Hilout, Saïd Superlinear convergence of a family of two-step Steffensen-type methods for generalized equations. (English) Zbl 1144.65036 Int. J. Pure Appl. Math. 40, No. 1, 1-10 (2007). The author discusses the approximate solution of a so called “abstract generalized equation”, that means, an equation the right hand side of which consists of a sum of a continuous function and a set-valued map between two Banach spaces. For approximating locally the unique solution of such equation, the two-step Steffensen method (the classical secant method) is suitably generalized. After a preliminaries section, recalling the definitions and the concepts (the pseudo-Lipschitz set-valued map, the divided difference, the Hölder continuous operator, Aubin continuity), the author is concerned with the construction of a family of generalized two-step Steffensen-type algorithms and with the convergence analysis of the algorithm. The main result of this paper is the proof that the algorithm is locally superlinear convergent. Reviewer: Iulian Coroian (Baia Mare) MSC: 65J15 Numerical solutions to equations with nonlinear operators 47H04 Set-valued operators 47J25 Iterative procedures involving nonlinear operators Keywords:Steffensen method; set-valued map; generalized equation; Aubin continuity; divided difference; superlinear convergence PDFBibTeX XMLCite \textit{S. Hilout}, Int. J. Pure Appl. Math. 40, No. 1, 1--10 (2007; Zbl 1144.65036)