Skip to main content

Advertisement

SpringerLink
Log in

Generalized vector mixed general quasi-variational-like inequalities in Hausdorff topological vector spaces

  • Original Paper
  • Published:
Optimization Letters Aims and scope Submit manuscript

Abstract

In this paper, we consider a generalized vector mixed general quasi-variational-like inequality in Hausdorff topological vector spaces. By using maximal element theorem, we prove existence theorems for two types of generalized vector mixed general quasi-variational-like inequalities without monotonicity and compactness.

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. Baiochi C., Capelo A.: Variational and Quasi Variational Inequalities. Wiley, New York (1984)

    Google Scholar 

  2. Cottle R.W., Giannessi F., Lions J.L.: Variational Inequalities and Complementarity Problems. John Wiley and Sons, New York (1980)

    MATH  Google Scholar 

  3. Ding X.P., Tarafdar E.: Generalized vector variational-like inequalities without monotonicity. In: Giannessi, F. (eds) Vector Variational Inequalities and Vector Equilibrium, Mathematical Theories, pp. 113–123. Kluwer, Dordrecht (2000)

    Chapter  Google Scholar 

  4. Ding X.P., Tarafdar E.: Generalized vector variational-like inequalities with C x -η-pseudomonotone mappings. In: Giannessi, F. (eds) Vector Variational Inequalities and Vector Equilibrium, Mathematical Theories, pp. 125–140. Kluwer, Dordecht (2000)

    Chapter  Google Scholar 

  5. Lee G.M., Lee B.S., Chang S.S.: On vector quasivariational inequalities. J. Math. Anal. Appl. 203, 626–639 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  6. Luo Q.: Generalized vector variational-like inequalities. In: Giannessi, F. (eds) Vector Variational Inequalities and Vector Equilibrium, Mathematical Theories, pp. 363–369. Kluwer, Dordrecht (2000)

    Google Scholar 

  7. Patriksson M.: Nonlinear Programming and Variational Inequality problems: A unified approach. Kluwer, Dordrecht (1999)

    MATH  Google Scholar 

  8. Chen, G.Y., Cheng, G.M.: Vector variational inequalities and vector optimization. In: Lecture Notes in Economics and Mathematical Systems, vol. 285, pp. 408–456 (1987)

  9. Lee G.M., Kim D.S., Cho S.J.: Generalized vector variational inequalities and fuzzy extensions. Appl. Math. Lett. 6(6), 47–51 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  10. Khan M.F., Khan M.F.: On generalized vector variational like inequalities. Nonlinear Anal. 59, 879–889 (2004)

    MathSciNet  MATH  Google Scholar 

  11. Yang X.Q.: Generalized convex functions and vector variational inequality. J. Optim. Theory Appl. 79, 563–580 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  12. Ding X.P.: The generalized vector quasi-variational-like inequalities. Comput. Math. Appl. 37, 57–67 (1999)

    Article  MATH  Google Scholar 

  13. Peng J.W.: Equilibrium problems on W space. Math. Appl. 12(3), 81–87 (1999)

    MathSciNet  MATH  Google Scholar 

  14. Usman F., Khan S.A.: A generalized mixed vector variational-like inequality problem. Nonlinear Anal. 71(11), 5354–5362 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  15. Irfan S.S., Ahmad R.: Generalized multivalued variational-like inequalities. J. Glob. Optim. 46(1), 25–30 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  16. Chen G.Y., Li S.J.: Existence of solutions for a generalized vector variational inequality. J. Optim Th. Appl. 90, 321–334 (1996)

    Article  MATH  Google Scholar 

  17. Lee G.M., Kim D.S., Lee B.S.: Generalized vector variational inequalities. Appl. Math. Lett. 9(1), 39–42 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  18. Noor M.A.: Multivalued general equilibrium problems. J. Math. Anal. Appl. 283, 140–149 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  19. Moudafi. A., Théra, M.: Proximal and dynamical approaches to equilibrium problems. In: Lecture Notes in Economics and Mathematical Systems, vol. 477, pp. 187–201, Springer, Berlin (1999)

  20. Kazmi K.R., Khan F.A.: Existence and iterative approximation of solutions of generalized mixed equilibrium problems. Comput. Math. Appl. 56, 1314–1321 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  21. 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)

