Inverse mapping theorem in Frechet spaces

We consider the classical inverse mapping theorem of Nash and Moser from the angle of some recent development by Ekeland and the authors. Geometrisation of tame estimates coupled with certain ideas coming from variational analysis when applied to a directionally differentiable mapping produces very general surjectivity result and, if injectivity can be ensured, inverse mapping theorem with the expected Lipschitz-like continuity of the inverse. We also present a brief application to differential equations.