Local analytic solutions of a functional differential equation with a deviating argument depending on the state derivative near resonance

This paper is concerned with a functional differential equation x(n)(z)=1x(az+bx′(z)) with a deviating argument depending on the state derivative, where a,ba,b are two 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 another functional differential 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, besides the case 0<|λ|<10<|λ|<1, we focus on those λλ on the unit circle S1S1, i.e., |λ|=1|λ|=1. We discuss not only those λλ’s at resonance, i.e. at a root of the unity, but also those λλ near resonance under the Brjuno condition.