Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
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NONEMPTY INTERSECTION THEOREMS AND SYSTEM OF GENERALIZED VECTOR EQUILIBRIUM PROBLEMS IN FC-SPACES

  • He, Rong-Hua (Department of Mathematics Chengdu University of Information Technology) ;
  • Li, Hong-Xu (Department of Mathematics Sichuan University)
  • Received : 2010.09.04
  • Published : 2013.01.31
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Abstract

By using some existence theorems of maximal elements for a family of set-valued mappings involving a better admissible set-valued mapping under noncompact setting of FC-spaces, we present some non-empty intersection theorems for a family $\{G_i\}_{i{\in}I}$ in product FC-spaces. Then, as applications, some new existence theorems of equilibrium for a system of generalized vector equilibrium problems are proved in product FC-spaces. Our results improve and generalize some recent results.

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References

  1. Q. H. Ansari, I. V. Konnov, and J.-C. Yao, On generalized vector equilibrium problems, Nonlinear Anal. 47 (2001), no. 1, 543-554. https://doi.org/10.1016/S0362-546X(01)00199-7
  2. Q. H. Ansari, S. Schaible, and J. Y. Yao, System of vector equilibrium problems and its applications, J. Optim. Theory Appl. 107 (2000), no. 3, 547-557. https://doi.org/10.1023/A:1026495115191
  3. Q. H. Ansari, A. H. Siddiqi, and S. Y. Wu, Existence and duality of generalized vector equilibrium problems, J. Math. Anal. Appl. 259 (2001), no. 1, 115-126. https://doi.org/10.1006/jmaa.2000.7397
  4. C. M. Chen, KKM property and fixed point theorems in metric spaces, J. Math. Anal. Appl. 323 (2006), no. 2, 1231-1237. https://doi.org/10.1016/j.jmaa.2005.11.030
  5. C. M. Chen and T. H. Chang, Some results for the family KKM(X, Y ) and the ${\Phi}-mapping$, J. Math. Anal. Appl. 329 (2007), no. 1, 92-101. https://doi.org/10.1016/j.jmaa.2006.06.040
  6. C. M. Chen, T. H. Chang, and Y. P. Liao, Coincidence theorems, generalized variational inequality theorems and minimax inequality theorems for the ${\Phi}-mapping$ on G-convex spaces, Fixed Point Theory Appl. 2007 (2007), Art. ID 78696, 13 pp.
  7. P. Deguire and M. Lassonde, Families selectantes, Topol. Methods Nonlinear Anal. 5 (1995), no. 2, 261-269. https://doi.org/10.12775/TMNA.1995.017
  8. P. Deguire, K. K. Tan, and X. Z. Yuan, The study of maximal elements, fixed points for $L_S-majorized$ mappings and their applications to minimax and variational inequalities in product topological spaces, Nonlinear Anal. 37 (1999), no. 7, 933-951. https://doi.org/10.1016/S0362-546X(98)00084-4
  9. X. P. Ding, New H − KKM theorems and their applications to geometric property, coincidence theorems, minimax inequality and maximal elements, Indian J. Pure Appl. Math. 26 (1995), no. 1, 1-19.
  10. X. P. Ding, Fixed points, minimax inequalities and equilibria of noncompact abstract economies, Taiwanese J. Math. 2 (1998), no. 1, 25-55. https://doi.org/10.11650/twjm/1500406893
  11. X. P. Ding, Maximal elements for GB-majorized mappings in product G-convex spaces. II, Appl. Math. Mech. 24 (2003), no. 9, 1017-1024. https://doi.org/10.1007/BF02437634
  12. X. P. Ding, Nonempty intersection theorems and system of generalized vector equilibrium problems in product G-convex spaces, Appl. Math. Mech. 25 (2004), no. 6, 618-626. https://doi.org/10.1007/BF02438204
  13. X. P. Ding, Maximal elements theorems in product FC-spaces and generalized games, J. Math. Anal. Appl. 305 (2005), no. 1, 29-42. https://doi.org/10.1016/j.jmaa.2004.10.060
  14. X. P. Ding, Nonempty intersection theorems and generalized multi-objective games in product FC-spaces, J. Global Optim. 37 (2007), no. 1, 63-73.
  15. X. P. Ding and J. Y. Park, Fixed points and generalized vector equilibrium problems in generalized convex spaces, Indian J. Pure Appl. Math. 34 (2003), no. 6, 973-990.
  16. Ky Fan, Some properties of convex sets related to fixed point theorems, Math. Ann. 266 (1984), no. 4, 519-537. https://doi.org/10.1007/BF01458545
  17. F. Giannessi, Theorems of alternative, quadratic programs and complementarity problems, In Variational inequalities and complementarity problems (Proc. Internat. School, Erice, 1978), pp. 151-186, Wiley, Chichester, 1980.
  18. F. Giannessi, Vector Variational Inequalities and Vector Equilibria Mathematics Theories, Kiuwer Academic Publishers, London, 2000.
  19. J. Guillerme, Nash equilibrium for set-valued maps, J. Math. Anal. Appl. 187 (1994), no. 3, 705-715. https://doi.org/10.1006/jmaa.1994.1384
  20. R. H. He and Y. Zhang, Some maximal elements theorems in FC-spaces, J. Inequal. Appl., doi:10.1155/2009/905605.
  21. M. Lassonde, Fixed point for Kakutani factorizable multifunctions, J. Math. Anal. Appl. 152 (1990), no. 1, 46-60. https://doi.org/10.1016/0022-247X(90)90092-T
  22. L. J. Lin, Z. T. Yu, and G. Kassay, Existence of equilibria for multivalued mappings and its application to vectorial equilibria, J. Optim. Theory Appl. 114 (2002), no. 1, 189-208. https://doi.org/10.1023/A:1015420322818
  23. S. Park and H. Kim, Coincidence theorems on a product of generalized convex spaces and applications to equilibria, J. Korean. Math. Soc. 36 (1999), no. 4, 813-828.