The equivalence between the Krasnoselskij, Mann and Ishikawa iterations

Authors

  • B. E. Rhoades Indiana University, USA
  • Ştefan M. Şoltuz Tiberiu Popoviciu, Institute of Numerical Analysis, Romanian Academy, Romania

DOI:

https://doi.org/10.33993/jnaat352-846

Keywords:

Krasnoselskij, Mann and Ishikawa iterations, strongly pseudocontractive map
Abstract views: 240

Abstract

We shall prove that Krasnoselskij iteration converges if and only if Mann-Ishikawa iteration converges, for certain classes of strongly pseudocontractive mappings.

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References

Berinde,V.,Berinde, M.,The fastest Krasnoselskij iteration for approximating fixed points of strictly pseudo-contractive mappings, Carpatian J. Math., 21, pp. 13-20, 2005.

Chang, S.S., Cho, Y.J., Lee, B.S., Jung, J.S., Kang, S. M., Iterative approximations of fixed points and solutions for strongly accretive and strongly pseudo-contractive mappings in Banach spaces, J. Math. Anal. Appl., 224, pp. 149-165, 1998,https://doi.org/10.1006/jmaa.1998.5993 DOI: https://doi.org/10.1006/jmaa.1998.5993

Deimling,K., Nonlinear Functional Analysis, Springer-Verlag, 1985, https://doi.org/10.1007/978-3-662-00547-7 DOI: https://doi.org/10.1007/978-3-662-00547-7

Ishikawa, S., Fixed points by a new iteration method, Proc. Amer. Math. Soc., 44, pp. 147-150, 1974, https://doi.org/10.1090/s0002-9939-1974-0336469-5 DOI: https://doi.org/10.1090/S0002-9939-1974-0336469-5

Krasnoselskij, M.A., Two remarks on the method of succesive approximations, Uspehi Mat. Nauk., 10, pp. 123-127, 1955.

Mann, W.E., Mean value in iteration, Proc. Amer. Math. Soc., 4, pp. 506-510, 1953, https://doi.org/10.1090/s0002-9939-1953-0054846-3 DOI: https://doi.org/10.1090/S0002-9939-1953-0054846-3

Park, J.A., Mann-iteration process for the fixed point of strictly pseudocontractive mapping in some Banach spaces, J. Korean Math. Soc., 31, pp. 333-337, 1994.

Rhoades, B.E., Şoltuz, Ştefan M., On the equivalence of Mann and Ishikawa iteration methods, Int. J. Math. Math. Sci., 2003, pp. 451-459, 2003, https://doi.org/10.1155/s0161171203110198 DOI: https://doi.org/10.1155/S0161171203110198

Rhoades, B.E., Şoltuz, Ştefan M., The equivalence of Mann iteration and Ishikawa iteration for non-Lipschitzian operators, Int. J. Math. Math. Sci., 2003, pp. 2645-2652, 2003, https://doi.org/10.1155/s0161171203211418 DOI: https://doi.org/10.1155/S0161171203211418

Weng, X., Fixed point iteration for local strictly pseudocontractive mapping, Proc. Amer. Math. Soc., 113, pp. 727-731, 1991, https://doi.org/10.1090/s0002-9939-1991-1086345-8 DOI: https://doi.org/10.1090/S0002-9939-1991-1086345-8

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Published

2006-08-01

How to Cite

Rhoades, B. E., & Şoltuz, Ştefan M. (2006). The equivalence between the Krasnoselskij, Mann and Ishikawa iterations. Rev. Anal. Numér. Théor. Approx., 35(2), 199–205. https://doi.org/10.33993/jnaat352-846

Issue

Vol. 35 No. 2 (2006)

Section

Articles

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Copyright (c) 2015 Journal of Numerical Analysis and Approximation Theory

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