Lipschitz Condition

Lipschitz condition

[′lip‚shits kən‚dish·ən]
A function ƒ satisfies such a condition at a point b if |ƒ(x) - ƒ(b)| ≤ K | x-b |, with K a constant, for all x in some neighborhood of b.

Lipschitz Condition


a restriction on the behavior of an increment of a function. If for any points x and x″ in the interval [a, b ] the increment of a function satisfies the inequality

ǀf(x) - f(x′) ≤ Mǀ x - x′ǀα

where 0 < α ≤ 1 and where M is some constant, a function f(x) is said to satisfy a Lipschitz condition of order α on the interval [a, b ]; this is written as f(x) ∈ Lip a. Every function satisfying a Lipschitz condition on the interval [a, b ] for some α > 0 is uniformly continuous on [a, b ]. A function having a bounded derivative on [a, b ] satisfies a Lipschitz condition on [a, b ] for any α ≤ 1.

The Lipschitz condition was first examined in 1864 by the German mathematician R. Lipschitz (1832–1903) as a sufficient condition for the convergence of the Fourier series of a function f(x). Although it is historically inaccurate, some mathematicians associate only the most important case of the Lipschitz condition, that of α = 1, with the name of Lipschitz; for the case α < 1 they speak of the Hölder condition.

References in periodicals archive ?
In [33] an estimate of the Lipschitz condition was derived for a univariate black box problem.
i) g(x, t) in (1) and (2) satisfies the Lipschitz condition about state vector x(t); namely, there exists constant [[delta].
In the final section of the paper, we assert the notion of multiplicative Lipschitz condition on the closed interval [x, y] [subset] (0, m).
Under assumption that f (t,x) satisfies the Lipschitz condition in the variable x, we will prove that
In Lipschitz optimization the upper bound for f (x) over a sub-region I [subset] D is evaluated by exploiting Lipschitz condition (1).
If [OMEGA] is bounded, then the uniform cone condition, assumed here, implies the Lipschitz condition, see [11].
He used the concept of H-differentiability which was introduced by Puri and Ralescu [13], and obtained the existence and uniqueness theorem for a solution of FDE under the Lipschitz condition.
Other topics include primitives of p-adic meromorphic functions, the Lipschitz condition for rational functions on ultrametric valued functions, the geometry of p-adic fractal strings, identities and congruencies for Genocchi numbers, and ultrametric calculus in field K.
A1) g(*) satisfies the Lipschitz condition of order [gamma] and g(*) [member of] [H.
T], the functions f (t, x) satisfy a Lipschitz condition with a constant K [member of] R such that
We further assume that y(x) is sufficiently differentiable and that the solution of (1) exists and satisfy the Lipschitz condition.
assuming a one-sided Lipschitz condition in funcion f, two monotone sequences that start at the lower solution a and the upper solution 0 and converge to a solutions [phi] and [PHI], are constructed; moreover every solution U [member of] [[alpha],[beta]] [a, /0] of problem (3.