Geodesic
An Implicit-Function Theorem for Inclusions
Matematičeskie zametki, Tome 91 (2012) no. 6, pp. 813-818.

Voir la notice de l'article provenant de la source Math-Net.Ru

We consider the question of the solvability of an inclusion $F(x,\sigma)\in A$, i.e., of determining a mapping (implicit function) $\sigma\mapsto x(\sigma)$ defined on a set such that $F(x(\sigma),\sigma)\in A$ for any $\sigma$ from this set. Results of this kind play a key role in the different branches of analysis and, especially, in the theory of extremal problems, where they are the main tool for deriving conditions for an extremum.
Keywords: implicit-function theorem, nonlinear equation, Newton's method, Banach space, multivalued mapping, continuous selector. 
@article{MZM_2012_91_6_a1,
     author = {E. R. Avakov and G. G. Magaril-Il'yaev},
     title = {An {Implicit-Function} {Theorem} for {Inclusions}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {813--818},
     publisher = {mathdoc},
     volume = {91},
     number = {6},
     year = {2012},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2012_91_6_a1/}
}
TY  - JOUR
AU  - E. R. Avakov
AU  - G. G. Magaril-Il'yaev
TI  - An Implicit-Function Theorem for Inclusions
JO  - Matematičeskie zametki
PY  - 2012
SP  - 813
EP  - 818
VL  - 91
IS  - 6
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MZM_2012_91_6_a1/
LA  - ru
ID  - MZM_2012_91_6_a1
ER  - 
%0 Journal Article
%A E. R. Avakov
%A G. G. Magaril-Il'yaev
%T An Implicit-Function Theorem for Inclusions
%J Matematičeskie zametki
%D 2012
%P 813-818
%V 91
%N 6
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MZM_2012_91_6_a1/
%G ru
%F MZM_2012_91_6_a1
E. R. Avakov; G. G. Magaril-Il'yaev. An Implicit-Function Theorem for Inclusions. Matematičeskie zametki, Tome 91 (2012) no. 6, pp. 813-818. http://geodesic.mathdoc.fr/item/MZM_2012_91_6_a1/

[1] Zh.-P. Oben, I. Ekland, Prikladnoi nelineinyi analiz, Mir, M., 1988 | MR | Zbl

[2] E. Michael, “Continuous selections. I”, Ann. of Math. (2), 63:2 (1956), 361–382 | DOI | MR | Zbl

[3] V. M. Tikhomirov, “Teorema o kasatelnom prostranstve i nekotorye ee modifikatsii”, Optimalnoe upravlenie, 7, Izd-vo Mosk. un-ta, M., 1977, 22–30

[4] S. M. Robinson, “Regularity and stability of convex multivalued functions”, Math. Oper. Res., 1:2 (1976), 130–143 | DOI | MR | Zbl

[5] E. H. Chabi, H. Zouaki, “Existence of a continuous solution of parametric nonlinear equation with constraints”, J. Convex Anal., 7:2 (2000), 413–426 | MR | Zbl

[6] A. V. Arutyunov, E. R. Avakov, A. F. Izmailov, “Directional regularity and metric regularity”, SIAM J. Optim., 18:3, 810–833 | MR | Zbl

[7] A. V. Arutyunov, “Teorema o neyavnoi funktsii bez apriornykh predpolozhenii normalnosti”, Zh. vychisl. matem. i matem. fiz., 46:2 (2006), 205–215 | MR | Zbl

[8] G. G. Magaril-Ilyaev, V. M. Tikhomirov, “Metod Nyutona, differentsialnye uravneniya i printsip Lagranzha dlya neobkhodimykh uslovii ekstremuma”, Optimalnoe upravlenie, Sbornik statei. K 60-letiyu so dnya rozhdeniya professora Viktora Ivanovicha Blagodatskikh, Tr. MIAN, 262, MAIK, M., 2008, 156–177 | MR | Zbl