Analytic solutions of iterative functional equations

Let r be a given positive number. Denote byD = Dr 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 1, write A(Dm,K) = {f | f :Dm→K is a continuous map, and f |(Dm)◦ is analytic}. Suppose G ∈ A(Dn+1,C), and Hk ∈ A(Dk,C), k = 2, . . . , n. In this paper, we study the iterative functional equation G(z, f (z), f 2(H2(z, f (z))), . . . , f n(Hn(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).  2002 Elsevier Science (USA). All rights reserved