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Zbl 1045.47058
Rhoades, B. E.; Şoltuz, Ştefan M.
The equivalence of Mann iteration and Ishikawa iteration for non-Lipschitzian operators. (English)
[J] Int. J. Math. Math. Sci. 2003, No. 42, 2645-2651 (2003). ISSN 0161-1712; ISSN 1687-0425/e

It is shown that the convergence of Mann iteration (one-step) is equivalent to the convergence of Ishikawa (two-step) iteration for various classes of non-Lipschitzian operators. Using essentially the technique of this paper, one can prove that the convergence of Mann-Ishikawa iteration is equivalent to the convergence of three-step iterations, which are known as Noor iteraions for non-Lipschitzian operators. Note that Noor iterations [introduced by {\it M. Aslam Noor}, J. Math. Anal. Appl. 251, 217--229 (2000; Zbl 0964.49007)] include Mann-Ishikawa iterations as special cases.
[Muhammad Aslam Noor (Sharjah)]

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

Keywords: Mann and Ishikawa iterations; non-Lipschitz operators; convergence; equivalence between one-step and two-step iterations

Citations: Zbl 0964.49007

Cited in: Zbl 1175.47064 Zbl 1215.47076 Zbl 1045.47057

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