Location via proxy:   [ UP ]  
[Report a bug]   [Manage cookies]                
Skip to main content
Springer Nature Link
Log in

Continuity of the solution mapping to parametric generalized vector equilibrium problems

  • Published:
Journal of Global Optimization Aims and scope Submit manuscript

Abstract

In this paper, two kinds of parametric generalized vector equilibrium problems in normed spaces are studied. The sufficient conditions for the continuity of the solution mappings to the two kinds of parametric generalized vector equilibrium problems are established under suitable conditions. The results presented in this paper extend and improve some main results in Chen and Gong (Pac J Optim 3:511–520, 2010), Chen and Li (Pac J Optim 6:141–152, 2010), Chen et al. (J Glob Optim 45:309–318, 2009), Cheng and Zhu (J Glob Optim 32:543–550, 2005), Gong (J Optim Theory Appl 139:35–46, 2008), Li and Fang (J Optim Theory Appl 147:507–515, 2010), Li et al. (Bull Aust Math Soc 81:85–95, 2010) and Peng et al. (J Optim Theory Appl 152(1):256–264, 2011).

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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

    Google Scholar 

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

    Article  Google Scholar 

  3. Anh L.Q., Khanh P.Q.: Various kinds of semicontinuity of the solution sets of parametric multivalued symmetric vector quasiequilibrium problems. J. Glob. Optim. 41, 539–558 (2008)

    Article  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  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  Google Scholar 

  6. Berge C.: Topological Spaces. Oliver and Boyd, London (1963)

    Google Scholar 

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

    Google Scholar 

  8. Chai, Y.F., Cho, Y.J., Li, J.: Some characterizations of ideal point in vector optimization problems. J. Inequal. Appl. 2008(231845):10 (2008)

    Google Scholar 

  9. Chen B., Gong X.H.: Continuity of the solution set to parametric set-valued weak vector equilibrium problems. Pac. J. Optim. 3, 511–520 (2010)

    Google Scholar 

  10. Chen, B., Gong, X.H., Yuan, S.M.: Connectedness and compactness of weak efficient solutions for set-valued vector equilibrium problems. J. Inequal. Appl. 2008(581849):15 (2008)

    Google Scholar 

  11. Chen C.R., Li S.J.: On the solution continuity of parametric generalized systems. Pac. J. Optim. 6, 141–152 (2010)

    Google Scholar 

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

    Article  Google Scholar 

  13. Chen C.R., Li S.J., Fang Z.M.: On the solution semicontinuity to a parametric generalized vector quasivariational inequality. Comput. Math. Appl. 60, 2417–2425 (2010)

    Article  Google Scholar 

  14. 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  Google Scholar 

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

    Google Scholar 

  16. Chen J.W., Cho Y.J., Kim J.K., Li J.: Multiobjective optimization problems with modified objective functions and cone constraints and applications. J. Glob. Optim. 49, 137–147 (2011)

    Article  Google Scholar 

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

    Article  Google Scholar 

  18. Fan K.: Some properties of convex sets related to fixed point theorems. Math. Ann. 266, 519–537 (1984)

    Article  Google Scholar 

  19. Fang Y.P., Huang N.J.: Feasibility and solvability of vector variational inequalities with moving cones in Banach spaces. Nonlinear Anal. 70, 2024–2034 (2009)

    Article  Google Scholar 

  20. Ferro F.: A minimax theorem for vector-valued functions. J. Optim. Theory Appl. 60, 19–31 (1989)

    Article  Google Scholar 

  21. Giannessi F.: Theorems of the alternative quadratic programs and complementarity problems. In: Cottle, R.W., Giannessi, F., Lions, J.L. (eds) Variational Inequalities and Complementarity Problems, Wiley, New York (1980)

    Google Scholar 

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

    Article  Google Scholar 

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

    Article  Google Scholar 

  24. 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  Google Scholar 

  25. Huang N.J., Fang Y.P.: On vector variational-like inequalities in reflexive Banach spaces. J. Glob. Optim. 32, 495–505 (2005)

    Article  Google Scholar 

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

    Article  Google Scholar 

  27. Huang N.J., Li J., Yao J.C.: Gap functions and existence of solutions to a system of vector equilibrium problems. J. Optim. Theory Appl. 133, 201–212 (2007)

    Article  Google Scholar 

  28. Khanh P.Q., Luu L.M.: Upper semicontinuity of the solution set to parametric vector quasivariational inequalities. J. Glob. Optim. 32, 569–580 (2005)

    Article  Google Scholar 

  29. Kimura K., Yao J.C.: Sensitivity analysis of solution mappings of parametric vector quasiequilibrium problems. J. Glob. Optim. 41, 187–202 (2008)

    Article  Google Scholar 

  30. 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  Google Scholar 

  31. Li S.J., Liu H.M., Chen C.R.: Lower semicontinuity of parametric generalized weak vector equilibrium problems. Bull. Aust. Math. Soc 81, 85–95 (2010)

    Article  Google Scholar 

  32. Li X.B., Li S.J.: Continuity of approximate solution mappings for parametric equilibrium problems. J. Glob. Optim. 51, 541–548 (2011)

    Article  Google Scholar 

  33. Peng, Z.Y., Yang, X.M., Peng, J.W.: On the lower semicontinuity of the solution mappings to parametric weak generalized Ky Fan inequality. J. Optim. Theory Appl. 152, 256–264 (2011)

    Article  Google Scholar 

  34. Xu S., Li S.J.: A new proof approach to lower semicontinuity for parametric vector equilibrium problems. Optim. Lett. 3, 459–543 (2009)

    Article  Google Scholar 

  35. Zhong R.Y., Huang N.J.: Lower semicontinuity for parametric weak vector variational inequalities in reflexive Banach spaces. J. Optim. Theory Appl. 149, 564–579 (2011)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Sichuan University, Chengdu, 610064, Sichuan, People’s Republic of China

    Bin Chen & Nan-jing Huang

Authors
  1. Bin Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Nan-jing Huang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Nan-jing Huang.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Chen, B., Huang, Nj. Continuity of the solution mapping to parametric generalized vector equilibrium problems. J Glob Optim 56, 1515–1528 (2013). https://doi.org/10.1007/s10898-012-9904-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10898-012-9904-5

Keywords

Mathematics Subject Classification

Search

Navigation