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Hölder continuity of the unique solution to quasiequilibrium problems in metric spaces. (English) Zbl 1176.90584

The authors study the local Hölder continuity of the solution sets of quasiequilibrium problems in metric spaces. The article generalizes and sharpens some results of M. Ait Mansour and L. Scrimali [J. Glob. Optim. 40, No. 1–3, 175–184 (2008; Zbl 1151.90008)] for Hilbert spaces. There are some applications to quasivariational inequalities, traffic network problems and quasioptimization problem (its constraints depend on the minimizer).

MSC:

90C31 Sensitivity, stability, parametric optimization
47J20 Variational and other types of inequalities involving nonlinear operators (general)
90B20 Traffic problems in operations research

Citations:

Zbl 1151.90008
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References:

[1] Ait Mansour, M., Scrimali, L.: Hölder continuity of solutions to elastic traffic network models. J. Glob. Optim. 40, 175–184 (2008) · Zbl 1151.90008 · doi:10.1007/s10898-007-9190-9
[2] Yen, N.D.: Hölder continuity of solutions to parametric variational inequalities. Appl. Math. Optim. 31, 245–255 (1995) · Zbl 0821.49011 · doi:10.1007/BF01215992
[3] Yen, N.D., Lee, G.M.: Solution sensitivity of a class of variational inequalities. J. Math. Anal. Appl. 215, 48–55 (1997) · Zbl 0906.49002 · doi:10.1006/jmaa.1997.5607
[4] Ait Mansour, M., Riahi, H.: Sensitivity analysis for abstract equilibrium problems. J. Math. Anal. Appl. 306, 684–691 (2005) · Zbl 1068.49005 · doi:10.1016/j.jmaa.2004.10.011
[5] Anh, L.Q., Khanh, P.Q.: On the Hölder continuity of solutions to parametric multivalued vector equilibrium problems. J. Math. Anal. Appl. 321, 308–315 (2006) · Zbl 1104.90041 · doi:10.1016/j.jmaa.2005.08.018
[6] Anh, L.Q., Khanh, P.Q.: Uniqueness and Hölder continuity of the solution to multivalued equilibrium problems in metric spaces. J. Glob. Optim. 37, 449–465 (2007) · Zbl 1156.90025 · doi:10.1007/s10898-006-9062-8
[7] Bianchi, M., Pini, R.: A note on stability for parametric equilibrium problems. Oper. Res. Lett. 31, 445–450 (2003) · Zbl 1112.90082 · doi:10.1016/S0167-6377(03)00051-8
[8] Bianchi, M., Pini, R.: Sensitivity for parametric vector equilibria. Optimization 55, 221–230 (2006) · Zbl 1149.90156 · doi:10.1080/02331930600662732
[9] Anh, L.Q., Khanh, P.Q.: Semicontinuity of the solution set of parametric multivalued vector quasiequilibrium problems. J. Math. Anal. Appl. 294, 699–711 (2004) · Zbl 1048.49004 · doi:10.1016/j.jmaa.2004.03.014
[10] Anh, L.Q., Khanh, P.Q.: On the stability of the solution sets of general multivalued vector quasiequilibrium problems. J. Optim. Theory Appl. 135, 271–284 (2007) · Zbl 1146.90516 · doi:10.1007/s10957-007-9250-9
[11] Anh, L.Q., Khanh, P.Q.: Semicontinuity of the approximate solution sets of multivalued quasiequilibrium problems. Numer. Funct. Anal. Optim. 29, 24–42 (2008) · Zbl 1211.90243 · doi:10.1080/01630560701873068
[12] Anh, L.Q., Khanh, P.Q.: Various kinds of semicontinuity and the solution sets of parametric multivalued symmetric vector quasiequilibrium problems. J. Glob. Optim. 41, 539–558 (2008) · Zbl 1165.90026 · doi:10.1007/s10898-007-9264-8
[13] Huang, N.J., Li, J., Thompson, H.B.: Stability for parametric implicit vector equilibrium problems. Math. Comput. Model. 43, 1267–1274 (2006) · Zbl 1187.90286 · doi:10.1016/j.mcm.2005.06.010
[14] Blum, E., Oettli, W.: From optimization and variational inequalities to equilibrium problems. Math. Stud. 63, 123–145 (1994) · Zbl 0888.49007
[15] Bensoussan, A., Goursat, M., Lions, J.-L.: Contrôle impulsionnel et inéquations quasivariationnelle. C. R. Acad. Sci. Paris. Sér. A 276, 1279–1284 (1973) · Zbl 0264.49004
[16] Chen, G.Y., Yang, X.Q., Yu, H.: A nonlinear scalarization function and generalized quasi-vector equilibrium problems. J. Glob. Optim. 32, 451–466 (2005) · Zbl 1130.90413 · doi:10.1007/s10898-003-2683-2
[17] Hai, N.X., Khanh, P.Q.: Systems of multivalued quasiequilibrium problems. Adv. Nonlinear Var. Inequal. 9, 97–108 (2006) · Zbl 1181.49009
