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In this article, the local convergence analysis of the multi-step seventhorder method is presented for solving nonlinear equations. The point worthnoting in our paper is that our analysis requires a weak hypothesis where theFr\'echet derivative of the nonlinear operator satisfies the $\psi$-continuitycondition and extends the applicability of the computation when both Lipschitzand H\"{o}lder conditions fail. The convergence in this study is shown underthe hypotheses on the first order derivative without involving derivatives ofthe higher-order. To find a subset of the original convergence domain, astrategy is devised. As a result, the new Lipschitz constants are at least astight as the old ones, allowing for a more precise convergence analysis in thelocal convergence case. Some numerical examples are provided to show theperformance of the method presented in this contribution over some existingschemes.

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