Multisummability of formal solutions of singular perturbation problems
type: Item
instance of: scholarly article
title: Multisummability of formal solutions of singular perturbation problems
zbMATH Open document ID: 1015.34080
author: Werner Balser
author: Jorge Mozo-Fernández
published in: Journal of Differential Equations
publication date: 2002-11-28T00:00:00Z
review text: Linear perturbation problems of the form \((*)\) \(\varepsilon z^{r+1}x'=A(z,\varepsilon)x-f(z,\varepsilon)\) are considered where \(x'=\partial x/\partial z\), \(r\in \mathbb{N}\), \(A\) is a \(\nu \times \nu\)-matrix and \(f\) is a \(\nu\)-vector, both holomorphic for \(|z|\), \(|\varepsilon |<\rho\). The matrix \(A(0,0)\) is assumed to be invertible. The authors study (mainly in the case \(r\geq 1\)) formal power series solutions to \((*)\) (in one of the variables \((z,\varepsilon)\), with coefficients depending holomorphically on the other, or in both variables), the Gevrey orders of the formal solutions, their integral representations and (multi)summability properties (in sectorial domains, w.r.t. one of the variables \((z,\varepsilon)\), or w.r.t. both variables). In the proofs, the authors use the notion of Nagumo norm of order \(p\) of a function \(f\) holomorphic in a disc of radius \(R\) about the origin: \(\|f\|_p=\sup _{|z|<R}((R-|z|)^p\|f(z)\|)\).
Mathematics Subject Classification ID: 34M30
Mathematics Subject Classification ID: 34M60
zbMATH DE Number: 1836805
zbMATH Keywords: (multi)summability
zbMATH Keywords: asymptotic expansion
zbMATH Keywords: formal power series solution
zbMATH Keywords: Gevrey order
zbMATH Keywords: Nagumo norm
zbMATH Keywords: integral representation
reviewed by: Vladimir P. Kostov
MaRDI profile type: Publication
full work available at URL: jdeq.2001.4143
OpenAlex ID: W1993173444
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DOI: 10.1006/JDEQ.2001.4143
description: scientific article; zbMATH DE number 1836805