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Parallel and sequential hybrid methods for a finite family of asymptotically quasi \(\phi \)-nonexpansive mappings

  • Pham Ky Anh
  • Dang Van Hieu
Original Research

Abstract

In this paper we study some novel parallel and sequential hybrid methods for finding a common fixed point of a finite family of asymptotically quasi \(\phi \)-nonexpansive mappings. The results presented here modify and extend some previous results obtained by several authors.

Keywords

Asymptotically quasi-\(\phi \)-nonexpansive mapping Common fixed point Hybrid method Parallel and sequential computation 

Mathematics Subject Classification

47H09 47H10 47J25 65J15 65Y05 

Notes

Acknowledgments

The authors are greateful to the referees for their useful comments to improve this article. We thank V. T. Dzung for performing computation on the LINUX cluster 1350. The research of the first author was partially supported by Vietnam Institute for Advanced Study in Mathematics (VIASM) and Vietnam National Foundation for Science and Technology Development (NAFOSTED).

Supplementary material

12190_2014_801_MOESM1_ESM.rar (331 kb)
Supplementary material 1 (rar 331 KB)

References

  1. 1.
    Alber, Ya.I.: Metric and generalized projection operators in Banach spaces: properties and applications. In: Kartosator, A.G. (ed.) Theory and Applications of Nonlinear Operators of Accretive and Monotone Type, vol. 178 of Lecture Notes in Pure and Applied Mathematics, pp. 15–50. Dekker, New York (1996)Google Scholar
  2. 2.
    Alber, Y.I., Ryazantseva, I.: Nonlinear Ill-Posed Problems of Monotone Type. Spinger, Dordrecht (2006)zbMATHGoogle Scholar
  3. 3.
    Anh, P.K., Chung, C.V.: Parallel hybrid methods for a finite family of relatively nonexpansive mappings. Numer. Funct. Anal. Optim. 35(6), 649–664 (2014)Google Scholar
  4. 4.
    Cioranescu, I.: Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems, vol. 62 of Mathematics and Its Applications. Kluwer, Dordrecht (1990)CrossRefGoogle Scholar
  5. 5.
    Chang, S.S., Kim, J.K., Wang, X.R.: Modified block iterative algorithm for solving convex feasibility problems in Banach spaces. J. Inequal. Appl. 2010, 869684 (2010). doi: 10.1155/2010/869684
  6. 6.
    Cho, Y.J., Qin, X., Kang, S.M.: Strong convergence of the modified Halpern-type iterative algorithms in B anach spaces. An. Stiint. Univ. Ovidius Constanta Ser. Mat. 17, 51–68 (2009)zbMATHMathSciNetGoogle Scholar
  7. 7.
    Chang, Q.W., Yan, H.: Strong convergence of a modified Halpern-type iteration for asymptotically quasi-\(\phi \)-nonexpansive mappings. An. Univ. Ovidius Constanta Ser. Mat. 21(1), 261–276 (2013). doi:  10.2478/auom-2013-0017 Google Scholar
  8. 8.
    Diestel, J.: Geometry of Banach Spaces—Selected Topics. Lecture Notes in Mathematics, p. 485. Springer, Berlin (1975)Google Scholar
  9. 9.
    Deng, W.Q.: Relaxed Halpern-type iteration method for countable families of totally quasi -\(\phi \)-asymptotically nonexpansive mappings. J. Inequal. Appl. 2013, 367 (2013). doi: 10.1186/1029-242X-2013-367
  10. 10.
    Deng, W.Q.: Strong convergence to common fixed points of a countable family of asymptotically strictly quasi-pseudocontractions. Math. Probl. Eng. 2013, Article ID 752625 (2013). doi: 10.1155/2013/752625
  11. 11.
    Deng, W.Q., Bai, P.: An implicit iteration process for common fixed points of two infinite families of asymptotically nonexpansive mappings in Banach spaces. J. Appl. Math. 2013 Article ID 602582 (2013)Google Scholar
  12. 12.
    Huang, N.J., Lan, H.Y., Kim, J.K.: A new iterative approximation of fixed points for asymptotically contractive type mappings in Banach spaces. Indian J. Pure Appl. Math. 35(4), 441–453 (2004)zbMATHMathSciNetGoogle Scholar
  13. 13.
    Kang, J., Su, Y., Zhang, X.: Hybrid algorithm for fixed points of weak relatively nonexpansive mappings and applications. Nonlinear Anal. Hybrid Syst. 4, 755–765 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Kim, J.K., Kim, C.H.: Convergence theorems of iterative schemes for a finite family of asymptotically quasi- nonexpansive type mappings in metric spaces. J. Comput. Anal. Appl. 14(6), 1084–1095 (2012)zbMATHMathSciNetGoogle Scholar
  15. 15.
    Kimura, Y., Takahashi, W.: On a hybrid method for a family of relatively nonexpansive mappings in a Banach space. J. Math. Anal. Appl. 357, 356–363 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Kim, T.H., Takahashi, W.: Strong convergence of modified iteration processes for relatively asymptotically nonexpansive mappings. Taiwanese J. Math. 14(6), 2163–2180 (2010)zbMATHMathSciNetGoogle Scholar
