Skip to main content
Log in

Existence and stability of solutions for a class of generalized vector equilibrium problems

  • Published:
Positivity Aims and scope Submit manuscript

Abstract

In this paper, two existence theorems concerning the strong efficient solutions and the weakly efficient solutions of generalized vector equilibrium problems are derived by using the Fan-KKM Theorem and an existence theorem for the efficient solutions of generalized vector equilibrium problems is established by using the scalarization method. Moreover, the lower semicontinuity of the strong efficient solution mapping and the weakly efficient solution mapping to parametric generalized vector equilibrium problems are showed under suitable conditions with neither monotonicity nor any information of the solution mappings. Finally, some applications to the vector optimization problems and the Stackelberg equilibrium problem are also given.

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

Similar content being viewed by others

References

  1. Anh, L.Q., Khanh, P.Q.: Semicontinuity of the solution set of parametric multivalued vector quasiequilibrium problems. J. Math. Anal. Appl. 294, 699–711 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  2. Anh, L.Q., Khanh, P.Q.: On the stability of the solution sets of general multivalued vector quasiequilibrium problems. J. Optim. Theory Appl. 135, 271–284 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  3. Anh, L.Q., Khanh, P.Q.: Continuity of solution maps of parametric quasiequilibrium problems. J. Global Optim. 46, 247–259 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  4. Ansari, Q.H.: Existence of solutions of systems of generalized implicit vector quasi-equilibrium problems. J. Math. Anal. Appl. 341, 1271–1283 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  5. Ansari, Q.H., Konnov, I.V., Yao, J.C.: Existence of a solution and variational principles for vector equilibrium problems. J. Optim. Theory Appl. 110, 481–492 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  6. Ansari, Q.H., Yao, J.C.: An existence result for the generalized vector equilibrium problem. Appl. Math. Lett. 12, 53–56 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  7. Ansari, Q.H., Yao, J.C. (eds.): Recent Developments in Vector Optimization. Springer, Berlin, New York, Heidelberg (2012)

  8. Aubin, J.P., Ekeland, I.: Applied Nonlinear Analysis. Wiley, New York (1984)

    MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  10. Chen, B., Huang, N.J.: Continuity of the solution mapping to parametric generalized vector equilibrium problems. J. Global Optim. 56, 1515–1528 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  11. Chen, B., Huang, N.J.: Connectedness of the solution set for symmetric vector quasiequilibrium problems. Pacific J. Optim. 9, 29–45 (2013)

    MathSciNet  MATH  Google Scholar 

  12. Chen, C.R., Li, S.J., Teo, K.L.: Solution semicontinuity of parametric generalized vector equilibrium problems. J. Global Optim. 45, 309–318 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  13. Chen, G.Y.: Existence of solutions for a vector variational inequality: an extension of Hartman–Stampacchia theorem. J. Optim. Theory Appl. 74, 445–456 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  14. Chen, G.Y., Huang, X.X., Yang, X.Q.: Vector Optimization: Set-Valued and Variational Analysis. Lecture Notes in Economics and Mathematical Systems, vol. 541. Springer, Berlin (2005)

  15. Chen, G.Y., Yang, X.Q., Yu, H.: A nonlinear scalarization function and generalized quasi-vector equilibrium problems. J. Global Optim. 32, 456–466 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  16. Cheng, Y.H., Zhu, D.L.: Global stability results for the weak vector variational inequality. J. Global Optim. 32, 543–550 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  17. Chiang, Y., Chadli, O., Yao, J.C.: Generalized vector equilibrium problems with trifunctions. J. Global Optim. 30, 135–154 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  18. Ding, X.P., Park, J.Y.: Generalized vector equilibrium problems in generalized convex spaces. J. Optim. Theory Appl. 120, 327–353 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  19. Farajzadeh, A.P., Amini-Harandi, A.: On the generalized vector equilibrium problems. J. Math. Anal. Appl. 344, 999–1004 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  20. Fan, X.D., Cheng, C.Z., Wang, H.J.: Stability analysis for vector quasiequilibrium problems. Positivity 17, 365–379 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  21. Fang, Y.P., Huang, N.J.: Strong vector variational inequalities in Banach spaces. Appl. Math. Lett. 19, 362–368 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  22. Fang, Y.P., Huang, N.J.: Vector equilibrium problems, minimal element problems and least element problems. Positivity 11, 251–268 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  23. Fu, J.Y.: Generalized vector quasi-equilibrium problems. Math. Methods Oper. Res. 52, 57–64 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  24. Giannessi, F.: Theorems of Alternative, Quadratic Programs and Complementarity Problems. In: Cottle, R.W., Giannessi, F., Lions, J.L. (eds.) Variational Inequalities and Complementarity Problems, pp. 151–186. Wiley, New York (1980)

