Analytic solutions for a functional differential equation related to a traffic flow model (Q1938131)

Summary: We study the existence of analytic solutions to the functional differential equation \[ (z(s) + \alpha)^2 z'(s) = \beta(z(s + z(s)) - z(s)) \] which comes from a traffic flow model. By reducing the equation by means of the Schröder transformation to an auxiliary equation, we discuss not only the constant \(\lambda\) at resonance, that is, at a root of unity, but also those \(\lambda\) near resonance under the Brjuno condition.