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

Houyu Zhao, and Jianguo Si. "Analytic solutions of a second-order iterative functional differential equation near resonance." Annales Polonici Mathematici 96.3 (2009): 209-226. <http://eudml.org/doc/280832>.

@article{HouyuZhao2009,
abstract = {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.},
author = {Houyu Zhao, Jianguo Si},
journal = {Annales Polonici Mathematici},
keywords = {iterative functional differential equation; analytic solution; resonance; Diophantine condition; Brjuno condition},
language = {eng},
number = {3},
pages = {209-226},
title = {Analytic solutions of a second-order iterative functional differential equation near resonance},
url = {http://eudml.org/doc/280832},
volume = {96},
year = {2009},
}

TY - JOUR
AU - Houyu Zhao
AU - Jianguo Si
TI - Analytic solutions of a second-order iterative functional differential equation near resonance
JO - Annales Polonici Mathematici
PY - 2009
VL - 96
IS - 3
SP - 209
EP - 226
AB - 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.
LA - eng
KW - iterative functional differential equation; analytic solution; resonance; Diophantine condition; Brjuno condition
UR - http://eudml.org/doc/280832
ER -