A new approximation method for finding common fixed points of families of demicontractive operators and applications

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Abstract

In this paper, we introduce a new strongly convergent algorithm for solving common fixed point problems in Hilbert spaces for a finite (infinite) family of demicontractive operators. As a consequence, we obtain strong convergence theorems for split common fixed point problems for a finite family of demicontractive operators, split variational inequality problems for inverse strongly monotone operators and split common null point problems for maximal monotone operators. Finally, the performance of the proposed algorithm is also illustrated by several preliminary numerical experiments.

Keywords

Fixed point problem split common fixed point problem split feasibility problem split variational inequality problem split null point problem 

Mathematics Subject Classification

47H10 47J25 47H45 65J15 

Notes

Acknowledgements

The authors would like to thank the referees for their valuable comments and suggestions which helped us very much in improving and presenting the original version of this paper. The authors would like to thank Professor Pham Ky Anh for drawing our attention to the subject and for many useful discussions.

References

  1. 1.
    Aoyama, K., Kimura, Y., Takahashi, W., Toyoda, M.: Approximation of common fixed points of a countable family of nonexpansive mappings in a Banach space. Nonlinear Anal. 67, 2350–2360 (2007)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Bruck, E.R., Reich, S.: Nonexpansive projections and resolvents of accretive operators in Banach spaces. Houston J. Math. 3, 459–470 (1977)MathSciNetMATHGoogle Scholar
  3. 3.
    Boonchari, D., Saejung, S.: Weak and strong convergence theorems of an implicit iteration for a countable family of continuous pseudocontractive mappings. J. Comput. Appl. Math. 233, 1108–1116 (2009)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Boonchari, D., Saejung, S.: Construction of common fixed points of a countable family of \(\lambda \)-demicontractive mappings in arbitrary Banach spaces. Appl. Math. Comput. 216, 173–178 (2010)MathSciNetMATHGoogle Scholar
  5. 5.
    Byrne, C.: Iterative oblique projection onto convex sets and the split feasibility problem. Inverse Prob. 18, 441–453 (2002)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Byrne, C.: A unified treatment of some iterative algorithms in signal processing and image reconstruction. Inverse Prob. 18, 103–120 (2004)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Byrne, C., Censor, Y., Gibali, A., Reich, S.: The split common null point problem. J. Nonlinear Convex Anal. 13, 759–775 (2012)MathSciNetMATHGoogle Scholar
  8. 8.
    Censor, Y., Elfving, T.: A multiprojection algorithm using Bregman projection in a product space. Numer. Algorithms 8, 221–239 (1994)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Censor, Y., Elfving, T., Kopf, N., et al.: The multiple-sets split feasibility problem and its applications for inverse problems. Inverse Prob. 21, 2071–2084 (2005)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Censor, Y., Segal, A.: The split common fixed point problem for directed operators. Convex J. Anal. 16, 587–600 (2009)MathSciNetMATHGoogle Scholar
  11. 11.
    Censor, Y., Gibali, A., Reich, S.: Algorithms for the split variational inequality problem. Numer. Algorithms 59, 301–323 (2012)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Chidume, C.E., Maruster, S.: Iterative methods for the computation of fixed points of demicontractive mappings. J. Comput. Appl. Math. 234, 861–882 (2010)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Cegielski, A.: General method for solving the split common fixed point problem. J. Optim. Theory Appl. 165, 385–404 (2015)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Cegielski, A., Al-Musallam, F.: Strong convergence of a hybrid steepest descent method for the split common fixed point problem. Optimization 65, 1463–1476 (2016)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Dang, Y., Gao, Y.: The strong convergence of a KM-CQ-like algorithm for a split feasibility problem. Inverse Probl. 27, 015007 (2011)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Eslamian, M., Eslamian, P.: Strong convergence of a split common fixed point problem. Numer. Funct. Anal. Optim. 37, 1248–1266 (2016)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Goebel, K., Reich, S.: Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings. Marcel Dekker, New York (1984)MATHGoogle Scholar
  18. 18.
    Hicks, T.L., Kubicek, J.D.: On the Mann iteration process in a Hilbert space. J. Math. Anal. Appl. 59, 498–504 (1997)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Hieu, D.V., Anh, P.K., Muu, L.D.: Modified hybrid projection methods for finding common solutions to variational inequality problems. Comput. Optim. Appl. 66, 75–96 (2017)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Hieu, D.V., Muu, L.D., Anh, P.K.: Parallel hybrid extragradient methods for pseudomonotone equilibrium problems and nonexpansive mappings. Numer. Algorithms 73, 197–217 (2016)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Hieu, D.V.: Parallel extragradient-proximal methods for split equilibrium problems. Math. Model. Anal. 21, 478–501 (2016)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Hieu, D.V.: An explicit parallel algorithm for variational inequalities. Bull. Malays. Math. Soc. (2017).  https://doi.org/10.1007/s40840-017-0474-z
