Semilocal convergence results for Newton-like methods using Kantorovich quasi-majorant functions involving the first derivative (Q2782914)

The author again continues his work on Newton-type methods for solving nonlinear operator equations \(F(x)= 0\) in a Banach space setting via an iteration \(x_{n+1}= x_n- A(x_n)^{- 1}F(x_n)\) (\(n= 0,1,\dots\); \(x_0\) given). This article is concerned with convergence results that are based upon Kantorovich's quasi-majorant functions. As these results cover different methods (in dependence of the choice for \(A\)) and provide uniqueness, the author generalizes results of \textit{J. Appell}, \textit{E. de Pascale}, \textit{J. V. Lysenko} and \textit{P. P. Zabrejko} [Numer. Funct. Anal. Optimization 18, No. 1-2, 1-17 (1997; Zbl 0881.65049)].NEWLINENEWLINENEWLINEAlthough other methods are mentioned, the author only discusses \(A(x)= F'(x)\) in more detail. The example given (\(F(x)= x^4/2400+ x-3\) in the Banach space \(\mathbb{R}\), starting the iteration with \(x_0= 0\)) seems to be somewhat simple when considering nonlinear equations in a Banach space setting.