# Lipschitz Condition

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## Lipschitz condition

[′lip‚shits kən‚dish·ən]
(mathematics)
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.
McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.
The following article is from The Great Soviet Encyclopedia (1979). It might be outdated or ideologically biased.

## 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.

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Moreover, to prove the contraction of the proximal mapping [U.sub.[lambda]], we introduce a new type of the Lipschitz continuity, which is a relaxation of the one used in .
In the last section, we demonstrate the Lipschitz continuity of the gradient and state the necessary condition for optimal solution.
By the virtue of properties of mollifier, Lipschitz continuity of the remaining functions [F.sup.j.sub.[epsilon]], j = 2, 3,4, can be obtained with constant C/[epsilon].
More precisely, the most important issue in numerical solutions of inverse problems is the Lipschitz continuity of the Frechet gradient.
Kahler, "Weighted Lipschitz continuity and harmonic Bloch and BESov spaces in the real unit ball," Proceedings of the Edinburgh Mathematical Society, vol.
In this paper, iteration (6) is considered for solving (1) along with its semilocal and local convergence analysis under weaker Lipschitz continuity condition on divided differences of order one on the involved operator G in Banach space setting.
In Section 4 we present a sufficient condition for Lipschitz continuity of the generated states and provide Lipschitz constants explicitly.
This equilibrium exists and is unique in case of Lipschitz continuity of the activation functions.
The nonlinear function [PSI](%, u, t) in the system dynamics (6) can be divided into [[PSI].sub.1](v, u, t) and [[PSI].sub.2](x, u, t) to analyze the Lipschitz continuity as follows:
Among the topics are definitions of quasiconformal maps, infinitesimal space, points of maximal stretching, Lipschitz continuity of quasiconformal maps, and criteria of univalence.
Nevertheless, due to the necessity of Lipschitz continuity in x of [psi](S(t, x)) appearing in the sequel, we need f to be essentially bounded, in the sense that

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