Recurrence relations for a family of iterations assuming Holder continuous second order Frechet derivative

摘要

The semilocal convergence using recurrence relations of a family of iterations for solving nonlinear equations in Banach spaces is established. It is done under the assumption that the second order Frechet derivative satisfies the Holder continuity condition. This condition is more general than the usual Lipschitz continuity condition used for this purpose. Examples can be given forwhich the Lipschitz continuity condition fails but the Holder continuity condition works on the second order Frechet derivative. Recurrence relations based on three parameters are derived. A theorem for existence and uniqueness along with the error bounds for the solution is provided. The R-order of convergence is shown to be equal to 3 + q when theta = +/- 1; otherwise it is 2 + q, where q epsilon (0, 1]. Numerical examples involving nonlinear integral equations and boundary value problems are solved and improved convergence balls are found for them. Finally, the dynamical study of the family of iterations is also carried out.