The Secant method and divided differences Hölder continuous

摘要

We apply the Secant method to solve non-linear operator equations in Banach spaces. A semilocal convergence result is obtained, where the first-order divided difference of the non-linear operator is Hölder continuous. For that, we use a technique based on a new system of recurrence relations to obtain domains of existence and uniqueness of the solution and give an explicit expression for the a priori error bounds. Moreover, we apply our results to the numerical solution of a non-linear boundary value problem of second-order.