Analytic Solutions of an Iterative Functional Differential Equation near Resonance - Tongbo Liu; Hong Li (International Journal of Differential Equations)
We investigate the existence of analytic solutions of a class of second-order differential equations involving iterates of the unknown function 𝑥(𝑧)+𝑐𝑥(𝑧)=𝑥(𝑎𝑧+𝑏𝑥(𝑧)) in the complex field ℂ. By reducing the equation with the Schröder transformation to the another functional differential equation without iteration of the unknown function 𝜆2𝑔(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.