Geodesic
An implicit function theorem without a priori assumptions about normality
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 46 (2006) no. 2, pp. 205-215 Cet article a éte moissonné depuis la source Math-Net.Ru

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The equation $F(x,\sigma)=0$, $x\in K$, in which $\sigma$ is a parameter and $x$ is an unknown taking values in a given convex cone in a Banach space $X$, is considered. This equation is examined in a neighborhood of a given solution $(x^*,\sigma^*)$ for which the Robinson regularity condition may be violated. Under the assumption that the 2-regularity condition (defined in the paper), which is much weaker than the Robinson regularity condition, is satisfied, an implicit function theorem is obtained for this equation. This result is a generalization of the known implicit function theorems even for the case when the cone $K$ coincides with the entire space $X$. 
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A. V. Arutyunov. An implicit function theorem without a priori assumptions about normality. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 46 (2006) no. 2, pp. 205-215. http://geodesic.mathdoc.fr/item/ZVMMF_2006_46_2_a1/

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

[2] Tikhomirov V. M., “Teorema Lyusternika o kasatelnom prostranstve i nekotorye ee modifikatsii”, Optimalnoe upravlenie. Matem. vopr. upravleniya proizvodstvom, 7, Izd-vo MGU, M., 1977, 22–30

[3] Robinson S., “Stability theory for systems of inequalities. Part II: differentiable nonlinear systems”, SIAM J. Numer Analys., 13:4 (1976), 497–513 | DOI | MR | Zbl

[4] Bonnans J. F., Shapiro A., Perturbation analysis of optimization problems, Springer, New York, 2000 | MR

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

[6] Izmailov A. F., “Teoremy o predstavlenii semeistv nelineinykh otobrazhenii i teoremy o neyavnoi funktsii”, Matem. zametki, 67:1 (2000), 57–68 | MR | Zbl

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

[8] Arutyunov A. B., “Nakryvanie nelineinykh otobrazhenii na konuse v okrestnosti anormalnoi tochki”, Matem. zametki, 77:4 (2005), 483–497 | MR | Zbl

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

[10] Michael E. A., “Continuous selections”, Ann. Math., 63:2 (1956), 361–382 ; 64:3, 562–580 | DOI | MR | Zbl | DOI | MR | Zbl

[11] Gelman B. D., “Obobschennaya teorema o neyavnom otobrazhenii”, Funkts. analiz i ego prilozh., 35:3 (2001), 28–35 | MR