Smoothness of solutions of nonlinear ODE
type: Item
instance of: scholarly article
title: Smoothness of solutions of nonlinear ODE
zbMATH Open document ID: 1020.34083
author: Plamen Djakov
author: Boris Mityagin
published in: Mathematische Annalen
publication date: 2002-12-01T00:00:00Z
review text: A Carleman class \(C^\infty_M\) of \(C^\infty\)-functions is defined as a set of functions \(f\) such that for some \(C\), \(R> 0\), we have \[ |f^{(k)}_{(t)}|\leq C\cdot R^K\cdot M_K,\quad k= 0,1,\dots, t\in I= [0,1], \] where \(M= \{M_K\}^\infty_0\) is a fixed sequence such that \(\log M_K\) is convex. Here, the following statement is proved: Let \(u\) be \(C^\infty\)-function on \([0,1]\) such that \(u'= \Phi(u,t)\), where \(\Phi(z,t)\) is an entire function of order 1, finite type in \(z\) and of a nonanalytic Carleman class in \(t\). Then \(u\) is in the same Carleman class. This is the extension of the authors' results in [Smoothness of solutions of a nonlinear ODE, Int. Equs. Operator Theory 44, No. 2, 149-171 (2002)].
Mathematics Subject Classification ID: 34M05
Mathematics Subject Classification ID: 34M20
Mathematics Subject Classification ID: 34A34
zbMATH DE Number: 1837396
zbMATH Keywords: nonlinear ordinary differential equations
zbMATH Keywords: solution
zbMATH Keywords: Carleman class
reviewed by: A. Yu. Timofeev
MaRDI profile type: Publication
full work available at URL: s00208-002-0335-3
OpenAlex ID: W2043225431
DOI: 10.1007/S00208-002-0335-3
description: scientific article; zbMATH DE number 1837396