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The equivalence between Mann-Ishikawa iterations and multistep iteration. (English) Zbl 1064.47070
In this interesting paper, the authors consider the equivalence between the one-step, two-step, three-step and multistep-iteration process for solving the nonlinear operator equations Tu=0 in a Banach space for pseudocontractive operators T. It is worth mentioning that three-step iterative schemes were introduced by M. A. Noor [J. Math. Anal. Appl. 251, 217–229 (2000; Zbl 0964.49007)]. Three-step iterations are usually called Noor iterations. The present authors also discuss the stability problems for these iterations. An open problem is also mentioned. Is there a map for which, namely: Noor iteration converges to a fixed point, but for which the Ishikawa iteration fails to converge?

47J25Iterative procedures (nonlinear operator equations)
47H10Fixed point theorems for nonlinear operators on topological linear spaces