Local convergence analysis of modified Newton-like method under majorant condition

This article has devoted to study of local convergence analysis of a modified Newton-like method of order three to approximate a locally unique zero of a non-linear operator in $\mathbb{B}$-space (Banach space). Here, we mainly proved the convergence of the method by using a new type of majorant condition instead of usual majorizing sequences and recurrence relations and provide a computable error bounds. Furthermore, we will establish the relationship between majorant function with special cases like Smale-type and Kantorovich-type functions. A few favorable numerical examples has also provided to show applicability of our analysis.