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.