Banach fixed-point theorem: Difference between revisions

Revision as of 17:26, 15 April 2024

Latest revision as of 19:58, 29 January 2025

(13 intermediate revisions by 8 users not shown)

Line 1:

{{Short description|Theorem about metric spaces}}

In [[mathematics]], the '''Banach fixed-point theorem''' (also known as the '''contraction mapping theorem''' or '''contractive mapping theorem''' or '''Banach-CaccioppoliBanach–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>

==Statement==

Line 14:

:<math>

\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}

Line 34:

:<math>d(x_{n+1}, x_n) \le 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>:

: <math>\begin{align}

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>

Let ε<math>\varepsilon > 0</math> be arbitrary. Since <math>q \in [0,1)</math>, we can find a large <math>N \in \N</math> so that

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

Line 52:

:<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>

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>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>

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 ''p<submath>1p_1</submath>'' and ''p<submath>2p_2</submath>'' would contradict the contraction of ''<math>T''</math>:

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

Line 64:

* 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-lipschitzLipschitz 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.

Line 84:

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 ''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.

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>

==Example==

==Example of numerical application - calculating high accuracy [[Pi|{{pi}}]]==

Banach theorem allows for example fast and accurate calculation of the {{pi}} number using the trigonometric

functions which numerically are the power [[Taylor_series|Taylor series]].

Because <math>\sin(\pi)=0</math> and the {{pi}} is the fixed point of for example the function

<math>f(x)=\sin(x)+x</math>

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

i.e.

:<math>f(0\pileq1+\cos(x)=\pileq1-\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.

and also the function

<math>f</math> is around {{pi}} the contraction mapping from the obvious reasons because its derivative in {{pi}} vanishes therefore {{pi}} can be obtained from the infinite superposition for example for the argument value 3:

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(f(3)...))))</math>

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

Already the triple superposition of this function at <math>3</math> gives {{pi}} with accuracy to 33 digits:

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

<math>f(f(f(3)))=3.141592653589793238462643383279502</math>.

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

==See also==

Line 137 ⟶ 133:

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

{{Metric spaces}}

{{Topology}}