Open Access Research Article

Comparison of fastness of the convergence among Krasnoselskij, Mann, and Ishikawa iterations in arbitrary real Banach spaces

GVR Babu1* and KNVV Vara Prasad2

Author Affiliations

1 Department of Mathematics, Andhra University, Visakhapatnam 530 003, India

2 Department of Mathematics, Dr. L. Bullayya College, Visakhapatnam 530 013, India

For all author emails, please log on.

Fixed Point Theory and Applications 2006, 2006:35704 doi:10.1155/FPTA/2006/35704


The electronic version of this article is the complete one and can be found online at: http://www.fixedpointtheoryandapplications.com/content/2006/1/35704


Received:25 April 2006
Accepted:4 September 2006
Published:25 January 2007

© 2006 Babu and Vara Prasad

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.

Let be an arbitrary real Banach space and a nonempty, closed, convex (not necessarily bounded) subset of . If is a member of the class of Lipschitz, strongly pseudocontractive maps with Lipschitz constant , then it is shown that to each Mann iteration there is a Krasnosleskij iteration which converges faster than the Mann iteration. It is also shown that the Mann iteration converges faster than the Ishikawa iteration to the fixed point of .

References

  1. Babu, GVR, Vara Prasad, KNVV: Mann iteration converges faster than Ishikawa iteration for the class of Zamfirescu operators. Fixed Point Theory and Applications. 2006, 6 pages (2006)

  2. Berinde, V: Approximating fixed points of Lipschitzian generalized pseudo-contractions. Mathematics & Mathematics Education (Bethlehem, 2000), pp. 73–81. World Scientific, New Jersey (2002).

  3. Berinde, V: Iterative Approximation of Fixed Points,p. xii+283. Editura Efemeride, Baia Mare (2002)

  4. Berinde, V: Comparing Krasnoselskij and Mann iterative methods for Lipschitzian generalized pseudo-contractions. Proceedings of International Conference on Fixed Point Theory and Applications. Valencia(Spain), 2003, pp. 15–26. Yokohama Publishers, Yokohama (2004)

  5. Berinde, V: Picard iteration converges faster than Mann iteration for a class of quasi-contractive operators. Fixed Point Theory and Applications. 2004(2), 97–105 (2004). Publisher Full Text OpenURL

  6. Bogin, J: On strict pseudo-contractions and fixed point theorems. Technion preprint, series no: MT-219, Haifa, Israel, 1974

  7. Chidume, CE, Osilike, MO: Nonlinear accretive and pseudo-contractive operator equations in Banach spaces. Nonlinear Analysis. Theory, Methods & Applications. 31(7), 779–789 (1998). PubMed Abstract | Publisher Full Text OpenURL

  8. Kato, T: Nonlinear semigroups and evolution equations. Journal of the Mathematical Society of Japan. 19, 508–520 (1967). Publisher Full Text OpenURL

  9. Verma, RU: A fixed-point theorem involving Lipschitzian generalised pseudo-contractions. Proceedings of the Royal Irish Academy. Section A. 97(1), 83–86 (1997)