Iterative methods with perturbations for the sum of two accretive operators in q-uniformly smooth Banach spaces

  • Suthep Suantai
  • Prasit Cholamjiak
  • Pongsakorn SunthrayuthEmail author
Original Paper


In this work, we introduce implicit and explicit iteration processes with perturbations for solving the fixed point problem of nonexpansive mappings and the quasi-variational inclusion problem. We then prove its strong convergence under some suitable conditions. In the last section of the paper, some applications are given also. The results obtained in this paper extend and improve some known others presented in the literature.


Variational inclusion Banach space Strong convergence Iterative method m-Accretive operator 

Mathematics Subject Classification

47H09 47H10 47H17 47J25 49J40 



S. Suantai would like to thank Chiang Mai University for financial supports, P. Cholamjiak was supported by the Thailand Research Fund and the Commission on Higher Education under Grant MRG5980248 and P. Sunthrayuth was supported by RMUTT research foundation scholarship of Rajamangala University of Technology Thanyaburi under Grant NRF04066005.


  1. 1.
    Noor, M.A., Noor, K.I.: Sensitivity analysis for quasivariational inclusions. J. Math. Anal. Appl. 236(2), 290–299 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Kamimura, S., Takahashi, W.: Approximating solutions of maximal monotone operators in Hilbert spaces. J. Approx. Theory 106(2), 226–240 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Yao, Y., Cho, Y.J., Liou, Y.-C.: Algorithms of common solutions for variational inclusions, mixed equilibrium problems and fixed point problems. Eur. J. Oper. Res. 212(2), 242–250 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Takahashi, S., Takahashi, W., Toyada, M.: Strong convergence theorems for maximal monotone operators with nonlinear mappings in Hilbert spaces. J. Optim. Theory Appl. 147, 27–41 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Manaka, H., Takahashi, W.: Weak convergence theorems for maximal monotone operators with nonspreading mappings in a Hilbert space. Cubo 13(1), 11–24 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Lopez, G., Martin-Marquez, V., Wang, F., Xu, H. K.: Forward–backward splitting methods for accretive operators in Banach spaces. In: Abstract and Applied Analysis, vol 2012, Article ID 109236Google Scholar
  7. 7.
    Gossez, J.P., Lami Dozo, E.: Some geometric properties related to the fixed point theory for nonexpansive mappings. Pac. J. Math. 40, 565–573 (1972)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Takahashi, W.: Convex Analysis and Approximation of Fixed Points. Yokohama Publishers, Yokohama (2000). (Japanese)zbMATHGoogle Scholar
  9. 9.
    Chidume, C.: Geometric Properties of Banach Spaces and Nonlinear Iterations. Springer, London (2009)Google Scholar
  10. 10.
    Xu, H.K.: Inequalities in Banach spaces with applications. Nonlinear Anal. Theory Methods Appl. 16, 1127–1138 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Browder, F.E.: Fixed-point theorems for noncompact mappings in Hilbert space. Proc. Natl. Acad. Sci. USA. 53, 1272–1276 (1965)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Yao, Y., Liou, Y.-C., Kang, S.M., Yu, Y.: Algorithms with strong convergence for a system of nonlinear variational inequalities in Banach spaces. Nonlinear Anal. 74, 6024–6034 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Mitrinović, D.S.: Analytic Inequalities. Springer, New York (1970)CrossRefzbMATHGoogle Scholar
  14. 14.
    Suzuki, T.: Strong convergence of Krasnoselskii and Mann’s type sequence for one-parameter nonexpansive semigroup without Bochner integrals. J. Math. Anal. Appl. 305, 227–239 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Xu, H.K.: Iterative algorithms for nonlinear operators. J. Lond. Math. Soc. 66(1), 240–256 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Barbu, V.: Nonlinear Semigroups and Differential Equations in Banach Spaces. Noordhoff, Leiden (1976)CrossRefzbMATHGoogle Scholar
  17. 17.
    Rockafellar, R.T.: On the maximal monotonicity of subdifferential mappings. Pac. J. Math. 33, 209–216 (1970)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Takahashi, S., Takahashi, W., Toyoda, M.: Strong convergence theorems for maximal monotone operators with nonlinear mappings in Hilbert spaces. J. Optim. Theory Appl. 147(1), 27–41 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Blum, E., Oettli, W.: From optimization and variational inequalities to equilibrium problems. Math. Stud. 63, 123–145 (1994)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Combettes, P.L., Hirstoaga, S.A.: Equilibrium programming in Hilbert spaces. J. Nonlinear Convex Anal. 6, 117–136 (2005)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Baillon, J.B., Haddad, G.: Quelques proprietes des operateurs angle-bornes et cycliquement monotones. Isr. J. Math. 26(2), 137–150 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Byrne, C.: A unified treatment of some iterative algorithms in signal processing and image reconstruction. Inverse Prob. 20, 103–120 (2004)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Italia S.r.l. 2017

Authors and Affiliations

  • Suthep Suantai
    • 1
  • Prasit Cholamjiak
    • 2
  • Pongsakorn Sunthrayuth
    • 3
    Email author
  1. 1.Department of Mathematics, Faculty of ScienceChiang Mai UniversityChiang MaiThailand
  2. 2.School of ScienceUniversity of PhayaoPhayaoThailand
  3. 3.Department of Mathematics and Computer Science, Faculty of Science and TechnologyRajamangala University of Technology Thanyaburi (RMUTT)ThanyaburiThailand

Personalised recommendations