One specification of Steffensen's method for solving nonlinear operator equations

We consider an analogue of Steffensen's method for solving nonlinear operator equations. The proposed method is a two-step iterative process. We study the convergence of the proposed method, prove the uniqueness of the solution and find the order of convergence. The proposed method uses no derivative operators. The convergence order is greater than that in Newton's method and some generalizations of the method of chords and Aitken–Steffensen's method. The method is applied to some test systems of nonlinear equations and the problem of curves intersection which are defined implicitly as solutions of differential equations. Numerical results are compared with the results obtained by Newton's method, the modified Newton method, and modifications of Wegstein's and Aitken's methods which were proposed by the author in previous works.