Third-order family of methods in Banach spaces

Recently, Parida and Gupta [P.K. Parida, D.K. Gupta, Recurrence relations for a Newton-like method in Banach spaces, J. Comput. Appl. Math. 206 (2007) 873–877] used Rall’s recurrence relation approach (from 1961) to approximate roots of nonlinear equations, by developing several methods, the latest of which is free of second derivative and it is of third order. In this paper, we use an idea of Kou and Li [J.-S. Kou, Y.-T. Li, Modified Chebyshev’s method free from second derivative for non-linear equations, Appl. Math. Comput. 187 (2007) 1027–1032] and modify the approach of Parida and Gupta, obtaining yet another third-order method to approximate a solution of a nonlinear equation in a Banach space. We give several applications to our method.