Skip to main content
SpringerLink
Log in

A splitting algorithm for finding fixed points of nonexpansive mappings and solving equilibrium problems

  • Published:
Journal of Fixed Point Theory and Applications Aims and scope Submit manuscript

Abstract

We consider the problem of finding a fixed point of a nonexpansive mapping, which is also a solution of a pseudo-monotone equilibrium problem, where the bifunction in the equilibrium problem is the sum of two ones. We propose a splitting algorithm combining the gradient method for equilibrium problem and the Mann iteration scheme for fixed points of nonexpansive mappings. At each iteration of the algorithm, two strongly convex subprograms are required to solve separately, one for each of the component bifunctions. Our main result states that, under paramonotonicity property of the given bifunction, the algorithm converges to a solution without any Lipschitz-type condition as well as Hölder continuity of the bifunctions involved.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Anh, P.K., Hai, T.N.: Splitting extragradient-like algorithms for strongly pseudomonotone equilibrium problems. Numer. Algorithm 76(1), 67–91 (2017)

    Article  MathSciNet  Google Scholar 

  2. Anh, P.N.: A hybrid extragradient method extended to fixed point problems and equilibrium problems. Optimization 62(2), 271–283 (2013)

    Article  MathSciNet  Google Scholar 

  3. Anh, P.N., Muu, L.D.: A hybrid subgradient algorithm for nonexpansive mappings and equilibrium problems. Optim. Lett. 8(2), 727–738 (2014)

    Article  MathSciNet  Google Scholar 

  4. Bigi, G., Castellani, M., Pappalardo, M., Passacantando, M.: Existence and solution methods for equilibria. Eur. J. Oper. Res. 227(1), 1–11 (2013)

    Article  MathSciNet  Google Scholar 

  5. Blum, E., Oettli, W.: From optimization and variational inequalities to equilibrium problems. Math. Stud. 63(1–4), 123–145 (1994)

    MathSciNet  MATH  Google Scholar 

  6. Censor, Y., Elfving, T.: A multiprojection algorithm using Bregman projections in a product space. Numer. Algorithm 8(2), 221–239 (1994)

    Article  MathSciNet  Google Scholar 

  7. Cruz, J.Y.B., Millán, R.D.: A direct splitting method for nonsmooth variational inequalities. J. Optim. Theory Appl. 161(3), 728–737 (2014)

    Article  MathSciNet  Google Scholar 

  8. Duc, P.M., Muu, L.D.: A splitting algorithm for a class of bilevel equilibrium problems involving nonexpansive mappings. Optimization 65(10), 1855–1866 (2016)

    Article  MathSciNet  Google Scholar 

  9. Duc, P.M., Muu, L.D., Quy, N.V.: Solution-existence and algorithms with their convergence rate for strongly pseudomonotone equilibrium problems. Pac. J. Optim. 12(4), 833–845 (2016)

    MathSciNet  MATH  Google Scholar 

  10. Eckstein, J., Svaiter, A.F.: General projective splitting methods for sums of maximal monotone operators. SIAM J. Control Optim. 48(2), 787–811 (2009)

    Article  MathSciNet  Google Scholar 

  11. Fan, K.: A minimax inequality and applications. In: Shisha, O. (ed.) Inequality, vol. III, pp. 103–113. Academic Press, New York (1972)

    Google Scholar 

  12. Hai, T.N., Vinh, N.T.: Two new splitting algorithms for equilibrium problems. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A. Mat. 111(4), 1051–1069 (2017)

    Article  MathSciNet  Google Scholar 

  13. Hieu, D.V., Moudafi, A.: A barycentric projected-subgradient algorithm for equilibrium problems. J. Nonlinear Var. Anal. 1(1), 43–59 (2017)

    MATH  Google Scholar 

  14. Hieu, D.V., Muu, L.D., Anh, P.K.: Parallel hybrid extragradient methods for pseudomonotone equilibrium problems and nonexpansive mappings. Numer. Algorithms 73(1), 197–217 (2016)

    Article  MathSciNet  Google Scholar 

  15. Hung, P.G., Muu, L.D.: The Tikhonov regularization extended to equilibrium problems involving pseudomonotone bifunctions. Nonlinear Anal. 74(17), 6121–6129 (2011)

    Article  MathSciNet  Google Scholar 

  16. Iiduka, H., Yamada, I.: A subgradient-type method for the equilibrium problem over the fixed point set and its applications. Optimization 58(2), 251–261 (2009)

    Article  MathSciNet  Google Scholar 

  17. Iusem, A.N.: On some properties of paramonotone operators. J. Convex Anal. 5(2), 269–278 (1998)

    MathSciNet  MATH  Google Scholar 

  18. Iusem, A.N., Sosa, W.: Iterative algorithms for equilibrium problems. Optimization 52(3), 301–316 (2003)

    Article  MathSciNet  Google Scholar 

  19. Mastroeni, G.: Gap functions for equilibrium problems. J. Glob. Optim. 27(4), 411–426 (2003)

    Article  MathSciNet  Google Scholar 

  20. Moudafi, A.: On the convergence of splitting proximal methods for equilibrium problems in Hilbert spaces. J. Math. Anal. Appl. 359(2), 508–513 (2009)

