Modifications of higher-order convergence for solving nonlinear equations

In [Liang Fang, Guoping He, Some modifications of Newton’s method with higher-order convergence for solving nonlinear equations, J. Comput. Appl. Math. 228 (2009) 296–303], the authors pointed out that the iteration constructed in [Y.M. Ham, C.B. Chun and S.G. Lee, Some higher-order modifications of Newton’s method for solving nonlinear equations, J. Comput. Appl. Math. 222 (2008) 477–486] failed when p=2p=2. They gave some counterexamples and obtained a modified result. However, they did not show the essential reason which leads to the incorrect result. In this paper, we shall show that reason and present more general results than the above-mentioned papers.