Analytic Solutions of a Second-Order Iterative Functional Differential Equations

In this paper, the existence of analytic solutions of an iterative functional differential equation is studied. We reduce this problem to finding analytic solutions of a functional differential equation without iteration of the unknown function. For technical reasons, in previous work the constant alpha given in Schroder transformation is required to fulfill that alpha is off the unit circle or lies on the circle with the Diophantine condition. In this paper, we break the restraint of the Diophantine condition and obtain results of analytic solutions in the case of alpha at resonance, i.e.. at a root of the unity and the case of alpha near resonance under the Brjuno condition.