A Fréchet derivative-free cubically convergent method for set-valued maps

Abstract

We introduce a new iterative method in order to approximate a locally unique solution of variational inclusions in Banach spaces. The method uses only divided differences operators of order one. An existenceconvergence theorem and a radius of convergence are given under some conditions on divided difference operator and Lipschitz-like continuity property of set-valued mappings. Our method extends the recent work related to the resolution of nonlinear equation in Argyros (J Math Anal Appl 332:97-108, 2007) and has the following advantages: faster convergence to the solution than all the previous known ones in Argyros and Hilout (Appl Math Comput, 2008 in press), Hilout (J Math Anal Appl 339: 53-761, 200853-761, , Positivity 10:673-700, 2006, and we do not need to evaluate any Fréchet derivative. We provide also an improvement of the ratio of our algorithm under some center-conditions and less computational cost. Numerical examples are also provided.