Extending the Newton-Kantorovich hypothesis for solving equations

  • Authors:

  • Ioannis K. Argyros;Saïd Hilout

  • Affiliations:

  • Cameron University, Department of Mathematics Sciences, Lawton, OK 73505, USA;Poitiers University, Laboratoire de Mathématiques et Applications, Bd. Pierre et Marie Curie, Téléport 2, B.P. 30179, 86962 Futuroscope Chasseneuil Cedex, France

  • Venue:
  • Journal of Computational and Applied Mathematics
  • Year:
  • 2010

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Abstract

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 Frechet-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 Hernandez, 2002 [11]; [3]; Proinov 2009, 2010 [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.