On the convergence radius of the modified Newton method for multiple roots under the center–Hölder condition

It is very important to enlarge the convergence ball of an iterative method. Recently, the convergence radius of the modified Newton method for finding multiple roots of nonlinear equations has been presented by Ren and Argyros when the involved function is Hölder and center–Hölder continuous. Different from the technique and the hypothesis used by them, in this paper, we also investigate the convergence radius of the modified Newton method under the condition that the derivative $f^{(m)}$ of function f satisfies the center–Hölder continuous condition. The radius given here is larger than that given by Ren and Argyros. The uniqueness ball of solution is also discussed. Some examples are given to show applications of our theorem.

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