LocalCrStability for Iterative Roots of Orientation-Preserving Self-Mappings on the Interval

Abstract

Stability of iterative roots is important in their numerical computation. It is known that under some conditions iterative roots of orientation-preserving self-mappings are both globallyC0stable and locallyC1stable but globallyC1unstable. Although the globalC1instability implies the general globalCr(r≥2) instability, the localC1stability does not guarantee the localCr(r≥2) stability. In this paper we generally prove the localCr(r≥2) stability for iterative roots. For this purpose we need a uniform estimate for the approximation to the conjugation inCrlinearization, which is given by improving the method used for theC1case.