The majorant method in the theory of Newton–Kantorovich approximations and generalized Lipschitz conditions

We provide a semilocal as well as a local convergence analysis for Newton’s and modified Newton’s methods in order to approximate a locally unique solution of a nonlinear equation in a Banach space setting. We use more precise majorizing sequences than in earlier studies such as Appell et al. (1997), Appell et al. (1991), Argyros (2004), Argyros and Hilout (2009), Kantorovich and Akilov (1982), Ortega and Rheinboldt (1970) and generalized Lipschitz continuity conditions. Our sufficient convergence conditions are weaker than before and our convergence analysis is tighter. Special cases and numerical examples are also given in this study.