m at h . FA ] 1 2 A ug 2 00 5 Differentiation , implicit functions , and applications to generalized well - posedness

This article is centered around generalizing a previous implicit function theorem of the author to be applicable for maps f : E⊓F → F which can be lifted to Keller C π –maps fi : E ⊓ Fi → Fi with Fi Banach and F = proj limFi . We prove theorems about existence and differentiability of functions g satisfying f (x, g (x)) = b constant. In addition to these abstract theorems, we give several examples of applications to proving smooth dependence of the solution on initial/boundary values and the nonlinearity in nonlinear (partial) differential equations.