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A Viscosity Approximation Method for the Split Feasibility Problems

  • Jitsupa Deepho
  • Poom Kumam
Chapter

Abstract

In this paper, we discuss the strong convergence of the viscosity approximation method for solving the split feasibility problem in Hilbert spaces. Consider also the iteration process \( \{ x_{n} \} \), where \( x_{0} \in C \) is arbitrary and \( x_{n + 1} = (1 - \alpha_{n} )P_{C} (I - \xi A^{*} (I - P_{Q} )A)x_{n} + \alpha_{n} f(x_{n} ),n \ge 1 \) where \( \alpha_{n} \in (0,1) \). The main result present in this paper improve and extend some recent result done by Xu [Iterative methods for the split feasibility problem in infinite-dimensional Hilbert space, Inverse Problem 26 (2010) 105018] and some others.

Keywords

CQ algorithm Metric projection Split feasibility problem Strong convergence theorem Variational inequality problem Viscosity approximation method 

Notes

Acknowledgment

The first author was supported by the Thailand Research Fund through the Royal Golden Jubilee Ph.D. Program (Grant No. PHD/0033/2554) and the King Mongkut’s University of Technology Thonburi. Moreover, this study was supported by the Higher Education Research Promotion and National Research University Project of Thailand, Office of the Higher Education Commission.

References

  1. 1.
    B. Halpern, Fixed points of nonexpanding maps. Bull. Amer. Math. Soc. 73, 957–961 (1967)CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    P.L. Lions, Approximation de points fixes de contractions. C. R. Acad. Sci. Ser. A-B Paris 284, 1357–1359 (1977)MATHGoogle Scholar
  3. 3.
    A. Moudafi, Viscosity approximations methods for fixed point problems. J. Math. Anal. Appl. 241, 46–55 (2000)CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    R. Wittman, Approximation of fixed points of nonexpansive mappings. Arch. Math. 58, 486–491 (1992)CrossRefGoogle Scholar
  5. 5.
    H.K. Xu, Viscosity approximations methods for nonexpansive mappings. J. Math. Anal. Appl. 298, 279–291 (2004)CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Y. Censor, T. Elving, A multiprojection algorithm using Bregman projections in product space. Numer. Algorithm 8(2–4), 221–239 (1994)CrossRefMATHGoogle Scholar
  7. 7.
    C. Byrne, Iterative oblique projection onto convex subsets and the split feasibility problem. Inverse Prob. 18(2), 441–453 (2002)CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Q. Yang, The relaxed CQ algorithm solving the split feasibility problem. Inverse Prob. 20(4), 1261–1266 (2004)CrossRefMATHGoogle Scholar
  9. 9.
    H.K. Xu, Iterative methods for the split feasibility problem in infinite-dimensional Hilbert spaces. Inverse Prob. 26 (2010) 105018 (p. 17)Google Scholar
  10. 10.
    J. Deepho, P. Kumam, The hybrid extragradient method for the split feasibility and fixed point problems, in Proceedings of The International Multiconference of Engineers and Computer Scientists 2014, vol I, IMECS, 12–14 Mar 2014, Hong Kong, pp. 558–563Google Scholar
  11. 11.
    H.K. Xu, Averaged mappings and the gardient-projection algorithm. J. Optim. Theory Appl. 150(2), 360–378 (2011)CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    H.K. Xu, Viscosity approximation methods for nonexpansive mapping. J. Math. Anal. Appl. 298(1), 279–291 (2004)CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    F.E. Browder, Fixed point theorems for noncompact mappings in Hilbert spaces. Proc. Natl. Acad. Sci. USA 53(6), 1272–1276 (1965)CrossRefMATHMathSciNetGoogle Scholar
  14. 14.
    J. Deepho and P. Kumam, A modified Halpern’s iterative scheme for solving split feasibility problems. Abstract Appl. Anal. 2012(876069), 8 (2012)Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of ScienceKing Mongkut’s University of Technology ThonburiBangkokThailand

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