Extended Local Convergence for Seventh order method with $\psi$-continuity condition in Banach Spaces

TL;DR

This paper presents a local convergence analysis of a seventh order multi-step method for solving nonlinear equations in Banach spaces, utilizing a weak $\psi$-continuity condition to extend applicability beyond traditional Lipschitz and Hölder assumptions.

Contribution

It introduces a convergence analysis under a $\psi$-continuity condition, requiring only first-order derivatives, and devises a strategy to identify a subset of the convergence domain.

Findings

Convergence is established under $\psi$-continuity without higher derivatives.

The method outperforms existing schemes in numerical tests.

New Lipschitz constants improve convergence precision.

Abstract

In this article, the local convergence analysis of the multi-step seventh order method is presented for solving nonlinear equations. The point worth noting in our paper is that our analysis requires a weak hypothesis where the Fr\'echet derivative of the nonlinear operator satisfies the $\psi$-continuity condition and extends the applicability of the computation when both Lipschitz and H\"{o}lder conditions fail. The convergence in this study is shown under the hypotheses on the first order derivative without involving derivatives of the higher-order. To find a subset of the original convergence domain, a strategy is devised. As a result, the new Lipschitz constants are at least as tight as the old ones, allowing for a more precise convergence analysis in the local convergence case. Some numerical examples are provided to show the performance of the method presented in this contribution over some existing schemes.