On Steffensen’s method on Banach spaces

We present a modification of Steffensen’s method as a predictor–corrector iterative method, so that we can use Steffensen’s method to approximate a solution of a nonlinear equation in Banach spaces from the same starting points from which Newton’s method converges. We study the semilocal convergence of the predictor–corrector method by using the majorant principle. We illustrate the method with an application to a discrete problem.