Analytic solutions of a second-order nonautonomous iterative functional differential equation

In this paper a second-order nonautonomous iterative functional differential equation is considered. 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. We first discuss the case that the constant α given in the Schröder transformation does not lie on the unit circle in C and the case that the constant lies on the circle but fulfills the Diophantine condition. Then we further study the case that the constant is a unit root in C but the Diophantine condition is offended. Finally, we investigate analytic solutions of the form of power functions.