On the method of tangent parabolas (Q1370572)

The author establishes new sufficient conditions for the convergence of the method of tangent parabolas to a locally unique solution of the equation (1) \(F(x)= 0\), where \(F\) is a Fréchet-differentiable operator from a convex subset of a Banach space \(B_1\) into another Banach space \(B_2\). To obtain convergence results it is usually assumed that the second Fréchet-derivative \(F''\) satisfies a Lipschitz condition with a positive constant \(N\). He proves the convergence for this method under the majorant method and under Newton-Kantorovich-type assumptions including the value \(N= 0\). He derives some new error bounds and compares his results with the ones obtained so far and shows that in some cases his error bounds are sharper under easier to check hypothesis. Some applications are also given to the solution of nonlinear integral equations occurring in radiative transfer and where the constant \(N\) is zero.