Extending the Newton–Kantorovich hypothesis for solving equations

✍ Scribed by Ioannis K. Argyros; Saïd Hilout

• Publisher: Elsevier Science
• Year: 2010
• Tongue: English
• Weight: 378 KB
• Volume: 234
• Category: Article
• ISSN: 0377-0427
• DOI: 10.1016/j.cam.2010.04.014

✦ Synopsis

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]; [2,7]; Ezquerro and Hernández, 2002 [11]; [3];Proinov 2009 Proinov , 2010 [16,17] [16,17]).

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.