Advertisement

Systems of variational inequalities with hierarchical variational inequality constraints for asymptotically nonexpansive and pseudocontractive mappings

  • Lu-Chuan Ceng
  • Ching-Feng WenEmail author
Original Paper
  • 17 Downloads

Abstract

The purpose of this paper is to introduce and analyze a hybrid extragradient-like implicit rule for finding a common solution of a general system of variational inequalities (GSVI) and a common fixed point problem (CFPP) of a countable family of uniformly Lipschitzian pseudocontractive mappings and an asymptotically nonexpansive mapping in Hilbert spaces. Here the hybrid extragradient-like implicit rule is based on Korpelevich’s extragradient method, the viscosity approximation method and the Mann iteration method. We prove the strong convergence of this method to a common solution of the GSVI and the CFPP, which solves a certain variational inequality on their common solution set. As an application, we give an algorithm to solve common fixed point problems of nonexpansive and pseudocontractive mappings, variational inequality problems and generalized mixed equilibrium problems in Hilbert spaces.

Keywords

Implicit rule General system of variational inequalities Fixed Point Asymptotically nonexpansive mapping Lipschitzian pseudocontractive mapping Hilbert spaces 

Mathematics Subject Classifications

49J30 47H09 47J20 

Notes

References

  1. 1.
    Ceng, L.C., Latif, A., Yao, J.C.: On solutions of a system of variational inequalities and fixed point problems in Banach spaces. Fixed Point Theory Appl. 176, 34 (2013)Google Scholar
  2. 2.
    Ceng, L.C., Wen, C.F.: Three-step Mann iterations for a general system of variational inequalities and an infinite family of nonexpansive mappings in Banach spaces. J. Inequal. Appl. 539, 27 (2013)Google Scholar
  3. 3.
    Ceng, L.C., Latif, A., Ansari, Q.H., Yao, J.C.: Hybrid extragradient method for hierarchical variational inequalities. Fixed Point Theory Appl. 222, 35 (2014)Google Scholar
  4. 4.
    Yao, Y.H., Liou, Y.C., Kang, S.M.: Approach to common elements of variational inequality problems and fixed point problems via a relaxed extragradient method. Comput. Math. Appl. 59, 3472–3480 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Ceng, L.C., Plubtieng, S., Wong, M.M., Yao, J.C.: System of variational inequalities with constraints of mixed equilibria, variational inequalities, and convex minimization and fixed point problems. J. Nonlinear Convex Anal. 16, 385–421 (2015)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Yao, Y.H., Liou, Y.C., Kang, S.M.: Two-step projection methods for a system of variational inequality problems in Banach spaces. J. Global Optim. 55, 801–811 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Ceng, L.C., Guu, S.M., Yao, J.C.: Hybrid viscosity CQ method for finding a common solution of a variational inequality, a general system of variational inequalities, and a fixed point problem. Fixed Point Theory Appl. 313, 25 (2013)Google Scholar
  9. 9.
    Gibali, A.: Two simple relaxed perturbed extragradient methods for solving variational inequalities in Euclidean spaces. J. Nonlinear Var. Anal. 2, 49–61 (2018)CrossRefGoogle Scholar
  10. 10.
    Verma, R.U.: General convergence analysis for two-step projection methods and applications to variational problems. Appl. Math. Lett. 18, 1286–1292 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Xu, H.K., Kim, T.H.: Convergence of hybrid steepest-descent methods for variational inequalities. J. Optim. Theory Appl. 119, 185–201 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Lim, T.C., Xu, H.K.: Fixed point theorems for asymptotically nonexpansive mappings. Nonlinear Anal. 22, 1345–1355 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Cai, G., Shehu, Y., Iyiola, O.S.: Strong convergence results for variational inequalities and fixed point problems using modified viscosity implicit rules. Numer. Algorithms 77, 535–558 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Ceng, L.C., Yao, J.C.: A hybrid iterative scheme for mixed equilibrium problems and fixed point problems. J. Comput. Appl. Math. 214, 186–201 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Chancelier, J.P.: Iterative schemes for computing fixed points of nonexpansive mappings in Banach spaces. J. Math. Anal. Appl. 353, 141–153 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Ceng, L.C., Hadjisavvas, N., Wong, N.C.: Strong convergence theorem by a hybrid extragradient-like approximation method for variational inequalities and fixed point problems. J. Global Optim. 46, 635–646 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Ceng, L.C., Petrusel, A., Wong, M.M., Yu, S.J.: Strong convergence of implicit viscosity approximation methods for pseudocontractive mappings in Banach spaces. Optimization 60, 659–670 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Ansari, Q.H., Wong, N.C., Yao, J.C.: The existence of nonlinear inequalities. Appl. Math. Lett. 12, 89–92 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Gwinner, J.: Stability of monotone variational inequalities with various applications. In: Giannessi, F., Maugeri, A. (eds.) Variational inequalities and network equilibrium problems, pp. 123–142. Plenum, New York (1995)Google Scholar
  20. 20.
    Takahashi, S., Takahashi, W.: Strong convergence theorem for a generalized equilibrium problem and a nonexpansive mapping in a Hilbert space. Nonlinear Anal. 69, 1025–1033 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Blum, E., Oettli, W.: From optimization and variational inequalities to equilibrium problems. Math. Stud. 63, 123–145 (1994)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Ceng, L.C., Ansari, Q.H., Schaible, S.: Hybrid extragradient-like methods for generalized mixed equilibrium problems, systems of generalized equilibrium problems and optimization problems. J. Global Optim. 53, 69–96 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Ceng, L.C., Yao, J.C.: A relaxed extragradient-like method for a generalized mixed equilibrium problem, a general system of generalized equilibria and a fixed point problem. Nonlinear Anal. 72, 1922–1937 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Ceng, L.C., Guu, S.M., Yao, J.C.: Hybrid iterative method for finding common solutions of generalized mixed equilibrium and fixed point problems. Fixed Point Theory Appl. 92, 19 (2012)Google Scholar
  25. 25.
    Deimling, K.: Zeros of accretive operators. Manuscr. Math. 13, 365–374 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Martin Jr., R.H.: Differential equations on closed subsets of a Banach space. Trans. Am. Math. Soc. 179, 399–414 (1973)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Reich, S.: Weak convergence theorems for nonexpansive mappings in Banach spaces. J. Math. Anal. Appl. 67, 274–276 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Xu, H.K.: Iterative algorithms for nonlinear operators. J. Lond. Math. Soc. 66, 240–256 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Ceng, L.C., Petruşel, A., Yao, J.C.: Composite viscosity approximation methods for equilibrium problem, variational inequality and common fixed points. J Nonlinear Convex Anal. 15, 219–240 (2014)MathSciNetzbMATHGoogle Scholar
  30. 30.
    Takahashi, W., Wen, C.F., Yao, J.C.: Split common fixed point problems and hierarchical variational inequality problems in Hilbert spaces. J. Nonlinear Convex Anal. 18(5), 925–935 (2017)MathSciNetzbMATHGoogle Scholar
  31. 31.
    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(8), 2350–2360 (2007)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© The Royal Academy of Sciences, Madrid 2019

Authors and Affiliations

  1. 1.Department of MathematicsShanghai Normal UniversityShanghaiChina
  2. 2.Center for Fundamental Science and Research Center for Nonlinear Analysis and OptimizationKaohsiung Medical UniversityKaohsiungTaiwan
  3. 3.Department of Medical ResearchKaohsiung Medical University HospitalKaohsiungTaiwan

Personalised recommendations