Analytic solutions of a second-order iterative functional differential equation near resonance

• Autorzy: Houyu Zhao , Jianguo Si
• Języki publikacji: EN

Abstrakty

EN

We study existence of analytic solutions of a second-order iterative functional differential equation
$x''(z) = ∑_{j=0}^{k}∑_{t=1}^{∞}C_{t,j}(z)(x^{[j]}(z))^{t} + G(z)$
in the complex field ℂ. By constructing an invertible analytic solution y(z) of an auxiliary equation of the form
$α²y''(αz)y'(z) = αy'(αz)y''(z) + [y'(z)]³[∑_{j=0}^{k}∑_{t=1}^{∞}C_{t,j}(y(z))(y(α^{j}z))^{t} + G(y(z))]$
invertible analytic solutions of the form $y(αy^{-1}(z))$ for the original equation are obtained. Besides the hyperbolic case 0 < |α| < 1, we focus on α on the unit circle S¹, i.e., |α|=1. We discuss not only those α at resonance, i.e. at a root of unity, but also near resonance under the Brjuno condition.

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Annales Polonici Mathematici
• Rocznik: 2009
• Tom: 96
• Numer: 3
• Strony: 209-226

Daty

wydano

2009

Twórcy

autor

Houyu Zhao

• School of Mathematics, Shandong University, Jinan, Shandong 250100, P.R. China

autor

Jianguo Si

• School of Mathematics, Shandong University, Jinan, Shandong 250100, P.R.China

Identyfikatory

DOI

10.4064/ap96-3-2