Implicit functions from topological vector spaces to Fr\'echet spaces in the presence of metric estimates

Abstract

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.