Third-order convergence theorem by using majorizing function for a modified Newton method in Banach space

Following the ideas of Frontini and Sormani, we present a modified Newton method in Banach space which is used to solve the nonlinear operator equation. We establish the Newton–Kantorovich convergence theorem for the modified Newton method with third-order convergence in Banach space by using majorizing function. We also get the error estimate. Finally, two examples are provided to show the application of our theorem.