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 analytic}. 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∈A(D 2n+1,C), and H k,K k∈A(D k,C), k=2,...,n. In this paper, the system of functional equations F(z,f(z),f 2(Ψ H 2(f)(z)),...,f n(Ψ H n(f)(z)),g(z),g 2(Ψ K 2(g)(z)),..., g n(Ψ K n(g)(z)))=0 G(z,f(z),f 2(Ψ H 2(f)(z)),...,f n(Ψ H n(f)(z)),g(z),g 2(Ψ K 2(g)(z)),..., g n(Ψ K n(g)(z)))=0(z∈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.
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