Analytic Solutions of an Iterative Functional Differential Equation Near Resonance

In this paper existence of local analytic solutions of an iterative functional differential equation is studied. As well as in previous works, we reduce this problem with the Schroder transformation to finding analytic solutions of a functional 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 s¹ or lies on the circle with the Diophantine condition. In this paper, we obtain analytic solutions in the case of α at resonance, i.e., at a root of the unity and the case of near resonance under the Brjuno condition.