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Equilibrium problems with lower and upper bounds. (English) Zbl 1175.90411
Summary: We obtain some existence results of equilibrium problems with lower and upper bounds by employing a fixed-point theorem due to Ansari and Yao and Ky Fan Lemma, respectively. Our results give answers to the open problem raised by Isac, Sehgal and Singh.

MSC:
90C47 Minimax problems in mathematical programming
47H10 Fixed-point theorems
47N10 Applications of operator theory in optimization, convex analysis, mathematical programming, economics
49J40 Variational inequalities
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[1] Ansari, Q.H.; Yao, J.C., A fixed point theorem and its applications to the system of variational inequalities, Bulletin of the Australian mathematical society, 59, 433-442, (1999) · Zbl 0944.47037
[2] Bourbaki, N., Espaces vectoriels topologiques, (1996), Hermann Paris · Zbl 0066.35301
[3] Isac, G.; Sehgal, V.M.; Singh, S.P., An alternate version of a variational inequality, Indian J. of math., 41, 1, 25-31, (1999) · Zbl 1034.49005
[4] Ansari, Q.H.; Wong, N.G.; Yao, J.C., The existence of nonlinear inequalities, Appl. math. lett., 12, 5, 89-92, (1999) · Zbl 0940.49010
[5] Aubin, J.P., L’analyse non linéaire et ses motivations économiques, (1984), Masson Paris · Zbl 0551.90001
[6] Antipin, A.S., On convergence of proximal methods to fixed point of extremal mappings and estimates of their rate of convergence, Computational mathematics and mathematical physics, 35, 539-551, (1995) · Zbl 0852.65046
[7] Bianchi, M.; Schaible, S., Generalized monotone bifunctions and equilibrium problems, Journal of optimization theory and applications, 90, 31-43, (1996) · Zbl 0903.49006
[8] Blum, E.; Oettli, W., From optimization and variational inequalities to equilibrium problems, The mathematics student, 63, 123-145, (1994) · Zbl 0888.49007
[9] Brézis, H.; Nirenberg, L.; Stampacchia, G., A remark on Ky Fan’s minimax principle, Bulletin uni. mat. italiana, 6, 4, 293-300, (1972) · Zbl 0264.49013
[10] Chadli, O.; Chbani, Z.; Riahi, H., Recession methods for equilibrium problems and applications to variational and hemivariational inequalities, Discrete and continuous dynamical systems, 5, 185-195, (1999) · Zbl 0949.49008
[11] Chadli, O.; Chbani, Z.; Riahi, H., Equilibrium problems and noncoercive variational inequalities, Optimization, 49, 1-12, (1999) · Zbl 0949.49008
[12] O. Chadli, Z. Chbani and H. Riahi, Equilibrium problems with generalized monotone bifunctions and applications to variational inequalities, Journal of Optimization Theory and Applications (to appear). · Zbl 0966.91049
[13] Hadjisavvas, N.; Schaible, S., From scalar to vector equilibrium problems quasimonotone case, Journal of optimization theory and applications, 96, 297-309, (1998) · Zbl 0903.90141
[14] Husain, T.; Tarafdar, E., Simultaneous variational inequalities, minimization problems and related results, Mathematica japonica, 39, 221-231, (1994) · Zbl 0802.47059
[15] Konnov, I.V., A general approach to finding stationary point and the solution of related problems, Computational mathematics and mathematical physics, 36, 585-593, (1996) · Zbl 1161.90491
[16] L.J. Lin and Z.T. Yu, Fixed-point theorems and equilibrium problems, Nonlinear Analysis, Theory, Methods and Applications (to appear). · Zbl 0989.47051
[17] Tarafdar, E.; Yuan, G.X.Z., Generalized variational inequalities and its applications, Nonlinear analysis, theory, method and applications, 30, 4171-4181, (1997) · Zbl 0912.49004
[18] Yuan, G.X.Z., KKM theory and applications in nonlinear analysis, (1999), Marcel Dekker New York
[19] Li, J., A lower and upper bounds version of a variational inequality, Appl. math. lett., 13, 5, 47-51, (2000) · Zbl 1023.49003
[20] N. Bourbaki, Espaces Vectoriels Topologiques, Chapitres 1 à 5, Masson. · Zbl 1106.46003
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