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Journal of Global Optimization

, Volume 52, Issue 4, pp 711–728 | Cite as

On the existence of solutions to generalized quasi-equilibrium problems

  • Truong Thi Thuy Duong
  • Nguyen Xuan TanEmail author
Article

Abstract

In this paper, we apply new results on variational relation problems obtained by D. T. Luc (J Optim Theory Appl 138:65–76, 2008) to generalized quasi-equilibrium problems. Some sufficient conditions on the existence of its solutions of generalized quasi-equilibrium problems are shown. As special cases, we obtain several results on the existence of solutions of generalized Pareto and weak quasi-equilibrium problems concerning C-pseudomonotone multivalued mappings. We deduce also some results on the existence of solutions to generalized vector Pareto and weakly quasivariational inequality and vector Pareto quasi-optimization problems with multivalued mappings.

Keywords

Generalized quasi-equilibrium problems Upper and lower quasivariational inclusions Quasi-optimization problems Upper and lower \({\mathcal{C}}\)-quasiconvex Upper and lower-quasiconvex-like multivalued mappings Upper and lower \({\mathcal{C}}\)-continuous multivalued mappings C-pseudomonotone C-strong pseudomonotone multivalued mappings 

Mathematics Subject Classification (2000)

49J27 49J53 91B50 90C48 

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References

  1. 1.
    Bianchi M., Pini R.: Coercivity conditions for equilibrium problems. J. Optim. Theory Appl. 124, 79–92 (2005)CrossRefGoogle Scholar
  2. 2.
    Blum E., Oettli W.: From optimization and variational inequalities to equilibrium problems. Math. Student 63(1–4), 123–145 (1994)Google Scholar
  3. 3.
    Duong T.T.T., Tan N.X.: On the existence of solutions to generalized quasi-equilibrium problems of type II and Related Problems. Adv. Nonlinear Var Inequal. 13(1), 29–47 (2010)Google Scholar
  4. 4.
    Fang Y.P., Huang N.J.: Existence results for generalized implicit vector variational inequalities with multivalued mappingpings. Indian J. Pure Appl. Math. 36, 629–640 (2005)Google Scholar
  5. 5.
    Ferro F.: A minimax theorem for vector-valued functions. J. Optim. Theory Appl. 60, 19–31 (1989)CrossRefGoogle Scholar
  6. 6.
    Hadjisavvas N.: Continuity and maximality properties of pseudomonotone operators. J. Convex Anal. 10, 465–475 (2003)Google Scholar
  7. 7.
    Li S.J., Zeng J.: Existenceof solutions for generalized vector quasi-equilibrium problems. Optim. Lett. 2(3), 341–349 (2008)CrossRefGoogle Scholar
  8. 8.
    Li X.B., Li S.J.: Existence of solutions for generalized vector quasi-equilibrium problems. Optim. Lett. 4(1), 17–28 (2010)CrossRefGoogle Scholar
  9. 9.
    Jian-Wen P., Soon-Yi W.: The generalized Tykhonov well-posedness for system of vector quasi-equilibrium problems. Optim. Lett. 4(4), 501–512 (2010)CrossRefGoogle Scholar
  10. 10.
    Pardalos P.M., Rassias T.M., Khan A.A.: Nonlinear Analysis and Variational Problems. Springer, Berlin (2010)CrossRefGoogle Scholar
  11. 11.
    Lin, L.J., Tan, N.X.: On quasivariational inclusion problems of type I and related problems. J. Global Optim. 39 (2007)Google Scholar
  12. 12.
    Luc D.T.: An abstract problem in variational analysis. J. Optim. Theory Appl. 138, 65–76 (2008)CrossRefGoogle Scholar
  13. 13.
    Luc D.T., Tan N.X.: Existence conditions in variational inclusions with constraints. Optimization 53, 505–515 (2004)CrossRefGoogle Scholar
  14. 14.
    Luc D.T, Sarabi E., Soubeyran A.: Existence of solutions in variational relation problems without convexity. J. Math. Anal. Appl. 364(2), 544–555 (2010)CrossRefGoogle Scholar
  15. 15.
    Minh N.B., Tan N.X.: Some sufficient conditions for the existence of equilibrium points concerning multivalued. Vietnam J. Math. 28, 295–310 (2000)Google Scholar
  16. 16.
    Minh N.B., Tan N.X.: On the existence of solutions of quasivariational inclusion problems of Stampacchia type. Adv. Nonlinear Var. Inequal. 8, 1–16 (2005)Google Scholar
  17. 17.
    Park S.: Fixed points and quasi-equilibrium problems. Nonlinear Oper. Theory. Math. Com. Model. 32, 1297–1304 (2000)Google Scholar
  18. 18.
    Tan N.X.: On the existence of solutions of quasi-variational inclusion problems. J. Opt. Theory Appl. 123, 619–638 (2004)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC. 2011

Authors and Affiliations

  1. 1.Vinh Technical Teachers Training UniversityVinh CityVietnam
  2. 2.Institute of MathematicsHanoiVietnam

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