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On the convergence of a new two-step iteration in the class of quasi-contractive operators. (English) Zbl 1236.47081
Summary: In the class of quasi-contractive operators satisfying Zamfirescu’s conditions, the most used fixed point iterative methods, that is, the Krasnoselskij, Mann and Ishikawa iterations, are all known to be convergent to the unique fixed point. In this paper, we consider a new two-step iterative scheme for approximating fixed points of quasi-contractive operators and we show that the Krasnoselskij, Mann, Ishikawa and new two-step iterations are equivalent for quasi-contractive operators satisfying Zamfirescu’s conditions.

MSC:
47J25Iterative procedures (nonlinear operator equations)
47H09Mappings defined by “shrinking” properties