A new Hybrid Projection Algorithm for Solving the Split Generalized Equilibrium Problems and the System of Variational Inequality Problems

  • Jitsupa Deepho
  • Wiyada KumamEmail author
  • Poom Kumam


In this paper, we introduced modified Mann iterative algorithms by the new hybrid projection method for finding a common element of the set of fixed points of a countable family of nonexpansive mappings, the set of the split generalized equilibrium problem and the set of solutions of the general system of the variational inequality problem for two-inverse strongly monotone mappings in real Hilbert spaces. The strong convergence theorem of the iterative algorithm in Hilbert spaces under certain mild conditions are provided.


Split generalized equilibrium problem Nonexpansive mapping Inverse-strongly monotone mapping General system of the variational inequality problem 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Mahdioui, H., Chadli, O.: On a system of generalized mixed equilibrium problems involving variational-like inequalities in Banach spaces: existence and algorithmic aspects, Advances in Operations Research 843486 (2012)Google Scholar
  2. 2.
    Censor, Y., Elfving, T.: A multiprojection algorithm using Bregman projections in product space. Numer. Algo. 8, 221–239 (1994)CrossRefMathSciNetzbMATHGoogle Scholar
  3. 3.
    Cianciaruso, F., Marino, G., Muglia, L., Yao, Y.: A hybrid projection algorithm for finding solutions of mixed equilibrium problem and variational inequality problem. Fixed Point Theory Appl. 2010 (2010)Google Scholar
  4. 4.
    Suzuki, T.: Strong convergence theorems for an infinite family of nonexpansive mappings in general Banach spaces. Fixed Point Theory Appl. 1, 103–123 (2005)Google Scholar
  5. 5.
    Shimoji, K., Takahashi, W.: Strong convergence to common fixed points of infinite nonexpansive mappings and applications. Taiwanese J. Math. 5(2), 87–404 (2001)MathSciNetGoogle Scholar
  6. 6.
    Opial, Z.: Weak convergence of the sequence of successive approximation for nonexpansive mappings. Bull. Amer. Math. Soc. 73, 561–597 (1967)CrossRefMathSciNetGoogle Scholar
  7. 7.
    Browder, F.E.: Nonlinear Operators and Nonlinear Equations of Evolution in Banach Spaces. American Mathematical Society, Washington (1976)CrossRefzbMATHGoogle Scholar
  8. 8.
    Goebel, K., Kirk, W.A.: Topics in Metric Fixed Point Theory. Cambridge University Press, Cambridge (1990)CrossRefzbMATHGoogle Scholar
  9. 9.
    Plubtieng, S., Thammathiwat, T.: A viscosity approximation method for finding a common fixed point of nonexpansive and firmly nonexpansive mappings in hilbert spaces. Thai. J. Math. 6, 377–390 (2008)zbMATHGoogle Scholar
  10. 10.
    Ceng, L.C., Wang, C.Y., Yao, J.C.: Strong Convergence Theorems by a Relaxed Extragradient Method for a General System of Variational Inequalities. Math. Method Oper. Res. 67, 375–390 (2008)CrossRefMathSciNetzbMATHGoogle Scholar
  11. 11.
    Censor, Y., Elfving, T.: A multiprojection algorithm using Bregman projections in a product space. Numer. Algo. 18(24), 221–239 (1994)CrossRefMathSciNetGoogle Scholar
  12. 12.
    Byrne, C.: Iterative oblique projection onto convex sets and the split feasibility problem. Inverse. probl. 18, 441–453 (2002)CrossRefMathSciNetzbMATHGoogle Scholar
  13. 13.
    Censor, Y., Elving, T., Kopf, N., Bortfeld, T.: The multiple-set split feasibility problem and its applications for inverse problem. Inverse Problem 21(6), 2071–2084 (2005)CrossRefzbMATHGoogle Scholar
  14. 14.
    Censor, Y., Motova, A., Segal, A.: Perturbed projections and subgardient projections for the multiple-set split feasibility problem. J. Math. Anal. Appl. 327, 1244–1256 (2007)CrossRefMathSciNetzbMATHGoogle Scholar
