An investigation of smooth maps in a neighbourhood of an abnormal point

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. © 2014 Russian Academy of Sciences (DoM), London Mathematical Society, Turpion Ltd.