Analytic Solutions of a Second-Order Functional Differential Equation

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 alpha given in the Schroder transformation is required to fulfil that a is off the unite circle or lies on the circle with the Diophantine condition alpha In this paper, we focus on those a on the unit circle, i.e. vertical bar alpha vertical bar = 1. We discuss not only those alpha's at resonance, i.e. at a root of the unity, but also those alpha's near resonance under the Brjuno condition, where the Diophantine condition is not required.