An Implicit-Function Theorem for a Class of Nonsmooth Functions

In this paper we introduce the concept of strong approximation of functions, and a related concept of strong Bouligand (B-) derivative, and we employ these ideas to prove an implicit-function theorem for nonsmooth functions. This theorem provides the same kinds of information as does the classical implicit-function theorem, but with the classical hypothesis of strong Fréchet differentiability replaced by strong approximation, and with Lipschitz continuity replacing Fréchet differentiability of the implicit function. Therefore it is applicable to a considerably wider class of functions than is the classical theorem.

In the last part of the paper we apply this implicit function result to analyze local solvability and stability of perturbed generalized equations.