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In the theory of differential equations, Lipschitz continuity is the central condition of the Picard–Lindelöf theorem which guarantees the existence and uniqueness of the solution to an initial value problem. A special type of Lipschitz continuity, called contraction, is used in the Banach fixed-point theorem. [2]
A Lipschitz-continuous function with constant is called contractive if <; it is called weakly-contractive if . Every contractive function satisfying Brouwer's conditions has a unique fixed point. Moreover, fixed-point computation for contractive functions is easier than for general functions.
For a Lipschitz continuous function, there is a double cone (shown in white) whose vertex can be translated along the graph so that the graph always remains entirely outside the cone. The concept of continuity for functions between metric spaces can be strengthened in various ways by limiting the way δ {\displaystyle \delta } depends on ε ...
The Peano existence theorem shows only existence, not uniqueness, but it assumes only that f is continuous in y, instead of Lipschitz continuous. For example, the right-hand side of the equation dy / dt = y 1 / 3 with initial condition y(0) = 0 is continuous but not Lipschitz continuous.
A sublinear modulus of continuity can easily be found for any uniformly continuous function which is a bounded perturbation of a Lipschitz function: if f is a uniformly continuous function with modulus of continuity ω, and g is a k Lipschitz function with uniform distance r from f, then f admits the sublinear module of continuity min{ω(t), 2r ...
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.
is a Lipschitz-continuous map, then there is a Lipschitz-continuous map : that extends f and has the same Lipschitz constant as f. Note that this result in particular applies to Euclidean spaces E n and E m, and it was in this form that Kirszbraun originally formulated and proved the theorem. [1]
Lipschitz domains in the sense above are sometimes called strongly Lipschitz by contrast with weakly Lipschitz domains. A domain Ω {\displaystyle \Omega } is weakly Lipschitz if for every point p ∈ ∂ Ω , {\displaystyle p\in \partial \Omega ,} there exists a radius r > 0 {\displaystyle r>0} and a map ℓ p : B r ( p ) → Q {\displaystyle ...