Newton's Method and Frobenius-Dieudonné Theorem in Nonnormable Spaces

We establish, via auxiliary categories, a version of Newton's method applicable to nonnormable spaces. We illustrate it with an application to perturbation of functional equations on unbounded domains. Under suitable conditions, the known solution of the unperturbed equation initiates a sequence of Newton iterates that converges to the solution of the perturbed equation. The conditions posed are those required to make Newton's method work. We also generalize the Frobenius-Dieudonné existence-uniqueness theorem for initial value problems for a vector variable.