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SOME NOTES ON ISHIKAWA ITERATION FOR MULTI-VALUED MAPPINGS

  • Song, Yisheng (College of Mathematics and Information Science Henan Normal University) ;
  • Cho, Yeol-Je (Department of Mathematics Education and the RINS Gyeongsang National University)
  • Received : 2009.09.24
  • Published : 2011.05.31

Abstract

In Shahzad and Zegeye [Nonlinear Anal. 71 (2009), no. 3-4, 838-844], the authors introduced several Ishikawa iterative schemes for xed points of multi-valued mappings in Banach spaces, and proved some strong convergence theorems by using their iterations. In their proofs of the main results, it seems reasonable and simpler to prove for the iteration {$x_n$} to be a Cauchy sequence. In this paper, we modify and improve the proofs of the main results given by Shahzad and Zegeye. Two concrete examples also are given.

References

  1. B. Panyanak, Mann and Ishikawa iterative processes for multivalued mappings in Banach spaces, Comput. Math. Appl. 54 (2007), no. 6, 872-877. https://doi.org/10.1016/j.camwa.2007.03.012
  2. K. P. R. Sastry and G. V. R. Babu, Convergence of Ishikawa iterates for a multi-valued mapping with a fixed point, Czechoslovak Math. J. 55(130) (2005), no. 4, 817-826.
  3. C. Shiau, K. K. Tan, and C. S. Wong, Quasi-nonexpansive multi-valued maps and selections, Fund. Math. 87 (1975), 109-119. https://doi.org/10.4064/fm-87-2-109-119
  4. N. Shahzad and H. Zegeye, On Mann and Ishikawa iteration schemes for multi-valued maps in Banach spaces, Nonlinear Anal. 71 (2009), no. 3-4, 838-844. https://doi.org/10.1016/j.na.2008.10.112
  5. Y. Song and H.Wang, Erratum to \Mann and Ishikawa iterative processes for multivalued mappings in Banach spaces, Comput. Math. Appl. 54 (2007), no. 6, 872-877", Comput. Math. Appl. 55 (2008), no. 12, 2999-3002. https://doi.org/10.1016/j.camwa.2007.03.012
  6. Y. Song and H.Wang , Convergence of iterative algorithms for multivalued mappings in Banach spaces, Nonlinear Anal. 70 (2009), no. 4, 1547-1556. https://doi.org/10.1016/j.na.2008.02.034

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