Geodesic
Inverse function theorem and conditions of extremum for abnormal problems with non-closed range
Sbornik. Mathematics, Tome 196 (2005) no. 9, pp. 1251-1269 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

The following two classical problems are considered: the existence and the estimate of a solution of an equation defined by a map $F$ in the neighbourhood of a point $x^*$; necessary conditions for an extremum at $x^*$ of a smooth function under equality-type constraints defined in terms of a non-linear map $F$. If the range of the first derivative of $F$ at $x^*$ is not closed, then one cannot use classical methods of analysis based on inverse function theorems and Lagrange's principle. The results on these problems obtained in this paper are of interest in the case when the range of the first derivative of $F$ at $x^*$ is non-closed; these are a further development of classical results extending them to abnormal problems with non-closed range. 
@article{SM_2005_196_9_a0,
     author = {E. R. Avakov and A. V. Arutyunov},
     title = {Inverse function theorem and conditions of extremum for abnormal problems with non-closed range},
     journal = {Sbornik. Mathematics},
     pages = {1251--1269},
     year = {2005},
     volume = {196},
     number = {9},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_2005_196_9_a0/}
}
TY  - JOUR
AU  - E. R. Avakov
AU  - A. V. Arutyunov
TI  - Inverse function theorem and conditions of extremum for abnormal problems with non-closed range
JO  - Sbornik. Mathematics
PY  - 2005
SP  - 1251
EP  - 1269
VL  - 196
IS  - 9
UR  - http://geodesic.mathdoc.fr/item/SM_2005_196_9_a0/
LA  - en
ID  - SM_2005_196_9_a0
ER  - 
%0 Journal Article
%A E. R. Avakov
%A A. V. Arutyunov
%T Inverse function theorem and conditions of extremum for abnormal problems with non-closed range
%J Sbornik. Mathematics
%D 2005
%P 1251-1269
%V 196
%N 9
%U http://geodesic.mathdoc.fr/item/SM_2005_196_9_a0/
%G en
%F SM_2005_196_9_a0
E. R. Avakov; A. V. Arutyunov. Inverse function theorem and conditions of extremum for abnormal problems with non-closed range. Sbornik. Mathematics, Tome 196 (2005) no. 9, pp. 1251-1269. http://geodesic.mathdoc.fr/item/SM_2005_196_9_a0/

[1] Alekseev V. M., Tikhomirov V. M., Fomin S. V., Optimalnoe upravlenie, Nauka, M., 1979 | MR

[2] Avakov E. R., “Teoremy ob otsenkakh v okrestnosti osoboi tochki otobrazheniya”, Matem. zametki, 47:5 (1990), 3–13 | MR | Zbl

[3] Avakov E. R., Arutyunov A. V., “Ob anormalnykh zadachakh s nezamknutym obrazom”, Dokl. RAN, 399:5 (2004), 583–586 | MR

[4] Sussmann H. J., “High-order open mapping theorems”, Directions in mathematical systems theory and optimization, eds. A. Rantzer, C. I. Byrnes, Springer-Verlag, Berlin, 2003, 293–316 | MR | Zbl

[5] Arutyunov A. V., “Teorema o neyavnoi funktsii kak realizatsiya printsipa Lagranzha. Anormalnye tochki”, Matem. sb., 191:1 (2000), 3–26 | MR | Zbl

[6] Dmitruk A. V., Milyutin A. A., Osmolovskii N. P., “Teorema Lyusternika i teoriya ekstremuma”, UMN, 35:6 (1980), 11–46 | MR | Zbl

[7] Ioffe A. D., Tikhomirov V. M., Teoriya ekstremalnykh zadach, Nauka, M., 1974 | MR | Zbl

[8] Avakov E. R., “Usloviya ekstremuma v gladkikh zadachakh s ogranicheniyami tipa ravenstv”, ZhVM i MF, 25:5 (1985), 680–693 | MR | Zbl

[9] Avakov E. R., “Neobkhodimye usloviya minimuma dlya neregulyarnykh zadach v banakhovykh prostranstvakh. Printsip maksimuma dlya anormalnykh zadach optimalnogo upravleniya”, Trudy MIAN, 185, 1988, 3–29 | MR

[10] Arutyunov A. V., Usloviya ekstremuma. Anormalnye i vyrozhdennye zadachi, Faktorial, M., 1997 | MR | Zbl

[11] Sukhinin M. F., Izbrannye glavy nelineinogo analiza, Izd-vo RUDN, M., 1992