Abstract
We show that the Mann and Ishikawa iterations are equivalently used to approximate fixed points of generalized contractions.
Open Access
The equivalence between the Mann and Ishikawa iterations dealing with generalized contractions
Received25 Jan 2006
Revised10 May 2006
Accepted11 May 2006
Published03 Jul 2006
References
S. Ishikawa, “Fixed points by a new iteration method,” Proceedings of the American Mathematical Society, vol. 44, no. 1, pp. 147–150, 1974.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNetW. R. Mann, “Mean value methods in iteration,” Proceedings of the American Mathematical Society, vol. 4, no. 3, pp. 506–510, 1953.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNetB. E. Rhoades, “Convergence of an Ishikawa-type iteration scheme for a generalized contraction,” Journal of Mathematical Analysis and Applications, vol. 185, no. 2, pp. 350–355, 1994.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNetB. E. Rhoades and Ş. M. Şoltuz, “On the equivalence of Mann and Ishikawa iteration methods,” International Journal of Mathematics and Mathematical Sciences, vol. 2003, no. 7, pp. 451–459, 2003.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNetB. E. Rhoades and Ş. M. Şoltuz, “The equivalence between the convergences of Ishikawa and Mann iterations for an asymptotically pseudocontractive map,” Journal of Mathematical Analysis and Applications, vol. 283, no. 2, pp. 681–688, 2003.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNetB. E. Rhoades and Ş. M. Şoltuz, “The equivalence of Mann iteration and Ishikawa iteration for non-Lipschitzian operators,” International Journal of Mathematics and Mathematical Sciences, vol. 2003, no. 42, pp. 2645–2651, 2003.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNetB. E. Rhoades and Ş. M. Şoltuz, “The equivalence between the convergences of Ishikawa and Mann iterations for an asymptotically nonexpansive in the intermediate sense and strongly successively pseudocontractive maps,” Journal of Mathematical Analysis and Applications, vol. 289, no. 1, pp. 266–278, 2004.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNetB. E. Rhoades and Ş. M. Şoltuz, “The equivalence of Mann iteration and Ishikawa iteration for a Lipschitzian -uniformly pseudocontractive and -uniformly accretive maps,” Tamkang Journal of Mathematics, vol. 35, no. 3, pp. 235–245, 2004.
View at: Google Scholar | Zentralblatt MATH | MathSciNetŞ. M. Şoltuz, “The equivalence of Picard, Mann and Ishikawa iterations dealing with quasi-contractive operators,” Mathematical Communications, vol. 10, no. 1, pp. 81–88, 2005.
View at: Google ScholarX. Weng, “Fixed point iteration for local strictly pseudo-contractive mapping,” Proceedings of the American Mathematical Society, vol. 113, no. 3, pp. 727–731, 1991.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
Copyright
Copyright © 2006 Hindawi Publishing Corporation. This is an open access article distributed under the
Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.