Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
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CONNECTEDNESS AND COMPACTNESS OF WEAK EFFICIENT SOLUTIONS FOR VECTOR EQUILIBRIUM PROBLEMS

  • Long, Xian Jun (College of Mathematics and Statistics Chongqing Technology and Business University) ;
  • Peng, Jian Wen (Department of Mathematics Chongqing Normal University)
  • Received : 2010.07.07
  • Published : 2011.11.30
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Abstract

In this paper, without assumption of monotonicity, we study the compactness and the connectedness of the weakly efficient solutions set to vector equilibrium problems by using scalarization method in locally convex spaces. Our results improve the corresponding results in [X. H. Gong, Connectedness of the solution sets and scalarization for vector equilibrium problems, J. Optim. Theory Appl. 133 (2007), 151-161].

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References

  1. Q. H. Ansari and J. C. Yao, An existence result for the generalized vector equilibrium, Appl. Math. Lett. 12 (1999), no. 8, 53-56. https://doi.org/10.1016/S0893-9659(99)00121-4
  2. J. P. Aubin and I. Ekeland, Applied Nonlinear Analysis, John Wiley, New York, 1984.
  3. M. Bianchi, N. Hadjisavvas, and S. Schaibles, Vector equilibrium problems with generalized monotone bifunctions, J. Optim. Theory Appl. 92 (1997), no. 3, 527-542. https://doi.org/10.1023/A:1022603406244
  4. Y. H. Cheng, On the connectedness of the solution set for the weak vector variational inequality, J. Math. Anal. Appl. 260 (2001), no. 1, 1-5. https://doi.org/10.1006/jmaa.2000.7389
  5. K. Fan, A generalization of Tychonoffs fixed point theorem, Math. Ann. 142 (1961), 305-310. https://doi.org/10.1007/BF01353421
  6. F. Giannessi, (ed.), Vector Variational Inequilities and Vector Equilibria: Mathematical Theories, Kluwer, Dordrechet, 2000.
  7. X. H. Gong, Efficiency and Henig efficiency for vector equilibrium problems, J. Optim. Theory Appl. 108 (2001), no. 1, 139-154. https://doi.org/10.1023/A:1026418122905
  8. X. H. Gong, Connectedness of the solution sets and scalarization for vector equilibrium problems, J. Optim. Theory Appl. 133 (2007), no. 2, 151-161. https://doi.org/10.1007/s10957-007-9196-y
  9. X. H. Gong and J. C. Yao, Connectedness of the set of efficient solutions for generalized systems, J. Optim. Theory Appl. 138 (2008), no. 2, 189-196. https://doi.org/10.1007/s10957-008-9378-2
  10. N. Hadjisavvas and S. Schaibles, From scalar to vector equilibrium problems in the quasimonotone case, J. Optim. Theory Appl. 96 (1998), no. 2, 297-309. https://doi.org/10.1023/A:1022666014055
  11. Y. D. Hu, The efficiency Theory of Multiobjective Programming, Shanghai: Shanghai Science and Technology Press, 1994.
  12. G. M. Lee, D. S. Kim, B. S. Lee, and N. D. Yun, Vector variational inequality as a tool for studying vector optimization problems, Nonlinear Anal. 34 (1998), no. 5, 745-765. https://doi.org/10.1016/S0362-546X(97)00578-6
  13. X. J. Long, N. J. Huang, and K. L. Teo, Existence and stability of solutions for generalized strong vector quasi-equilibrium problems, Math. Comput. Modelling 47 (2008), no. 3-4, 445-451. https://doi.org/10.1016/j.mcm.2007.04.013
  14. D. T. Luc, Theory of Vector Optimization, Lecture Notes in Economics and Mathematics Systems, Vol. 319, Springer-Verlag, New York, 1989.

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