Analytic solutions of a second order iterative functional differential equation

This paper is concerned with an iterative functional differential equationc1x(z)+c2x′(z)+c3x″(z)=x(az+bx′(z))with the delay depends on the argument of the unknown function and the state derivative. By reducing the equation with the Schröder transformation to another functional differential equation without iteration of the unknown function, we give existence of its local analytic solutions which extend the known results in related literature.