- Contraction mapping
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 k < 1 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 for k ≤ 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 k such that
for 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.
Firmly non-expansive mapping
A non-expansive mapping with k = 1 can be strengthened to a firmly non-expansive mapping in a Hilbert space H if the following holds for all x and y in H:
- Short map
- Contraction (operator theory)
- Vasile I. Istratescu, Fixed Point Theory, An Introduction, D.Reidel, Holland (1981). ISBN 90-277-1224-7 provides an undergraduate level introduction.
- Andrzej Granas and James Dugundji, Fixed Point Theory (2003) Springer-Verlag, 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
Wikimedia Foundation. 2010.
Look at other dictionaries:
Contraction — may refer to: In physiology: Muscle contraction, one that occurs when a muscle fiber lengthens or shortens Uterine contraction, contraction of the uterus, such as during childbirth Contraction, a stage in wound healing In linguistics: Synalepha,… … Wikipedia
Contraction principle — In mathematics, contraction principle may refer to: the Banach fixed point theorem, also known as the contraction mapping theorem/principle; the contraction principle in large deviations theory This disambiguation page lists mathematics articles… … Wikipedia
Banach fixed-point theorem — In mathematics, 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 spaces; it guarantees the existence and uniqueness of fixed points of… … Wikipedia
Banach fixed point theorem — 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 spaces; it guarantees the existence and uniqueness of fixed points of certain self maps… … Wikipedia
List of mathematics articles (C) — NOTOC C C closed subgroup C minimal theory C normal subgroup C number C semiring C space C symmetry C* algebra C0 semigroup CA group Cabal (set theory) Cabibbo Kobayashi Maskawa matrix Cabinet projection Cable knot Cabri Geometry Cabtaxi number… … Wikipedia
Metric space — In mathematics, a metric space is a set where a notion of distance (called a metric) between elements of the set is defined. The metric space which most closely corresponds to our intuitive understanding of space is the 3 dimensional Euclidean… … Wikipedia
Brouwer fixed point theorem — In mathematics, the Brouwer fixed point theorem is an important fixed point theorem that applies to finite dimensional spaces and which forms the basis for several general fixed point theorems. It is named after Dutch mathematician L. E. J.… … Wikipedia
Comparametric equation — A comparametric equation is an equation that describes a parametric relationship between a function and a dilated version of the same function, where the equation does not involve the parameter. For example, ƒ(2t) = 4ƒ(t) is a comparametric… … Wikipedia
Schwarz–Ahlfors–Pick theorem — In mathematics, the Schwarz–Ahlfors–Pick theorem is an extension of the Schwarz lemma for hyperbolic geometry, such as the Poincaré half plane model. It states that the Poincaré metric is distance decreasing on harmonic functions.The theorem… … Wikipedia
Ricci decomposition — In semi Riemannian geometry, the Ricci decomposition is a way of breaking up the Riemann curvature tensor of a pseudo Riemannian manifold into pieces with useful individual algebraic properties. This decomposition is of fundamental importance in… … Wikipedia
- There Is No Third Choice., 1979 — The film about the armament drive in the countries of Western Europe, about the beginning of the Cold War and peaceful initiatives of theUSSR.
- Defeat Of Militaristic Japan. 1945., 1985 — About the war between the USSR and the militaristic Japan.
- From the Time Distance. Soviet Union And Chinese Revolution., 1986 — A film tells about the most essential and important periods in relations between the USSR and People's Republic of China based on the unique and little-known facts.