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In this paper, the local convergence analysis of the multi-step seventh ordermethod is presented for solving nonlinear equations assuming that thefirst-order Fr\'echet derivative belongs to the Lipschitz class. Thesignificance of our work is that it avoids the standard practice of Taylorexpansion thereby, extends the applicability of the scheme by applying thetechnique based on the first-order derivative only. Also, this study providesradii of balls of convergence, the error bounds in terms of distances inaddition to the uniqueness of the solution. Furthermore, generalization of thisanalysis satisfying H\"{o}lder continuity condition is provided since it ismore relaxed than Lipschitz continuity condition. We have considered somenumerical examples and computed the radii of the convergence balls.

Broadening the convergence domain of Seventh-order method satisfying Lipschitz and H\"{o}lder conditions