Stability for iterative roots of piecewise monotonic functions

Lin Li,Wei Song,Yingying Zeng

doi:10.1186/s13660-015-0925-8

Abstract

Global $C^{0}$ and local $C^{1}$ stability of iterative roots for monotonic functions defined on a compact interval, as well as global $C^{1}$ instability under some assumptions, are well-known facts. In this paper, we investigate the stability of iterative roots for piecewise monotonic functions with nonmonotonicity height equal to 1. We prove the roots are $C^{1}$ locally stable and $C^{0}$ global stable with the same extension.

Highlights

Given a Banach space X and r ≥, Cr(X) is defined as the set of all Cr self-mappings on X
Iterative roots is connected to the research of embedding flow and topological conjugacy in dynamical systems [, ], which is involved in the study of functional equations [, ]
In addition to monotonic mappings [, ], plenty results were obtained for the iterative roots of piecewise monotonic functions [ – ]

Summary

Introduction

Given a Banach space X and r ≥ , Cr(X) is defined as the set of all Cr self-mappings on X. For every continuous iterative root f of F of order k ≥ there exists a corresponding natural number (f ), which maps I into K(F). Such f is called a root of -extension (see [ ]). The following result gives the iterative roots of those functions with -extension. We consider the stability of iterative roots for piecewise monotonic functions with nonmonotonicity height equal to We prove that those roots with the same extension are locally C stable and globally C stable. Each function F and Fm has a kth order C iterative root fand fm on their characteristic interval, respectively.

Hyers-Ulam stability

Examples