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.