Analytic Solutions of an Iterative Functional Differential Equation

This paper is concerned with an iterative functional differential equation x x r z = c0z +c1x z +c2x x z +· · ·+cmx m z , where r and m are nonnegative integers, x 0 z = z x 1 z = x z x 3 z = x x x z , etc. are the iterates of the function x z , and m j=0 cj = 0. By constructing a convergent power series solution y z of a companion equation of the form αy αr+1z = y αrz m j=0 cjy αjz , analytic solutions of the form y αy−1 z for the original differential equation are obtained.