Convergence results for a fast iterative method in linear spaces (Q1809745)

Recently, the author proposed a fast iterative method for solving the nonlinear equation \(F(x)= 0\), where \(F\) is a twice Fréchet differentiable operator defined on some convex subset \(D\) of a Banach space \(E_1\) with values in a Banach space \(E_2\). Using standard Newton-Kantorovich assumption he proved that under Lipschitz-type hypotheses on an upper bound of the same derivative the iterative method under review has the order of convergence equal to four. In this paper, he relaxed these conditions using Lipschitz-hypotheses on the first Fréchet-derivative only and examines the monotone convergence of the same method in a partially ordered topological space setting.