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?


47J25 Iterative procedures involving nonlinear operators
47H10 Fixed-point theorems


Zbl 0964.49007
Full Text: DOI


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