Local convergence of a relaxed two-step Newton like method with applications

We present a local convergence analysis for a relaxed two-step Newton-like method. We use this method to approximate a solution of a nonlinear equation in a Banach space setting. Hypotheses on the first Fréchet derivative and on the center divided-difference of order one are used. In earlier studies such as Amat et al. (Numer Linear Algebra Appl 17:639–653, 2010, Appl Math Lett 25(12):2209–2217, 2012, Appl Math Comput 219(24):11341–11347, 2013, Appl Math Comput 219(15):7954–7963, 2013, Reducing Chaos and bifurcations in Newton-type methods. Abstract and applied analysis. Hindawi Publishing Corporation, Cairo, 2013) these methods are analyzed under hypotheses up to the second Fréchet derivative and divided differences of order one. Numerical examples are also provided in this work.