Local convergence for some high convergence order Newton-like methods with frozen derivatives

Abstract

We present a local convergence analysis of some families of Newton-like methods with frozen derivatives in order to approximate a locally unique solution of an equation in a Banach space setting. In earlier studies such as Amat et al. (Appl Math Lett. 25:2209–2217, 2012), Petkovic (Multipoint methods for solving nonlinear equations, Elsevier, Amsterdam, 2013), Traub (Iterative methods for the solution of equations, AMS Chelsea Publishing, Providence, 1982) and Xiao and Yin (Appl Math Comput, 2015) the local convergence was proved based on hypotheses on the derivative of order higher than two although only the first derivative appears in these methods. In this paper we expand the applicability of these methods using only hypotheses on the first derivative and Lipschitz constants. Numerical examples are also presented in this study.