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Acta Mathematica Sinica, English Series

, Volume 32, Issue 11, pp 1357–1376 | Cite as

Further investigation into split common fixed point problem for demicontractive operators

  • Yekini ShehuEmail author
  • Oluwatosin T. Mewomo
Article

Abstract

Our contribution in this paper is to propose an iterative algorithm which does not require prior knowledge of operator norm and prove strong convergence theorem for approximating a solution of split common fixed point problem of demicontractive mappings in a real Hilbert space. So many authors have used algorithms involving the operator norm for solving split common fixed point problem, but as widely known the computation of these algorithms may be difficult and for this reason, authors have recently started constructing iterative algorithms with a way of selecting the step-sizes such that the implementation of the algorithm does not require the calculation or estimation of the operator norm. We introduce a new algorithm for solving the split common fixed point problem for demicontractive mappings with a way of selecting the step-sizes such that the implementation of the algorithm does not require the calculation or estimation of the operator norm and then prove strong convergence of the sequence in real Hilbert spaces. Finally, we give some applications of our result and numerical example at the end of the paper.

Keywords

Demicontractive mappings split common fixed point problems iterative scheme strong convergence Hilbert spaces 

MR(2010) Subject Classification

47H06 47H09 47J05 47J25 

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References

  1. [1]
    Aoyama, K., Kohsaka, F.: Viscosity approximation process for a sequence of quasinonexpansive mappings. Fixed Point Theory Appl., 2014:17, 11pp. (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    Bauschke, H. H., Combettes, P. L.: A weak-to-strong convergence principle for Fejér-monotone methods in Hilbert spaces. Math. Oper. Res., 26 2, 248–264 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    Bréziz, H.: Operateur maximaux monotones, in mathematics studies, vol. 5, North-Holland, Amsterdam, The Netherlands, 1973Google Scholar
  4. [4]
    Byrne, C.: Iterative oblique projection onto convex sets and the split feasibility problem. Inverse Problems, 18 2, 441–453 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    Byrne, C.: A unified treatment of some iterative algorithms in signal processing and image reconstruction. Inverse Problems, 20 1, 103–120 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    Byrne, C., Censor, Y., Gibali, A., et al.: The Split Common Null Point Problem. J. Nonlinear Convex Anal., 13 4, 759–775 (2012)MathSciNetzbMATHGoogle Scholar
  7. [7]
    Cegielski, A.: Iterative Methods for Fixed Point Problems in Hilbert Spaces. Lecture Notes in Mathematics, 2057, Springer, Heidelberg, 2012zbMATHGoogle Scholar
  8. [8]
    Cegielski, A., Zalas, R.: Methods for variational inequality problem over the intersection of fixed point sets of quasi-nonexpansive operators. Numer. Funct. Anal. Optim., 34, 255–283 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    Censor, Y., Elfving, T.: A multiprojection algorithm using Bregman projections in a product space. Numerical Algorithms, 8(2–4), 221–239 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    Censor, Y., Gibali, A., Reich, S.: Algorithms for the split variational inequality problem. Numerical Algorithms, 59, 301–323 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    Censor, Y., Segal, A.: The split common fixed point problem for directed operators. J. Convex Anal., 16 2, 587–600 (2009)MathSciNetzbMATHGoogle Scholar
  12. [12]
    Chang, S. S., Joseph Lee, H. W., Chan, C. K., et al.: Split feasibility problem for quasi-nonexpansive multi-valued mappings and total asymptotically strict pseudo-contractive mapping. Appl. Math. Comput., 219, 10416–10424 (2013)MathSciNetzbMATHGoogle Scholar
  13. [13]
    Chidume, C. E., Maruster, S.: Iterative methods for the computation of fixed points of demicontractive mappings. J. Comput. Appl. Math., 234, 861–882 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  14. [14]
    Cui, H., Su, M., Wang, F.: Damped projectionmethod for split common fixed point problems. J. Inequal. Appl., 2013, 2013:123MathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    Hicks, T. L., Kubicek, J. D.: On the Mann Iteration Process in a Hilbert Space. J. Math. Anal. Appl., 59, 498–504 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  16. [16]
    Hirstoaga, S. A.: Iterative selection methods for common fixed point problems. J. Math. Anal. Appl., 324, 1020–1035 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  17. [17]
    Lemaire, B.: Which fixed point does the iteration method select? In: Recent Advances in Optimization, vol. 452, Springer, Berlin, Germany, 1997, 154–157CrossRefGoogle Scholar
  18. [18]
    Lopez, G., Martin-Marquez, V., Wang, F., et al.: Solving the split feasibility problem without prior knowledge of matrix norms. Inverse Probl., 28(8), 085004, 18pp (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  19. [19]
