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

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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. 
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     author = {E. R. Avakov and G. G. Magaril-Il'yaev},
     title = {An {Implicit-Function} {Theorem} for {Inclusions}},
     journal = {Matemati\v{c}eskie zametki},
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     url = {http://geodesic.mathdoc.fr/item/MZM_2012_91_6_a1/}
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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/