Convergence theorems of a new multiparametric family of Newton-like method in Banach space

In this work, we have considered a new multi-parametric family of modified Newton-like methods(MNL) of order three to approximate a zero of a nonlinear operator in \mathbb{B}-space (Banach space). Here, we studied the semilocal convergence analysis of this family of methods by using a new type of majorant condition. Note that this majorant condition generalizes the earlier majorant conditions used for studying convergence analysis of third order methods. Moreover, by using second-order directional derivative of the majorizing function we obtained an error estimate. We also established relations between our majorant condition and assumption based on Kantorovich, Smale-type and Nesterov-Nemirovskii-type, that will show our result generalize these earlier convergence results.

Convergence theorems of a new multiparametric family of Newton-like method in Banach space Keywords:

Multi-parametric family of modified Newton-like (MNL) methods , Majorant conditions , Majorizing function , Nesterov-Nemirovskii condition , Kantorovich-type assumption , Smale-type assumption