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Vector Variational Inequalities in a Hausdorff Topological Vector Space

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Vector Variational Inequalities and Vector Equilibria

Part of the book series: Nonconvex Optimization and Its Applications ((NOIA,volume 38))

Abstract

Using Fan-Browder type fixed point theorem, we prove two theorems on the existence of solutions of Vector Variational Inequality in a Hausdorff topological vector space. Our results are fairly general enough to sharpen and cover earlier corresponding results of many authors. In particular, our results genaralize recent results of Lai and Yao, Yu and Yao. In addition, the equivalent relation between solutions of Generalized Minty Vector Variational Inequality and generalized vector-minimum points of Vector Optimization Problems is shown.

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References

  1. Baiocchi C. and Capelo A., “Variational and Quasi-Variational Inequalities. Applications to Free-Boundary Problems”. J. Wiley, 1994.

    Google Scholar 

  2. Browder. F.E., “The fixed point theory of multivalued mappings in topological vector space”. Mathem. Ann., Vol. 177, 1968, pp. 283–301.

    Article  MathSciNet  MATH  Google Scholar 

  3. Chen G.-Y., “Existence of solutions for a vector variational inequality: an extension of the Hartman-Stampacchia Theorem”. Jou. Optimiz. Theory Appls., Vol. 74, 1992, pp. 445–456.

    Article  MATH  Google Scholar 

  4. Chen G.-Y. and Li S.J., “Existence of solutions for generalized vector quasivariational inequality”. Jou. Optimiz. Theory Appls., Vol. 90, 1996, pp. 321–334.

    Article  MATH  Google Scholar 

  5. Fan Ky, “A generalization of Tychonoff’s fixed point theorem”. Mathem. Ann., Vol. 142, 1961, pp. 305–310.

    Article  MATH  Google Scholar 

  6. Giannessi F., “Theorems of alternative, quadratic programs and complementarity problems”. In “Variational Inequalities and Complementarity Problems”, R. W. Cottle, F. Giannessi and J.-L. Lions Eds., J. Wiley and Sons, Chichester, England 1980, pp. 151–186.

    Google Scholar 

  7. Giannessi F., “On Minty variational principle”. In “New Trends in Mathematical Programming”, Giannessi-Komlosi-Rapcsák Eds., Kluwer, New York, 1997, pp. 93–99.

    Google Scholar 

  8. Kelly J.L. and Namioka I., “Linear Topological Spaces”. Springer-Verlag, Berlin/ Heidelberg/New York, 1963.

    Google Scholar 

  9. Kinderlehrer D. and Stampacchia G., “An Introduction to Variational Inequalities”. Academic Press, New York, 1980.

    Google Scholar 

  10. Lai T.C. and Yao J.C., “Existence results for VVIP”. Appl. Mathem. Lett., Vol. 9, 1996, pp. 17–19.

    Article  MathSciNet  MATH  Google Scholar 

  11. Lee G.M., Kim D.S. and Kuk H., “Existence of solutions for vector optimization problems”. Jou. Mathem. Anal. Appls., Vol. 220, 1998, pp. 90–98.

    Article  MathSciNet  MATH  Google Scholar 

  12. Lee G.M., Kim D.S., Lee B.S. and Cho S.J., “Generalized vector variational inequality and fuzzy extension”. Appl. Mathem. Lett., Vol. 6, 1993, pp. 47–51.

    Article  MathSciNet  MATH  Google Scholar 

  13. Lee G.M., Kim D.S., Lee B.S. and Yen N.D., “Vector variational inequality as a tool for studying vector optimization problems”. Nonlinear Anal., Vol. 34, 1998, pp. 745–765.

    Article  MathSciNet  MATH  Google Scholar 

  14. Lee G.M., Lee B.S. and Chang S.S., “On vector quasivariational inequalities”. Jou. Mathem. Anal., Appls., Vol. 203, 1996, pp. 626–638.

    Article  MathSciNet  MATH  Google Scholar 

  15. Lin L.J., “Pre-vector variational inequalities”. Bull. Austral. Mathem. Soc., Vol. 53, 1996, pp. 63–70.

    Article  MATH  Google Scholar 

  16. Park S., “Foundations of the KKM theory via coincidences of composites of upper semicontinuous maps”. Jou. Korean Mathem. Soc., Vol. 31, 1994, pp. 493–519.

    MATH  Google Scholar 

  17. Rudin W., “Functional Analysis”. McGraw-Hill, New York, 1973.

    Google Scholar 

  18. Sawaragi Y., Nakayama H. and Tanino T., “Theory of Multiobjective Optimization”. Academic Press, New York, 1985.

    Google Scholar 

  19. Siddiqi A.H., Ansari A.H. and Khaliq A., “On vector variational inequalities”. Jou. Optimiz. Theory Appls., Vol. 84, 1995, pp. 171–180.

    Article  MathSciNet  MATH  Google Scholar 

  20. Yang X.Q., “Vector variational inequality and its duality”. Nonlinear Anal., Vol. 21, 1993, pp. 869–877.

    Article  MathSciNet  MATH  Google Scholar 

  21. Yu S.J. and Yao J.C., “On vector variational inequalities”. Jou. Optimiz. Theory Appls., Vol. 89, 1996, pp. 749–769.

    Article  MathSciNet  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Applied Mathematics, Pukyong National University, Pusan, Korea

    Gue Myung Lee

  2. Department of Applied Mathematics, Korea Maritime University, Pusan, Korea

    Sangho Kum

Authors
  1. Gue Myung Lee
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  2. Sangho Kum
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Editor information

Editors and Affiliations

  1. Department of Mathematics, University of Pisa, Pisa, Italy

    Franco Giannessi

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© 2000 Kluwer Academic Publishers

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Lee, G.M., Kum, S. (2000). Vector Variational Inequalities in a Hausdorff Topological Vector Space. In: Giannessi, F. (eds) Vector Variational Inequalities and Vector Equilibria. Nonconvex Optimization and Its Applications, vol 38. Springer, Boston, MA. https://doi.org/10.1007/978-1-4613-0299-5_17

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  • DOI: https://doi.org/10.1007/978-1-4613-0299-5_17

  • Publisher Name: Springer, Boston, MA

  • Print ISBN: 978-1-4613-7985-0

  • Online ISBN: 978-1-4613-0299-5

  • eBook Packages: Springer Book Archive

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