A note on “Convergence radius of Osada's method under Hölder continuous condition”

In this paper we revise the proofs of the results obtained in “Convergence radius of Osada's method under Hölder continuous condition” [4], because the remainder of the Taylor's expansion used for the obtainment of the local convergence radius is not correct. So we perform the complete study in order to modify the equation for getting the local convergence radius, the uniqueness radius and the error bounds. Moreover a dynamical study for the third order Osada's method is also developed.