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Strict feasibility for generalized mixed variational inequality in reflexive Banach spaces
1. | Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, China |
References:
[1] |
Ya. Alber, Generalized projection operators in Banach spaces: properties and applications, in "Proceedings of the Israel Seminar Ariel," Israel. Funct. Differ. Equ., 1 (1994), 1-21. |
[2] |
J. P. Aubin, "Optima and Equilibria," Springer-Verlag, Berlin, 1993. |
[3] |
C. Baiocchi, G. Buttazzo and F. Gastaldi, General existence theorems for unilateral problems in continuum mechanics, Archive Rational Mechanics Anal., 100 (1988), 149-189.
doi: doi:10.1007/BF00282202. |
[4] |
M. Bianchi, N. Hadjisavvas and S. Schaible, Minimal coercivity conditions and exceptional families of elements in quasimonotone variational inequalities, J. Optim. Theory Appl., 122 (2004), 1-17.
doi: doi:10.1023/B:JOTA.0000041728.12683.89. |
[5] |
I. Cioranescu, "Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems," Kluwer, Dordrecht, 1990. |
[6] |
J. P. Crouzeix, Pseudomonotone variational inequality problems: Existence of solutions, Math. Program., 78 (1997), 305-314.
doi: doi:10.1007/BF02614358. |
[7] |
A. Daniilidis and N. Hadjisavvas, Coercivity conditions and variational inequalities, Math. Program., 86 (1999), 433-438.
doi: doi:10.1007/s101070050097. |
[8] |
F. Facchinei and J. S. Pang, "Finite-dimensional Variational Inequalities and Complementarity Problems," Springer-Verlag, New York, 2003. |
[9] |
J. H. Fan, X. Liu and J. L. Li, Iterative schemes for approximating solutions of generalized variational inequalities in Banach spaces, Nonlinear Anal. TMA, 70 (2009) 3997-4007.
doi: doi:10.1016/j.na.2008.08.008. |
[10] |
Y. P. Fang and N. J. Huang, Feasibility and solvability for vector complementarity problems, J. Optim. Theory Appl., 129 (2006), 373-390.
doi: doi:10.1007/s10957-006-9073-0. |
[11] |
S. C. Fang and E. L. Peterson, Generalized variational inequalities, J. Optim. Theory Appl., 38 (1982), 363-383.
doi: doi:10.1007/BF00935344. |
[12] |
O. Güler, Existence of interior points and interior paths in nonlinear monotone complementarity problems, Math. Oper. Res., 18 (1993), 128-147.
doi: doi:10.1287/moor.18.1.128. |
[13] |
P.T. Harker and J. S. Pang, Finite-dimensional Variational and Nonlinear Complementarity Problems: A survey of theory, algorithms and applications, Math. Program., 48 (1990), 161-220.
doi: doi:10.1007/BF01582255. |
[14] |
Y. R. He, Stable pseudomonotone variational inequality in reflexive Banach spaces, J. Math. Anal. Appl., 330 (2007), 352-363.
doi: doi:10.1016/j.jmaa.2006.07.063. |
[15] |
Y. R. He, A new projection algorithm for mixed variational inequalities, Acta Math. Sci., 27 (2007), 215-220. |
[16] |
Y. R. He, X. Z. Mao and M. Zhou, Strict feasibility of variational inequalities in reflexive Banach spaces, Acta Math. Sinica (English Series), 23 (2007), 563-570.
doi: doi:10.1007/s10114-005-0918-5. |
[17] |
R. Hu and Y. P. Fang, Feasibility-solvability theorem for a generalized system, J. Optim. Theory Appl., 142 (2009), 493-499.
doi: doi:10.1007/s10957-009-9510-y. |
[18] |
R. Hu and Y. P. Fang, Strict feasibility and solvability for vector equilibrium problems in reflexive Banach spaces, Optim. Lett., DOI 10.1007/s11590-010-0215-9.
doi: doi:10.1007/s11590-010-0215-9. |
[19] |
V. Jeyakumar, W. Oettli and M. Natividad, A solvability theorem for a class of quasiconvex mappings with applications to optimization, J. Math. Anal. Appl., 179 (1993), 537-546.
doi: doi:10.1006/jmaa.1993.1368. |
[20] |
S. Karamardian, Complementarity problems over cones with monotone and pseudomonotone maps, J. Optim. Theory Appl., 18 (1976), 445-454.
