Isolated zeros of Lipschitzian metrically regular \(\mathbb{R}^n\)-functions. (English) Zbl 0987.49009

Let \(f: \mathbb{R}^n\to \mathbb{R}^n\) be a locally Lipschitz function. The goal of the paper is to find conditions ensuring that the preimage sets \(f^{-1}(y)\) (i.e., the solution sets of the equations \(f(x)= y\), \(y\) fixed) are locally finite.
The most important notion of the paper is the pseudo-regularity of \(f\). It is defined as the pseudo-Lipschitz property of the inverse mapping and can be described equivalently by a surjectivity condition for the contingent derivative of \(f\).
The first part of the paper contains some important properties of pseudo-regular functions. So the author gives some results regarding the Lipschitz behavior of the contingent derivative of \(f\) and regarding the differentiability properties of the directional derivative \(f'(x;\cdot)\).
The second part of the paper is devoted to the topological structure of the preimage sets of such functions. Using Rademachers theorem it is pointed out that these sets are locally finite almost everywhere, i.e., that for all \(x\in \mathbb{R}^n\) there exist neighborhoods \(U\) of \(x\) and \(V\) of \(f(x)\) such that for almost all \(y\in V\) the sets \(f^{-1}(y)\cap U\) are finite. In the main theorem, \(f\) is assumed additionally to be directionally differentiable. It is shown that in this case for every point \(x^0\in \mathbb{R}^n\) there exist neighborhoods \(U\) of \(x^0\) and \(V\) of \(f(x)\) such that
1. \(\forall x\in U\), \(\forall u\neq 0: f'(x; u)\neq 0\),
2. \(x\in U\) is an isolated preimage of \(f(x)\),
3. \(f^{-1}(y)\cap U\) is finite for all \(y\in V\),
4. \(\inf_{x\in U} \min_{\|u\|= 1}\|f'(x; u)\|> 0\).
The results are demonstrated by means of selected examples.


49J52 Nonsmooth analysis
26B10 Implicit function theorems, Jacobians, transformations with several variables
26E25 Set-valued functions
49J53 Set-valued and variational analysis
Full Text: DOI


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