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- Contraction_mapping abstract "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.".
- Contraction_mapping wikiPageID "6239".
- Contraction_mapping wikiPageRevisionID "582155783".
- Contraction_mapping hasPhotoCollection Contraction_mapping.
- Contraction_mapping subject Category:Fixed_points_(mathematics).
- Contraction_mapping subject Category:Metric_geometry.
- Contraction_mapping comment "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.".
- Contraction_mapping label "Application contractante".
- Contraction_mapping label "Contracción (espacio métrico)".
- Contraction_mapping label "Contraction mapping".
- Contraction_mapping label "Contrazione (spazio metrico)".
- Contraction_mapping label "Kontrakcja (matematyka)".
- Contraction_mapping label "Kontraktion (Mathematik)".
- Contraction_mapping label "Сжимающее отображение".
- Contraction_mapping label "压缩映射".
- Contraction_mapping label "収縮写像".
- Contraction_mapping sameAs Kontrakce_(matematika).
- Contraction_mapping sameAs Kontraktion_(Mathematik).
- Contraction_mapping sameAs Contracción_(espacio_métrico).
- Contraction_mapping sameAs Application_contractante.
- Contraction_mapping sameAs Contrazione_(spazio_metrico).
- Contraction_mapping sameAs 収縮写像.
- Contraction_mapping sameAs Kontrakcja_(matematyka).
- Contraction_mapping sameAs m.01vqm.
- Contraction_mapping sameAs Q515173.
- Contraction_mapping sameAs Q515173.
- Contraction_mapping wasDerivedFrom Contraction_mapping?oldid=582155783.
- Contraction_mapping isPrimaryTopicOf Contraction_mapping.