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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.
If K = 1 the function is called a short map, and if 0 ≤ K < 1 and f maps a metric space to itself, the function is called a contraction. In particular, a real-valued function f : R → R is called Lipschitz continuous if there exists a positive real constant K such that, for all real x 1 and x 2 ,
In mathematics, the Banach fixed-point theorem (also known as the contraction mapping theorem or contractive mapping theorem or Banach–Caccioppoli theorem) is an important tool in the theory of metric spaces; it guarantees the existence and uniqueness of fixed points of certain self-maps of metric spaces and provides a constructive method to find those fixed points.
A domain is weakly Lipschitz if for every point , there exists a radius > and a map : such that ℓ p {\displaystyle \ell _{p}} is a bijection ; ℓ p {\displaystyle \ell _{p}} and ℓ p − 1 {\displaystyle \ell _{p}^{-1}} are both Lipschitz continuous functions;
Thus metric spaces together with metric maps form a category Met. Met is a subcategory of the category of metric spaces and Lipschitz functions. A map between metric spaces is an isometry if and only if it is a bijective metric map whose inverse is also a metric map. Thus the isomorphisms in Met are precisely the isometries.
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In mathematics, a chaotic map is a map (an evolution function) that exhibits some sort of chaotic behavior. Maps may be parameterized by a discrete-time or a continuous-time parameter. Discrete maps usually take the form of iterated functions. Chaotic maps often occur in the study of dynamical systems.
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