Inverse function theorem and conditions of extremum for abnormal problems with non-closed range

The following two classical problems are considered: the existence and the estimate of a solution of an equation defined by a map $F$ in the neighbourhood of a point $x^*$; necessary conditions for an extremum at $x^*$ of a smooth function under equality-type constraints defined in terms of a non-linear map $F$. If the range of the first derivative of $F$ at $x^*$ is not closed, then one cannot use classical methods of analysis based on inverse function theorems and Lagrange's principle. The results on these problems obtained in this paper are of interest in the case when the range of the first derivative of $F$ at $x^*$ is non-closed; these are a further development of classical results extending them to abnormal problems with non-closed range.