Geodesic
A Generalized Implicit Function Theorem
Funkcionalʹnyj analiz i ego priloženiâ, Tome 35 (2001) no. 3, pp. 28-35

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We prove a theorem generalizing the classical implicit function theorem to the case in which the derivative of the map is a surjective continuous linear operator. We do not assume that the kernel of the derivative is a complemented subspace. 
@article{FAA_2001_35_3_a2,
     author = {B. D. Gel'man},
     title = {A {Generalized} {Implicit} {Function} {Theorem}},
     journal = {Funkcionalʹnyj analiz i ego prilo\v{z}eni\^a},
     pages = {28--35},
     publisher = {mathdoc},
     volume = {35},
     number = {3},
     year = {2001},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FAA_2001_35_3_a2/}
}
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B. D. Gel'man. A Generalized Implicit Function Theorem. Funkcionalʹnyj analiz i ego priloženiâ, Tome 35 (2001) no. 3, pp. 28-35. http://geodesic.mathdoc.fr/item/FAA_2001_35_3_a2/