    Google Scholar 

  25. Giannessi, F. (ed.): Vector Variational Inequalities and Vector Equilibria. Kluwer Academic Publishers, Dordrecht/Boston/London (2000)

  26. Gong, X.H.: Strong vector equilibrium problems. J. Global Optim. 36, 339–349 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  27. Gong, X.H.: Continuity of the solution set to parametric weak vector equilibrium problems. J. Optim. Theory Appl. 139, 35–46 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  28. Gong, X.H., Yao, J.C.: Lower semicontinuity of the set of efficient solutions for generalized systems. J. Optim. Theory Appl. 138, 197–205 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  29. G\(\ddot{\rm {o}}\)linescu, C.: Variational Methods in Partially Ordered Spaces. Springer, Berlin Heidelberg/New York (2003)

  30. G\(\acute{{\rm {o}}}\)rniewicz, L.: Topological Fixed Point Theorey of Multivalued Mappings. Kluwer Academic Publishers, Dordrecht/Boston/London (1999)

  31. Guu, S.M., Li, J.: Vector variational-like inequalities with generalized bifunctions defined on nonconvex sets. Nonlinear Anal. TMA 71, 2847–2855 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  32. Han, Y., Gong, X.H.: Lower semicontinuity of solution mapping to parametric generalized strong vector equilibrium problems. Appl. Math. Lett. 28, 38–41 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  33. Hou, S.H., Yu, H., Chen, G.Y.: On vector quasi-equilibrium problems with set-valued maps. J. Optim. Theory Appl. 119, 485–498 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  34. Hu, R., Fang, Y.P.: A characterization of nonemptiness and boundedness of the solution sets for equilibrium problems. Positivity 17, 431–441 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  35. Huang, N.J., Li, J., Thompson, H.B.: Stability for parametric implicit vector equilibrium problems. Math. Comput. Model. 43, 1267–1274 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  36. Konnov, I.V., Yao, J.C.: Existence of solutions for generalized vector equilibrium problems. J. Math. Anal. Appl. 233, 328–335 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  37. Li, X.B., Li, S.J.: Existence of solutions for generalized vector quasi-equilibrium problems. Optim. Lett. 4, 17–28 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  38. Li, S.J., Teo, K.L., Yang, X.Q.: Generalized vector quasi-equilibrium problems. Math. Methods Oper. Res. 61, 385–397 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  39. Li, S.J., Fang, Z.M.: Lower semicontinuity of the solution mappings to a parametric generalized Ky Fan inequality. J. Optim. Theory Appl. 147, 507–515 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  40. Li, S.J., Liu, H.M., Zhang, Y., Fang, Z.M.: Continuity of the solution mappings to parametric generalized strong vector equilibrium problems. J. Global Optim. 55, 597–610 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  41. Lin, L.J., Chen, H.L.: The study of KKM theorems with applications to vector equilibrium problems and implict vector variational inequalities problems. J. Global Optim. 32, 135–157 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  42. Nagy, S.: Stackelberg equilibia via variational inequalities and projections. J. Global Optim. 57, 821–828 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  43. Xu, Y.D., Li, S.J.: On the lower semicontinuity of the solution mappings to a parametric generalized strong vector equilibrium problem. Positivity 17, 341–353 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  44. Zeng, L.C., Yao, J.C.: An existence result for generalized vector equilibrium problems without pseudomonotonicity. Appl. Math. Lett. 19, 1320–1326 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  45. Zhang, W.Y., Fang, Z.M., Zhang, Y.: A note on the lower semicontinuity of efficient solutions for parametric vector equilibrium problems. Appl. Math. Lett. 26, 469–472 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  46. Zhou, J.X., Chen, G.: Diagonal convexity conditions for problems in convex analysis and quasi-variational inequalities. J. Math. Anal. Appl. 132, 213–225 (1988)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

The authors are grateful to the editor and the referees for their valuable comments and suggestions.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Nan-jing Huang.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Han, Y., Huang, Nj. Existence and stability of solutions for a class of generalized vector equilibrium problems. Positivity 20, 829–846 (2016). https://doi.org/10.1007/s11117-015-0389-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11117-015-0389-6

Keywords

Mathematics Subject Classification

Navigation