[18] Hai, N.X., Khanh, P.Q.: Existence of solutions to general quasiequilibrium problems and applications. J. Optim. Theory Appl. 133, 317–327 (2007) · Zbl 1146.49004 · doi:10.1007/s10957-007-9170-8
[19] Hai, N.X., Khanh, P.Q.: The solution existence of general variational inclusion problems. J. Math. Anal. Appl. 328, 1268–1277 (2007) · Zbl 1108.49020 · doi:10.1016/j.jmaa.2006.06.058
[20] Khaliq, A.: Implicit vector quasiequilibrium problems with applications to variational inequalities. Nonlinear Anal. 63, 1823–1831 (2005) · Zbl 1224.90195 · doi:10.1016/j.na.2005.01.070
[21] Li, S.J., Teo, K.L., Yang, X.Q., Wu, S.Y.: Gap functions and existence of solutions to generalized vector quasiequilibrium problems. J. Glob. Optim. 34, 427–440 (2006) · Zbl 1090.49014 · doi:10.1007/s10898-005-2193-5
[22] Lin, L.J., Chen, H.L.: The Study of KKM theorems with applications to vector equilibrium problems and implicit vector variational inequality problems. J. Glob. Optim. 32, 135–157 (2005) · Zbl 1079.90153 · doi:10.1007/s10898-004-2119-7
[23] Zeng, L.C., Jao, J.C.: An existence result for generalized vector equilibrium problems without pseudomonotonicity. Appl. Math. Lett. 19, 1320–1326 (2006) · Zbl 1190.90244 · doi:10.1016/j.aml.2005.09.010
[24] 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
[25] Oettli, W., Yen, N.D.: Quasicomplementarity problems of type R 0. J. Optim. Theory Appl. 89, 467–474 (1996) · Zbl 0851.90121 · doi:10.1007/BF02192540
[26] Wardrop, J.G.: Some theoretical aspects of road traffic research. In: Proceedings of the Institute of Civil Engineers, Part II, pp. 325–378 (1952)
[27] De Luca, M., Maugeri, A.: Quasivariational inequality and applications to equilibrium problems with elastic demands. In: Clarke, F.M., Dem’yanov, V.F., Giannessi, F. (eds.) Proceedings of the IVth Course of the International School of Mathematics, Erice, June 19–July 1, 1988. Nonsmooth Optimization and Related Topics, vol. 43, pp. 61–77. Plenum, New York (1989) · Zbl 0746.90018
[28] De Luca, M.: Existence of solutions for a time-dependent quasivariational inequality. Suppl. Rend. Circ. Mat. Palermo Ser. 2 48, 101–106 (1997) · Zbl 0958.49006
[29] Giannessi, F.: Vector Variational Inequalities and Vector Equilibria, Mathematical Theories. Nonconvex Optimization and Its Application, vol. 38, Kluwer Academic, Dordrecht (2000) · Zbl 0952.00009
[30] Goh, C.J., Yang, X.Q.: Vector equilibrium problem and vector optimization. Eur. J. Oper. Res. 116, 615–628 (1999) · Zbl 1009.90093 · doi:10.1016/S0377-2217(98)00047-2
[31] 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
[32] Khanh, P.Q., Luu, L.M.: Some existence results for vector quasivariational inequalities involving multifunctions and applications to traffic equilibrium problems. J. Glob. Optim. 32, 551–568 (2005) · Zbl 1097.49012 · doi:10.1007/s10898-004-2693-8
[33] Nagurney, A.: Network Economics: A Variational Inequality Approach. Kluwer Academic, Dordrecht (1993) · Zbl 0873.90015
[34] Smith, M.J.: The existence, uniqueness and stability of traffic equilibrium. Transp. Res. 138, 295–304 (1979) · doi:10.1016/0191-2615(79)90022-5
[35] Yang, X.Q., Goh, C.J.: On vector variational inequalities: Application to vector equilibria. J. Optim. Theory Appl. 95, 431–443 (1997) · Zbl 0892.90158 · doi:10.1023/A:1022647607947
[36] De Luca, M.: Generalized quasivariational inequalities and traffic equilibrium problems. In: Giannessi, F., Maugeri, A. (eds.) Variational Inequalities and Network Equilibrium Problems, pp. 45–54. Plenum, New York (1995) · Zbl 0847.49007
[37] Maugeri, A.: Variational and quasivariational inequalities in network flow models: recent developments in theory and algorithms. In: Giannessi, F., Maugeri, A. (eds.) Variational Inequalities and Network Equilibrium Problems, pp. 195–211. Plenum, New York (1995) · Zbl 0847.49010
[38] Giannessi, F.: Theorems of the alternative, quadratic programs, and complementarity problems. In: Cottle, R.W., Giannessi, F., Lions, J.L. (eds.) Variational Inequalities and Complementarity Problems, pp. 151–186. Wiley, New York (1980) · Zbl 0484.90081
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