×

zbMATH — the first resource for mathematics

Existence of solutions to general quasiequilibrium problems and applications. (English) Zbl 1146.49004
Summary: A general quasiequilibrium problem is proposed including, among others, equilibrium problems, implicit variational inequalities, and quasivariational inequalities involving multifunctions. Sufficient conditions for the existence of solutions with and without relaxed pseudomonotonicity are established. Even semicontinuity may not be imposed. These conditions improve several recent results in the literature.

MSC:
49J40 Variational inequalities
91B50 General equilibrium theory
47H10 Fixed-point theorems
47N10 Applications of operator theory in optimization, convex analysis, mathematical programming, economics
47J20 Variational and other types of inequalities involving nonlinear operators (general)
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Bianchi, M., Hadjisavvas, N., Schaible, S.: Vector equilibrium problems with generalized monotone bifunctions. J. Optim. Theory Appl. 92, 527–542 (1997) · Zbl 0878.49007 · doi:10.1023/A:1022603406244
[2] Bianchi, M., Schaible, S.: Generalized monotone bifunctions and equilibrium problems. J. Optim. Theory Appl. 90, 31–43 (1996) · Zbl 0903.49006 · doi:10.1007/BF02192244
[3] Chadli, O., Chbani, Z., Riahi, H.: Equilibrium problems with generalized monotone bifunctions and applications to variational inequalities. J. Optim. Theory Appl. 105, 299–323 (2000) · Zbl 0966.91049 · doi:10.1023/A:1004657817758
[4] Chadli, O., Chbani, Z., Riahi, H.: Equilibrium problems and noncoercive variational inequalities. Optimization 50, 17–27 (2001) · Zbl 1022.49013 · doi:10.1080/02331930108844551
[5] Chadli, O., Riahi, H.: On generalized vector equilibrium problems. J. Glob. Optim. 16, 33–41 (2000) · Zbl 0966.60003 · doi:10.1023/A:1008393715273
[6] Lin, L.J., Ansari, Q.H., Wu, J.Y.: Geometric properties and coincidence theorems with applications to generalized vector equilibrium problems. J. Optim. Theory Appl. 117, 121–137 (2003) · Zbl 1063.90062 · doi:10.1023/A:1023656507786
[7] Kum, S., Lee, G.M.: Remarks on implicit vector variational inequalities. Taiwan. J. Math. 6, 369–382 (2002) · Zbl 1035.49005
[8] Lee, G.M., Kum, S.: On implicit vector variational inequalities. J. Optim. Theory Appl. 104, 409–425 (2000) · Zbl 0970.47052 · doi:10.1023/A:1004617914993
[9] Khanh, P.Q., Luu, L.M.: On the existence of solutions to vector quasivariational inequalities and quasicomplementarity problems with applications to traffic network equilibria. J. Optim. Theory Appl. 123, 533–548 (2004) · Zbl 1059.49017 · doi:10.1007/s10957-004-5722-3
[10] Fan, K.: Some properties of convex sets related to fixed point theorems. Math. Ann. 266, 519–537 (1984) · Zbl 0532.47043 · doi:10.1007/BF01458545
[11] Tarafdar, E.: A fixed point theorem equivalent to the Fan–Knaster–Kuratowski–Mazurkiewicz theorem. J. Math. Anal. Appl. 128, 475–479 (1987) · Zbl 0644.47050 · doi:10.1016/0022-247X(87)90198-3
[12] Lin, L.J.: Applications of fixed point theorem in G-convex space. Nonlinear Anal. Theory Methods Appl. 46, 601–608 (2001) · Zbl 1001.47041 · doi:10.1016/S0362-546X(99)00456-3
[13] Guerraggio, A., Tan, N.X.: On general vector quasioptimization problems. Math. Methods Oper. Res. 55, 347–358 (2002) · Zbl 1031.90033 · doi:10.1007/s001860200212
[14] Fu, J.Y.: Generalized vector quasiequilibrium problems. Math. Methods Oper. Res. 52, 57–64 (2000) · Zbl 1054.90068 · doi:10.1007/s001860000058
[15] Fu, J.Y., Wan, A.H.: Generalized vector equilibrium problems with set-valued mappings. Math. Methods Oper. Res. 56, 259–268 (2002) · Zbl 1023.90057 · doi:10.1007/s001860200208
[16] Kristály, A., Varga, C.: Set-valued versions of Ky Fan’s inequality with application to variational inclusion theory. J. Math. Anal. Appl. 282, 8–20 (2003) · Zbl 1031.49008 · doi:10.1016/S0022-247X(02)00335-9
[17] Lin, L.J., Yu, Z.T., Kassay, G.: Existence of equilibria for multivalued mappings and its application to vectorial equilibria. J. Optim. Theory Appl. 114, 189–208 (2002) · Zbl 1023.49014 · doi:10.1023/A:1015420322818
[18] Ansari, Q.H., Flores-Bazán, F.: Generalized vector quasiequilibrium problems with applications. J. Math. Anal. Appl. 277, 246–356 (2003) · Zbl 1022.90023 · doi:10.1016/S0022-247X(02)00535-8
[19] Kum, S., Lee, G., Yao, J.C.: An existence result for implicit vector variational inequalities with multifunctions. Appl. Math. Lett. 16, 453–458 (2003) · Zbl 1096.49004 · doi:10.1016/S0893-9659(03)00019-3
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.