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Sensitivity analysis for multivalued quasiequilibrium problems in metric spaces: Hölder continuity of solutions. (English) Zbl 1188.90274

Summary: Hölder continuity and uniqueness of the solutions of general multivalued vector quasiequilibrium problems in metric spaces are established. The results are shown to be extensions of recent ones for equilibrium problems with some improvements. Applications in quasivariational inequalities, vector quasioptimization and traffic network problems are provided as examples for others in various optimization-related problems.

MSC:

90C48 Programming in abstract spaces
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[1] Ait Mansour, M., Scrimali, L.: Hölder continuity of solutions to elastic traffic network models. J. Glob. Optim., online. · Zbl 1151.90008
[2] Ait Mansour M. and Riahi H. (2005). Sensitivity analysis for abstract equilibrium problems. J. Math. Anal. Appl. 306: 684–691 · Zbl 1068.49005 · doi:10.1016/j.jmaa.2004.10.011
[3] Anh L.Q. and Khanh P.Q. (2004). Semicontinuity of the solution set of parametric multivalued vector quasiequilibrium problems. J. Math. Anal. Appl. 294: 699–711 · Zbl 1048.49004 · doi:10.1016/j.jmaa.2004.03.014
[4] Anh L.Q. and Khanh P.Q. (2006). On the Hölder continuity of solutions to parametric multivalued vector equilibrium problems. J. Math. Anal. Appl. 321: 308–315 · Zbl 1104.90041 · doi:10.1016/j.jmaa.2005.08.018
[5] Anh L.Q. and Khanh P.Q. (2007a). Uniqueness and Hölder continuity of the solution to multivalued equilibrium problems in metric spaces. J. Glob. Optim. 37: 449–465 · Zbl 1156.90025
[6] Anh L.Q. and Khanh P.Q. (2007b). On the stability of the solution sets of general multivalued vector quasiequilibrium problems. J. Optim. Theory Appl. 135: 271–284 · Zbl 1146.90516 · doi:10.1007/s10957-007-9250-9
[7] Anh, L.Q., Khanh, P.Q.: Various kinds of semicontinuity and the solution sets of parametric multivalued symmetric vector quasiequilibrium problems. J. Glob. Optim. (in press) · Zbl 1165.90026
[8] Bensoussan A., Goursat M. and Lions J.L. (1973). Contrôle impulsionnel et inéquations quasivariationnelle. C. R. Acad. Sci. Paris, Sér. A 276: 1279–1284 · Zbl 0264.49004
[9] Bianchi M. and Pini R. (2003). A note on stability for parametric equilibrium problems. Oper. Res. Lett. 31: 445–450 · Zbl 1112.90082 · doi:10.1016/S0167-6377(03)00051-8
[10] Blum E. and Oettli W. (1994). From optimization and variational inequalities to equilibrium problems. Math. Stud. 63: 123–145 · Zbl 0888.49007
[11] 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 Press, New York (1995) · Zbl 0847.49007
[12] 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. John Wiley and Sons, New York (1980) · Zbl 0484.90081
[13] Giannessi F. (2000). Vector Variational Inequalities and Vector Equilibria. Kluwer, Dordrecht · Zbl 0952.00009
[14] Goh C.J. and Yang X.Q. (1999). Vector equilibrium problem and vector optimization. European J. Oper. Res. 116: 615–628 · Zbl 1009.90093 · doi:10.1016/S0377-2217(98)00047-2
[15] Hai N.X. and Khanh P.Q. (2006). Systems of multivalued quasiequilibrium problems. Adv. Nonlinear Variat. Inequal. 9: 97–108 · Zbl 1181.49009
[16] Hai N.X. and Khanh P.Q. (2007a). The solution existence of general variational inclusion problems. J. Math. Anal. Appl. 328: 1268–1277 · Zbl 1108.49020 · doi:10.1016/j.jmaa.2006.06.058
[17] Hai N.X. and Khanh P.Q. (2007b). Existence of solutions to general quasiequilibrium problems and applications. J. Optim. Theory Appl. 133: 317–327 · Zbl 1146.49004 · doi:10.1007/s10957-007-9170-8
[18] Khaliq A. (2005). Implicit vector quasiequilibrium problems with applications to variational inequalities. Nonlinear Anal. 63: 1823–1831 · Zbl 1224.90195 · doi:10.1016/j.na.2005.01.070
[19] Khanh P.Q. and Luu L.M. (2004). 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 · Zbl 1059.49017 · doi:10.1007/s10957-004-5722-3
[20] Khanh P.Q. and Luu L.M. (2005). Some existence results for vector quasivariational inequalities involving multifunctions and applications to traffic equilibrium problems. J. Glob. Optim. 32: 551–568 · Zbl 1097.49012 · doi:10.1007/s10898-004-2693-8
[21] Kluge R. (1979). Nichtlineare Variationsungleichungen und Extremalaufgaben. Wissenschaften, Berlin · Zbl 0452.49001
[22] 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 Press, New York (1995) · Zbl 0847.49010
[23] Smith M.J. (1979). The existence, uniqueness and stability of traffic equilibrium. Transport. Res. 138: 295–304 · doi:10.1016/0191-2615(79)90022-5
[24] Wardrop, J.G.: Some theoretical aspects of road traffic research. Proc. Inst. Civil Eng. Part II, 325–378 (1952)
[25] Yen N.D. (1995). Hölder continuity of solutions to parametric variational inequalities. Appl. Math. Optim. 31: 245–255 · Zbl 0821.49011 · doi:10.1007/BF01215992
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