Continuity of iteration and approximation of iterative roots

摘要

Motivated by computing iterative roots for general continuous functions, in this paper we prove the continuity of the iteration operators Tn, defined by Tnf=fn. We apply the continuity and introduce the concept of continuity degree to answer positively the approximation question: If limm→∞Fm=F, can we find an iterative root fm of Fm of order n for each m∈N such that the sequence (fm) tends to the iterative root of F of order n associated with a given initial function? We not only give the construction of such an approximating sequence (fm) but also illustrate the approximation of iterative roots with an example. Some remarks are presented in order to compare our approximation with the Hyers–Ulam stability.