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

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Zhao H., Si J.

Annales Polonici Mathematici

2009

tom 96

nr 3

209-226

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.

Ograniczanie wyników

1 Annales Polonici Mathematici

1 Si J.

1 Zhao H.

1 2009

Wyniki wyszukiwania

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