  23. 23.
    Kraikaew, R., Saejung, S.: On split common fixed point problems. J. Math. Anal. Appl. 415, 513–524 (2014)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Li, M., Yao, Y.: Strong convergence of an iterative algorithm for \(\lambda \)-strictly pseudo-contractive mappings in Hilbert spaces. An. St. Univ. Ovidius Constanta. 18, 219–228 (2010)MathSciNetMATHGoogle Scholar
  25. 25.
    Liu, L.S.: Ishikawa and Mann iteration process with errors for nonlinear strongly accretive mappings in Banach spaces. J. Math. Anal. Appl. 194, 114–125 (1995)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Maingé, P.E.: Regularized and inertial algorithms for common fixed points of nonlinear operators. J. Math. Anal. Appl. 34, 876–887 (2008)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Maingé, P.E.: A hybrid extragradient-viscosity method for monotone operators and fixed point problems. SIAM J. Control Optim. 47, 1499–1515 (2008)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Maingé, P.E.: The viscosity approximation process for quasi-nonexpansive mappings in Hilbert spaces. Comput. Math. Appl. 59, 74–79 (2010)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Masad, E., Reich, S.: A note on the multiple-set split convex feasibility problem in Hilbert space. J. Nonlinear Convex Anal. 8, 367–371 (2007)MathSciNetMATHGoogle Scholar
  30. 30.
    Moudafi, A.: The split common fixed point problem for demicontractive mappings. Inverse Prob. 26, 055007 (2010)MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Reich, S.: Constructive Techniques for Accretive and Monotone Operators. Applied Nonlinear Analysis, pp. 335–345. Academic Press, New York (1979)Google Scholar
  32. 32.
    Rockafellar, R.T.: Monotone operators and the proximal point algorithm. SIAM J. Control Optim. 14, 877–898 (1976)MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Shehu, S.: New convergence theorems for split common fixed point problems in Hilbert spaces. J. Nonlinear Convex Anal. 16, 167–181 (2015)MathSciNetMATHGoogle Scholar
  34. 34.
    Shehu, Y., Cholamjiak, P.: Another look at the split common fixed point problem for demicontractive operators. RACSAM. 110, 201–218 (2016)MathSciNetCrossRefMATHGoogle Scholar
  35. 35.
    Thong, D.V.: Viscosity approximation methods for solving fixed point problems and split common fixed point problems. J. Fixed Point Theory Appl. 19, 1481–1499 (2017)MathSciNetCrossRefMATHGoogle Scholar
  36. 36.
    Thong, D.V., Hieu, D.V.: An inertial method for solving split common fixed point problems. J. Fixed Point Theory Appl. 19, 3029–3051 (2017)MathSciNetCrossRefMATHGoogle Scholar
  37. 37.
    Thong, D.V., Hieu, D.V.: Weak and strong convergence theorems for variational inequality problems. Numer. Algorithms (2017).  https://doi.org/10.1007/s11075-017-0412-z
  38. 38.
    Thong, D.V., Hieu, D.V.: Modified subgradient extragradient method for inequality variational problems. Numer. Algorithms (2017).  https://doi.org/10.1007/s11075-017-0452-4
  39. 39.
    Thong, D.V., Hieu, D.V.: Modified subgradient extragradient algorithms for variational inequality problems and fixed point problems. Optimization 67, 83–102 (2017)MathSciNetCrossRefGoogle Scholar
  40. 40.
    Thong D.V., Hieu, D.V.: Inertial extragradient algorithms for strongly pseudomonotone variational inequalities. J. Comput. Appl. Math. (accepted) Google Scholar
  41. 41.
    Tang, Y.C., Liu, L.W.: Several iterative algorithms for solving the split common fixed point problem of directed operators with applications. Optimization 65, 53–65 (2016)MathSciNetCrossRefMATHGoogle Scholar
  42. 42.
    Takahashi, W.: Nonlinear Functional Analysis—Fixed Point Theory and its Applications. Yokohama Publishers Inc., Yokohama (2000)MATHGoogle Scholar
  43. 43.
    Wang, F., Xu, H.K.: Cyclic algorithms for split feasibility problems in Hilbert spaces. Nonlinear Anal. 74, 4105–4111 (2011)MathSciNetCrossRefMATHGoogle Scholar
  44. 44.
    Xu, H.K.: Iterative algorithms for nonlinear operators. J. Lond. Math. Soc. 66, 240–256 (2002)MathSciNetCrossRefMATHGoogle Scholar
  45. 45.
    Xu, H.K.: A variable Krasnosel’skii-Mann algorithm and the multiple-set split feasibility problem. Inverse Prob. 22, 2021–2034 (2006)CrossRefMATHGoogle Scholar
  46. 46.
    Xu, H.K.: Iterative methods for the split feasibility problem in infinite dimensional Hilbert spaces. Inverse Prob. 26, 105018 (2010)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Applied Analysis Research Group, Faculty of Mathematics and StatisticsTon Duc Thang UniversityHo Chi Minh CityVietnam
  2. 2.Department of MathematicsCollege of Air ForceNha TrangVietnam

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