## 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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### References:

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