In A Kantorovich–type analysis for a fast iterative method for solving nonlinear
equations, and Convergence and applications of Newton–type iterations, Argyros introduced a new derivative–free quadratically convergent method
for solving a nonlinear equation in Banach space. In this paper, we extend this method to generalized
equations in order to approximate a locally unique solution. The method uses only divided differences
operators of order one. Under some Lipschitz–type conditions on the first and second order divided
differences operators and Lipschitz–like property of set–valued maps, an existence–convergence theorem
and a radius of convergence are obtained. Our method has the following advantages: we extend
the applicability of this method than all the previous ones, and we do not need to evaluate
any Fréchet derivative. We provide also an improvement on the radius of convergence for our
algorithm, under some center–condition and less computational cost. Numerical examples are also
provided.