The equivalence between Mann-Ishikawa iterations and multistep iteration. (English) Zbl 1064.47070

Rhoades, B. E.; Soltuz, Stefan M.

Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 58, No. 1-2, 219-228 (2004).

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?

Reviewer: Muhammad Aslam Noor (Islamabad)

MSC:

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

Keywords:

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

Citations:

Zbl 0964.49007

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: Springer Berlin · Zbl 0559.47040
[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
[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