Semilocal Convergence for a Fifth‐Order Newton′s Method Using Recurrence Relations in Banach Spaces

Abstract

We study a modified Newton′s method with fifth‐order convergence for nonlinear equations in Banach spaces. We make an attempt to establish the semilocal convergence of this method by using recurrence relations. The recurrence relations for the method are derived, and then an existence‐uniqueness theorem is given to establish the R‐order of the method to be five and a priori error bounds. Finally, a numerical application is presented to demonstrate our approach.