On an extension of a global implicit function theorem

Abstract

We study the existence of global implicit functions for equations defined on open subsets of Banach spaces. The partial derivative with respect to the second variable is only required to have a left inverse instead of being invertible. Generalizing known results, we provide sufficient criteria which are easy to check. These conditions essentially rely on the existence of diffeomorphisms between the respective projections of the set of zeros and appropriate Banach spaces, as well as a corresponding growth bound. The projections further allow to consider cases where the global implicit function is not defined on all of the open subset corresponding to the first variable.