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

2005/11/08 by Helge Glöckner, Helge Glockner, Glockner, Helge · 1 citation
Mathematics · #26E15 #26E20 #26E30 #46S10 #47H10 #58C15 #58C20 #Advanced Banach Space Theory #FOS: Mathematics #Fixed Point Theorems Analysis #Functional Analysis (math.FA) #Functional Equations Stability Results #General Mathematics (math.GM) #math.FA #math.GM #msc:26E15 #msc:26E20 #msc:26E30 #msc:46S10 #msc:47H10 #msc:58C15 #msc:58C20

paper · pdf · doi:10.48550/arxiv.math/0511218

LaTeX, 59 pages, broadly written preprint (v2: new Appendix C extends C^1-case from locally compact fields to complete valued fields)

openalex publication_date 2005/11/08 · arxiv created 2007/04/27 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

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