A Steffensen type method of two steps in Banach spaces with applications

This paper is devoted to the analysis of a Steffensen-type of two steps with order of convergence at least three. The main advantage of this method is that it does not need to evaluate any Fréchet derivative or any bilinear operator. The method includes extra parameters in the divided difference in order to ensure a good approximation to the first derivative in each iteration. We prove, using recurrence relations, a semilocal convergence result in Banach spaces and do a detailed study of the domain of parameters associated to this result. Finally, some numerical results, including differentiable and nondifferentiable operators, are presented. Special attention is paid in the approximation of solutions of boundary problems using the multiple shooting method and in the approximation of a nonlinear model related with image processing.