    Maingé, P. E.: The viscosity approximation process for quasi-nonexpansive mappings in Hilbert spaces. Comput. Math. Appl., 59 1, 74–79 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  20. [20]
    Maingé, P. E.: Strong convergence of projected subgradient methods for nonsmooth and nonstrictly convex minimization. Set-Valued Anal., 16, 899–912 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  21. [21]
    Maruster, S., Popirlan, C.: On the Mann-type iteration and the convex feasibility problem. J. Comput. Appl. Math., 212(2), 390–396 (2008MathSciNetCrossRefzbMATHGoogle Scholar
  22. [22]
    Moudafi, A.: A note on the split common fixed-point problem for quasi-nonexpansive operators. Nonlinear Anal., 74 12, 4083–4087 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  23. [23]
    Moudafi, A.: The split common fixed-point problem for demicontractive mappings. Inverse Problems, 26, 587–600 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  24. [24]
    Moudafi, A.: Viscosity-type algorithms for the split common fixed-point problem. Adv. Nonlinear Var. Ineq., 16, 61–68 (2013)MathSciNetzbMATHGoogle Scholar
  25. [25]
    Qu, B., Xiu, N.: A note on the CQ algorithm for the split feasibility problem. Inverse Problems, 21 5, 1655–1665 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  26. [26]
    Shehu, Y., Cholamjiak, P.: Another look at the split common fixed point problem for demicontractive operators. Rev. R. Acad. Cienc. Exactas Fs. Nat. Ser. A Math. RACSAM, 110 1, 201–218 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  27. [27]
    Tang, Y.-C., Peng, J.-G., Liu, L.-W.: A cyclic algorithm for the split common fixed point problem of demicontractive mappings in Hilbert spaces. Math. Model. Anal., 17, 457–466 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  28. [28]
    Wang, F., Cui, H.: Convergence of a cyclic algorithm for the split common fixed point problem without continuity assumption. Math. Model. Anal., 18, 537–542 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  29. [29]
    Xu, H.-K.: Iterative methods for the split feasibility problem in infinite-dimensional Hilbert spaces. Inverse Problems, 26(10), Article ID 105018, (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  30. [30]
    Xu, H.-K.: A variable Krasnoselskii-Mann algorithm and the multiple-set split feasibility problem. Inverse Problems, 22 6, 2021–2034 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  31. [31]
    Xu, H.-K.: Iterative algorithm for nonlinear operators. J. London Math. Soc., 66 2, 1–17 (2002)MathSciNetGoogle Scholar
  32. [32]
    Yamada, I., Ogura, N.: Hybrid steepest descent method for variational inequality problem over the fixed point set of certain quasi-nonexpansive mapping. Numer. Funct. Anal. Optimiz., 25, 619–655 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  33. [33]
    Yang, Q.: The relaxed CQ algorithm solving the split feasibility problem. Inverse Problems, 20 4, 1261–1266 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  34. [34]
    Yang, Q., Zhao, J.: Generalized KM theorems and their applications. Inverse Problems, 22 3, 833–844 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  35. [35]
    Yao, Y., Chen, R., Liou, Y.-C.: A unified implicit algorithm for solving the triple-hierarchical constrained optimization problem. Math. Comput. Model., 55(3–4), 1506–1515 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  36. [36]
    Yao, Y., Cho, Y.-J., Liou, Y.-C.: Hierarchical convergence of an implicit doublenet algorithm for nonexpansive semigroups and variational inequalities. Fixed Point Theory Appl., vol. 2011, article 101 (2011)CrossRefzbMATHGoogle Scholar
  37. [37]
    Yao, Y., Cho, Y.-J.: A strong convergence of a modified krasnoselskii-mann method for non-expansive mappings in hilbert spaces. Math. Model. Anal., 15 2, 265–274 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  38. [38]
    Yao, Y., Jigang, W., Liou, Y.-C.: Regularized methods for the split feasibility problem. Abstr. Appl Anal., vol. 2012, Article ID 140679, 13 pages (2012)MathSciNetzbMATHGoogle Scholar
  39. [39]
    Yao, Y., Liou, Y.-C., Kang, S. M.: Two-step projection methods for a system of variational inequality problems in Banach spaces. J. Global Optim., 55 4, 801–811 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  40. [40]
    Zhao, J., He, S.: Strong convergence of the viscosity approximation process for the split common fixed-point problem of quasi-nonexpansive mappings. J. Appl. Math., Vol. 2012, Article ID 438023, 12 pages (2012)MathSciNetzbMATHGoogle Scholar
  41. [41]
    Zhao, J., Yang, Q.: Several solution methods for the split feasibility problem. Inverse Problems, 21 5, 1791–1799 (2005)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Chinese Mathematical Society and Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of NigeriaNsukkaNigeria
  2. 2.School of Mathematics, Statistics and Computer ScienceUniversity of Kwazulu NatalDurbanSouth Africa

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