# The system of generalized set-valued equilibrium problems

- Jian-Wen Peng
^{1}Email author

**2006**:16764

https://doi.org/10.1155/JIA/2006/16764

© Peng 2006

**Received: **14 September 2004

**Accepted: **18 November 2004

**Published: **8 February 2006

## 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.

## Keywords

## Authors’ Affiliations

## References

- 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)00202-4MATHMathSciNetView ArticleGoogle Scholar - Ansari QH, Konnov IV, Yao J-C:
**On generalized vector equilibrium problems.***Nonlinear Analysis*2001,**47**(1):543–554. 10.1016/S0362-546X(01)00199-7MATHMathSciNetView ArticleGoogle Scholar - 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:1026495115191MATHMathSciNetView ArticleGoogle Scholar - 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.MATHMathSciNetView ArticleGoogle Scholar - 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/S0004972700033116MATHMathSciNetView ArticleGoogle Scholar - 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)00121-4MATHMathSciNetView ArticleGoogle Scholar - Ansari QH, Yao J-C:
**Systems of generalized variational inequalities and their applications.***Applicable Analysis*2000,**76**(3–4):203–217. 10.1080/00036810008840877MATHMathSciNetView ArticleGoogle Scholar - Aubin J-P, Ekeland I:
*Applied Nonlinear Analysis, Pure and Applied Mathematics (New York)*. John Wiley & Sons, New York; 1984:xi+518.Google Scholar - 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 - 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:1022603406244MATHMathSciNetView ArticleGoogle Scholar - Bianchi M, Schaible S:
**Generalized monotone bifunctions and equilibrium problems.***Journal of Optimization Theory and Applications*1996,**90**(1):31–43. 10.1007/BF02192244MATHMathSciNetView ArticleGoogle Scholar - Blum E, Oettli W:
**From optimization and variational inequalities to equilibrium problems.***The Mathematics Student*1994,**63**(1–4):123–145.MATHMathSciNetGoogle Scholar - 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)00079-3MATHMathSciNetView ArticleGoogle Scholar - Chen GY, Yu H:
**Existence of solutions to a random equilibrium system.***Journal of Systems Science and Mathematical Sciences*2002,**22**(3):278–284.MATHMathSciNetGoogle Scholar - 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/BF00940305MATHMathSciNetView ArticleGoogle Scholar - 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)00084-4MATHMathSciNetView ArticleGoogle Scholar - 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.38.2.121MATHMathSciNetView ArticleGoogle Scholar - 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/s001860200208MATHMathSciNetView ArticleGoogle Scholar - 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:1022666014055MATHMathSciNetView ArticleGoogle Scholar - Kelley JL, Namioka I:
*Linear topological spaces, The University Series in Higher Mathematics*. D. Van Nostrand, New Jersey; 1963:xv+256.View ArticleGoogle Scholar - Konnov IV:
**Relatively monotone variational inequalities over product sets.***Operations Research Letters*2001,**28**(1):21–26. 10.1016/S0167-6377(00)00063-8MATHMathSciNetView ArticleGoogle Scholar - 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.1999.6312MATHMathSciNetView ArticleGoogle Scholar - 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:1015420322818MATHMathSciNetView ArticleGoogle Scholar - Michael E:
**A note on paracompact spaces.***Proceedings of the American Mathematical Society*1953,**4:**831–838. 10.1090/S0002-9939-1953-0056905-8MATHMathSciNetView ArticleGoogle Scholar - Oettli W:
**A remark on vector-valued equilibria and generalized monotonicity.***Acta Mathematica Vietnamica*1997,**22**(1):213–221.MATHMathSciNetGoogle Scholar - Oettli W, Schläger D:
**Existence of equilibria for monotone multivalued mappings.***Mathematical Methods of Operations Research*1998,**48**(2):219–228. 10.1007/s001860050024MATHMathSciNetView ArticleGoogle Scholar - Pang J-S:
**Asymmetric variational inequality problems over product sets: applications and iterative methods.***Mathematical Programming*1985,**31**(2):206–219. 10.1007/BF02591749MATHMathSciNetView ArticleGoogle Scholar - Peng JW:
**Equilibrium problems for W-spaces.***Mathematica Applicata (Wuhan)*1999,**12**(3):81–87.MATHMathSciNetGoogle Scholar - 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/S0002-9939-1975-0380530-7MATHMathSciNetView ArticleGoogle Scholar - 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.1993.1025MATHMathSciNetView ArticleGoogle Scholar - Zhang SS:
*Variational inequalities and complementarity problem theory with applications*. Shanghai Science and Technology, Shanghai; 1991.Google Scholar - 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)90054-6MATHMathSciNetView ArticleGoogle Scholar

## Copyright

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.