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The system of generalized set-valued equilibrium problems

Abstract

We introduce new and interesting model of system of generalized set-valued equilibrium problems which generalizes and unifies the system of set-valued equilibrium problems, the system of generalized implicit vector variational inequalities, the system of generalized vector and vector-like variational inequalities introduced by Ansari et al. (2002), the system of generalized vector variational inequalities presented by Allevi et al. (2001), the system of vector equilibrium problems and the system of vector variational inequalities given by Ansari et al. (2000), the system of scalar variational inequalities presented by Ansari Yao (1999, 2000), Bianchi (1993), Cohen and Caplis (1988), Konnov (2001), and Pang (1985), the system of Ky-Fan variational inequalities proposed bt Deguire et al. (1999) as well as a variety of equilibrium problems in the literature. Several existence results of a solution for the system of generalized set-valued equilibrium problems will be shown.

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References

  1. Allevi E, Gnudi A, Konnov IV: Generalized vector variational inequalities over product sets. Nonlinear Analysis 2001,47(1):573–582. 10.1016/S0362-546X(01)00202-4

    Article  MATH  MathSciNet  Google Scholar 

  2. Ansari QH, Konnov IV, Yao J-C: On generalized vector equilibrium problems. Nonlinear Analysis 2001,47(1):543–554. 10.1016/S0362-546X(01)00199-7

    Article  MATH  MathSciNet  Google Scholar 

  3. Ansari QH, Schaible S, Yao J-C: System of vector equilibrium problems and its applications. Journal of Optimization Theory and Applications 2000,107(3):547–557. 10.1023/A:1026495115191

    Article  MATH  MathSciNet  Google Scholar 

  4. Ansari QH, Schaible S, Yao J-C: The system of generalized vector equilibrium problems with applications. Journal of Global Optimization 2002,22(1–4):3–16.

    Article  MATH  MathSciNet  Google Scholar 

  5. Ansari QH, Yao J-C: A fixed point theorem and its applications to a system of variational inequalities. Bulletin of the Australian Mathematical Society 1999,59(3):433–442. 10.1017/S0004972700033116

    Article  MATH  MathSciNet  Google Scholar 

  6. Ansari QH, Yao J-C: An existence result for the generalized vector equilibrium problem. Applied Mathematics Letters 1999,12(8):53–56. 10.1016/S0893-9659(99)00121-4

    Article  MATH  MathSciNet  Google Scholar 

  7. Ansari QH, Yao J-C: Systems of generalized variational inequalities and their applications. Applicable Analysis 2000,76(3–4):203–217. 10.1080/00036810008840877

    Article  MATH  MathSciNet  Google Scholar 

  8. Aubin J-P, Ekeland I: Applied Nonlinear Analysis, Pure and Applied Mathematics (New York). John Wiley & Sons, New York; 1984:xi+518.

    Google Scholar 

  9. Bianchi M: Pseudo P-monotone operators and variational inequalities. In Report No. 6. Istituto di Econometria e Matematica per le Decisioni Economiche, Universita Cattolica del Sacro Cuore, Milan; 1993.

    Google Scholar 

  10. Bianchi M, Hadjisavvas N, Schaible S: Vector equilibrium problems with generalized monotone bifunctions. Journal of Optimization Theory and Applications 1997,92(3):527–542. 10.1023/A:1022603406244

    Article  MATH  MathSciNet  Google Scholar 

  11. Bianchi M, Schaible S: Generalized monotone bifunctions and equilibrium problems. Journal of Optimization Theory and Applications 1996,90(1):31–43. 10.1007/BF02192244

    Article  MATH  MathSciNet  Google Scholar 

  12. Blum E, Oettli W: From optimization and variational inequalities to equilibrium problems. The Mathematics Student 1994,63(1–4):123–145.

    MATH  MathSciNet  Google Scholar 

  13. Chadli O, Chiang Y, Huang S: Topological pseudomonotonicity and vector equilibrium problems. Journal of Mathematical Analysis and Applications 2002,270(2):435–450. 10.1016/S0022-247X(02)00079-3

    Article  MATH  MathSciNet  Google Scholar 

  14. Chen GY, Yu H: Existence of solutions to a random equilibrium system. Journal of Systems Science and Mathematical Sciences 2002,22(3):278–284.

