Abstract:

The secondorder differential equation involving iterates of the unknown function λ2x″(z)+λ1x′(z)+λ0x(z)=f(∑mj=0cjxj(z))+G(z) is investigated in the complex field C for the existence of analytic solutions. By reducing the equation with the Schrder transformation,　x(z)=y(αy-1(z)), to another functional differential equation without iteration of the unknown function λ2［α2y″(αz)y′(z)-αy′(αz)y″(z)］+λ1αy′(αz)(y′(z))2+λ0y(αz)(y′(z))3=(y′(z))3［f(∑mj=0cjy(αjz))+G(y(z))］, we give the existence of its local invertible analytic solutions. We discuss not only those α given in the Schrder transformation in the hyperbolic case 0<|α|<1　and resonance, but also those α near resonance under Brjuno　condition.

Key words: iteration functional differential equation, analytic solution, resonance, power series