An Implicit-Function Theorem for Inclusions

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.