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WEAK AND STRONG CONVERGENCE THEOREMS FOR THE MODIFIED ISHIKAWA ITERATION FOR TWO HYBRID MULTIVALUED MAPPINGS IN HILBERT SPACES

  • Cholamjiak, Watcharaporn (School of Science University of Phayao) ;
  • Chutibutr, Natchaphan (School of Science University of Phayao) ;
  • Weerakham, Siwanat (School of Science University of Phayao)
  • Received : 2017.02.28
  • Accepted : 2017.11.02
  • Published : 2018.07.31

Abstract

In this paper, we introduce new iterative schemes by using the modified Ishikawa iteration for two hybrid multivalued mappings in a Hilbert space. We then obtain weak convergence theorem under suitable conditions. We use CQ and shrinking projection methods with Ishikawa iteration for obtaining strong convergence theorems. Furthermore, we give examples and numerical results for supporting our main results.

Keywords

weak convergence;strong convergence;common fixed point;hybrid multivalued mapping;Ishikawa iteration

Acknowledgement

Supported by : Thailand Research Fund

References

  1. R. P. Agarwal, D. O'Regan, and D. R. Sahu, Fixed point theory for Lipschitzian-type mappings with applications, Topological Fixed Point Theory and Its Applications, 6, Springer, New York, 2009.
  2. H. H. Bauschke, E. Matouskova, and S. Reich, Projection and proximal point methods: convergence results and counterexamples, Nonlinear Anal. 56 (2004), no. 5, 715-738. https://doi.org/10.1016/j.na.2003.10.010
  3. F. E. Browder, Nonexpansive nonlinear operators in a Banach space, Proc. Nat. Acad. Sci. USA 54 (1965), 1041-1044. https://doi.org/10.1073/pnas.54.4.1041
  4. A. Buangern, A. Aeimrun, and W. Cholamjiak, Iterative methods for a generalized equilibrium problem and a nonexpansive multi-valued mapping, Vietnam J. Math. 45 (2017), no. 3, 477-492. https://doi.org/10.1007/s10013-016-0225-8
  5. P. Cholamjiak and W. Cholamjiak, Fixed point theorems for hybrid multivalued mappings in Hilbert spaces, J. Fixed Point Theory Appl. 18 (2016), no. 3, 673-688. https://doi.org/10.1007/s11784-016-0302-3
  6. K. Goebel and W. A. Kirk, Topics in Metric Fixed Point Theory, Cambridge Studies in Advanced Mathematics, 28, Cambridge University Press, Cambridge, 1990.
  7. K. Goebel and S. Reich, Uniform Convexity, Hyperbolic Geometry, and Nonexpansive mappings, Monographs and Textbooks in Pure and Applied Mathematics, 83, Marcel Dekker, Inc., New York, 1984.
  8. D. Gohde, Zum Prinzip der kontraktiven Abbildung, Math. Nachr. 30 (1965), 251-258. https://doi.org/10.1002/mana.19650300312
  9. S. Iemoto and W. Takahashi, Approximating common fixed points of nonexpansive mappings and nonspreading mappings in a Hilbert space, Nonlinear Anal. 71 (2009), no. 12, e2082-e2089. https://doi.org/10.1016/j.na.2009.03.064
  10. S. Ishikawa, Fixed points by a new iteration method, Proc. Amer. Math. Soc. 44 (1974), 147-150. https://doi.org/10.1090/S0002-9939-1974-0336469-5
  11. W. A. Kirk, A fixed point theorem for mappings which do not increase distances, Amer. Math. Monthly 72 (1965), 1004-1006. https://doi.org/10.2307/2313345
  12. F. Kohsaka and W. Takahashi, Existence and approximation of fixed points of firmly nonexpansive-type mappings in Banach spaces, SIAM J. Optim. 19 (2008), no. 2, 824-835. https://doi.org/10.1137/070688717
  13. F. Kohsaka, Fixed point theorems for a class of nonlinear mappings related to maximal monotone operators in Banach spaces, Arch. Math. (Basel) 91 (2008), no. 2, 166-177. https://doi.org/10.1007/s00013-008-2545-8
  14. W. R. Mann, Mean value methods in iteration, Proc. Amer. Math. Soc. 4 (1953), 506-510. https://doi.org/10.1090/S0002-9939-1953-0054846-3
  15. G. Marino and H.-K. Xu, Weak and strong convergence theorems for strict pseudocontractions in Hilbert spaces, J. Math. Anal. Appl. 329 (2007), no. 1, 336-346. https://doi.org/10.1016/j.jmaa.2006.06.055
  16. C. Martinez-Yanes and H.-K. Xu, Strong convergence of the CQ method for fixed point iteration processes, Nonlinear Anal. 64 (2006), no. 11, 2400-2411. https://doi.org/10.1016/j.na.2005.08.018
  17. K. Nakajo and W. Takahashi, Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups, J. Math. Anal. Appl. 279 (2003), no. 2, 372-379. https://doi.org/10.1016/S0022-247X(02)00458-4
  18. H. Piri, S. Rahrovi, and P. Kumam, Generalization of Khan fixed point theorem, J. Math. Computer Sci. 17 (2017), 76-83. https://doi.org/10.22436/jmcs.017.01.07
  19. S. Reich, Research Problems: The fixed point property for non-expansive mappings, Amer. Math. Monthly 83 (1976), no. 4, 266-268. https://doi.org/10.1080/00029890.1976.11994096
  20. S. Reich, Approximate selections, best approximations, fixed points, and invariant sets, J. Math. Anal. Appl. 62 (1978), no. 1, 104-113. https://doi.org/10.1016/0022-247X(78)90222-6
  21. S. Reich, Weak convergence theorems for nonexpansive mappings in Banach spaces, J. Math. Anal. Appl. 67 (1979), no. 2, 274-276. https://doi.org/10.1016/0022-247X(79)90024-6
  22. S. Reich, Research Problems: The fixed point property for non-expansive mappings. II, Amer. Math. Monthly 87 (1980), no. 4, 292-294. https://doi.org/10.1080/00029890.1980.11995019
  23. S. Suantai, Weak and strong convergence criteria of Noor iterations for asymptotically nonexpansive mappings, J. Math. Anal. Appl. 311 (2005), no. 2, 506-517. https://doi.org/10.1016/j.jmaa.2005.03.002
  24. S. Suantai, P. Cholamjiak, Y. J. Cho, and W. Cholamjiak, On solving split equilibrium problems and fixed point problems of nonspreading multi-valued mappings in Hilbert spaces, Fixed Point Theory Appl. 2016 (2016), Paper No. 35, 16 pp. https://doi.org/10.1186/s13663-016-0501-z
  25. W. Takahashi, Nonlinear Functional Analysis, Yokohama Publishers, Yokohama, 2000.
  26. W. Takahashi, Fixed point theorems for new nonlinear mappings in a Hilbert space, J. Nonlinear Convex Anal. 11 (2010), no. 1, 79-88.
  27. W. Takahashi, Y. Takeuchi, and R. Kubota, Strong convergence theorems by hybrid methods for families of nonexpansive mappings in Hilbert spaces, J. Math. Anal. Appl. 341 (2008), no. 1, 276-286. https://doi.org/10.1016/j.jmaa.2007.09.062