Increasing the applicability of Steffensenʹs method

We present an original alternative to the majorant principle of Kantorovich to study the semilocal convergence of Steffensenʹs method when it is applied to solve nonlinear systems which are differentiable. This alternative allows choosing starting points from which the convergence of Steffensenʹs method is guaranteed, but it is not from the majorant principle. Moreover, this study extends the applicability of Steffensenʹs method to the solution of nonlinear systems which are nondifferentiable and improves a previous result given by the authors.