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NEW HYBRID ALGORITHM FOR WEAK RELATIVELY NONEXPANSIVE MAPPING AND INVERSE-STRONGLY MONOTONE MAPPING IN BANACH SPACE

  • Zhang, Xin (Department of Mathematics, Tianjin Polytechnic University) ;
  • Su, Yongfu (Department of Mathematics, Tianjin Polytechnic University) ;
  • Kang, Jinlong (Department of Mathematics, Tianjin Polytechnic University)
  • Received : 2010.02.21
  • Accepted : 2010.05.31
  • Published : 2011.01.30

Abstract

The purpose of this paper is to prove strong convergence theorems for finding a common element of the set of fixed points of a weak relatively nonexpansive mapping and the set of solutions of the variational inequality for an inverse-strongly-monotone mapping by a new hybrid method in a Banach space. We shall give an example which is weak relatively nonexpansive mapping but not relatively nonexpansive mapping in Banach space $l^2$. Our results improve and extend the corresponding results announced by Ying Liu[Ying Liu, Strong convergence theorem for relatively nonexpansive mapping and inverse-strongly-monotone mapping in a Banach space, Appl. Math. Mech. -Engl. Ed. 30(7)(2009), 925-932] and some others.

Keywords

References

  1. J.L. Lions and G.Stampacchia, Variational inequalities Comm. Pure Appl. Math. 20(3), 493-517 (1967). https://doi.org/10.1002/cpa.3160200302
  2. H. Iiduka, W. Takahashi, Strong convergence studied by a hybrid type method for monotone operators in a Banach space, Nonlinear Analysis 68(12), 3679-3688 (2008). https://doi.org/10.1016/j.na.2007.04.010
  3. H. Iiduka, W. Takahashi, Weak convergence of a projection algorithm for variational inequalities in a Banach space, J. Math. Anal. Appl. 339(1), 668-679 (2008). https://doi.org/10.1016/j.jmaa.2007.07.019
  4. H. Iiduka, W. Takahashi, Strong convergence theorems for nonexpansive mappings and inverse-strongly monotone mappings, Nonlinear Analysis 61(4), 341-350 (2005).
  5. S.Y. Matsushita, W. Takahashi, A strong convergence theorem for relatively nonexpansive mappings in a Banach space, Journal of Approximation Theory 134(2), 257-266 (2005). https://doi.org/10.1016/j.jat.2005.02.007
  6. K. Ball, E.A. Carlen and E.H. Lieb, Sharp uniform convexity and smoothness inequalities for trace norms. Invent. Math, 115(1), 463-482 (1994). https://doi.org/10.1007/BF01231769
  7. Y.I. Alber, S. Reich, An iterative method for solving a class of nonlinear operator equations in Banach spaces, Panamer. Math. J. 4(2), 39-54 (1994).
  8. R.T. Rockafellar, On the maximality of sums of nonlinear monotone operators, Trans. Amer. Math. Soc. 149(1), 75-88 (1970). https://doi.org/10.1090/S0002-9947-1970-0282272-5
  9. Ying Liu, Strong convergence theorem for relatively nonexpansive mapping and inverse-strongly-monotone mapping in a Banach space, Appl. Math. Mech. -Engl. Ed. 30(7)(2009), 925-932. https://doi.org/10.1007/s10483-009-0711-y
  10. D. Butnariu, S. Reich, A.J. Zaslavski, Asymptotic behavior of relatively nonexpansive operators in Banach spaces, J. Appl. Anal. 7(2001)151-174.
  11. Y. Censor, S. Reich, Iterations of paracontractions and firmly nonexpansive operators with applications to feasibility and optimization, Optimization 37 (1996) 323-339. https://doi.org/10.1080/02331939608844225
  12. S.Y. Matsushita, W. Takahashi, A strong convergence theorem for relatively nonexpansive mappings in a Banach space, J. Approx. Theory 134 (2005) 257-266. https://doi.org/10.1016/j.jat.2005.02.007
  13. Y. Su, D. Wang, M. Shang, Strong convergence of monotone hybrid algorithm for hemi-relatively nonexpansive mappings, Fixed Point Theory Appl. (2008) doi:10.1155/2008/284613.
  14. L. Wei, Y. Cho, H. Zhou, A strong convergence theorem for common fixed points of two relatively nonexpansive mappings, J. Appl. Math. Comput. (2008) doi:10.1007/s12190-008-0092-x.
  15. H. Zegeye, N. Shahzad, Strong convergence for monotone mappings and relatively weak nonexpansive mappings, Nonlinear Anal. (2008).
  16. Yongfu. Su, Junyu Gao, Haiyun Zhou, Monotone CQ algorithm of fixed points for weak relatively nonexpansive mappings and applications, Journal of Mathematical Research and Exposition, 28:4 (2008), 957-967.
  17. Habtu. Zegeye, Naseer Shahzad, Strong convergence theorems for monotone mappings and relatively weak nonexpansive mappings, Nonlinear Analysis, 70:7 (2009), 2707-2716. https://doi.org/10.1016/j.na.2008.03.058
  18. Y. Su, D. Wang and M. Shang, Strong Convergence of Monotone Hybrid Algorithm for Hemi-Relatively Nonexpansive Mappings, Fixed Point Theory and Applications Volume 2008, Article ID 284613, 8 pages doi:10.1155/2008/284613
  19. Yongfu. Su, Ziming Wang, Hongkun Xu, Strong convergence theorems for a common fixed point of two hemi-relatively nonexpansive mappings, Nonlinear Analysis, 71 (2009) 5616-5628. https://doi.org/10.1016/j.na.2009.04.053
  20. M.Y. Carlos, H.K. Xu, Strong convergence of the CQ method for fixed point iteration processes, Nonlinear Anal. 64 (2006), 2240-2411.
  21. S.Y. Matsushita, W. Takahashi, A strong convergence theorem for relatively nonexpansive mappings in a Banach space, Journal of Approximation Theory, 134 (2005), 257-266. https://doi.org/10.1016/j.jat.2005.02.007
  22. Ya.I. Alber, Metric and generalized projection operators in Banach spaces: properties and applications, in: A.G. Kartsatos (Ed.) Theory and Applications of Nonlinear Operators of Accretive and Monotone Type, Marcel Dekker, New York, 1996, pp. 15-50.
  23. I. Cioranescu, Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems, Kluwer, Dordrecht, 1990.
  24. W. Takahashi, Nonlinear Functional Analysis, Yokohama-Publishers, 2000.