Journal of Optimization Theory and Applications

, Volume 146, Issue 2, pp 347–357 | Cite as

Auxiliary Principle and Algorithm for Mixed Equilibrium Problems and Bilevel Mixed Equilibrium Problems in Banach Spaces

  • X. P. Ding


A new class of bilevel mixed equilibrium problems is introduced and studied in real Banach spaces. By using the auxiliary principle technique, new iterative algorithms for solving the mixed equilibrium problems and bilevel mixed equilibrium problems are suggested and analyzed. Strong convergence of the iterative sequences generated by the algorithms is proved under suitable conditions. The behavior of the solution set of the bilevel mixed equilibrium problem is also discussed.


Mixed equilibrium problems Bilevel mixed equilibrium problems Monotonicity Auxiliary principle Banach spaces 


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  1. 1.
    Moudafi, A.: Proximal methods for a class of bilevel monotone equilibrium problems. J. Glob. Optim. (2009). doi: 10.1007/s10898-009-9476-1 Google Scholar
  2. 2.
    Peng, J.W., Yao, J.C.: Some new iterative algorithms for generalized mixed equilibrium problems with strict pseudo-contractions and monotone mappings. Taiwan. J. Math. 13, 1537–1582 (2009) MathSciNetMATHGoogle Scholar
  3. 3.
    Peng, J.W., Yao, J.C.: A modified CQ method for equilibrium problems, fixed points and variational inequality. Fixed Point Theory 9, 515–531 (2008) MathSciNetMATHGoogle Scholar
  4. 4.
    Al-Homidan, S., Ansari, Q.H., Yao, J.C.: Some generalizations of Ekeland-type variational principle with applications to equilibrium problems and fixed point theory. Nonlinear Anal. Ser. A, Theory Methods Appl. 69, 126–139 (2008) CrossRefMathSciNetMATHGoogle Scholar
  5. 5.
    Zeng, L.C., Wu, S.Y., Yao, J.C.: Generalized KKM theorem with applications to generalized minimax inequalities and generalized equilibrium problems. Taiwan. J. Math. 10, 1497–1514 (2006) MathSciNetMATHGoogle Scholar
  6. 6.
    Ceng, L.C., Al-Homidan, S., Ansari, Q.H., Yao, J.C.: An iterative scheme for equilibrium problems and fixed point problems of strict pseudo-contraction mappings. J. Comput. Appl. Math. 223, 967–974 (2009) CrossRefMathSciNetMATHGoogle Scholar
  7. 7.
    Peng, J.W., Yao, J.C.: A new extragradient method For mixed equilibrium problems, fixed point problems and variational inequality problems. Math. Comput. Model. 49, 1816–1828 (2009) CrossRefMathSciNetMATHGoogle Scholar
  8. 8.
    Ceng, L.C., Mastroeni, G., Yao, J.C.: Hybrid proximal-point method for common solutions of equilibrium problems and zeros of maximal monotone operators. J. Optim. Theory Appl. 142, 431–449 (2009) CrossRefMathSciNetMATHGoogle Scholar
  9. 9.
    Peng, J.W., Yao, J.C.: A viscosity approximation scheme for system of equilibrium problems, nonexpansive mappings and monotone mappings. Nonlinear Anal., Ser. A, Theory Methods Appl. 71, 6001–6010 (2009) CrossRefMathSciNetMATHGoogle Scholar
  10. 10.
    Ding, X.P.: Iterative algorithm of solutions for generalized mixed implicit equilibrium-like problems. Appl. Math. Comput. 162(2), 799–809 (2005) CrossRefMathSciNetMATHGoogle Scholar
  11. 11.
    Ding, X.P., Lin, Y.C., Yao, J.C.: Predictor-corrector algorithms for solving generalized mixed implicit quasi-equilibrium problems. Appl. Math. Mech. 27(9), 1157–1164 (2006) CrossRefMathSciNetMATHGoogle Scholar
  12. 12.
    Xia, F.Q., Ding, X.P.: Predictor-corrector algorithms for solving generalized mixed implicit quasi-equilibrium problems. Appl. Math. Comput. 188(1), 173–179 (2007) CrossRefMathSciNetMATHGoogle Scholar
  13. 13.
    Moudafi, A.: Mixed equilibrium problems: sensitivity analysis and algorithmic aspects. Comput. Math. Appl. 44, 1099–1108 (2002) CrossRefMathSciNetMATHGoogle Scholar
  14. 14.
    Huang, N.J., Lan, H.Y., Cho, Y.J.: Sensitivity analysis for nonlinear generalized mixed implicit equilibrium problems with non-monotone set-valued mappings. J. Comput. Appl. Math. 196, 608–618 (2006) CrossRefMathSciNetMATHGoogle Scholar
  15. 15.
    Kazmi, K.R., Khan, F.A.: Existence and iterative approximation of solutions of generalized mixed equilibrium problems. Comput. Math. Appl. 56, 1314–1321 (2008) CrossRefMathSciNetMATHGoogle Scholar
  16. 16.
    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) CrossRefMathSciNetMATHGoogle Scholar
  17. 17.
    Bigi, G., Castellani, M., Kassay, G.: A dual view of equilibrium problems. J. Math. Anal. Appl. 342, 17–26 (2008) MathSciNetMATHGoogle Scholar
  18. 18.
    Peng, J.W.: Iterative algorithms for mixed equilibrium problems, strict pseudocontractions and monotone mappings. J. Optim. Theory Appl. (2009). doi: 10.1007/s10957-009-9585-5 Google Scholar
  19. 19.
    Antipin, A.S.: Iterative gradient prediction-type methods for computing fixed-point of extremal mappings. In: Guddat, J., Jonden, H.Th., Nizicka, F., Still, G., Twitt, F. (eds.) Parametric Optimization and Related Topics IV[C], pp. 11–24. Peter Lang, Frankfurt (1997) Google Scholar
  20. 20.
    Chang, S.S.: Variational Inequalities and Complementary Problems: Theory and Applications. Shanghai Scientific and Technical Press, Shanghai (1991). (In Chinese) Google Scholar

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© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.College of Mathematics and Software ScienceSichuan Normal UniversityChengduChina

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