Analytic solutions of systems of functional equations

Abstract

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⩾1, write A(D ^m, K)={f‖f: D ^m→K is a continuous map, and f‖(D ^m)° is analytit}. For H∈A(D ^m,C)(m⩾2), f∈A(D, D) and z∈D, write Ψ _H(f)(z)=H(z, f(z),...,f^m−1(z)). Suppose F,G H∈A(D ²ⁿ⁺¹,C), and H _k, K_k H∈A(D ^k,C), k=2,…,n. In this paper, the system of functional equations $\left\{ \begin{gathered} F(z,f(z),f^2 (\Psi _{H_2 } (f)(z)).....f^n (\Psi _{H_n } (f)(z)),g(z),g^2 (\Psi _{K_2 } (g)(z))..... \hfill \\ g^n (\Psi _{K_n } (g)(z))) = 0 \hfill \\ G(z,f(z),f^2 (\Psi _{H_2 } (f)(z)),.....f^n (\Psi _{H_n } (f)(z)),g(z),g^2 (\Psi _{K_2 } (g)(z)),..... \hfill \\ g^n (\Psi _{K_n } (g)(z))) = 0 \hfill \\ \end{gathered} \right.$ (z H∈D) is studied and some conditions for the system of equations to have a solution or a unique solution in A(D,D) × A(D,D) are given.