DOI QR코드

DOI QR Code

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.

Keywords

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

Cited by

  1. Iterative approximation for split equality fixed point problem for family of multivalued mappings vol.109, pp.2, 2015, https://doi.org/10.1007/s13398-014-0207-1
  2. Krasnoselskii-type algorithm for fixed points of multi-valued strictly pseudo-contractive mappings vol.2013, pp.1, 2013, https://doi.org/10.1186/1687-1812-2013-58
  3. Fixed point theorems for hybrid multivalued mappings in Hilbert spaces vol.18, pp.3, 2016, https://doi.org/10.1007/s11784-016-0302-3
  4. On Mann-type iteration method for a family of hemicontractive mappings in Hilbert spaces vol.2013, pp.1, 2013, https://doi.org/10.1186/1029-242X-2013-41
  5. Convergence theorems for generalized nonexpansive multivalued mappings in hyperbolic spaces vol.5, pp.1, 2016, https://doi.org/10.1186/s40064-016-2557-y
  6. Strong convergence theorems by Halpern-Mann iterations for multi-valued relatively nonexpansive mappings in Banach spaces with applications vol.2012, pp.1, 2012, https://doi.org/10.1186/1029-242X-2012-73
  7. A NEW ITERATIVE SCHEME FOR MULTIVALUED MAPPINGS IN CAT(0) SPACES vol.06, pp.04, 2013, https://doi.org/10.1142/S1793557113500538
  8. A new iteration scheme for a hybrid pair of generalized nonexpansive mappings vol.2014, pp.1, 2014, https://doi.org/10.1186/1687-1812-2014-205
  9. Convergence Theorems for Fixed Points of Multivalued Strictly Pseudocontractive Mappings in Hilbert Spaces vol.2013, 2013, https://doi.org/10.1155/2013/629468
  10. Convergence theorems for new classes of multivalued hemicontractive-type mappings vol.2014, pp.1, 2014, https://doi.org/10.1186/1687-1812-2014-93
  11. Viscosity Approximation Methods for Multivalued Nonexpansive Mappings vol.13, pp.5, 2016, https://doi.org/10.1007/s00009-015-0644-x
  12. On solving split equilibrium problems and fixed point problems of nonspreading multi-valued mappings in Hilbert spaces vol.2016, pp.1, 2016, https://doi.org/10.1186/s13663-016-0509-4
  13. On an Iterative Process for Generalized Nonexpansive Multi-valued Mappings in Banach Spaces vol.44, pp.4, 2016, https://doi.org/10.1007/s10013-016-0194-y
  14. Fixed points of multivalued quasi-nonexpansive mappings using a faster iterative process vol.30, pp.7, 2014, https://doi.org/10.1007/s10114-014-3590-9
  15. Δ-convergence theorems for multi-valued nonexpansive mappings in hyperbolic spaces vol.249, 2014, https://doi.org/10.1016/j.amc.2014.10.076
  16. On approximation of fixed points of multivalued pseudocontractive mappings in Hilbert spaces vol.2016, pp.1, 2016, https://doi.org/10.1186/s13663-016-0548-x
  17. An implicit iteration process for solving a fixed point problem of a finite family of multi-valued mappings in Banach spaces vol.25, pp.11, 2012, https://doi.org/10.1016/j.aml.2012.01.032
  18. Mann and Ishikawa-Type Iterative Schemes for Approximating Fixed Points of Multi-valued Non-Self Mappings vol.13, pp.6, 2016, https://doi.org/10.1007/s00009-016-0750-4
  19. Demiclosedness principle and approximation theorems for certain classes of multivalued mappings in Hilbert spaces vol.2013, pp.1, 2013, https://doi.org/10.1186/1687-1812-2013-61
  20. On the Ishikawa iteration processes for multivalued mappings in some CAT(κ) spaces vol.2014, pp.1, 2014, https://doi.org/10.1186/1687-1812-2014-1
  21. Mixed type iterations for multivalued nonexpansive mappings in hyperbolic spaces vol.2014, pp.1, 2014, https://doi.org/10.1186/1687-1812-2014-140
  22. Common fixed points of two multivalued nonexpansive maps in Kohlenbach hyperbolic spaces vol.2014, pp.1, 2014, https://doi.org/10.1186/1687-1812-2014-181
  23. A NEW ITERATION SCHEME FOR A HYBRID PAIR OF NONEXPANSIVE MAPPINGS vol.38, pp.1, 2016, https://doi.org/10.5831/HMJ.2016.38.1.127
  24. Some convergence results for multivalued quasi-nonexpansive mappings in CAT ( κ ) spaces vol.2015, pp.1, 2015, https://doi.org/10.1186/s13663-014-0251-8
  25. On split inclusion problem and fixed point problem for multi-valued mappings 2018, https://doi.org/10.1007/s40314-017-0426-0
  26. The split common fixed point problem for multivalued demicontractive mappings and its applications pp.1579-1505, 2018, https://doi.org/10.1007/s13398-018-0496-x
  27. A New S-Type Iteration Scheme for Generalized Nonexpansive Mappings pp.2250-1762, 2018, https://doi.org/10.1007/s40010-017-0420-3
  28. Split Null Point Problems and Fixed Point Problems for Demicontractive Multivalued Mappings vol.15, pp.5, 2018, https://doi.org/10.1007/s00009-018-1251-4
  29. On the convergence of a new iterative algorithm of three infinite families of generalized nonexpansive multi-valued mappings vol.128, pp.4, 2018, https://doi.org/10.1007/s12044-018-0424-1
  30. A modified inertial shrinking projection method for solving inclusion problems and quasi-nonexpansive multivalued mappings pp.1807-0302, 2018, https://doi.org/10.1007/s40314-018-0661-z
  31. Fixed points of multivalued nonexpansive mappings in Banach spaces vol.2012, pp.1, 2012, https://doi.org/10.1186/1687-1812-2012-73