Geodesic
An investigation of smooth maps in a~neighbourhood of an abnormal point
Izvestiya. Mathematics , Tome 78 (2014) no. 2, pp. 213-250

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We study the solubility of systems of non-linear equations in a neighbourhood of an abnormal point and prove inverse function theorems that guarantee the existence of solutions satisfying a linear-root estimate in the neighbourhood of the abnormal point. We also address the related issue of necessary conditions for an extremum at an abnormal point in a finite-dimensional constrained problem. We obtain second-order necessary optimality conditions that improve the known results.
Keywords: inverse function theorem, abnormal point, conditional extremum problem. 
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     author = {E. R. Avakov and A. V. Arutyunov and D. Yu. Karamzin},
     title = {An investigation of smooth maps in a~neighbourhood of an abnormal point},
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E. R. Avakov; A. V. Arutyunov; D. Yu. Karamzin. An investigation of smooth maps in a~neighbourhood of an abnormal point. Izvestiya. Mathematics , Tome 78 (2014) no. 2, pp. 213-250. http://geodesic.mathdoc.fr/item/IM2_2014_78_2_a0/