Concerning the Convergence of a Modified Newton-Like Method

Abstract

We provide sufficient convergence conditions for a certain Newton-like method to a locally unique solution of a nonlinear equation in a Banach space. We assume that the Fréchet-derivative of the operator involved satisfies in some sense uniformly continuous conditions, which are weaker than earlier ones. We show that our results apply where earlier ones fail. Finally, we solve a nonlinear integral equation of Uryson-type that cannot be solved using Proposition 2 in [10].