Continuity of iteration and approximation of iterative roots

✍ Scribed by Wenmeng Zhang; Weinian Zhang

• Book ID: 104007195
• Publisher: Elsevier Science
• Year: 2011
• Tongue: English
• Weight: 286 KB
• Volume: 235
• Category: Article
• ISSN: 0377-0427
• DOI: 10.1016/j.cam.2010.08.010

✦ Synopsis

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→∞ F m = F , can we find an iterative root f m of F m of order n for each m ∈ 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.