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DOI: 10.23671/VNC.2018.4.9171
An Implicit Function Theorem in Non-Smooth Case
Khachatryan R. A.
Vladikavkaz Mathematical Journal 2017. Vol. 19. Issue 4.
Abstract: In this paper, we consider an equation of the form \(F(x,y)=0\), \(x\in X\), \(y\in M\), where \(M\) is a set. By the method of tents (tangent cones), when the set \(M\) is given by a nonsmooth restriction of equality type, the existence of a differentiable function \(y(\cdot)\) such that \(F(x, y(x))=0\), \(y(x)\in M\), \(y(x_0)=y_0\) is proved. In particular, the existence of smooth local selections for multivalued mappings of the form \(a(x) = \{y \in \mathbb{R}^m:\, f_i(x, y) = 0,\, i \in I,\, g(y) = 0\}\), \(x \in \mathbb{R}^n\), \(y \in \mathbb{R}^m\), is studied by the method of tents. It is assumed that the functions \(f_i(x, y)\), \(i \in I\), are strictly differentiable, and the function \(g (y)\) is locally Lipschitzian. Under certain additional conditions it is proved that through any point of the graph of a set-valued mapping there passes a differentiable selection of this mapping. These assertion can be interpreted as an implicit function theorem in the nonsmooth analysis. Strongly differentiable tents for the sets defined by nonsmooth constraints of the equality type are also constructed in the article. A sufficient condition is provided for the intersection of strictly differentiable tents to be a strictly differentiable tent. It is also shown that the Clark tangent cones are Boltiansky tents for sets defined by locally Lipschitz functions.
For citation: Khachatryan R. A. An Implicit Function Theorem in Non-Smooth Case //
Vladikavkazskii matematicheskii zhurnal [Vladikavkaz Math. J.], 2017, vol.
19, no. 1, pp. 86-96. DOI 10.23671/VNC.2018.4.9171
1. Avakov E. R., Magaril-Il'yaev G. G. An imlicit-function theorem
for inclusion, Mathematical Notes, 2012, vol. 91, no. 5-6, pp.
764-769. DOI: 10.1134/S0001434612050227.
2. Alekseev V. M., Tikhomirov V. M., Fomin S. V. Optimalnoe
upravleniye [Optimal Control], Moscow, Nauka, 1979, 408 p. (in
Russian).
3. Arutyunov A. V. Implicit functiont theorem without a priori
assumptions about normality, Computational Mathematics and
Mathematical Physics, 2006, vol. 46, no. 2, pp. 195-205. DOI:
10.1134/S0965542506020023.
4. Gel'man B. D. A Generalized Implicit Function Theorem, Functional
Analysis and its Applications, 2001, vol. 35, no. 3, pp. 183-188.
DOI: 10.1023/A:1012322727547.
5. Aubin J. P., Ekland I. Applied Nonlinear Analysis, Courier
Corporation, 2006, 518 p.
6. Boltyanskii V. G. Optimalnoe upravleniye diskretnimi sistemami
[Optimal Control of Discrete Systems], Moscow, Nauka, 1973, 446 p.
(in Russian).
7. Boltyanskii V. G. The method of tents in the theory of extremal
problems, Russian Mathematical Surveys, 1975, vol. 30, no. 3, pp.
1-54. DOI: 10.1070/RM1975v030n03ABEH001411.
8. Kolmogorov A. N., Fomin S. V. Elements of the Theory of
Functions and Functional Analysis. Graylok Press, 1965, 257 p.
9. Clarke F. H. Optimization and Nonsmooth Analysis. New York,
Awiley-Intercience Publication John Wiley & Sons, 1983, 296 p.
10. Polovinkin E. S. Mnogoznachnij analiz i differencialnye
wklucheniya [Multivalued Analysis and Differential Inclusions],
Moscow, Fizmatlit, 2014, 606 p. (in Russian).
11. Pshenichnii B. N. Wipuklij analiz i ekstremalniye zadachi
[Convex Analysis and Extremal Problems], Moscow, Nauka, 1980, 320 p.
(in Russian).
12. Magaril-Il'yaev G. G. The implicit function theorem for
Lipschitz maps, Russian Mathematical Surveys, 1978, vol. 33, no. 1,
pp. 209-210. DOI: 10.1070/RM1978v033n01ABEH002249.
13. Clarke F. H. On the inverse fuction theorem, Pacific Journal of
Mathematics, 1976, vol. 64, no. 1, pp. 97-102.
14. Michael E. Continous selections 1, Ann. Math., 1956, vol. 64,
no. 1, pp. 361-381.
15. Khachatryan R. A. On set-valued mappings with starlike graphs,
Journal of Contemporary Mathematical Analysis, 2012, vol. 47, no. 1,
pp. 28-44.
16. Khachatryan R. A., Arutyan F. G. Intersection of tents in
infinite dimensional spaces, Journal of Contemporary Mathematical
Analysis, 2001, vol. 36, no. 2, pp. 27-34.
17. Khachatryan R. A. On the existence of continous and smooth
selections for multivalued mappings, Journal of Contemporary
Mathematical Analysis, 2002, vol. 37, no. 2, pp. 30-40.
