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Lipschitzian stability of constraint systems and generalized equations. (English) Zbl 0805.93044

A class of multifunctions \(\phi\) from \(\mathbb{R}^ n\) into \(\mathbb{R}^ m\) given in the form \[ \phi(x):= \{y\in \mathbb{R}^ m: F(x,y)\in \Lambda,\;(x,y)\in \Omega\},\tag{1} \] where \(\Lambda\subset \mathbb{R}^ q\) and \(\Omega\subset \mathbb{R}^{n+m}\) are closed sets and \(F: \mathbb{R}^ n\times \mathbb{R}^ m\to \mathbb{R}^ q\) is a continuous vector function, is considered.
The goal is to establish refined sufficient conditions, on the initial data \(F\), \(\Lambda\) and \(\Omega\) ensuring the pseudo-Lipschitzian property of the multifunction (1).
The paper contains several necessary and sufficient conditions for such a multifunction being pseudo-Lipschitzian. It is also presented background material in nonsmooth analysis connected with nonconvex constructions of the normal cone for sets, the coderivative for multifunctions and the first- and the second-order subdifferentials for extended real-valued functions.

MSC:

93D09 Robust stability
34A60 Ordinary differential inclusions
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