Two Point Iterative Schemes for Nondifferentiable Equations in Banach Space

Abstract

The local as well as the semi-local convergence analysis is established for a certain single step-two point iterative scheme defined on a Banach space setting. These schemes converge to a locally unique solution of a nonlinear equation. Both types of convergence are based on w-type continuity and majorizing functions and sequences. An auxiliary fixed linear operator is utilized to assure the existence of inverses of the linear operators involved as well as the initial points of the iterative scheme. The local analysis provides the radius of convergence, error estimates and information on the uniqueness of the solution. Moreover, the semi local analysis provides sufficient convergence conditions, error estimates and uniqueness of the solution results. Numerical examples further validate the theoretical results.