Nonlocal Generalized Implicit Function Theorems in Hilbert Spaces

Abstract

A nonlocal generalized implicit function theorem is obtained for mappings acting in Hilbert spaces. Its proof is based on the theory of ordinary differential equations. Under natural assumptions, we use this theorem to derive a global generalized implicit function theorem as well as a global implicit function theorem and a global inverse function theorem as particular cases of the former. Using the estimates produced, we prove an assertion on the continuation of an implicit function from a given closed set to the entire parameter space.