Recurrence relations for semilocal convergence of a fifth-order method in Banach spaces
The article deals with the following iterative of fifth-order scheme for the approximative solving a nonlinear operator equation \(F(x) = 0\) with an operator \(F\) between Banach spaces \(X\) and \(Y\): \[ y_n = x_n - \Gamma_nF(x_n) \] \[ z_n = x_n - \bigg[I + \frac12 L_F(x_n)(I - L_F(x_n))^{-1}\bigg]\Gamma_nF(x_n), \] \[ x_{n+1} = z_n - [I + F''(x_n)\Gamma_n(z_n - x_n)]^{-1}\Gamma_nF(z_n) \] (\(\Gamma_n = [F'(x_n)]^{-1}\), \(L_F(x_n) = \Gamma_nF''(x_n)\Gamma_nF(x_n)\)). The authors formulate some new conditions that guarantee the existence of the Halley approximations in a ball \(B(x_0,R)\), the convergence of these approximations to a solution of the equation \(F(x) = 0\) (lying in the ball \(B(x_0,R)\)), and \textit{a priori} and \textit{a posteriori} error estimates. In the end of the article, two numerical results (for a simple scalar equation and a nonlinear integral equation) are presented. It should be noted that the iterative scheme under consideration is formally ill-posed (it is not clear the role of \(y_n\) for the definition of \(x_{n+1}\)).
scientific article; zbMATH DE number 6022120
scientific article