Convergence of Steffensen’s method for non-differentiable operators

Abstract

We present a semilocal as well as a local convergence analysis of Steffensen's method in order to approximate a locally unique solution of a nonlinear equation in a Banach space setting. The novelty of this paper is twofold. On the one hand, we show convergence under general conditions including earlier ones as special cases. In particular, these conditions allow the important study when the operator involved is not differentiable. On the other hand, we use a combination of conditions that allow a more precise computation of the upper bounds on the norms of the inverses involved leading to a tighter convergence analysis. Finally, we provide numerical examples involving nonlinear integral equations on mixed Hammerstein type that appear in chemistry, vehicular traffic theory, biology, and queuing theory.