Improved results on estimating and extending the radius of an attraction ball

Ostrowski provided the sharp sufficient condition ρ(F′(x∗))<1ρ(F′(x∗))<1 for x∗x∗ to be an attraction point, for a nonlinear mapping differentiable at a fixed point x∗x∗[1]. This result provides no estimate for the size of the attraction ball. Recently, Cătinaş [2] provided such an estimate in terms of ‖F′(x∗)‖<1‖F′(x∗)‖<1 in a Hölder continuity setting. We show that the results by Cătinaş remain valid in a weaker setting by simply replacing the Hölder by the center-Hölder continuity assumption. The radius of convergence of Picard’s iteration is extended, which allows a wider choice of initial guesses. Moreover the estimates of the distances ‖x0−x∗‖‖x0−x∗‖ are more precise, which lead to the computation of fewer iterates to achieve a desired accuracy. We also provide examples where our results apply, whereas those by Cătinaş [2] do not, or where our results are better.