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

Houyu Zhao ; Jianguo Si

Annales Polonici Mathematici, Tome 95 (2009), p. 209-226 / Harvested from The Polish Digital Mathematics Library

Résumé

We study existence of analytic solutions of a second-order iterative functional differential equation $x^{\prime \prime }(z) = ∑_{j=0}^{k}∑_{t=1}^{∞}C_{t,j}(z)(x^{[j]}(z))^{t} + G(z)$ in the complex field ℂ. By constructing an invertible analytic solution y(z) of an auxiliary equation of the form $α²y^{\prime \prime }(αz)y^{\prime }(z) = αy^{\prime }(αz)y^{\prime \prime }(z) + [y^{\prime }(z)]³[∑_{j=0}^{k}∑_{t=1}^{∞}C_{t,j}(y(z))(y(α^{j}z))^{t} + G(y(z))]$ 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.

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Publié le : 2009-01-01

Zbl 1178.34074

EUDML-ID : urn:eudml:doc:280832