A new type of recurrence relations for the secant method*

Abstract

We apply the Secant method to solve nonlinear operator equations in Banach spaces. We establish a Newton-Kantorovich convergence theorem using a new system of recurrence relations and give an explicit expression for the a priori error bounds. Moreover, we apply our results to the numerical resolution of a nonlinear boundary value problem of second order and improve the error bounds obtained by other authors.