Analytic solutions of iterative functional equations

Let r be a given positive number. Denote by D = D-r the closed disc in the complex plane C whose center is the origin and radius is r. For any subset K of C and any integer m greater than or equal to 1, write A(D-m, K) = {f \ f : D-m --> K is a continuous map, and f \ (D-m)degrees is analytic}. Suppose G is an element of A(Dn+1, C), and H-k is an element of A(D-k, C), k = 2,..., n. In this paper, we study the iterative functional equation G(z, f (z), f(2) (H-2(z, f(z))),...,f(n)(H-n(z, f(z),...,f(n-1)(z)))) = 0, and give some conditions for the equation to have a solution and a unique solution in A(D, D). (C) 2002 Elsevier Science (USA). All rights reserved.