Analytic Solutions of an Iterative Differential Equation under Brjuno Condition

<正> In this paper,the differential equation involving iterates of the unknown function,x'（z） = [a2 - x2（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 analyticsolution for the above equation,by making use of Banach fixed point theorem.Then,as well as inmany previous works,we reduce the equation with the Schroder transformation x（z） = y(αy-1（z）) tothe following another functional differential equation without iteration of the unknown functionαy'（αz） = [α2 - y2（αz）]y'（z）y（αmz）,which is called an auxiliary equation.By constructing local invertible analytic solutions of the auxiliaryequation,analytic solutions of the form y(αy-1（z）) for the original iterative differential equation areobtained.We discuss not only these α given in Schroder transformation in the hyperbolic case 0 <lαl < 1 and resonance,i.e.,at a root of the unity,but also those α near resonance （i.e.,near a rootof the unity） under Brjuno condition.Finally,we introduce explicit analytic solutions for the originaliterative differential equation by means of a recurrent formula,and give some particular solutions inthe form of power functions when a = 0.