New convergence criteria for the Newton-Kantorovich method and some applications to nonlinear integral equations
Starting from the fact that the Newton-Kantorovich method in the classical setting does not apply to Urysohn integral equations in some spaces (for instance, in \(L^p\)-spaces, for \(1<p<2\)), the main aim of the paper is to obtain convergence criteria for the Newton-Kantorovich method, by modifying the smoothness hypotheses in a suitable way. One merit of these new conditions is that they may be checked rather effectively for Urysohn integral equations. To emphasize this, the new convergence criteria are then systematically applied to nonlinear integral equations of Urysohn type, taking as the ambient Banach space \(X\) various particular spaces: the Chebyshev space \(C\), the Lebesgue space \(L^p\) and the Orlicz space \(LM\). It is also remarked that all the obtained results carry over the ideal spaces of measurable functions.
nonlinear integral equations of Urysohn type
scientific article; zbMATH DE number 1279136
scientific article