Gutiérrez, J. M.; Hernández, M. A. Newton’s method under weak Kantorovich conditions. (English) Zbl 0965.65081 IMA J. Numer. Anal. 20, No. 4, 521-532 (2000). For solving nonlinear equations \(x= F(x)\) in a Banach space, the Newton-Kantorovich method is well-known. Unfortunately, the classical theorem provides convergence only if the Fréchet derivative of \(F\) is Lipschitz continuous.The authors prove convergence results and error estimates under the weaker assumption \[ \|F'(x)- F'(x_0)\|\leq L\|x- x_0\|,\quad L>0, \] for some given \(x_0\). The results are then illustrated for a nonlinear integral equation. Reviewer: Etienne Emmrich (Berlin) Cited in 32 Documents MSC: 65J15 Numerical solutions to equations with nonlinear operators 47J25 Iterative procedures involving nonlinear operators 45G10 Other nonlinear integral equations 65R20 Numerical methods for integral equations Keywords:nonlinear operator equation; Banach space; Newton-Kantorovich method; convergence; error estimates; nonlinear integral equation PDFBibTeX XMLCite \textit{J. M. Gutiérrez} and \textit{M. A. Hernández}, IMA J. Numer. Anal. 20, No. 4, 521--532 (2000; Zbl 0965.65081) Full Text: DOI