    MATH  MathSciNet  Google Scholar 

  15. Cohen G, Chaplais F: Nested monotony for variational inequalities over product of spaces and convergence of iterative algorithms. Journal of Optimization Theory and Applications 1988,59(3):369–390. 10.1007/BF00940305

    Article  MATH  MathSciNet  Google Scholar 

  16. Deguire P, Tan KK, Yuan GX-Z: The study of maximal elements, fixed points for-majorized mappings and their applications to minimax and variational inequalities in product topological spaces. Nonlinear Analysis 1999,37(7):933–951. 10.1016/S0362-546X(98)00084-4

    Article  MATH  MathSciNet  Google Scholar 

  17. Fan K: Fixed-point and minimax theorems in locally convex topological linear spaces. Proceedings of the National Academy of Sciences of the United States of America 1952, 38: 121–126. 10.1073/pnas.38.2.121

    Article  MATH  MathSciNet  Google Scholar 

  18. Fu J-Y, Wan A-H: Generalized vector equilibrium problems with set-valued mappings. Mathematical Methods of Operations Research 2002,56(2):259–268. 10.1007/s001860200208

    Article  MATH  MathSciNet  Google Scholar 

  19. Hadjisavvas N, Schaible S: From scalar to vector equilibrium problems in the quasimonotone case. Journal of Optimization Theory and Applications 1998,96(2):297–309. 10.1023/A:1022666014055

    Article  MATH  MathSciNet  Google Scholar 

  20. Kelley JL, Namioka I: Linear topological spaces, The University Series in Higher Mathematics. D. Van Nostrand, New Jersey; 1963:xv+256.

    Book  Google Scholar 

  21. Konnov IV: Relatively monotone variational inequalities over product sets. Operations Research Letters 2001,28(1):21–26. 10.1016/S0167-6377(00)00063-8

    Article  MATH  MathSciNet  Google Scholar 

  22. Konnov IV, Yao J-C: Existence of solutions for generalized vector equilibrium problems. Journal of Mathematical Analysis and Applications 1999,233(1):328–335. 10.1006/jmaa.1999.6312

    Article  MATH  MathSciNet  Google Scholar 

  23. Lin LJ, Yu ZT, Kassay G: Existence of equilibria for multivalued mappings and its application to vectorial equilibria. Journal of Optimization Theory and Applications 2002,114(1):189–208. 10.1023/A:1015420322818

    Article  MATH  MathSciNet  Google Scholar 

  24. Michael E: A note on paracompact spaces. Proceedings of the American Mathematical Society 1953, 4: 831–838. 10.1090/S0002-9939-1953-0056905-8

    Article  MATH  MathSciNet  Google Scholar 

  25. Oettli W: A remark on vector-valued equilibria and generalized monotonicity. Acta Mathematica Vietnamica 1997,22(1):213–221.

    MATH  MathSciNet  Google Scholar 

  26. Oettli W, Schläger D: Existence of equilibria for monotone multivalued mappings. Mathematical Methods of Operations Research 1998,48(2):219–228. 10.1007/s001860050024

    Article  MATH  MathSciNet  Google Scholar 

  27. Pang J-S: Asymmetric variational inequality problems over product sets: applications and iterative methods. Mathematical Programming 1985,31(2):206–219. 10.1007/BF02591749

    Article  MATH  MathSciNet  Google Scholar 

  28. Peng JW: Equilibrium problems for W-spaces. Mathematica Applicata (Wuhan) 1999,12(3):81–87.

    MATH  MathSciNet  Google Scholar 

  29. Su CH, Sehgal VM: Some fixed point theorems for condensing multifunctions in locally convex spaces. Proceedings of the American Mathematical Society 1975, 50: 150–154. 10.1090/S0002-9939-1975-0380530-7

    Article  MATH  MathSciNet  Google Scholar 

  30. Tian GQ, Zhou J: Quasi-variational inequalities without the concavity assumption. Journal of Mathematical Analysis and Applications 1993,172(1):289–299. 10.1006/jmaa.1993.1025

    Article  MATH  MathSciNet  Google Scholar 

  31. Zhang SS: Variational inequalities and complementarity problem theory with applications. Shanghai Science and Technology, Shanghai; 1991.

    Google Scholar 

  32. Zhou J, Chen G: Diagonal convexity conditions for problems in convex analysis and quasi-variational inequalities. Journal of Mathematical Analysis and Applications 1988,132(1):213–225. 10.1016/0022-247X(88)90054-6

    Article  MATH  MathSciNet  Google Scholar 

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Correspondence to Jian-Wen Peng.

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Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License ( https://creativecommons.org/licenses/by/2.0 ), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Peng, JW. The system of generalized set-valued equilibrium problems. J Inequal Appl 2006, 16764 (2006). https://doi.org/10.1155/JIA/2006/16764

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