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We study the semi-local convergence of a three-step Newton-type method for solving nonlinear equations under the classical Lipschitz conditions for first-order derivatives. To develop a convergence analysis, we use the approach of restricted convergence regions in combination with majorizing scalar sequences and our technique of recurrent functions. Finally, a numerical example is given.

On the Semi-Local Convergence of a Fifth-Order Convergent Method for Solving Equations