Continuity of iteration and approximation of iterative roots

JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS

卷:235

Article

Zhang, Weinian¹ 

[1] Sichuan Univ, Yangtze Ctr Math, Chengdu 610064, Sichuan, Peoples R China

关键词: Iteration operator;    Iterative root;    Approximation;    Stability;    Equicontinuity;    Continuity degree;   

DOI  :  10.1016/j.cam.2010.08.010

来源: Elsevier

【 摘 要 】

Motivated by computing iterative roots for general continuous functions, in this paper we prove the continuity of the iteration operators T-n. defined by T(n)f = f(n). We apply the continuity and introduce the concept of continuity degree to answer positively the approximation question: If lim(m ->infinity)F(m) = F, can we find an iterative root f(m) of F-m of order n for each m is an element of N such that the sequence (f(m)) 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 (f(m)) 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. (C) 2010 Elsevier B.V. All rights reserved.

【 授权许可】

Free