一类迭代函数方程的解析解

Analytic Solutions of a Class of Iterative Functional Equations

关键词:  迭代函数方程  解析解  差商  函数空间  紧致凸集

Key words:  iterative functional equation  analytic solution  difference quotient  functional space  compact convex set

摘要:

设r是个给定的正数,用D=Dr表示复平面C内以原点为心r为半径的闭圆盘.令A(D,D)={f:f为从D到D的连续映射,并且f|D⁰解析}.设G:Dⁿ⁺¹ →C连续(n ≥2),并且G|(Dⁿ⁺¹)⁰解析,g₁,…,g_n ∈A(D,D),本文讨论了迭代函数方程G(z,f(g₁(z)),…,fⁿ(g_n(z)))=0,给出该方程在A(D,D)中有解和有唯一解的条件.

Abstract:

Let r be a given positive number.Denote by D=Dr the closed disc in the complex plane C whose center is the origin and radius is r.Write A(D,D)={f:f is a continuous map from D to itself,and f|D⁰ is analytic}.Suppose G:Dⁿ⁺¹ →C is a continuous map (n ≥ 2),and G|(Dⁿ⁺¹) ⁰ is analytic.Let g₁,…,g_n∈A(D,D) be given maps.In this paper, we study the iterative functional equation G(z,f(g₁(z)),…,fⁿ(g_n(z)))=0 and give some conditions for the equation to have a solution and a unique solution in A(D,D).