doi: doi:10.1007/BF00932654. |
[21] |
B. T. Kien, M. M. Wong, N. C. Wong and J. C. Yao, Degree theory for generalized variational inequalities and applications, Eur. J. Oper. Res., 193 (2009), 12-22.
doi: doi:10.1016/j.ejor.2007.10.028. |
[22] |
J. L. Li, The generalized projection operator on reflexive Banach spaces and its applications, J. Math. Anal. Appl., 306 (2005), 55-71.
doi: doi:10.1016/j.jmaa.2004.11.007. |
[23] |
F. S. Qiao and Y. R. He, Strict feasibility of pseudomonotone set-valued variational inequalities, Optimization, 60 (2011), 303-310.
doi: doi:10.1080/02331934.2010.507985. |
[24] |
R. T. Rockafellar, "Convex Analysis," Princeton University Press, Princeton, NJ, 1970. |
[25] |
M. Sion, On general minimax theorems, Pacific J. Math., 8 (1958), 171-176. |
[26] |
W. Takahashi, "Nonlinear Functional Analysis," Yokohama Publishers, Yokohama, 2000. |
[27] |
M. M. Vainberg, "Variational Methods and Method of Monotone Operators," Wiley, New York, 1973. |
[28] |
Z. B. Wang and N.J. Huang, Degree theory for a generalized set-valued variational inequality with an application in Banach spaces, J. Global Optim., 49 (2011), 343-357
doi: doi:10.1007/s10898-010-9547-3. |
[29] |
K. Q. Wu and N. J. Huang, The generalized $f$-projection operator with an application, Bull. Aust. Math. Soc., 73 (2006), 307-317.
doi: doi:10.1017/S0004972700038892. |
[30] |
H. Y. Yin, C. X. Xu and Z. X. Zhang, The $F$-complementarity problem and its equivalence with the least element problem, Acta Math. Sinica, 44 (2001), 679-686. |
[31] |
R. Y. Zhong and N. J. Huang, Stability analysis for Minty mixed variational inequality in reflexive Banach spaces, J. Optim. Theory Appl., 147 (2010), 454-472.
doi: doi:10.1007/s10957-010-9732-z. |
show all references
References:
[1] |
Ya. Alber, Generalized projection operators in Banach spaces: properties and applications, in "Proceedings of the Israel Seminar Ariel," Israel. Funct. Differ. Equ., 1 (1994), 1-21. |
[2] |
J. P. Aubin, "Optima and Equilibria," Springer-Verlag, Berlin, 1993. |
[3] |
C. Baiocchi, G. Buttazzo and F. Gastaldi, General existence theorems for unilateral problems in continuum mechanics, Archive Rational Mechanics Anal., 100 (1988), 149-189.
doi: doi:10.1007/BF00282202. |
[4] |
M. Bianchi, N. Hadjisavvas and S. Schaible, Minimal coercivity conditions and exceptional families of elements in quasimonotone variational inequalities, J. Optim. Theory Appl., 122 (2004), 1-17.
doi: doi:10.1023/B:JOTA.0000041728.12683.89. |
[5] |
I. Cioranescu, "Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems," Kluwer, Dordrecht, 1990. |
[6] |
J. P. Crouzeix, Pseudomonotone variational inequality problems: Existence of solutions, Math. Program., 78 (1997), 305-314.
doi: doi:10.1007/BF02614358. |
[7] |
A. Daniilidis and N. Hadjisavvas, Coercivity conditions and variational inequalities, Math. Program., 86 (1999), 433-438.
doi: doi:10.1007/s101070050097. |
[8] |
F. Facchinei and J. S. Pang, "Finite-dimensional Variational Inequalities and Complementarity Problems," Springer-Verlag, New York, 2003. |
[9] |
J. H. Fan, X. Liu and J. L. Li, Iterative schemes for approximating solutions of generalized variational inequalities in Banach spaces, Nonlinear Anal. TMA, 70 (2009) 3997-4007.
doi: doi:10.1016/j.na.2008.08.008. |
[10] |
Y. P. Fang and N. J. Huang, Feasibility and solvability for vector complementarity problems, J. Optim. Theory Appl., 129 (2006), 373-390.
