Skip to main content
Log in

An iterative algorithm for generalized mixed equilibrium problem

Afrika Matematika Aims and scope Submit manuscript

Abstract

In this paper, we consider a generalized mixed equilibrium problem and its related an auxiliary problem in real Hilbert space. We prove a result for the existence and uniqueness of solutions of the auxiliary problem. This result is then used to define proximal mapping for generalized mixed equilibrium problem. Further, based on this result, we give an iterative algorithm which consists of a proximal mapping technique step followed by a suitable orthogonal projection onto a moving half-space for solving generalized mixed equilibrium problem. Furthermore we prove that the sequences generated by iterative algorithm converge weakly to a solution of generalized mixed equilibrium problem. Finally, we discuss some special cases of the main result.

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

Access this article

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

Instant access to the full article PDF.

Institutional subscriptions

References

  1. Noor, M.A.: Auxiliary principle technique for equilibrium problems. J. Optim. Theory Appl. 122, 371–386 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  2. Moudafi, A., Théra, M.: Proximal and dynamical approaches to equilibrium problems, Lecture Notes in Economics and Mathematical Systems, Vol. 477, pp. 187–201, Springer, New York (1999)

  3. Moudafi, A.: Mixed equilibrium problems: Sensitivity analysis and algorithmic aspect. Comput. Math. Appl. 44, 1099–1108 (2002)

    Article  MATH  MathSciNet  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  5. Moudafi, A.: Second order differential proximal methods for equilibrium problems. J. Inequal. Pure Appl. Math. 4(1) (2003), Art. 18

    Google Scholar 

  6. Nadezhkina, N., Takahashi, W.: Strong convergence theorem by a hybrid method for nonexpansive mappings and Lipschitz continuous monotone mapping. SIAM J. Optim. 16(40), 1230–1241 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  7. Peng, J.W., Yao, J.C.: A new hybrid-extragradient method for generalized mixed equilibrium problems, fixed point problems and variational inequality problems. Taiwanese J. Math. 12(6), 1401–1432 (2008)

    MATH  MathSciNet  Google Scholar 

  8. Zhang, Q.B.: An algorithm for solving the general variational inclusion involving \(A\)-monotone operators. Comput. Math. Appl. 61, 1682–1686 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  9. Antipin, A.S.: Iterative gradient predictor type methods for computing fixed point of external mappings. In: Guddat, J., Jonden, H.Th., Nizicka, F., Still, G., Twitt F. (eds.) Parametric optimization and related topics IV[C]. Peter Lang, Frankfurt, Main, pp 11–24 (1997)

  10. Chang, S.S.: Variational Inequalities and Complementarity Problems: Theory and Applications (in Chinese). Shanghai Scientific and Technical Pres, Shanghai (1991)

    Google Scholar 

  11. Ding, X.P.: Auxiliary principle and algorithm for mixed equilibrium problems and bilevel mixed equilibrium problems in Banach Spaces. J. Optim Theory Appl 146, 347–357 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  12. Kazmi, K.R., Khaliq, A., Raouf, A.: Iterative approximation of solution of generalized mixed set-valued variational inequality problem. Math. Inequal. Appl. 10(3), 677–691 (2007)

    MATH  MathSciNet  Google Scholar 

Download references

Acknowledgments

The authors are very thankful to the referee for his valuable comments and suggestions toward the improvement of the manuscript.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to K. R. Kazmi.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Kazmi, K.R., Rizvi, S.H. An iterative algorithm for generalized mixed equilibrium problem. Afr. Mat. 25, 857–867 (2014). https://doi.org/10.1007/s13370-013-0159-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s13370-013-0159-1

Keywords

Mathematics Subject Classification (2000)

Navigation