Skip to main content

Advertisement

SpringerLink
Log in

Lower Semicontinuity for Parametric Weak Vector Variational Inequalities in Reflexive Banach Spaces

  • Published:
Journal of Optimization Theory and Applications Aims and scope Submit manuscript

Abstract

In this paper, we study the solution stability of parametric weak Vector Variational Inequalities with set-valued and single-valued mappings, respectively. We obtain the lower semicontinuity of the solution mapping for the parametric set-valued weak Vector Variational Inequality with strictly C-pseudomapping in reflexive Banach spaces. Moreover, under some requirements that the mapping satisfies the degree conditions, we establish the lower semicontinuity of the solution mapping for a parametric single-valued weak Vector Variational Inequality in reflexive Banach spaces, by using the degree-theoretic approach. The results presented in this paper improve and extend some known results due to Kien and Yao (Set-Valued Anal. 16:399–412, 2008) and Wong (J. Glob. Optim. 46:435–446, 2010).

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

Access this article

Log in via an institution

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. Kien, B.T., Yao, J.C.: Localization of generalized normal maps and stability of variational inequalities in reflexive Banach spaces. Set-Valued Anal. 16, 399–412 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  2. Wong, M.-M.: Lower semicontinuity of the solution map to a parametric vector variational inequality. J. Glob. Optim. 46, 435–446 (2010)

    Article  MATH  Google Scholar 

  3. Giannessi, F.: Theorems of alternative, quadratic programs and complementary problems. In: Cottle, R.W., Giannessi, F., Lions, J.C. (eds.) Variational Inequality and Complementary Problems. Wiley, New York (1980)

    Google Scholar 

  4. Chen, G.Y., Cheng, Q.M.: Vector variational inequality and vector optimization. In: Lecture Notes in Economics and Mathematical Systems, vol. 285, pp. 408–416. Springer, New York (1987)

    Google Scholar 

  5. Chen, G.Y., Yang, X.Q.: The vector complementarity problem and its equivalences with the weak minimal element. J. Math. Anal. Appl. 153, 136–158 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  6. 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)

    MATH  Google Scholar 

  7. Giannessi, F.: Vector Variational Inequalities and Vector Equilibria: Mathematical Theories. Kluwer Academic, Dordrecht (2000)

    MATH  Google Scholar 

  8. Huang, N.J., Li, J., Thompson, H.B.: Implicit vector equilibrium problems with applications. Math. Comput. Model. 37, 1343–1356 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  9. Huang, N.J., Rubinov, A.M., Yang, X.Q.: Vector optimization problems with nonconvex preferences. J. Glob. Optim. 40, 765–777 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  10. Huang, N.J., Gao, C.J.: Some generalized vector variational inequalities and complementarity problems for multivalued mappings. Appl. Math. Lett. 16, 1003–1010 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  11. Yang, X.Q.: Vector variational inequality and its duality. Nonlinear Anal. TMA 21, 729–734 (1993)

    Article  Google Scholar 

  12. Yang, X.Q.: Vector complementarity and minimal element problems. J. Optim. Theory Appl. 77, 483–495 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  13. 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 

  14. Anh, L.Q., Khanh, P.Q.: Semicontinuity of solution sets to parametric quasivariational inclusions with applications to traffic networks II: lower semicontinuities applications. Set-Valued Anal. 16, 943–960 (2008)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  16. 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 

  17. 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 

  18. 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 

  19. Li, S.J., Chen, G.Y., Teo, K.L.: On the stability of generalized vector quasivariational inequality problems. J. Optim. Theory Appl. 113, 283–295 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  20. Lee, G.M., Kim, D.S., Lee, B.S., Yen, N.D.: Vector variational inequality as a tool for studying vector optimization problems. Nonlinear Anal. TMA 34, 745–765 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  21. Kien, B.T., Wong, M.M.: On the solution stability of variational inequality. J. Glob. Optim. 39, 101–111 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  22. Kien, B.T.: Lower semicontinuity of the solution map to a parametric generalized variational inequality in reflexive Banach spaces. Set-Valued Anal. 16, 1089–1105 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  23. Brézis, H.: Équations et inéquations non linéaires dans les espaces vectoriel en dualité. Ann. Inst. Fourier 18, 115–175 (1968)

    MATH  Google Scholar 

  24. Dafermos, S.: Sensitivity analysis in variational inequalities. Math. Oper. Res. 13, 421–434 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  25. Domokos, A.: Solution sensitivity of variational inequalities. J. Math. Math. Appl. 230, 382–389 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  26. Levy, A.B.: Sensitivity of solutions to variational inequalities on Banach spaces. SIAM J. Control Optim. 38, 50–60 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  27. Robinson, S.M.: Localized normal maps and the stability of variational conditions. Set-Valued Anal. 12, 259–274 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  28. Robinson, S.M.: Solution continuity affine variational inequalities. SIAM. J. Optim. 18, 1046–1060 (2007)

    MATH  Google Scholar 

  29. Robinson, S.M., Lu, S.: Solution continuity in variational conditions. J. Glob. Optim. 40, 405–415 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  30. Yen, N.D.: Hölder continuity of solution to a parametric variational inequality. Appl. Math. Optim. 31, 245–255 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  31. Daniilidis, A., Hadjisavvas, N.: Coercivity conditions and variational inequalities. Math. Program. 86, 433–438 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  32. Rochafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1970)

    Google Scholar 

  33. Zeidler, E.: Nonlinear Functional Analysis and Its Application, II: Nonlinear Monotone Operators. Springer, Berlin (1990)

    Google Scholar 

  34. Cioranescu, I.: Geometry of Banach Spaces Duality Mappings and Nonlinear Problems. Kluwer, Norwell (1990)

    MATH  Google Scholar 

  35. Fonseca, I., Gangbo, W.: Degree Theory in Analysis and Applications. Oxford, New York (1995)

    MATH  Google Scholar 

  36. Lloyd, N.G.: Degree Theory. Cambridge University Press, Cambridge (1978)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Sichuan University, Chengdu, Sichuan, 610064, P.R. China

    Ren-you Zhong & Nan-jing Huang

Authors
  1. Ren-you Zhong
    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.

Additional information

Communicated by Nguyen Dong Yen.

This work was supported by the Key Program of NSFC (Grant No. 70831005) and the National Natural Science Foundation of China (10671135).

Rights and permissions

Reprints and permissions

About this article

Cite this article

Zhong, Ry., Huang, Nj. Lower Semicontinuity for Parametric Weak Vector Variational Inequalities in Reflexive Banach Spaces. J Optim Theory Appl 149, 564–579 (2011). https://doi.org/10.1007/s10957-011-9805-7

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10957-011-9805-7

Keywords

Search

Navigation