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Solution continuity of parametric generalized vector equilibrium problems with strictly pseudomonotone mappings
College of Mathematics and Statistics, Chongqing University, Chongqing, 401331, China |
In this paper, new continuity (both lower and upper semicontinuities) results of solution mappings to parametric generalized (strong) vector equilibrium problems are established by scalarization approaches, under $f$-strict pseudomonotonicity assumptions. Especially, based on this new kind of monotonicity, the compactness of the mapping $F$ is not required, which is different from the related literature. Some examples are also provided to illustrate main conclusions.
References:
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L. Q. Anh and P. Q. Khanh,
Semicontinuity of the solution set of parametric multivalued vector quasiequilibrium problems, J. Math. Anal. Appl., 294 (2004), 699-711.
doi: 10.1016/j.jmaa.2004.03.014. |
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L. Q. Anh and P. Q. Khanh,
Semicontinuity of solution sets to parametric quasivariational inclusions with applications to traffic networks Ⅰ: Upper semicontinuities, Set-Valued Anal., 16 (2008), 267-279.
doi: 10.1007/s11228-008-0074-z. |
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L. Q. Anh and P. Q. Khanh,
Semicontinuity of solution sets to parametric quasivariational inclusions with applications to traffic networks Ⅱ: Lower semicontinuities applications, Set-Valued Anal., 16 (2008), 943-960.
doi: 10.1007/s11228-008-0082-z. |
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J. P. Aubin and I. Ekeland,
Applied Nonlinear Analysis, John Wiley and Sons, New York, 1984. |
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C. Berge,
Topological Spaces, Oliver and Boyd, London, 1963. |
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B. Chen and N. J. Huang,
Continuity of the solution mapping to parametric generalized vector equilibrium problems, J. Glob. Optim., 56 (2013), 1515-1528.
doi: 10.1007/s10898-012-9904-5. |
[7] |
G. Y. Chen, X. X. Huang and X. Q. Yang,
Vector Optimization: Set-Valued and Variational Analysis, Springer, Berlin, 2005. |
[8] |
C. R. Chen and S. J. Li,
Semicontinuity of the solution set map to a set-valued weak vector variational inequality, J. Ind. Manag. Optim., 3 (2007), 519-528.
doi: 10.3934/jimo.2007.3.519. |
[9] |
C. R. Chen, S. J. Li and K. L. Teo,
Solution semicontinuity of parametric generalized vector equilibrium problems, J. Glob. Optim., 45 (2009), 309-318.
doi: 10.1007/s10898-008-9376-9. |
[10] |
C. R. Chen, S. J. Li, J. Zeng and X. B. Li,
Error analysis of approximate solutions to parametric vector quasiequilibrium problems, Optim. Lett., 5 (2011), 85-98.
doi: 10.1007/s11590-010-0192-z. |
[11] |
Y. H. Cheng and D. L. Zhu,
Global stability results for the weak vector variational inequality, J. Glob. Optim., 32 (2005), 543-550.
doi: 10.1007/s10898-004-2692-9. |
[12] |
F. Giannessi (ed. ),
Vector Variational Inequalities and Vector Equilibria: Mathematical Theories, Kluwer Academic Publishers, Dordrecht, 2000.
doi: 10.1007/978-1-4613-0299-5. |
[13] |
X. H. Gong,
Continuity of the solution set to parametric weak vector equilibrium problems, J. Optim. Theory Appl., 139 (2008), 35-46.
doi: 10.1007/s10957-008-9429-8. |
[14] |
X. H. Gong, K. Kimura and J. C. Yao,
Sensitivity analysis of strong vector equilibrium problems, J. Nonlinear Convex Anal., 9 (2008), 83-94.
|
[15] |
X. H. Gong and J. C. Yao,
Lower semicontinuity of the set of efficient solutions for generalized systems, J. Optim. Theory Appl., 138 (2008), 197-205.
doi: 10.1007/s10957-008-9379-1. |
[16] |
N. J. Huang, J. Li and H. B. Thompson,
Stability for parametric implicit vector equilibrium problems, Math. Comput. Modelling., 43 (2006), 1267-1274.
doi: 10.1016/j.mcm.2005.06.010. |
[17] |
K. Kimura, Y. C. Liou, S. Y. Wu and J. C. Yao,
Well-Posedness for parametric vector equilibrium problems with applications, J. Ind. Manag. Optim., 4 (2008), 313-327.
doi: 10.3934/jimo.2008.4.313. |
[18] |
K. Kimura, Y. C. Liou and J. C. Yao,
Semicontinuity of the solution mapping of $ε$-vector equilibrium problem, Pac. J. Optim., 3 (2007), 345-359.
|
[19] |
K. Kimura and J. C. Yao,
Semicontinuity of solution mappings of parametric generalized vector equilibrium problems, J. Optim. Theory Appl., 138 (2008), 429-443.
doi: 10.1007/s10957-008-9386-2. |
[20] |
K. Kimura and J. C. Yao,
Sensitivity analysis of solution mappings of parametric generalized quasi vector equilibrium problems, Taiwanese J. Math., 12 (2008), 2233-2268.
