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Zbl 1064.47070
The equivalence between Mann-Ishikawa iterations and multistep iteration.
(English)
[J] Nonlinear Anal., Theory Methods Appl. 58, No. 1-2, A, 219-228 (2004). ISSN 0362-546X

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 {\it 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?
MSC 2000:
*47J25 Methods for solving nonlinear operator equations (general)
47H10 Fixed point theorems for nonlinear operators on topol.linear spaces

Keywords: Noor iteration; Mann iteration; Ishikawa iteration; strongly pseudocontractive map; strongly accretive map

Citations: Zbl 0964.49007

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