A general semilocal convergence result for Newton’s method under centered conditions for the second derivative
University of La Rioja, Department of Mathematics and
Computation, C/ Luis de Ulloa
Revised: 26 March 2012
From Kantorovich’s theory we present a semilocal convergence result for Newton’s method which is based mainly on a modification of the condition required to the second derivative of the operator involved. In particular, instead of requiring that the second derivative is bounded, we demand that it is centered. As a consequence, we obtain a modification of the starting points for Newton’s method. We illustrate this study with applications to nonlinear integral equations of mixed Hammerstein type.
Mathematics Subject Classification: 45G10 / 47H99 / 65J15
Key words: Newton’s method / the Newton–Kantorovich theorem / semilocal convergence / majorizing sequence / a priori error estimates / Hammerstein’s integral equation
© EDP Sciences, SMAI, 2012