Analytic solutions of a second-order iterative functional differential equation

This paper is concerned with a second-order iterative functional differential equation x " (x([r])(z))= c(0)z + c(1)x(z) + ... + cmx([m])(z), where r and m are nonnegative integers, x([0])(z) = z, x([1])(z) = x(z), x([2])(z) = x(x(z)), etc., are the iterates of the function x(z). By constructing a convergent power series solution y(z) of a companion equation of form alpha (2)gamma "(alpha (r+1)z)y'(alpha (r)z) = alpha gamma'(alpha (r+1)z)gamma "(alpha (r)z) + [gamma'(alpha (r)z)](3) [Sigma (m)(i=0) c(i)gamma(alpha (i)z)] analytic solutions of the form y(alpha gamma (-1)(z)) for the original differential equation are obtained. (C) 2000 Elsevier Science B.V. All rights reserved. MSC. 34K25.