Extended Semi-Local Convergence of Newton's Method using the Center Lipschitz Condition and the Restricted Convergence Domain

Abstract

The objective of this study is to extend the usage of Newton's method for Banach space valued operators. We use our new idea of restricted convergence domain in combination with the center Lipschitz hypothesis on the Frechet-derivatives where the center is not necessarily the initial point. This way our semi-local convergence analysis is tighter than in earlier works (since the new majorizing function is at least as tight as the ones used before) leading to weaker criteria, better error bounds more precise information on the solution. These improvements are obtained under the same computational effort.