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On System of Generalized Vector Quasi-equilibrium Problems with Set-valued Maps

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Abstract

In this paper, we introduce four new types of the system of generalized vector quasi-equilibrium problems with set-valued maps which include system of vector quasi-equilibrium problems, system of vector equilibrium problems, system of variational inequality problems, and vector equilibrium problems in the literature as special cases. We prove the existence of solutions for such kinds of system of generalized vector quasi-equilibrium problems. Consequently, we derive some existence results of a solution for the system of vector quasi-equilibrium problems and the generalized Debreu type equilibrium problem for vector-valued functions.

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References

  1. Giannessi F. ed. (2000), Vector Variational Inequalities and Vector Equilibria. Mathematical Theories, Kluwer Academic Publishers, Dordrecht/Boston/London

    Google Scholar 

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

    Article  Google Scholar 

  3. Ansari Q.H., Yao J.C. (1999), An existence results for the generalized vector equilibrium problem. Applied Mathematics Letters 12, 53–56

    Article  Google Scholar 

  4. Konnov I.V., Yao J.C. (1999), Existence of solutions for generalized vector equilibrium Problems. Journal of Mathematical Analysis and Applications 233, 328–335

    Article  Google Scholar 

  5. Oettli W., Schlager D. (1998), Existence of equilibria for monotone multivalued mappings. Mathematical Methods and Operation Research 48, 219–228

    Article  Google Scholar 

  6. Bianchi M., Hadjisavvas N., Schaibles S. (1997), Vector equilibrium problems with generalized monotone bifunctions. Journal of Optimization Theory and Applications 92, 527–542

    Article  Google Scholar 

  7. Gong X.H. (2001), Efficiency and henig efficiency for vector equilibrium problems. Journal of Optimization Theory and Applications 108, 139–154

    Article  Google Scholar 

  8. Peng J.W. (2000), Generalized set-valued equilibrium problems in topological vector space. Journal of Chonqqing Normal University 17(4): 36–40

    Google Scholar 

  9. Hou S.H., Yu H., Chen G.Y. (2003), On vector quasi-equilibrium problems with set-valued maps. Journal of Optimization Theory and Applications 119, 485–498

    Article  Google Scholar 

  10. Ansari Q.H., Flores-Bazan F. (2003), Generalized vector quasi-equilibrium problems with applications. Journal of Mathematical Analysis and Applications 277, 246–256

    Article  Google Scholar 

  11. Peng J.W. (2002), Generalized vector quasi-equilibrium problems on W-space. Journal of Mathematical Research and Exposition 22(4): 519–524

    Google Scholar 

  12. Fu J.Y. (2000), Generalized vector quasi-equilibrium problems. Mathematical Methods and Operation Research 52, 57–64

    Article  Google Scholar 

  13. Chiang Y., Chadli O., Yao J.C. (2003), Existence of solutions to implicit vector variational inequalities. Journal of Optimization Theory and Applications 116, 251–264

    Article  Google Scholar 

  14. Ansari Q.H., Chan W.K., Yang X.Q. (2004), The system of vector quasi-equilibrium Problems with Applications. Journal of Global Optimization 29, 45–57

    Article  Google Scholar 

  15. Ionescu T.C. (1988), On the approximation of upper semi-continuous correspondences and the equilibriums of Generalized games. Journal of Mathematical Analysis and Applications 136, 267–289

    Article  Google Scholar 

  16. Yuan G.X.-Z., Isac G., Lai K.K., Yu J. (1998), The study of minimax inequalities, abstract economics and applications to variational inequalities and Nash equilibria. Acta Applicandae Mathematicae 54, 135–166

    Article  Google Scholar 

  17. Ansari Q.H., Khan Z. (2004). System of generalized vector quasi-equilibrium problems with applications. In: Nanda S., Rajasekhar G.P. (eds). Mathematical Analysis and Applications. Narosa Publishing House, New Delhi, India, pp. 1–13

    Google Scholar 

  18. Ansari Q.H., Schaible S., Yao J.C. (2002), The system of generalized vector equilibrium problems with applications. Journal of Global Optimization 23, 3–16