doi: doi:10.1007/s10957-006-9073-0. |
[11] |
S. C. Fang and E. L. Peterson, Generalized variational inequalities, J. Optim. Theory Appl., 38 (1982), 363-383.
doi: doi:10.1007/BF00935344. |
[12] |
O. Güler, Existence of interior points and interior paths in nonlinear monotone complementarity problems, Math. Oper. Res., 18 (1993), 128-147.
doi: doi:10.1287/moor.18.1.128. |
[13] |
P.T. Harker and J. S. Pang, Finite-dimensional Variational and Nonlinear Complementarity Problems: A survey of theory, algorithms and applications, Math. Program., 48 (1990), 161-220.
doi: doi:10.1007/BF01582255. |
[14] |
Y. R. He, Stable pseudomonotone variational inequality in reflexive Banach spaces, J. Math. Anal. Appl., 330 (2007), 352-363.
doi: doi:10.1016/j.jmaa.2006.07.063. |
[15] |
Y. R. He, A new projection algorithm for mixed variational inequalities, Acta Math. Sci., 27 (2007), 215-220. |
[16] |
Y. R. He, X. Z. Mao and M. Zhou, Strict feasibility of variational inequalities in reflexive Banach spaces, Acta Math. Sinica (English Series), 23 (2007), 563-570.
doi: doi:10.1007/s10114-005-0918-5. |
[17] |
R. Hu and Y. P. Fang, Feasibility-solvability theorem for a generalized system, J. Optim. Theory Appl., 142 (2009), 493-499.
doi: doi:10.1007/s10957-009-9510-y. |
[18] |
R. Hu and Y. P. Fang, Strict feasibility and solvability for vector equilibrium problems in reflexive Banach spaces, Optim. Lett., DOI 10.1007/s11590-010-0215-9.
doi: doi:10.1007/s11590-010-0215-9. |
[19] |
V. Jeyakumar, W. Oettli and M. Natividad, A solvability theorem for a class of quasiconvex mappings with applications to optimization, J. Math. Anal. Appl., 179 (1993), 537-546.
doi: doi:10.1006/jmaa.1993.1368. |
[20] |
S. Karamardian, Complementarity problems over cones with monotone and pseudomonotone maps, J. Optim. Theory Appl., 18 (1976), 445-454.
doi: doi:10.1007/BF00932654. |
[21] |
B. T. Kien, M. M. Wong, N. C. Wong and J. C. Yao, Degree theory for generalized variational inequalities and applications, Eur. J. Oper. Res., 193 (2009), 12-22.
doi: doi:10.1016/j.ejor.2007.10.028. |
[22] |
J. L. Li, The generalized projection operator on reflexive Banach spaces and its applications, J. Math. Anal. Appl., 306 (2005), 55-71.
doi: doi:10.1016/j.jmaa.2004.11.007. |
[23] |
F. S. Qiao and Y. R. He, Strict feasibility of pseudomonotone set-valued variational inequalities, Optimization, 60 (2011), 303-310.
doi: doi:10.1080/02331934.2010.507985. |
[24] |
R. T. Rockafellar, "Convex Analysis," Princeton University Press, Princeton, NJ, 1970. |
[25] |
M. Sion, On general minimax theorems, Pacific J. Math., 8 (1958), 171-176. |
[26] |
W. Takahashi, "Nonlinear Functional Analysis," Yokohama Publishers, Yokohama, 2000. |
[27] |
M. M. Vainberg, "Variational Methods and Method of Monotone Operators," Wiley, New York, 1973. |
[28] |
Z. B. Wang and N.J. Huang, Degree theory for a generalized set-valued variational inequality with an application in Banach spaces, J. Global Optim., 49 (2011), 343-357
doi: doi:10.1007/s10898-010-9547-3. |
[29] |
K. Q. Wu and N. J. Huang, The generalized $f$-projection operator with an application, Bull. Aust. Math. Soc., 73 (2006), 307-317.
doi: doi:10.1017/S0004972700038892. |
[30] |
H. Y. Yin, C. X. Xu and Z. X. Zhang, The $F$-complementarity problem and its equivalence with the least element problem, Acta Math. Sinica, 44 (2001), 679-686. |
[31] |
R. Y. Zhong and N. J. Huang, Stability analysis for Minty mixed variational inequality in reflexive Banach spaces, J. Optim. Theory Appl., 147 (2010), 454-472.
doi: doi:10.1007/s10957-010-9732-z. |
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