Holomorphic solutions to functional differential equations
type: Item
instance of: scholarly article
title: Holomorphic solutions to functional differential equations
zbMATH Open document ID: 1202.34159
author: Hong Oh Kim
author: Gregory Derfel
author: B. Van Brunt
published in: Journal of Mathematical Analysis and Applications
publication date: 2010-05-19T00:00:00Z
review text: This paper is concerned with the initial value problem \[ y'(z)+\alpha y(z)=\lambda y(z^n), \qquad y(\widetilde{z})=y_0, \] where \(\alpha\), \(\lambda\neq 0\) and \(y_0\) are constants, \(n>1\) is an integer and \(\widetilde{z}\) is a fixed point for \(g(z)=z^n\). The function \(g\) has one attracting fixed point at \(z_0=0\) and \(n-1\) repelling fixed points on the circle \(C(0,1)=\{z:|z|=1\}\). First, the authors study solutions that are holomorphic at the fixed points (attracting or repelling). More especially, they prove the existence of a natural boundary for solutions holomorphic at the attracting fixed point and also they show that the holomorphic solutions at repelling fixed points can be continued throughout the complex plane except at attracting fixed points. The special case \(\alpha=0\) is studied separately. This work is motivated by the paper of \textit{M. Heard} [J. Math. Anal. Appl. 44, 745--757 (1973; Zbl 0289.34115)] which deals with the famous pantograph equation.
reviewed by: Panagiotis Ch. Tsamatos
Mathematics Subject Classification ID: 34M99
zbMATH DE Number: 5710091
zbMATH Keywords: complex functional differential equations
zbMATH Keywords: pantograph equation
zbMATH Keywords: complex dynamics
MaRDI profile type: Publication
full work available at URL: j.jmaa.2010.02.035
OpenAlex ID: W2010687285
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DOI: 10.1016/J.JMAA.2010.02.035
description: scientific article; zbMATH DE number 5710091