Finite order differentiability properties, fixed points and implicit functions over valued fields

TL;DR

This paper establishes an implicit function theorem for C^k-maps over valued fields, extending classical results to more general topological vector spaces and Banach spaces, with applications to fixed point dependence.

Contribution

It introduces a new implicit function theorem for C^k-maps over valued fields, generalizing existing theorems to broader topological vector space contexts.

Findings

Proves an implicit function theorem for C^k-maps over valued fields.

Shows C^k-dependence of fixed points on parameters for contraction families.

Extends results to k times Lipschitz differentiable maps and real case Keller C^k_c-maps.

Abstract

We prove an implicit function theorem for C^k-maps from arbitrary topological vector spaces over valued fields to Banach spaces (for k at least 2). As a tool, we show the C^k-dependence of fixed points on parameters for suitable families of contractions of a Banach space. Similar results are obtained for k times strictly differentiable maps, and for k times Lipschitz differentiable maps. In the real case, our results subsume an implicit function theorem for Keller C^k_c-maps from arbitrary topological vector spaces to Banach spaces.