Analytic Solutions of a Second-Order Functional Differential Equation

Abstract

This paper is concerned with the existence of analytic solutions of a second-order functional differential equation. As well as in many previous works, we reduce this problem to finding analytic solutions of a functional differential equation without iteration of the unknown function x. For technical reasons, in previous works the constant α given in the Schroder transformation is required to fulfil that α is off the unite circle or lies on the circle with the Diophantine condition. In this paper, we focus on those α on the unit circle, i.e. | α|=1. We discuss not only those α′s at resonance, i.e. at a root of the unity, but also those α′s near resonance under the Brjuno condition, where the Diophantine condition is not required.