How to solve nonlinear equations when a third order method is not applicable (Q1293246)

For the solution of an equation \(F(x)=0\) involving a nonlinear operator \(F\) between Banach spaces \(X\), \(Y\), the authors consider the family of iterative methods \[ x_{n+1} = x_n - \big[ I + \tfrac{1}{2}L(x_n)[I - L(x_n)]^{-1} \big] \Gamma_n F(x_n),\quad n = 1,2,\dots , \] where \(\Gamma_n = F'(x_n)^{-1}\), \(L(x_n) = \Gamma_n A \Gamma_n F(x_n)\), and \(A\) represents a fixed bilinear operator between \(X\) and \(Y\). By means of a standard majorizing technique, second-order convergence of the method is proved under conditions of Newton-Kantorovich type and corresponding error bounds are given. Several numerical examples are included.