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Generalized vector implicit quasi complementarity problems

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Abstract

In this paper, we introduce and study a generalized class of vector implicit quasi complementarity problem and the corresponding vector implicit quasi variational inequality problem. By using Fan-KKM theorem, we derive existence of solutions of generalized vector implicit quasi variational inequalities without any monotonicity assumption and establish the equivalence between those problems in Banach spaces.

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References

  1. Baiocchi C., Capelo A.: Variational and Quasivariational Inequalities. Wiley, New York (1984)

    Google Scholar 

  2. Bnouhachem A., Noor M.A.: Numerical method for general mixed quasi variational inequalities. Appl. Math. Comput. 204, 27–36 (2008)

    Article  Google Scholar 

  3. Carbone A.: A note on complementarity problem. Int. J. Math. Math. Sci. 21, 621–623 (1998)

    Article  Google Scholar 

  4. Chang S.S., Huang N.J.: Generalized multivalued implicit complementarity problems in Hilbert spaces. Math. Japonica 36, 1093–1100 (1991)

    Google Scholar 

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

    Article  Google Scholar 

  6. Cottle R.W., Dantzig G.B.: Complementarity pivot theory of mathematical programming. Linear Algeb. Appl. 1, 163–185 (1968)

    Google Scholar 

  7. Fan K.: A generalization of Tychonoff’s fixed point theorem. Math. Ann. 142, 305–310 (1961)

    Article  Google Scholar 

  8. Farajzadeh A.P., Zafarani J.: Vector F-implicit complementarity problems in topological vector spaces. Appl. Math. Lett. 20, 1075–1081 (2007)

    Article  Google Scholar 

  9. Farajzadeh A.P., Noor M.A., Zainab S.: Mixed quasi complementarity problems in topological vector spaces. J. Global Optim. 45, 229–235 (2009)

    Article  Google Scholar 

  10. Giannessi F., Maugeri A., Pardalos P.M.: Equilibrium Problems: Nonsmooth Optimization and Variational Inequality Models. Kluwer Academic Publishers, Dordrecht (2001)

    Google Scholar 

  11. Huang N.J., Li J.: F-implicit complementarity problems in Banach spaces. Z. Anal. Anwendungen. 23, 293–302 (2004)

    Article  Google Scholar 

  12. Huang N.J., Fang Y.P.: Strong vector F-complementary problem and least element problem of feasible set. Nonlin. Anal. 61, 901–918 (2005)

    Article  Google Scholar 

  13. Huang N.J., Li J., O’ Regan D.: Generalized f-complementarity problems in Banach spaces. Nonlin. Anal. 68, 3828–3840 (2008)

    Article  Google Scholar 

  14. Huang N.J., Yang X.Q., Chan W.K.: Vector complementarity problems with a variable ordering relation. Europ. J. Oper. Res. 176, 15–26 (2007)

    Article  Google Scholar 

  15. Isac G.: Topological Methods in Complementarity Theory. Kluwer Academic Publishers, Dordrecht (2000)

    Google Scholar 

  16. Itoh S., Takahashi W., Yanagi K.: Variational inequalities and complementarity problems. J. Math. Soc. Jpn. 30, 23–28 (1978)

    Article  Google Scholar 

  17. Karamardian S.: Generalized complementarity problem. J. Optim. Theory Appl. 8, 161–168 (1971)

    Article  Google Scholar 

  18. Khan S.A.: Generalized vector complementarity-type problems in topological vector spaces. Comput. Math. Appl. 59, 3595–3602 (2010)

    Article  Google Scholar 

  19. Lemke C.E.: Bimatrix equilibrium points and mathematical programming. Management Sci. 11, 681–689 (1965)

    Article  Google Scholar 

  20. Lee B.S., Farajzadeh A.P.: Generalized vector implicit complementarity problems with corresponding variational inequality problems. Appl. Math. Lett. 21, 1095–1100 (2008)

    Article  Google Scholar 

  21. Lee B.S., Khan M.F., Salahuddin : Vector F-implicit complementarity problems with corresponding variational inequality problems. Appl. Math. Lett. 20, 433–438 (2007)

    Article  Google Scholar 

  22. Li J., Huang N.J.: Vector F-implicit complementarity problems in Banach spaces. Appl. Math. Lett. 19, 464–471 (2006)

    Article  Google Scholar 

  23. Mosco, U.: Implicit Variational Problems and Quasi Variational Inequalities. In: Nonlinear Operators and the Calculus of Variations. Lecture Notes in Mathematics, vol. 543, pp. 83–156. Springer, Berlin (1976)

  24. Noor M.A.: Mixed quasi variational inequalities. Appl. Math. Comput. 146, 553–578 (2003)

    Article  Google Scholar 

  25. Pardalos P.M.: The linear complementarity problem. In: Gomez, S., Hennart, J.P. (eds) Advances in Optimization and Numerical Analysis, pp. 39–49. Kluwer Academic Publishers, Dordrecht (1994)

    Google Scholar 

  26. Pardalos P.M., Rassias T.M., Khan A.A.: Nonlinear Analysis and Variational Problems. Springer, New York (2010)

    Book  Google Scholar 

  27. Thera M.: Existence results for the nonlinear complementarity problem and applications to nonlinear analysis. J. Math. Anal. Appl. 154, 572–584 (1991)

    Article  Google Scholar 

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

    Article  Google Scholar 

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

    Article  Google Scholar 

  30. Yin H.Y., Xu C.X., Zhang Z.X.: The F-complementarity problem and its equivalence with the least element problem. Acta Math. Sinica 44, 679–686 (2001)

    Google Scholar 

  31. Zhang S., Shu Y.: Complementarity problems with applications to mathematical programming. Acta Math. Appl. Sinica. 15, 380–388 (1992)

    Google Scholar 

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  1. Department of Mathematics, BITS, Pilani-Dubai, P.O. Box 345055, Dubai, UAE

    Suhel Ahmad Khan

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  1. Suhel Ahmad Khan
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Khan, S.A. Generalized vector implicit quasi complementarity problems. J Glob Optim 49, 695–705 (2011). https://doi.org/10.1007/s10898-010-9557-1

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  • DOI: https://doi.org/10.1007/s10898-010-9557-1

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