Analytic Solutions of an Iterative Functional Differential Equation near Resonance

摘要

We investigate the existence of analytic solutions of a class of second-order differential equations involving iterates of the unknown functionx′′(z)+cx′(z)=x(az+bx(z))in the complex fieldℂ. By reducing the equation with the Schröder transformation to the another functional differential equation without iteration of the unknown functionλ2g′′(λz)g′(z)−λg′(λz)g′′(z)+c(g′(z))2(λg′(λz)−ag′(z))=(g′(z))3(g(λ2z)−ag(λz)), we get its local invertible analytic solutions.