Analytic solutions of an iterative differential equation under Brjuno condition

Abstract

In this paper, the differential equation involving iterates of the unknown function, $$ x'(z) = [a^2 - x^2 (z)]x^{[m]} (z) $$ with a complex parameter a, is investigated in the complex field ℂ for the existence of analytic solutions. First of all, we discuss the existence and the continuous dependence on the parameter a of analytic solution for the above equation, by making use of Banach fixed point theorem. Then, as well as in many previous works, we reduce the equation with the Schroder transformation x(z) = y(αy−1(z)) to the following another functional differential equation without iteration of the unknown function $$ \alpha y'(\alpha z) = [a^2 - y^2 (\alpha z)]y'(z)y(\alpha ^m z), $$ which is called an auxiliary equation. By constructing local invertible analytic solutions of the auxiliary equation, analytic solutions of the form y(αy−1(z)) for the original iterative differential equation are obtained. We discuss not only these α given in Schroder transformation in the hyperbolic case 0 < |α| < 1 and resonance, i.e., at a root of the unity, but also those α near resonance (i.e., near a root of the unity) under Brjuno condition. Finally, we introduce explicit analytic solutions for the original iterative differential equation by means of a recurrent formula, and give some particular solutions in the form of power functions when a = 0.