  15. 15.
    Eicke, B.: Iteration methods for convexly constrained ill-posed problems in Hilbert space. Numer. Function. Anal. Optim. 13(5–6), 423–429 (1992)MathSciNetGoogle Scholar
  16. 16.
    Landweber, L.: An iterative formula for Fredholm integral equations of the first kind. Amer. J. Math. 73, 615–625 (1951)CrossRefMathSciNetzbMATHGoogle Scholar
  17. 17.
    Moudafi, A.: Split monotone variational inclusions. J. Optim. Theory Appl. 150, 275–283 (2011)CrossRefMathSciNetzbMATHGoogle Scholar
  18. 18.
    Kazmi, K.R., Rizvi, S.H.: Iterative approximation of a common solution of a split generalized equilibrium problem and a fixed point problem for nonexpansive semigroup. Math. Sci. 7, 1 (2013). doi: 10.1186/2251-7456-7-1 CrossRefMathSciNetGoogle Scholar
  19. 19.
    Kirk, W.A.: Fixed point theorem for mappings which do not increase distance, Amer. Math. Monthly 72, 1004–1006 (1965)CrossRefMathSciNetzbMATHGoogle Scholar
  20. 20.
    Takahashi, W., Toyoda, M.: Weak convergence theorems for nonexpansive mappings and monotone mappings. J. Optim. Theory Appl., 118 (2003)Google Scholar
  21. 21.
    Kumam, P.: A relaxed extragradient approximation method of two inverse-strongly monotone mappings for a general system of variational inequalities, fixed point and equilibrium problem. Bull. Iranian Math. Soc. 36, 227–250 (2010)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Kumam, W., Kumam, P.: Hybrid iterative scheme by relaxed extragradient method for solutions of equilibrium problems and a general system of variational inequalities with application to optimization. Nonlinear Anal. Hybrid. Syst. 3, 640–656 (2009)CrossRefMathSciNetzbMATHGoogle Scholar
  23. 23.
    Verma, R.U.: On a new system of nonlinear variational inequalities and associated iterative algorithms. Math. Sci. Res. 3(8), 65–68 (1999)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Verma, R.U.: Iterative algorithms a new system of nonlinear quasivariational inequalities. Adv. Nonlinear Var. Inequal. 4(1), 117–124 (2001)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Stampacchia, G.: Formes bilineaires coercitives sur les ensembles convexs. C. R. Acad. Sci. Paris. 258, 4413–4416 (1964)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Ceng, L.C., Wang, C.Y., Yao, J.C.: Strong convergence theorems by a relaxed extragradient method for a general system of variational inequalities. Math. Methods Oper. Res. 67, 375–390 (2008)CrossRefMathSciNetzbMATHGoogle Scholar
  27. 27.
    Mann, W.R.: Mean value methods in iteration. Proc. Amer. Math. Soc. 4, 506–510 (1953)CrossRefMathSciNetzbMATHGoogle Scholar
  28. 28.
    Reich, S.: Weak convergence theorems for nonexpansive mappings. J. Math. Anal. Appl. 67, 274–276 (1978)CrossRefGoogle Scholar
  29. 29.
    Genel, A., Lindenstrass, J.: An example concerning fixed points, Israel. J. Math. 22, 81–86 (1975)zbMATHGoogle Scholar
  30. 30.
    Burachik, R.S., Lopes, J.O., Svaiter, B.F.: An outer approximation method for the variational inequality problem. ISIAM J. Control Optim. 43, 2071–2088 (2005). 7–898 (1976)CrossRefMathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  1. 1.Department of MathematicsFaculty of Science King Mongkut’s University of Technology Thonburi (KMUTT)BangkokThailand
  2. 2.Department of Mathematics and Computer Science Faculty of Science and TechnologyRajamangala University of Technology Thanyaburi (RMUTT)PathumthaniThailand
  3. 3.Department of MathematicsFaculty of ScienceKing Mongkut’s University of Technology Thonburi (KMUTT)BangkokThailand

Personalised recommendations