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Superlinear convergence of a family of two-step Steffensen-type methods for generalized equations. (English) Zbl 1144.65036

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.

MSC:

65J15 Numerical solutions to equations with nonlinear operators
47H04 Set-valued operators
47J25 Iterative procedures involving nonlinear operators
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