  In this paper, the local convergence analysis of the multi-step seventh order
method is presented for solving nonlinear equations assuming that the
first-order Fr\'echet derivative belongs to the Lipschitz class. The
significance of our work is that it avoids the standard practice of Taylor
expansion thereby, extends the applicability of the scheme by applying the
technique based on the first-order derivative only. Also, this study provides
radii of balls of convergence, the error bounds in terms of distances in
addition to the uniqueness of the solution. Furthermore, generalization of this
analysis satisfying H\"{o}lder continuity condition is provided since it is
more relaxed than Lipschitz continuity condition. We have considered some
numerical examples and computed the radii of the convergence balls.
