Local invertible analytic solution of a functional differential equation with deviating arguments depending on the state derivative

This paper is concerned with a functional differential equation of the formx′(z)=1/x(az+bx′(z))x′(z)=1/x(az+bx′(z)) with the distinctive feature that the argument of the unknown function depends on the state derivative, where a,ba,b are two complex numbers. By reducing the equation with the Schröder transformation to another functional differential equation with proportional delay, we give existence of its local invertible analytic solutions. We discuss not only that the constant α given in Schröder transformation at resonance, i.e., at a root of the unity, but also those α near resonance (near a root of the unity) under the Brjuno condition.