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Local \(C^r\) stability for iterative roots of orientation-preserving self-mappings on the interval. (English) Zbl 1442.65448

Summary: 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 globally \(C^0\) stable and locally \(C^1\) stable but globally \(C^1\) unstable. Although the global \(C^1\) instability implies the general global \(C^r\) (\(r \geq 2\)) instability, the local \(C^1\) stability does not guarantee the local \(C^r\) (\(r \geq 2\)) stability. In this paper we generally prove the local \(C^r\) (\(r \geq 2\)) stability for iterative roots. For this purpose we need a uniform estimate for the approximation to the conjugation in \(C^r\) linearization, which is given by improving the method used for the \(C^1\) case.

MSC:

65Q30 Numerical aspects of recurrence relations
39B12 Iteration theory, iterative and composite equations
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