Abstract
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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Ait Mansour, M., Scrimali, L.: Hölder continuity of solutions to elastic traffic network models. J. Glob. Optim., online.
Ait Mansour M. and Riahi H. (2005). Sensitivity analysis for abstract equilibrium problems. J. Math. Anal. Appl. 306: 684–691
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
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
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
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
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)
Bensoussan A., Goursat M. and Lions J.L. (1973). Contrôle impulsionnel et inéquations quasivariationnelle. C. R. Acad. Sci. Paris, Sér. A 276: 1279–1284
Bianchi M. and Pini R. (2003). A note on stability for parametric equilibrium problems. Oper. Res. Lett. 31: 445–450
Blum E. and Oettli W. (1994). From optimization and variational inequalities to equilibrium problems. Math. Stud. 63: 123–145
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)
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)
Giannessi F. (2000). Vector Variational Inequalities and Vector Equilibria. Kluwer, Dordrecht
Goh C.J. and Yang X.Q. (1999). Vector equilibrium problem and vector optimization. European J. Oper. Res. 116: 615–628
Hai N.X. and Khanh P.Q. (2006). Systems of multivalued quasiequilibrium problems. Adv. Nonlinear Variat. Inequal. 9: 97–108
Hai N.X. and Khanh P.Q. (2007a). The solution existence of general variational inclusion problems. J. Math. Anal. Appl. 328: 1268–1277
Hai N.X. and Khanh P.Q. (2007b). Existence of solutions to general quasiequilibrium problems and applications. J. Optim. Theory Appl. 133: 317–327
Khaliq A. (2005). Implicit vector quasiequilibrium problems with applications to variational inequalities. Nonlinear Anal. 63: 1823–1831
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
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
Kluge R. (1979). Nichtlineare Variationsungleichungen und Extremalaufgaben. Wissenschaften, Berlin
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)
Smith M.J. (1979). The existence, uniqueness and stability of traffic equilibrium. Transport. Res. 138: 295–304
Wardrop, J.G.: Some theoretical aspects of road traffic research. Proc. Inst. Civil Eng. Part II, 325–378 (1952)
Yen N.D. (1995). Hölder continuity of solutions to parametric variational inequalities. Appl. Math. Optim. 31: 245–255
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Anh, L.Q., Khanh, P.Q. Sensitivity analysis for multivalued quasiequilibrium problems in metric spaces: Hölder continuity of solutions. J Glob Optim 42, 515–531 (2008). https://doi.org/10.1007/s10898-007-9268-4
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10898-007-9268-4