BALL CONVERGENCE FOR A TWO STEP METHOD WITH MEMORY AT LEAST OF ORDER $2+\SQRT{2}$ - Argyros, I. K.; Behl, R.; Motsa, S.S. (Journal of Nonlinear Analysis and Optimization: Theory & Applications (JNAO), 2018)
We present a local convergence analysis of at least $2+\sqrt{2}$ convergence order two-step method in order to approximate a locally unique solution of nonlinear equation in a Banach space setting. In the earlier study, [6,15] the authors of these paper did not discuss that studies. Furthermore, the order of convergence was shown using Taylor series expansions and hypotheses up to the sixth order derivative or or even higher of the function involved which restrict the applicability of the proposed scheme. However, only first order derivative appears in the proposed scheme. In order to overcome this problem, we proposed the hypotheses up to only first order derivative. In this way, we not only expand the applicability of the methods but also propose convergence domain. Finally, we present some numerical experiments where earlier studies cannot apply to solve nonlinear equations but our study does not exhibit this type of problem/restriction.