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An iterative method for solving proximal split feasibility problems and fixed point problems

  • Wongvisarut Khuangsatung
  • Pachara Jailoka
  • Suthep SuantaiEmail author
Article
  • 79 Downloads

Abstract

The purpose of this research is to introduce a regularized algorithm based on the viscosity method for solving the proximal split feasibility problem and the fixed point problem in Hilbert spaces. A strong convergence result of our proposed algorithm for finding a common solution of the proximal split feasibility problem and the fixed point problem for nonexpansive mappings is established. We also apply our main result to the split feasibility problem, and the fixed point problem of nonexpansive semigroups, respectively. Finally, we give numerical examples for supporting our main result.

Keywords

Fixed point problems Proximal split feasibility problems Nonexpansive mappings 

Mathematics Subject Classification

47H09 47H10 

Notes

Acknowledgements

The authors would like to thank the referees for valuable comments and suggestions for improving this work. W. Khuangsatung would like to thank Rajamangala University of Technology Thanyaburi and S. Suantai would like to thank Chiang Mai University for the financial support.

References

  1. Abbas M, AlShahrani M, Ansari QH, Iyiola OS, Shehu Y (2018) Iterative methods for solving proximal split minimization problems. Numer Algorithms 78:193–215MathSciNetCrossRefGoogle Scholar
  2. Aleyner A, Censor Y (2005) Best approximation to common fixed points of a semigroup of nonexpansive operator. J Nonlinear Convex Anal 6(1):137–151MathSciNetzbMATHGoogle Scholar
  3. Browder FE (1956) Nonexpansive nonlinear operators in a Banach space. Proc Natl Acad Sci USA 54:1041–1044MathSciNetCrossRefGoogle Scholar
  4. Browder FE (1976) Nonlinear operators and nonlinear equations of evolution in Banach spaces. Proc. Symp. Pure Math. 18:78–81Google Scholar
  5. Byrne C (2002) Iterative oblique projection onto convex sets and the split feasibility problem. Inverse Probl 18(2):441–453MathSciNetCrossRefGoogle Scholar
  6. Cegielski A (2012) Iterative methods for fixed point problems in Hilbert spaces. Lecture notes in mathematics, vol 2057. Springer, HeidelbergGoogle Scholar
  7. Censor Y, Elfving T (1994) A multiprojection algorithm using Bregman projections in a product space. Numer Algorithms 8:221–239MathSciNetCrossRefGoogle Scholar
  8. Censor Y, Bortfeld T, Martin B, Trofimov A (2006) A unified approach for inversion problems in intensity-modulated radiation therapy. Phys Med Biol 51:2353–2365CrossRefGoogle Scholar
  9. Chang SS, Kim JK, Cho YJ, Sim J (2014) Weak and strong convergence theorems of solutions to split feasibility problem for nonspreading type mapping in Hilbert spaces. Fixed Point Theory Appl 2014:11MathSciNetCrossRefGoogle Scholar
  10. Combettes PL, Pesquet JC (2011a) Proximal splitting methods in signal processing. In: Fixed-point algorithms for inverse problems in science and engineering. Springer, New York, pp 185–212Google Scholar
  11. Combettes PL, Pesquet JC (2011b) Proximal splitting methods in signal processing. Fixed Point Algorithms Inverse Probl Sci Eng 49:185–212MathSciNetCrossRefGoogle Scholar
  12. Lopez G, Martin-Marquez V, Wang F et al (2012) Solving the split feasibility problem without prior knowledge of matrix norms. Inverse Probl 28:085004MathSciNetCrossRefGoogle Scholar
  13. Mainge PE (2008) Strong convergence of projected subgradient methods for nonsmooth and nonstrictly convex minimization. Set Valued Anal 16:899–912MathSciNetCrossRefGoogle Scholar
  14. Moudafi A, Thakur BS (2014) Solving proximal split feasibility problems without prior knowledge of operator norms. Optim Lett 8:2099–2110MathSciNetCrossRefGoogle Scholar
  15. Qu B, Xiu N (2005) A note on the CQ algorithm for the split feasibility problem. Inverse Probl 21(5):1655–1665MathSciNetCrossRefGoogle Scholar
  16. Shehu Y, Iyiola OS (2015) Convergence analysis for proximal split feasibility problems and fixed point problems. J Appl Math Comput 48:221–239MathSciNetCrossRefGoogle Scholar
  17. Shehu Y, Iyiola OS (2017a) Convergence analysis for the proximal split feasibility problem using an inertial extrapolation term method. J Fixed Point Theory Appl 19(4):2483–2510MathSciNetCrossRefGoogle Scholar
  18. Shehu Y, Iyiola OS (2017b) Strong convergence result for proximal split feasibility problem in Hilbert spaces. Optimization 66(12):2275–2290MathSciNetCrossRefGoogle Scholar
  19. Shehu Y, Iyiola OS (2018) Accelerated hybrid viscosity and steepest-descent method for proximal split feasibility problems. Optimization 67(4):475–492MathSciNetCrossRefGoogle Scholar
  20. Shehu Y, Cai G, Iyiola OS (2015) Iterative approximation of solutions for proximal split feasibility problems. Fixed Point Theory Appl 2015:123MathSciNetCrossRefGoogle Scholar
  21. Shimizu T, Takahashi W (1997) Strong convergence to common fixed points of families of nonexpansive mappings. J Math Anal Appl 211(1):71–83MathSciNetCrossRefGoogle Scholar
  22. Takahashi W (2000) Nonlinear functional analysis. Yokohama Publishers, YokohamazbMATHGoogle Scholar
  23. Witthayarat U, Cho YJ, Cholamjiak P (2018) On solving proximal split feasibility problems and applications. Ann Funct Anal 9(1):111–122MathSciNetCrossRefGoogle Scholar
  24. Xu HK (2003) An iterative approach to quadric optimization. J Optim Theory Appl 116:659–678MathSciNetCrossRefGoogle Scholar
  25. Yao Z, Cho SY, Kang SM et al (2014) A regularized algorithm for the proximal split feasibility problem. Abstr Appl Anal 6:894272MathSciNetzbMATHGoogle Scholar

Copyright information

© SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2019

Authors and Affiliations

  1. 1.Department of Mathematics and Computer Science, Faculty of Science and TechnologyRajamangala University of Technology Thanyaburi (RMUTT)ThanyaburiThailand
  2. 2.Data Science Research Center, Department of Mathematics, Faculty of ScienceChiang Mai UniversityChiang MaiThailand

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