On a Steffensen-like method for solving nonlinear equations

We study a generalization of Steffensen’s method in Banach spaces. Our main aim is to obtain similar convergence as Newton’s method, but without evaluating the first derivative of the operator involved. As motivation, we analyse numerical solutions of boundary-value problems approximated by the multiple shooting method that uses the proposed iterative scheme. Sufficient conditions for the semilocal convergence analysis of the method, including error estimates and the \(R\)-order of convergence, are provided. Finally, the theoretical results are applied to a nonlinear system of equations related with the approximation of a Hammerstein-type integral equation.