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 Schrodƒtƒt er transformation to finding analytic solutions of a functional equation without iteration of the unknown function x. For technical reasons, in previous works the constant ƒÑ given in the Schroƒtƒtder transformation is required to fulfil that ƒÑ 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 ƒÑ at resonance, i.e., at a root of the unity and the case of near resonance under the Brjuno condition.