zbMATH — the first resource for mathematics

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
[1] Aubin J.P., Advances in Mathematics: Supplementary Studies pp 160– (1981)
[2] DOI: 10.1287/moor.9.1.87 · Zbl 0539.90085
[3] Aubin J.P., Applied nonlinear analysis · Zbl 1115.47049
[4] DOI: 10.1016/0022-247X(88)90034-0 · Zbl 0654.49004
[5] Clarke F.H., Optimization and Nonsmooth Analysis (1983) · Zbl 0582.49001
[6] DOI: 10.1007/BF01445166 · Zbl 0692.49018
[7] DOI: 10.1070/RM1980v035n06ABEH001973 · Zbl 0479.49015
[8] DOI: 10.1090/S0002-9939-1994-1215027-7
[9] DOI: 10.1287/moor.19.3.753 · Zbl 0835.49019
[10] DOI: 10.1137/S1052623495284029 · Zbl 0899.49004
[11] Sieklucki K., Topology. A geometric approach (1992)
[12] Federer H., Geometric Measure Theory (1969) · Zbl 0176.00801
[13] DOI: 10.1215/S0012-7094-50-01713-3 · Zbl 0037.20401
[14] DOI: 10.1090/S0002-9947-1981-0613784-7
[15] Kummer B., Math. programming with data perturbations 195 pp 201– (1997)
[16] DOI: 10.1080/02331939908844456 · Zbl 0991.90125
[17] Mordukhovich B.S., SIAM Proceedings in Applied Mathematics pp 32–
[18] DOI: 10.2307/2154544 · Zbl 0791.49018
[19] DOI: 10.1016/0362-546X(89)90083-7 · Zbl 0687.54015
[20] DOI: 10.1137/0713043 · Zbl 0347.90050
[21] DOI: 10.1287/moor.5.1.43 · Zbl 0437.90094
[22] Robinson S.M., Math. Prog. Study 30 pp 109– (1987)
[23] DOI: 10.1287/moor.16.2.292 · Zbl 0746.46039
[24] DOI: 10.1016/0362-546X(85)90024-0 · Zbl 0573.54011
[25] DOI: 10.1007/978-3-642-02431-3 · Zbl 0888.49001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.