On a characterization of some Newton-like methods of R-order at least three

In this paper, we extend to Banach spaces the result given by Gander in (Amer. Math. Monthly 92 (1985) 131) to obtain a characterization of Newton-like iterative process with R-order of convergence at least three. To do this, we consider the construction and the study of the semilocal convergence of a multiparametric family of iterative processes in Banach spaces for solving the nonlinear equation F ( x ) = 0 .