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The system of generalized set-valued equilibrium problems
Journal of Inequalities and Applications volume 2006, Article number: 16764 (2006)
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.
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
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
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
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.
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
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
Ansari QH, Yao J-C: Systems of generalized variational inequalities and their applications. Applicable Analysis 2000,76(3–4):203–217. 10.1080/00036810008840877
Aubin J-P, Ekeland I: Applied Nonlinear Analysis, Pure and Applied Mathematics (New York). John Wiley & Sons, New York; 1984:xi+518.
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.
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
Bianchi M, Schaible S: Generalized monotone bifunctions and equilibrium problems. Journal of Optimization Theory and Applications 1996,90(1):31–43. 10.1007/BF02192244
Blum E, Oettli W: From optimization and variational inequalities to equilibrium problems. The Mathematics Student 1994,63(1–4):123–145.
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
Chen GY, Yu H: Existence of solutions to a random equilibrium system. Journal of Systems Science and Mathematical Sciences 2002,22(3):278–284.
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
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
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
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
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
Kelley JL, Namioka I: Linear topological spaces, The University Series in Higher Mathematics. D. Van Nostrand, New Jersey; 1963:xv+256.
Konnov IV: Relatively monotone variational inequalities over product sets. Operations Research Letters 2001,28(1):21–26. 10.1016/S0167-6377(00)00063-8
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
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
Michael E: A note on paracompact spaces. Proceedings of the American Mathematical Society 1953, 4: 831–838. 10.1090/S0002-9939-1953-0056905-8
Oettli W: A remark on vector-valued equilibria and generalized monotonicity. Acta Mathematica Vietnamica 1997,22(1):213–221.
Oettli W, Schläger D: Existence of equilibria for monotone multivalued mappings. Mathematical Methods of Operations Research 1998,48(2):219–228. 10.1007/s001860050024
Pang J-S: Asymmetric variational inequality problems over product sets: applications and iterative methods. Mathematical Programming 1985,31(2):206–219. 10.1007/BF02591749
Peng JW: Equilibrium problems for W-spaces. Mathematica Applicata (Wuhan) 1999,12(3):81–87.
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
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
Zhang SS: Variational inequalities and complementarity problem theory with applications. Shanghai Science and Technology, Shanghai; 1991.
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
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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
- Equilibrium Problem
- Existence Result