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Hölder continuity of solutions to parametric weak generalized Ky Fan inequality. (English) Zbl 1229.90214
The authors study the problem of finding a point x ¯K(λ) such that f(x ¯,y,μ)-int(C) for all yK(λ), where f is a vector-valued function with values in a normed space in which a convex, pointed and closed cone C is given, K(·) is a set-valued mapping with values in a metric space X, λ and μ are parameters. They establish the Hölder continuity of the solution mapping, which is not necessarily single-valued, with respect to the parameters λ and μ under some assumptions on the Hölder continuity of the mapping K(·), the Hölder strong monotonicity, Hölder continuity and convexity of the function f.

MSC:
90C31Sensitivity, stability, parametric optimization
49J40Variational methods including variational inequalities
References:
[1]Fan, K.: Extensions of two fixed point theorems of F.E. Browder. Math. Z. 112, 234–240 (1969) · Zbl 0185.39503 · doi:10.1007/BF01110225
[2]Ansari, Q.H.: Vector equilibrium problems and vector variational inequalities. In: Giannessi, F. (ed.) Vector variational inequalities and vector equilibria. Mathematical Theories. Kluwer, Dordrecht, pp. 1–16 (2000)
[3]Bianchi, M., Hadjisavvas, M., 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
[4]Fan, K.: A minimax inequality and applications. In: Shisha, O. (ed.) Inequality III. Academic Press, New York, pp. 103–113 (1972)
[5]Li, X.B., Li, S.J.: Existences of solutions for generalized vector quasiequilibrium problems. Optim. Lett. 4, 17–28 (2010) · Zbl 1183.49006 · doi:10.1007/s11590-009-0142-9
[6]Anh, L.Q., Khanh, P.Q.: Various kinds of semicontinuity and 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
[7]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
[8]Chen, C.R., Li, S.J., Teo, K.L.: Solution semicontinuity of parametric generalized vector equilibrium problems. J. Glob. Optim. 45, 309–318 (2009) · Zbl 1213.54028 · doi:10.1007/s10898-008-9376-9
[9]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
[10]Mansour, M.A., 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
[11]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
[12]Anh, L.Q., Khanh, P.Q.: On the Hölder continuity of solutions to multivalued vector equilibrium problems. J. Math. Anal. Appl. 321, 308–315 (2006) · Zbl 1104.90041 · doi:10.1016/j.jmaa.2005.08.018
[13]Anh, L.Q., Khanh, P.Q.: Uniqueness and Hölder continuity of solution to multivalued vector equilibrium problems in metric spaces. J. Glob. Optim. 37, 449–465 (2007) · Zbl 1156.90025 · doi:10.1007/s10898-006-9062-8
[14]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) · Zbl 1188.90274 · doi:10.1007/s10898-007-9268-4
[15]Lee, G.M., Kim, D.S., Lee, B.S., Yen, N.D.: Vector variational inequality as a tool for studying vector optimization problems. Nonlinear Anal. 34, 745–765 (1998) · Zbl 0956.49007 · doi:10.1016/S0362-546X(97)00578-6
[16]Mansour, M.A., Ausssel, D.: Quasimonotone variational inequalities and quasiconvex programming: quantitative stability. Pac. J. Optim. 2, 611–626 (2006)
[17]Li, S.J., Li, X.B., Wang, L.N., Teo, K.L.: The Hölder continuity of solutions to generalized vector equilibrium problems. Eur. J. Oper. Res. 199, 334–338 (2009) · Zbl 1176.90643 · doi:10.1016/j.ejor.2008.12.024
[18]Aubin, J.P., Ekeland, I.: Applied Nonlinear Analysis. Wiley, New York (1984)
[19]Nadler, S.B.: Multivalued contraction mappings. Pac. J. Math. 30, 475–488 (1969) · Zbl 0187.45002 · doi:10.2140/pjm.1969.30.475
[20]Bianchi, M., Pini, R.: Sensitivity for parametric vector equilibria. Optimization 55, 221–230 (2006) · Zbl 1149.90156 · doi:10.1080/02331930600662732
[21]Gong, X.H., Yao, J.-C.: Connectedness of the set of efficient solutions for generalized systems. J. Optim. Theory Appl. 138, 189–196 (2008) · Zbl 05314924 · doi:10.1007/s10957-008-9378-2