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Locally Lipschitz Functions, Cofinal Completeness, and UC Spaces

Journal of Mathematical Analysis and Applications, 2015
Let X, d be a metric space. We find necessary and sufficient conditions on the space for the locally Lipschitz functions to coincide with each of two more restrictive classes of locally Lipschitz functions studied by several authors: the uniformly ...
Gerald Beer, M Isabel Garrido
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Locally Lipschitz Functionals

2021
In this chapter we focus our attention on the theory developed by Clarke for locally Lipschitz functionals. More precisely, we will investigate the properties of the generalized directional derivative and the Clarke subdifferential as well as the connection with the convex subdifferential.
Nicuşor Costea, Alexandru Kristály, Csaba Varga   +2 more
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On implicit function theorem for locally Lipschitz equations

Mathematical Programming, 2022
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Aram V. Arutyunov, Sergey E. Zhukovskiy
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Semigroups of locally Lipschitz operators.

Mathematical Journal of Okayama University, 2002
The infinitesimal generators of semigroups of locally Lipschitz operators are characterized. An application of the obtained result is given to the Cauchy problem for the Kirchhoff equation with real analytic initial data.
Kobayashi, Yoshikazu, Tanaka, Naoki
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Local Minimality of a Lipschitz Extremal

Canadian Journal of Mathematics, 1992
AbstractIn this paper the question of weak and strong local optimality of a Lipschitz (as opposed to C1 ) extremal is addressed. We show that the classical Jacobi sufficient conditions can be extended to the case of Lipschitz candidates. The key idea for this achievement lies in proving that the “generalized” strengthened Weierstrass condition is ...
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Lipschitz classes on local fields

Science in China Series A: Mathematics, 2007
The Lipschitz class Lipα on a local field K is defined in this note, and the equivalent relationship between the Lipschitz class Lipα and the Holder type space C α (K) is proved. Then, those important characteristics on the Euclidean space R n and the local field K are compared, so that one may interpret the essential ...
Wei-yi Su, Guo-xiang Chen
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Local Lipschitz-constant Functions and Maximal Subdifferentials

Set-Valued Analysis, 2003
Let \(X\) be a Banach space. For the study of generalized subdifferentials, the authors introduce the notion of topologically robust upper semicontinuity for real-valued functions. So the function \(k(x)\) is called to be topologically robust usc if it is upper semicontinuous and quasi lower semicontinuous.
Borwein, J. M., Vanderwerff, J., Wang, Xianfu   +2 more
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The Role of Local Convexity in Lipschitz Maps

Journal of Convex Analysis, 2011
Lipschitz maps between metric spaces (in particular, between Banach spaces) are abundant and afford a great deal of flexibility: they can be glued, pasted, and truncated without impairing the Lipschitz property. When their target space is the real line, they also can be extended to the whole space without increasing the Lipschitz ...
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An Effective Nonsmooth Optimization Algorithm for Locally Lipschitz Functions

Journal of Optimization Theory and Applications, 2012
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Nezam Mahdavi-Amiri, Rohollah Yousefpour
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New constructions for local approximation of Lipschitz functions. I

Nonlinear Analysis: Theory, Methods & Applications, 2003
In a former paper, the author presented a new generalized differentiability notion for Lipschitz functions \(f: \mathbb{R}^n\to\mathbb{R}\) according to \[ \begin{multlined} Df(x_0)= \text{conv}\Biggl\{v\in\mathbb{R}^n\mid\exists g\in \mathbb{R}^n,\| g\|= 1,\exists r(x_0,\cdot,g)\in \eta(x_0),\;\exists\alpha_k\to +0:\\ v= \lim_{k\to\infty}\,{1\over ...
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