    Article  MathSciNet  MATH  Google Scholar 

  22. Ding X.P.: Iterative algorithm of solutions for generalized mixed implicit equilibrium-like problems. Appl. Math. Comput. 162(2), 799–809 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  23. 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)

    Article  MathSciNet  MATH  Google Scholar 

  24. Ding X.P.: Approximation solvability of system of generalized mixed implicity equilibrium problems in Banach spaces. J. Sichuan Normal Univ. (N.S.) 34(1), 1–9 (2011)

    Google Scholar 

  25. László S.: Some existence results of solutions for general variational inequalities. J. Optim. Theory Appl. 150, 425–443 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  26. Peng, J.M., Yang, X.M.: Generalized vector quasi-variational-like inequalities. J. Inequal. Appl. doi:1029-242X-2006-59387 (2006)

  27. Huang N.J., Li J., Thompson H.B.: Generalized vector F-variational inequalities and vector F-completementarity problems for point to set mappings. Math. Comput. Modelling 48, 908–917 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  28. Yang X.Q., Yao J.C.: Gap functions and existence of solutions to set valued vector variational inequalities. J. Optim. Theory Appl. 11, 407–417 (2002)

    Article  MathSciNet  Google Scholar 

  29. Chadli O., Chiang Y., Huang S.: Topological pseudomonotonicity and vector equilibrium problems. J. Math. Anal. Appl. 270, 435–450 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  30. Chen G.Y., Huang X.X., Yang X.Q.: Vector Optimization Set-Valued and Variational Analysis. Springer, Berlin (2005)

    MATH  Google Scholar 

  31. Göpfert A., Tammer C., Riahi H., Zălinescu C.: Variational Methods in Partially Ordered Spaces. Springer, New York (2003)

    MATH  Google Scholar 

  32. Giannessi F., Maugeri A., Pardalos P.M.: Equilibrium Problems and Variational Models. Kluwer, Dordrecht (2001)

    Google Scholar 

  33. Pardalos, P.M., Rassias, T.M., Khan, A.A. (eds.): Nonlinear Analysis and Variational Problems. Springer. http://www.springer.com/mathematics/applications/book/978-1-4419-0157-6 (2010)

  34. Gibert R.P., Panagiotopoulos P.D., Pardalos P.M.: From Convexity to Nonconvexity. Kluwer, Dordrecht (2001)

    Book  Google Scholar 

  35. Peng J.W., Wu S.Y.: The generalized Tykhonov well-posedness for system of vector quasi-equilibrium problems. Optim. Lett. 4(4), 501–512 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  36. Mishra S.K., Wang S.Y., Lai K.K.: On non-smooth α-invex functions and vector variational-like inequality. Optim. Lett. 2(1), 91–98 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  37. Tian G.Q., Zhou J.X.: Quasi-variational inequalities without the concavity assumptions. J. Math. Anal. Appl. 132, 289–299 (1993)

    Article  MathSciNet  Google Scholar 

  38. Zhou Z.X., Chen G.: Diagonal convexity conditions for problems in convex analysis and quasi- variational inequalities. J. Math. Anal. Appl. 132, 219–228 (1988)

    Article  Google Scholar 

  39. Aubin J.P., Ekland I.: Applied Nonlinear Analysis. Wiley-Interscience, New York (1984)

    MATH  Google Scholar 

  40. Su C.H., Sehgal V.M.: Some fixed points theorems for condensing multifunctions in locally convex spaces. Proc. Am. Math. Soc. 50, 150–154 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  41. Kim W.K.: Existence of maximal element and equilibrium for a nonparacompact N-person game. Proc. Am. Math. Soc. 116, 797–807 (1992)

    MATH  Google Scholar 

  42. Giannessi F.: Vector Variational Inequalities and Vector Equilibrium, Mathematical Theories. Kluwer Academic Publishers, Dordrecht (2000)

    Book  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. College of Mathematics and Software Science, Sichuan Normal University, Chengdu, Sichuan, 610068, People’s Republic of China

    Xie Ping Ding

  2. Department of Mathematics, Aligarh Muslim University, Aligarh, 202002, India

    Salahuddin

Authors
  1. Xie Ping Ding
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Salahuddin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Xie Ping Ding.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Ding, X.P., Salahuddin Generalized vector mixed general quasi-variational-like inequalities in Hausdorff topological vector spaces. Optim Lett 7, 893–902 (2013). https://doi.org/10.1007/s11590-012-0464-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11590-012-0464-x

Keywords

Search

Navigation