En mathématiques et plus particulièrement en analyse, une application contractante, ou contraction, est une application qui rapproche les images, ou, plus rigoureusement, une application k-lipschitzienne avec 0 ≤ k < 1. Le théorème de point fixe le plus simple et le plus utilisé concerne les applications contractantes.↑ Jean-Pierre Bourguignon, Calcul variationnel, Éditions de l'École Polytechnique,‎ 2008 (ISBN 978-2-73021415-5, lire en ligne), p.

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dbpedia-owl:abstract
  • En mathématiques et plus particulièrement en analyse, une application contractante, ou contraction, est une application qui rapproche les images, ou, plus rigoureusement, une application k-lipschitzienne avec 0 ≤ k 1. Le théorème de point fixe le plus simple et le plus utilisé concerne les applications contractantes.
  • In mathematics, a contraction mapping, or contraction, on a metric space (M,d) is a function f from M to itself, with the property that there is some nonnegative real number such that for all x and y in M,The smallest such value of k is called the Lipschitz constant of f. Contractive maps are sometimes called Lipschitzian maps. If the above condition is instead satisfied fork ≤ 1, then the mapping is said to be a non-expansive map.More generally, the idea of a contractive mapping can be defined for maps between metric spaces. Thus, if (M,d) and (N,d') are two metric spaces, and , then there is a constant such thatfor all x and y in M.Every contraction mapping is Lipschitz continuous and hence uniformly continuous (for a Lipschitz continuous function, the constant k is no longer necessarily less than 1).A contraction mapping has at most one fixed point. Moreover, the Banach fixed point theorem states that every contraction mapping on a nonempty complete metric space has a unique fixed point, and that for any x in M the iterated function sequence x, f (x), f (f (x)), f (f (f (x))), ... converges to the fixed point. This concept is very useful for iterated function systems where contraction mappings are often used. Banach's fixed point theorem is also applied in proving the existence of solutions of ordinary differential equations, and is used in one proof of the inverse function theorem.
  • In matematica, una contrazione o applicazione di contrazione è una funzione da uno spazio metrico in se stesso tale che la distanza tra l'immagine di due elementi qualsiasi del dominio sia inferiore alla distanza delle relative controimmagini.
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  • ;Existence Soient ∈ E quelconque et pour tout entier naturel , . L'application — itérée fois de l'application — est -lipschitzienne donc : On en déduit, par application réitérée de l'inégalité triangulaire : : Ce majorant tend vers zéro quand tend vers l'infini, donc est une suite de Cauchy. Comme E est complet, cette suite de Cauchy converge vers une limite , vérifiant la majoration annoncée de l'erreur. Enfin, de , on déduit en passant à la limite et en utilisant la continuité de que . ;Unicité Soient et deux points fixes de . On a alors : d'où .
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  • Preuve classique
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  • En mathématiques et plus particulièrement en analyse, une application contractante, ou contraction, est une application qui rapproche les images, ou, plus rigoureusement, une application k-lipschitzienne avec 0 ≤ k < 1. Le théorème de point fixe le plus simple et le plus utilisé concerne les applications contractantes.↑ Jean-Pierre Bourguignon, Calcul variationnel, Éditions de l'École Polytechnique,‎ 2008 (ISBN 978-2-73021415-5, lire en ligne), p.
  • In matematica, una contrazione o applicazione di contrazione è una funzione da uno spazio metrico in se stesso tale che la distanza tra l'immagine di due elementi qualsiasi del dominio sia inferiore alla distanza delle relative controimmagini.
  • In mathematics, a contraction mapping, or contraction, on a metric space (M,d) is a function f from M to itself, with the property that there is some nonnegative real number such that for all x and y in M,The smallest such value of k is called the Lipschitz constant of f. Contractive maps are sometimes called Lipschitzian maps.
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  • Application contractante
  • Aplicació contractiva
  • Contracción (espacio métrico)
  • Contraction mapping
  • Contrazione (spazio metrico)
  • Kontrakce (matematika)
  • Kontrakcja (matematyka)
  • Kontraktion (Mathematik)
  • Сжимающее отображение
  • 収縮写像
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