Convergence of an iterative method in Banach spaces with Lipschitz continuous first derivative

摘要

In this paper, the convergence of a third order Newton-like method used for solving F(x) = 0 in Banach spaces is established by using recurrence relations, when the first Frechet derivative of F satisfies the Lipschitz condition. Here, we relaxed the necessary conditions on F in order to study the convergence. This work is useful when either second derivative of F may not exist or may not satisfy Lipschitz condition. An existence and uniqueness theorem is derived for the root x~* of F(x) = 0. A numerical example is given to demonstrate the applicability of the method.