Tools of differential calculus to solve funcional equations

Abstract

In this work we first prove the converse of Taylor’s Theorem. This allows us to prove next the Omega-lemma and the differentiability of the Evaluation map for certain Banach spaces of analytic functions. These two theorems together with the Implicit Function Theorem are applied to certain functional equations in order to prove Poincaré’s Linearization Theorem and the Analytic Stable Manifold Theorem.