|
[21] |
K. Kimura and J. C. Yao,
Semicontinuity of solution mappings of parametric generalized strong vector equilibrium problems, J. Ind. Manag. Optim., 4 (2008), 167-181.
doi: 10.3934/jimo.2008.4.167. |
[22] |
S. J. Li, H. M. Liu, Y. Zhang and Z. M. Fang,
Continuity of the solution mappings to parametric generalized strong vector equilibrium problems, J. Glob. Optim., 55 (2013), 597-610.
doi: 10.1007/s10898-012-9985-1. |
[23] |
Z. Y. Peng, X. M. Yang and J. W. Peng,
On the lower semicontinuity of the solution mappings to parametric weak generalized Ky Fan inequality, J. Optim. Theory Appl., 152 (2012), 256-264.
doi: 10.1007/s10957-011-9883-6. |
[24] |
P. H. Sach and L. A. Tuan,
New scalarizing approach to the stability analysis in parametric generalized Ky Fan inequality problems, J. Optim. Theory Appl., 157 (2013), 347-364.
doi: 10.1007/s10957-012-0105-7. |
[25] |
X. Wang and N. J. Huang,
Stability analysis for set-valued vector mixed variational inequalities in real reflexive Banach spaces, J. Ind. Manag. Optim., 9 (2013), 57-74.
doi: 10.3934/jimo.2013.9.57. |
[26] |
Q. L. Wang and S. J. Li,
Lower semicontinuity of the solution mapping to a parametric generalized vector equilibrium problem, J. Ind. Manag. Optim., 10 (2014), 1225-1234.
doi: 10.3934/jimo.2014.10.1225. |
[27] |
Q. L. Wang, Z. Lin and X. B. Li,
Semicontinuity of the solution set to a parametric generalized strong vector equilibrium problem, Positivity, 18 (2014), 733-748.
doi: 10.1007/s11117-014-0273-9. |
[28] |
R. Wangkeeree, R. Wangkeeree and P. Preechasilp,
Continuity of the solution mappings to parametric generalized vector equilibrium problems, Appl. Math. Lett., 29 (2014), 42-45.
doi: 10.1016/j.aml.2013.10.012. |
[29] |
Y. D. Xu and S. J. Li,
On the lower semicontinuity of the solution mappings to a parametric generalized strong vector equilibrium problem, Positivity, 17 (2013), 341-353.
doi: 10.1007/s11117-012-0170-z. |
[30] |
W. Y. Zhang, Z. M. Fang and Y. Zhang,
A note on the lower semicontinuity of efficient solutions for parametric vector equilibrium problems, Appl. Math. Lett., 26 (2013), 469-472.
doi: 10.1016/j.aml.2012.11.010. |
show all references
References:
[1] |
L. Q. Anh and P. Q. Khanh,
Semicontinuity of the solution set of parametric multivalued vector quasiequilibrium problems, J. Math. Anal. Appl., 294 (2004), 699-711.
doi: 10.1016/j.jmaa.2004.03.014. |
[2] |
L. Q. Anh and P. Q. Khanh,
Semicontinuity of solution sets to parametric quasivariational inclusions with applications to traffic networks Ⅰ: Upper semicontinuities, Set-Valued Anal., 16 (2008), 267-279.
doi: 10.1007/s11228-008-0074-z. |
[3] |
L. Q. Anh and P. Q. Khanh,
Semicontinuity of solution sets to parametric quasivariational inclusions with applications to traffic networks Ⅱ: Lower semicontinuities applications, Set-Valued Anal., 16 (2008), 943-960.
doi: 10.1007/s11228-008-0082-z. |
[4] |
J. P. Aubin and I. Ekeland,
Applied Nonlinear Analysis, John Wiley and Sons, New York, 1984. |
[5] |
C. Berge,
Topological Spaces, Oliver and Boyd, London, 1963. |
[6] |
B. Chen and N. J. Huang,
Continuity of the solution mapping to parametric generalized vector equilibrium problems, J. Glob. Optim., 56 (2013), 1515-1528.
doi: 10.1007/s10898-012-9904-5. |
[7] |
G. Y. Chen, X. X. Huang and X. Q. Yang,
Vector Optimization: Set-Valued and Variational Analysis, Springer, Berlin, 2005. |
[8] |
C. R. Chen and S. J. Li,
Semicontinuity of the solution set map to a set-valued weak vector variational inequality, J. Ind. Manag. Optim., 3 (2007), 519-528.
doi: 10.3934/jimo.2007.3.519. |
[9] |
C. R. Chen, S. J. Li and K. L. Teo,
Solution semicontinuity of parametric generalized vector equilibrium problems, J. Glob. Optim., 45 (2009), 309-318.
doi: 10.1007/s10898-008-9376-9. |
[10] |
C. R. Chen, S. J. Li, J. Zeng and X. B. Li,
Error analysis of approximate solutions to parametric vector quasiequilibrium problems, Optim. Lett., 5 (2011), 85-98.
