Convergence of an Iterative Method for Fixed Points in Banach Spaces

The aim of this paper is to study the semilocal convergence of a third order iterative method used for finding fixed points of nonlinear operator equations in Banach spaces.This convergence is established under the assumption that the first Frechet derivative of the involved operator satisfies the Lipschitz continuity condition. The existence and uniqueness regions along with a priori error bounds for a fixed point are derived. The R-order of the method is also shown to be equal to three. Finally, an integral equation is worked out with our method and with a Newton-like method and results are compared. It is observed that our method gives superior existence and uniqueness regions.