Abstract In this manuscript, we accelerate the local convergence of a third-order biparametric iterative method in ℝ or ℂ by assuming that the first-order Fréchet derivative satisfies the Generalized continuity condition. We extend this analysis by using the Hölder continuity condition, which allows us to solve more numerical problems. Our study also shows the sizes of the convergence balls, the smallest error bounds that can be computed, and the fact that the answer is unique. Several math tests show that this third-order method gives better results than the midpoint method established by I.K. Argyros and S. George 4. This method solves problems that earlier studies have not been able to solve.
Argyros et al. (Mon,) studied this question.