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

Houyu Zhao¹ ; Jianguo Si²

¹ School of Mathematics Shandong University Jinan, Shandong 250100, P.R. China
² School of Mathematics Shandong University Jinan, Shandong 250100, P.R.China

Annales Polonici Mathematici, Tome 96 (2009) no. 3, pp. 209-226

Résumé

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.

Détail

DOI : 10.4064/ap96-3-2

Keywords: study existence analytic solutions second order iterative functional differential equation sum sum infty complex field bbb constructing invertible analytic solution auxiliary equation form alpha alpha alpha alpha sum sum infty alpha invertible analytic solutions form alpha original equation obtained besides hyperbolic alpha focus alpha unit circle alpha discuss only those alpha resonance root unity near resonance under brjuno condition