    Article  Google Scholar 

  19. Ansari Q.H., Schaible S., Yao J.C. (2000), Systems of vector equilibrium problems and its applications. Journal of Optimization Theory and Applications 107, 547–557

    Article  Google Scholar 

  20. Allevi E., Gnudi A., Konnov I.V. (2001), Generalized vector variational inequalities over product sets. Nonlinear Analysis, Theory Methods and Applications 47, 573–582

    Article  Google Scholar 

  21. Ansari Q.H., Yao J.C. (1999), A fixed-point theorem and its applications to the systems of variational inequalities. Bulletin of the Australian Mathematical Society 59, 433–442

    Article  Google Scholar 

  22. Ansari Q.H., Yao J.C. (2000), System of generalized variational inequalities and their applications. Applicable Analysis 76(3–4): 203–217

    Article  Google Scholar 

  23. Cohen G., Chaplais F. (1988), Nested monotony for variational inequalities over a product of spaces and convergence of iterative algorithms. Journal of Optimization Theory and Applications 59, 360–390

    Article  Google Scholar 

  24. Pang J.S. (1985), Asymmetric variational inequality problems over product sets: applications and iterative methods. Mathematical Programming 31, 206–219

    Article  Google Scholar 

  25. Chen G.Y., Yu H. (2002), Existence of solutions to random equilibrium system. Journal of System Science and Mathematical Sciences 22(3): 278–284 (in Chinese).

    Google Scholar 

  26. Zhou J.X., Chen G. (1988), Diagonal convexity conditions for problems in convex analysis and quasi-variational inequalities. Journal of Mathematical Analysis and Applications 132, 213–225

    Article  Google Scholar 

  27. Luc D.T. (1989) Theory of Vector Optimization. Springer-Verlag, Berlin

    Google Scholar 

  28. Aubin, J.P. and Ekeland, I. (1984), Applied Nonlinear Analysis, John Wiley & Sons.

  29. Petryshyn W.V., Fitzpatrick P.M. (1974), Fixed point theorems of multivalued noncompact acyclic mappings. Pacific Journal of Mathematics 54, 17–23

    Google Scholar 

  30. Deguire P., Tan K.K., Yuan G.X.-Z. (1999), The study of maximal elements, fixed points for L S -majorized mappings and their applications to minimax and variational inequalities in the product topological spaces. Nonlinear Analysis, Theory Methods and Applications 37, 933–951

    Article  Google Scholar 

  31. Chebbi S., Florenzano M. (1999), Maximal elements and equilibria for condensing correspondences. Nonlinear Analysis, Theory Methods and Applications 38, 995–1002

    Article  Google Scholar 

  32. Tian G.Q., Zhou J.X. (1993), quasi-variational inequalities without the concavity assumption. Journal of Mathematical Analysis and Applications 172, 289–299

    Article  Google Scholar 

  33. Su C.H., Sehgal V.M. (1975), Some fixed point theorems for condensing multifunctions in locally convex spaces. Proceedings of the National Academy of Sciences of the USA 50, 150–154

    Google Scholar 

  34. Aubin, J.P. and Ekeland, I. (1984), Applied Nonlinear Analisis. John Wiley & Sons.

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Authors and Affiliations

  1. College of Mathematics and Computer Science, Chongqing Normal University, Chongqing, 400047, P.R. China

    Jian-Wen Peng & Xin-Min Yang

  2. Department of Management Science, School of Management, Fudan University, Shanghai, 200433, P. R.China

    Jian-Wen Peng

  3. Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong

    Heung-Wing Joseph Lee

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  1. Jian-Wen Peng
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  2. Heung-Wing Joseph Lee
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  3. Xin-Min Yang
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Correspondence to Jian-Wen Peng.

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Peng, JW., Lee, HW.J. & Yang, XM. On System of Generalized Vector Quasi-equilibrium Problems with Set-valued Maps. J Glob Optim 36, 139–158 (2006). https://doi.org/10.1007/s10898-006-9004-5

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  • DOI: https://doi.org/10.1007/s10898-006-9004-5

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