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

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.

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

References in corpus (4)

• Lie groups over non-discrete topological fields
• Implicit functions from topological vector spaces to Fréchet spaces in the presence of metric estimates
• Differentiation, implicit functions, and applications to generalized well-posedness
• Aspects of p-adic non-linear functional analysis

Cited by in corpus (11)

• The diffeomorphism group of a non-compact orbifold
• Fundamentals of submersions and immersions between infinite-dimensional manifolds
• Implicit functions from topological vector spaces to Fréchet spaces in the presence of metric estimates
• Contractible Lie groups over local fields
• Lectures on Lie groups over local fields
• Aspects of p-adic non-linear functional analysis
• Normal Form of Equivariant Maps and Singular Symplectic Reduction in Infinite Dimensions with Applications to Gauge Field Theory
• Difference Problems and Differential Problems
• A Lie group structure on the group of real analytic diffeomorphisms of a compact real analytic manifold with corners
• Fréchet differentiability in Fréchet spaces, and differential equations with unbounded variable delay
• Ultrametric and non-locally convex analogues of the general curve lemma of convenient differential calculus