  17. 17.
    Kim, T.H., Xu, H.K.: Strong convergence of modified Mann iterations for asymptotically mappings and semigroups. Nonlinear Anal. 64, 1140–1152 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Kim, J.K.: Strong convergence theorems by hybrid projection methods for equilibrium problems and fixed point problems of the asymptotically quasi-\(\phi \)-nonexpansive mappings. Fixed Point Theory Appl. 2011, 10 (2011)Google Scholar
  19. 19.
    Li, Y., Liu, H.B.: Strong convergence theorems for modifying Halpern-Mann iterations for a quasi-\(\phi \)-asymptotically nonexpansive multi-valued mapping in Banach spaces. Appl. Math. Comput. 218, 6489–6497 (2012)CrossRefMathSciNetGoogle Scholar
  20. 20.
    Liu, X.F.: Strong convergence theorems for a finite family of relatively nonexpansive mappings. Vietnam J. Math. 39(1), 63–69 (2011)zbMATHMathSciNetGoogle Scholar
  21. 21.
    Matsushita, S., Takahashi, W.: A strong convergence theorem for relatively nonexpansive mappings in a Banach space. J. Approx. Theory. 134, 257–266 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  22. 22.
    Plubtieng, S., Ungchittrakool, K.: Approximation of common fixed points for a countable family of relatively nonexpansive mappings in a Banach space and applications. Nonlinear Anal. 72, 2896–2908 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    Qin, X., Cho, Y.J., Kang, S.M., Zhou, H.: Convergence of a modified Halpern-type iteration algorithm for quasi-\(\phi \)-nonexpansive mappings. Appl. Math. Lett. 22, 1051–1055 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  24. 24.
    Reich, S., Sabach, S.: Two strong convergence theorems for a proximal method in reflexive Banach spaces. Numer. Funct. Anal. Optim. 31, 22–44 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  25. 25.
    Reich, S., Sabach, S.: Two strong convergence theorems for Bregman strongly nonexpansive operators in reflexive Banach spaces. Nonlinear Anal. 73, 122–135 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  26. 26.
    Su, Y., Li, M., Zhang, H.: New monotone hybrid algorithm for hemi-relatively nonexpansive mappings and maximal monotone operators. Appl. Math. Comput. 217(12), 5458–5465 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  27. 27.
    Su, Y.F., Wang, Z.M., Xu, H.K.: Strong convergence theorems for a common fixed point of two hemi-relatively nonexpansive mappings. Nonlinear Anal. 71, 5616–5628 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  28. 28.
    Tang, J.F., Chang, S.S., Liu, M., Liu, J.A.: Strong convergence theorem of a hybrid projection algorithm for a family of quasi-\(\phi \)-asymptotically nonexpansive mappings. Opuscula Math. 30(3), 341–348 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  29. 29.
    Takahashi, W., Takeuchi, Y., Kubota, R.: Strong convergence theorems by hybrid methods for families of nonexpansive mappings in Hilbert spaces. J. Math. Anal. Appl. 341, 276–286 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  30. 30.
    Takahashi, W., Zembayashi, K.: Strong convergence theorem by a new hybrid method for equilibrium p roblems and relatively nonexpansive mappings. Fixed Point Theory Appl. 2008, (2008); Article ID 528476. doi: 10.1155/2008/528476
  31. 31.
    Wang, Z.M., Kumam, P.: Hybrid projection algorithm for two countable families of hemirelatively nonexpansive mappings and applications. J. Appl. Math. 2013 (2013); Article ID 524795. doi: 10.1155/2013/524795
  32. 32.
    Wang, Y., Xuan, W.: Convergence theorems for common fixed points of a finite family of relatively nonexpansive mappings in banach spaces. Abstr. Appl. Anal. 2013 (2013); Article ID 259470. doi: 10.1155/2013/259470
  33. 33.
    Zhao, L., Chang, S., Kim, J.K.: Mixed type iteration for total asymptotically nonexpansive mappings in hyperbolic spaces. Fixed Point Theory Appl. 2013, 353 (2013). doi: 10.1186/1687-1812-2013-353
  34. 34.
    Zhou, H., Gao, X.: A strong convergence theorem for a family of quasi-\(\phi \)-nonexpansive mappings in a banach space. Fixed Point Theory Appl. 2009 (2009); Article ID 351265. doi: 10.1155/2009/351265

Copyright information

© Korean Society for Computational and Applied Mathematics 2014

Authors and Affiliations

  1. 1.Department of MathematicsVietnam National UniversityThanh Xuan, HanoiVietnam

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