Analytic solutions of a second-order iterative functional differential equation near resonance
Volume 96 / 2009
Abstract
We study existence of analytic solutions of a second-order iterative functional differential equation $$x' '(z)=\sum_{j=0}^{k}\sum_{t=1}^{\infty}C_{t,j}(z)(x^{[j]}(z))^{t}+G(z)$$ in the complex field $\Bbb C.$ By constructing an invertible analytic solution $y(z)$ of an auxiliary equation of the form $$ \alpha^2 y' '(\alpha z)y'(z)=\alpha y'(\alpha z)y' '(z)+[y'(z)]^3\Big[\sum_{j=0}^{k}\sum_{t=1}^{\infty}C_{t,j}(y(z))(y(\alpha^{j}z))^{t}+G(y(z))\Big]$$ invertible analytic solutions of the form $y(\alpha y^{-1}(z))$ for the original equation are obtained. Besides the hyperbolic case $0<|\alpha|<1$, we focus on $\alpha$ on the unit circle $S^1$, i.e., $|\alpha|=1$. We discuss not only those $\alpha$ at resonance, i.e. at a root of unity, but also near resonance under the Brjuno condition.
