  We prove an implicit function theorem for Keller C^k_c-maps from arbitrary
real or complex topological vector spaces to Frechet spaces, imposing only a
certain metric estimate on the partial differentials. As a tool, we show the
C^k-dependence of fixed points on parameters for suitable families of
contractions of a Frechet space. The investigations were stimulated by a recent
metric approach to differentiability in Frechet spaces by Olaf Mueller. Our
results also subsume generalizations of Mueller's Inverse Function Theorem for
mappings between Frechet spaces. As an application, we prove existence and
uniqueness of solutions to suitable ordinary differential equations in Frechet
spaces, and study their dependence on initial conditions and parameters.

Functional Analysis