Extending the Newton-Kantorovich hypothesis for solving equations

The famous Newton-Kantorovich hypothesis (Kantorovich and Akilov, 1982 3], Argyros, 2007 2], Argyros and Hilout, 2009 7]) has been used for a long time as a sufficient condition for the convergence of Newton’s method to a solution of an equation in connection with the Lipschitz continuity of the Fréchet-derivative of the operator involved. Here, using Lipschitz and center-Lipschitz conditions, and our new idea of recurrent functions, we show that the Newton-Kantorovich hypothesis can be weakened, under the same information. Moreover, the error bounds are tighter than the corresponding ones given by the dominating Newton-Kantorovich theorem (Argyros, 1998 1]; 2] and 7]; Ezquerro and Hernández, 2002 11]; 3]; Proinov 2009, 2010 16] and 17]).Numerical examples including a nonlinear integral equation of Chandrasekhar-type (Chandrasekhar, 1960 9]), as well as a two boundary value problem with a Green’s kernel (Argyros, 2007 2]) are also provided in this study.