泛函方程组的解析解

§ 1　IntroductionThe Feigenbaum functional equation plays an importantrole in the theory concerninguniversal properties of one-parameter families of maps of the interval that has the formf2 (λx) +λf(x) =0 ,0 <λ=-f(1 ) <1 ,f(0 ) =1 ,(1 .1 )where f is a map ofthe interval-1 ,1 ] into itself.Lanford1 ] exhibited a computer-assist-ed proof for the existence of an even analytic solution to Eq.(1 .1 ) .It was shown in2 ]that Eq.(1 .1 ) does not have an entire solution.Si3] discussed the it…

Analytic solutions of systems of 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⩾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 (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. Supported by the National Natural Science Foundation of China (10226014), Guangxi Science Foundation (0229001).