    Article  MathSciNet  Google Scholar 

  21. Muu, L.D., Oettli, W.: Convergence of an adaptive penalty scheme for finding constrained equilibria. Nonlinear Anal. 18(12), 1159–1166 (1992)

    Article  MathSciNet  Google Scholar 

  22. Muu, L.D., Quoc, T.D.: Regularization algorithms for solving monotone Ky Fan inequalities with application to a Nash–Cournot equilibrium model. J. Optim. Theory Appl. 142(1), 185–204 (2009)

    Article  MathSciNet  Google Scholar 

  23. Muu, L.D., Quy, N.V.: On existence and solution methods for strongly pseudomonotone equilibrium problems. Vietnam J. Math. 43, 229–238 (2015)

    Article  MathSciNet  Google Scholar 

  24. Nikaidô, H., Isoda, K.: Note on noncooperative convex games. Pac. J. Math. 5(5), 807–815 (1955)

    Article  Google Scholar 

  25. Passty, G.B.: Ergodic convergence to a zero of the sum of monotone operators in Hilbert space. J. Math. Anal. Appl. 72(2), 383–390 (1979)

    Article  MathSciNet  Google Scholar 

  26. Quoc, T.D., Anh, P.N., Muu, L.D.: Dual extragradient algorithms extended to equilibrium problems. J. Glob. Optim. 52(1), 139–159 (2012)

    Article  MathSciNet  Google Scholar 

  27. Quoc, T.D., Muu, L.D., Hien, N.V.: Extragradient algorithms extended to equilibrium problems. Optimization 57(6), 749–776 (2008)

    Article  MathSciNet  Google Scholar 

  28. Reich, S.: A limit theorem for projections. Linear Multilinear Algebra 13(3), 281–290 (1983)

    Article  MathSciNet  Google Scholar 

  29. Reich, S., Sabach, S.: Three strong convergence theorems regarding iterative methods for solving equilibrium problems in reflexive Banach spaces. Contemp. Math. 568, 225–240 (2012)

    Article  MathSciNet  Google Scholar 

  30. Rockafellar, R.T.: Monotone operators and the proximal point algorithm. SIAM J. Control Optim. 5, 877–890 (1976)

    Article  MathSciNet  Google Scholar 

  31. Santos, P.S.M., Scheimberg, S.: An inexact subgradient algorithm for equilibrium problems. Comput. Appl. Math. 30(1), 91–107 (2011)

    MathSciNet  MATH  Google Scholar 

  32. Sun, S.: An alternative regularization method for equilibrium problems and fixed point of nonexpansive mappings. J. Appl. Math. 2012, Article ID 202860 (2012). https://doi.org/10.1155/2012/202860

    MathSciNet  Google Scholar 

  33. Svaiter, B.F.: On weak convergence of the Douglas–Rachford method. SIAM J. Control Optim. 49(1), 280–287 (2011)

    Article  MathSciNet  Google Scholar 

  34. Tada, A., Takahashi, W.: Weak and strong convergence theorems for a nonexpansive mapping and an equilibrium problem. J. Optim. Theory Appl. 133(3), 359–370 (2007)

    Article  MathSciNet  Google Scholar 

  35. Takahashi, S., Takahashi, W.: Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces. J. Math. Anal. Appl. 331(1), 506–515 (2007)

    Article  MathSciNet  Google Scholar 

  36. Tan, K.-K., Xu, H.-K.: Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process. J. Math. Anal. Appl. 178, 301–308 (1993)

    Article  MathSciNet  Google Scholar 

  37. Tseng, P.: A modied forward-backward splitting method for maximal monotone mappings. SIAM J. Control Optim. 38(2), 431–446 (2000)

    Article  MathSciNet  Google Scholar 

  38. Tuy, H.: Convex Analysis and Global Optimization, 2nd edn. Springer, Berlin (2016)

    Book  Google Scholar 

  39. Vuong, P.T., Strodiot, J.-J., Nguyen, V.H.: On extragradient-viscosity methods for solving equilibrium and fixed point problems in a Hilbert space. Optimization 64(2), 429–451 (2015)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors would like to thank the associate editor and anonymous referee for their constructive comments and helpful remarks. This work is supported by the National Foundation for Science and Technology Development (NAFOSTED) of Vietnam under grant number 101.01-2017.315.

Author information

Authors and Affiliations

  1. Institute of Mathematics and Applied Sciences, Thang Long University, Hanoi, Vietnam

    Le Dung Muu

  2. Institute of Mathematics, Vietnam Academy of Science and Technology, Hanoi, Vietnam

    Xuan Thanh Le

Authors
  1. Le Dung Muu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Xuan Thanh Le
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Le Dung Muu.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Muu, L.D., Le, X.T. A splitting algorithm for finding fixed points of nonexpansive mappings and solving equilibrium problems. J. Fixed Point Theory Appl. 20, 130 (2018). https://doi.org/10.1007/s11784-018-0612-8

Download citation

  • Published:

  • DOI: https://doi.org/10.1007/s11784-018-0612-8

Keywords

Mathematics Subject Classification

Search

Navigation