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 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.