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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?

##### MSC:
 47J25 Iterative procedures involving nonlinear operators 47H10 Fixed-point theorems
Zbl 0964.49007
Full Text:
##### References:
 [1] Chidume, C.E; Mutangadura, S.A, An example on the Mann iteration method for Lipschitz pseudocontractions, Proc. am. math. soc., 129, 2359-2363, (2001) · Zbl 0972.47062 [2] Deimling, K, Zeroes of accretive operators, Manuscripta math., 13, 365-374, (1974) · Zbl 0288.47047 [3] Deimling, K, Nonlinear functional analysis, (1985), Springer Berlin [4] Ishikawa, S, Fixed points by a new iteration method, Proc. am. math. soc., 44, 147-150, (1974) · Zbl 0286.47036 [5] Mann, W.R, Mean value in iteration, Proc. am. math. soc., 4, 506-510, (1953) · Zbl 0050.11603 [6] Morales, C; Jung, J.S, Convergence of paths for pseudocontractive mappings in Banach spaces, Proc. am. math. soc., 128, 3411-3419, (2000) · Zbl 0970.47039 [7] Noor, M.A, New approximation schemes for general variational inequalities, J. math. anal. appl., 251, 217-229, (2000) · Zbl 0964.49007 [8] Noor, M.A; Rassias, T.M; Huang, Z, Three-step iterations for nonlinear accretive operator equations, J. math. anal. appl., 274, 59-68, (2002) · Zbl 1028.65063 [9] Rhoades, B.E; M. Şoltuz, Ştefan, On the equivalence of Mann and Ishikawa iteration methods, Int. J. math. math. sci., 2003, 451-459, (2003) · Zbl 1014.47052 [10] Rhoades, B.E; M. Şoltuz, Ştefan, The equivalence of Mann iteration and Ishikawa iteration for non-Lipschitzian operators, Int. J. math. math. sci., 2003, 2645-2652, (2003) · Zbl 1045.47058 [11] Rhoades, B.E; M. Şoltuz, Ştefan, The equivalence between the convergences of Ishikawa and Mann iterations for asymptotically pseudocontractive map, J. math. anal. appl., 283, 681-688, (2003) · Zbl 1045.47057 [12] B.E. Rhoades, Ştefan M. Şoltuz, The equivalence of Mann and Ishikawa iteration for a Lipschitzian psi-uniformly pseudocontractive and psi-uniformly accretive maps, Tamkang J. Math. 35 (2004), to appear. · Zbl 1078.47052 [13] B.E. Rhoades, Ştefan M. Şoltuz, The equivalence between T-stability of Mann and Ishikawa iterations, submitted for publication. [14] Rhoades, B.E; M. Şoltuz, Ştefan, The equivalence between the convergences of Ishikawa and Mann iterations for asymptotically nonexpansive in the intermediate sense and strongly successively pseudocontractive maps, J. math. anal. appl., 289, 266-278, (2004) · Zbl 1053.47055 [15] Weng, X, Fixed point iteration for local strictly pseudocontractive mapping, Proc. am. math. soc., 113, 727-731, (1991) · Zbl 0734.47042
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