |journal=The Computer Journal |volume=53 |issue=4 |pages=443–464 |year=2010 |doi=10.1093/comjnl/bxm108 }}</ref>

==Converses==

Several converses of the Banach contraction principle exist. The following is due to [[Czeslaw Bessaga]], from [[1959]]:

==Example==

Let <math>f:X\rightarrow X</math> be a map of an abstract [[set (mathematics)|set]] such that each [[iterated function|iterate]] ''f''^&nbsp;n has a unique fixed point. Let ''q'' be a real number, 0 &lt; q &lt; 1. Then there exists a complete metric on ''X'' such that ''f'' is contractive, and ''q'' is the contraction constant.

An application of the Banach fixed-point theorem and fixed-point iteration can be used to quickly obtain an approximation of {{pi}} with high accuracy. Consider the function <math>f(x)=\sin(x)+x</math>. It can be verified that {{pi}} is a fixed point of ''f'', and that ''f'' maps the interval <math>\left[3\pi/4,5\pi/4\right]</math> to itself. Moreover, <math>f'(x)=1+\cos(x)</math>, and it can be verified that

==Generalizations==

See the article on [[fixed point theorems in infinite-dimensional spaces]] for generalizations.

:<math>0\leq1+\cos(x)\leq1-\frac{1}{\sqrt{2}}<1</math>

on this interval. Therefore, by an application of the [[mean value theorem]], ''f'' has a Lipschitz constant less than 1 (namely <math>1-1/\sqrt{2}</math>). Applying the Banach fixed-point theorem shows that the fixed point {{pi}} is the unique fixed point on the interval, allowing for fixed-point iteration to be used.

For example, the value 3 may be chosen to start the fixed-point iteration, as <math>3\pi/4\leq3\leq5\pi/4</math>. The Banach fixed-point theorem may be used to conclude that

: <math>\pi=f(f(f(\cdots f(3)\cdots)))).</math>

Applying ''f'' to 3 only three times already yields an expansion of {{pi}} accurate to 33 digits:

: <math>f(f(f(3)))=3.141592653589793238462643383279502\ldots\,.</math>

==See also==

{{Div col|colwidth=30em}}

* [[Brouwer fixed-point theorem]]

* [[Caristi fixed-point theorem]]

* [[Contraction mapping]]

* [[Fichera's existence principle]]

* [[Fixed-point iteration]]

* [[Fixed-point theorem]]s

* [[Infinite compositions of analytic functions]]

* [[Kantorovich theorem]]

{{Div col end}}

==Notes==

<references/>

==References==

*{{cite book |first1=Praveen |last1=Agarwal |first2=Mohamed |last2=Jleli |first3=Bessem |last3=Samet |chapter=Banach Contraction Principle and Applications |title=Fixed Point Theory in Metric Spaces |publisher=Springer |___location=Singapore |year=2018 |isbn=978-981-13-2912-8 |pages=1–23 |doi=10.1007/978-981-13-2913-5_1 }}

* Vasile I. Istratescu, ''Fixed Point Theory, An Introduction'', D.Reidel, the Netherlands (1981). ISBN 90-277-1224-7 See chapter 7.

*{{cite book |first=Carmen |last=Chicone |title=Ordinary Differential Equations with Applications |___location=New York |publisher=Springer |edition=2nd |year=2006 |isbn=0-387-30769-9 |chapter=Contraction |chapter-url=https://books.google.com/books?id=yfY2uGROVrUC&pg=PA121 |pages=121–135 }}

* Andrzej Granas and James Dugundji, ''Fixed Point Theory'' (2003) Springer-Verlag, New York, ISBN 0-387-00173-5.

*{{cite book |first1=Andrzej |last1=Granas |first2=James |last2=Dugundji |author-link2=James Dugundji |title=Fixed Point Theory |year=2003 |publisher=Springer-Verlag |___location=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.

*{{cite book |first=Vasile I. |last=Istrăţescu |title=Fixed Point Theory: An Introduction |publisher=D. Reidel |___location=The Netherlands |year=1981 |isbn=90-277-1224-7 }} See chapter 7.

*{{cite book |last1=Kirk |first1=William A. |last2=Khamsi |first2=Mohamed A. |title=An Introduction to Metric Spaces and Fixed Point Theory |year=2001 |publisher=John Wiley |___location=New York |isbn=0-471-41825-0 }}

{{PlanetMath attribution |urlname=banachfixedpointtheorem |title=Banach fixed point theorem }}

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An earlier version of this article was posted on [http://planetmath.org/encyclopedia/BanachFixedPointTheorem.html Planet Math]. This article is [[open content]].

{{Metric spaces}}

[[Category:Topology]]

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[[de:Fixpunktsatz von Banach]]

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