doi: 10.1007/s11590-010-0192-z. |
[11] |
Y. H. Cheng and D. L. Zhu,
Global stability results for the weak vector variational inequality, J. Glob. Optim., 32 (2005), 543-550.
doi: 10.1007/s10898-004-2692-9. |
[12] |
F. Giannessi (ed. ),
Vector Variational Inequalities and Vector Equilibria: Mathematical Theories, Kluwer Academic Publishers, Dordrecht, 2000.
doi: 10.1007/978-1-4613-0299-5. |
[13] |
X. H. Gong,
Continuity of the solution set to parametric weak vector equilibrium problems, J. Optim. Theory Appl., 139 (2008), 35-46.
doi: 10.1007/s10957-008-9429-8. |
[14] |
X. H. Gong, K. Kimura and J. C. Yao,
Sensitivity analysis of strong vector equilibrium problems, J. Nonlinear Convex Anal., 9 (2008), 83-94.
|
[15] |
X. H. Gong and J. C. Yao,
Lower semicontinuity of the set of efficient solutions for generalized systems, J. Optim. Theory Appl., 138 (2008), 197-205.
doi: 10.1007/s10957-008-9379-1. |
[16] |
N. J. Huang, J. Li and H. B. Thompson,
Stability for parametric implicit vector equilibrium problems, Math. Comput. Modelling., 43 (2006), 1267-1274.
doi: 10.1016/j.mcm.2005.06.010. |
[17] |
K. Kimura, Y. C. Liou, S. Y. Wu and J. C. Yao,
Well-Posedness for parametric vector equilibrium problems with applications, J. Ind. Manag. Optim., 4 (2008), 313-327.
doi: 10.3934/jimo.2008.4.313. |
[18] |
K. Kimura, Y. C. Liou and J. C. Yao,
Semicontinuity of the solution mapping of $ε$-vector equilibrium problem, Pac. J. Optim., 3 (2007), 345-359.
|
[19] |
K. Kimura and J. C. Yao,
Semicontinuity of solution mappings of parametric generalized vector equilibrium problems, J. Optim. Theory Appl., 138 (2008), 429-443.
doi: 10.1007/s10957-008-9386-2. |
[20] |
K. Kimura and J. C. Yao,
Sensitivity analysis of solution mappings of parametric generalized quasi vector equilibrium problems, Taiwanese J. Math., 12 (2008), 2233-2268.
|
[21] |
K. Kimura and J. C. Yao,
Semicontinuity of solution mappings of parametric generalized strong vector equilibrium problems, J. Ind. Manag. Optim., 4 (2008), 167-181.
doi: 10.3934/jimo.2008.4.167. |
[22] |
S. J. Li, H. M. Liu, Y. Zhang and Z. M. Fang,
Continuity of the solution mappings to parametric generalized strong vector equilibrium problems, J. Glob. Optim., 55 (2013), 597-610.
doi: 10.1007/s10898-012-9985-1. |
[23] |
Z. Y. Peng, X. M. Yang and J. W. Peng,
On the lower semicontinuity of the solution mappings to parametric weak generalized Ky Fan inequality, J. Optim. Theory Appl., 152 (2012), 256-264.
doi: 10.1007/s10957-011-9883-6. |
[24] |
P. H. Sach and L. A. Tuan,
New scalarizing approach to the stability analysis in parametric generalized Ky Fan inequality problems, J. Optim. Theory Appl., 157 (2013), 347-364.
doi: 10.1007/s10957-012-0105-7. |
[25] |
X. Wang and N. J. Huang,
Stability analysis for set-valued vector mixed variational inequalities in real reflexive Banach spaces, J. Ind. Manag. Optim., 9 (2013), 57-74.
doi: 10.3934/jimo.2013.9.57. |
[26] |
Q. L. Wang and S. J. Li,
Lower semicontinuity of the solution mapping to a parametric generalized vector equilibrium problem, J. Ind. Manag. Optim., 10 (2014), 1225-1234.
doi: 10.3934/jimo.2014.10.1225. |
[27] |
Q. L. Wang, Z. Lin and X. B. Li,
Semicontinuity of the solution set to a parametric generalized strong vector equilibrium problem, Positivity, 18 (2014), 733-748.
doi: 10.1007/s11117-014-0273-9. |
[28] |
R. Wangkeeree, R. Wangkeeree and P. Preechasilp,
Continuity of the solution mappings to parametric generalized vector equilibrium problems, Appl. Math. Lett., 29 (2014), 42-45.
doi: 10.1016/j.aml.2013.10.012. |
[29] |
Y. D. Xu and S. J. Li,
On the lower semicontinuity of the solution mappings to a parametric generalized strong vector equilibrium problem, Positivity, 17 (2013), 341-353.
doi: 10.1007/s11117-012-0170-z. |
[30] |
W. Y. Zhang, Z. M. Fang and Y. Zhang,
A note on the lower semicontinuity of efficient solutions for parametric vector equilibrium problems, Appl. Math. Lett., 26 (2013), 469-472.
doi: 10.1016/j.aml.2012.11.010. |
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