Abstract
We provide a local convergence analysis for a unified family of methods in order to approximate a locally unique solution of a nonlinear equation in a Banach space setting. These methods have fourth-order convergence, when specialized in the finite dimensional Euclidean space. In contrast to the earlier studies using hypotheses up to the fifth Frechet-derivative, we only use hypotheses on the first Frechet-derivative and Lipschitz constants. The applicability of these methods is expanded this way. Numerical examples are presented to show that we can solve equations in cases not possible with earlier approaches.
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