Local analytic solutions of a functional differential equation αz+βx′(z)=x(az+bx″(z))αz+βx′(z)=x(az+bx″(z))

In this paper, we are concerned with the existence of analytic solution of a functional differential equation αz+βx′(z)=x(az+bx″(z))αz+βx′(z)=x(az+bx″(z)), where α,β,a,b are four complex numbers. We first discuss the existence of analytic solutions for some special cases of the above equation. Then, by reducing the equation with the Schröder transformation to the another functional equation with proportional delay, an existence theorem is established for analytic solutions of the original equation. For the constant λλ given in the Schröder transformation, we discuss the case 0<|λ|<10<|λ|<1 and λλ on the unit circle S1S1, i.e., |λ|=1|λ|=1. We study λλ is at resonance, i.e., at a root of the unity and λλ is near resonance under the Brjuno condition.