Contracting application

A contracting application is a application k-lipschitzienne with K ∈] 0,1 The contracting applications are the primary product of the theorem of point fixes simplest and more used.

Let us notice that a contracting function is uniformly continuous.

Theorem of the point fixes for a contracting application

A metric Espace Complet (not vacuum) is E and F a contracting application of E in E (with the coefficient k<1).

Then there exists a single fixed point $\backslash \; quad\; x^*$ of F in E, i.e. such as $\backslash \; quad\; F\; (x^*)\; =x^*$. Moreover any continuation of elements of E checking the recurrence $\backslash \; quad\; x\_\; \{n+1\}\; =f\; (x\_n)$ converges towards $\backslash \; quad\; x^*$.

Demonstration

• Let us show initially unicity fixed point (possible for the moment…).

If thus there were 2 fixed points $x^*$ and $x^\; \{**\}$ one would have because of " contractance" of F:

what is obviously contradictory.

• Is now an unspecified point $\backslash \; quad\; x\_0$ of E. Consider the continuation $(x\_n)\; \_\; \{N\; \backslash \; in\; \backslash \; mathbb\; NR\}$ defined by its first term ($x\_0$) and the recurrence $\backslash \; quad\; x\_\; \{n+1\}\; =f\; (x\_n)$. Let us show that it is about a Suite of Cauchy of E. Indeed one has immediately

$\backslash \; forall\; m\; \backslash \; in\; \backslash \; mathbb\; N^*\; \backslash \; quad\; D\; (x\_\; \{m+1\},\; x\_m)\; =d\; (F\; (x\_m),\; F\; (x\_\; \{M-1\}))\backslash \; the\; k.d\; (x\_m,\; x\_\; \{M-1\})$ and by recurrence

$D\; (x\_\; \{m+1\},\; x\_m)\; \backslash \; the\; k^m.d\; (x\_1,\; x\_0)$.

One from of deduced by application reiterated from the triangular inequality:

$\backslash \; forall\; N\; \backslash \; in\; \backslash \; mathbb\; NR\; \backslash \; quad\; \backslash \; forall\; p\; \backslash \; in\; \backslash \; mathbb\; N^*$

$\backslash \; quad\; D\; (x\_\; \{n+p\},\; x\_n)\; \backslash \; D\; (x\_\; \{n+1\},\; x\_n)\; +d\; (x\_\; \{n+2\},\; x\_\; \{n+1\})\; +\dots \; +d\; (x\_\; \{n+p\},\; x\_\; \{n+p-1\})$

$\backslash \; quad\; D\; (x\_\; \{n+p\},\; x\_n)\; \backslash \; it\; (k^n+k^\; \{n+1\}\; +\dots \; +k^\; \{n+p-1\})\; D\; (x\_1,\; x\_0)$

$\backslash \; quad\; D\; (x\_\; \{n+p\},\; x\_n)\; \backslash \; the\; k^n\; \backslash \; frac\; \{1-k^p\}\; \{1-k\}\; D\; (x\_1,\; x\_0)\; \backslash \; quad\; (*)$.

This last member tends towards 0 when N tends towards the infinite one and can thus be made as small as it is wanted it as soon as N is rather large (and p natural unspecified). There is thus well a Suite of Cauchy.

As E is Complet, this continuation of Cauchy converges towards a limit $\backslash \; quad\; x^*$. Of more than $\backslash \; quad\; x\_\; \{n+1\}\; =f\; (x\_n)$ one deduces while passing in extreme cases and by using continuity of F that $\backslash \; quad\; F\; (x^*)\; =x^*$, which completes the demonstration.

Remarquons that this result is all the more interesting that it gives us a calculation algorithm of the point fixes (it is " method of the approximations successives") contrary to other theorems of point fixes which ensure us only of the existence of this (these…) not without indicating how to determine them. Moreover while passing in extreme cases for p in the inequality (*) and by using the continuity of the distance D, one obtains (without knowing $\backslash \; quad\; x^*$ exactly) one raising (often " pessimiste") error:

$d\; (x^*,\; x\_n)\; \backslash \; Leq\; \backslash \; frac\; \{k^n\}\; \{1-k\}\; D\; (x\_1,\; x\_0)$.

Traditional applications

• Solution of numerical equations, to see in particular Method of Newton

• approximate Resolution of linear systems by iteration
• Resolution of differential equation: Theorem of Cauchy-Lipschitz
• Theorem of the implicit functions

Application

Random links:

Limp Bizkit | Large-Chastang | Lussac (the Gironde) | Carpinone | Gathering

© 2007-2008 speedlook.com; article text available under the terms of GFDL, from fr.wikipedia.org