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January  2017, 13(1): 477-488. doi: 10.3934/jimo.2016027

Solution continuity of parametric generalized vector equilibrium problems with strictly pseudomonotone mappings

Xin Zuo , Chun-Rong Chen , and  Hong-Zhi Wei

College of Mathematics and Statistics, Chongqing University, Chongqing, 401331, China

* Corresponding author: Chun-Rong Chen

Received  January 2015 Published  March 2016

Fund Project: This research was supported by the National Natural Science Foundation of China (Grant numbers: 11301567 and 11026144) and the Fundamental Research Funds for the Central Universities (Grant number: 106112015CDJXY100002).

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.

Keywords: Semicontinuity, parametric generalized (strong) vector equilibrium problems, scalarization, strict pseudomonotonicity.
Mathematics Subject Classification: Primary: 49K40, 90C31; Secondary: 47J20.
Citation: Xin Zuo, Chun-Rong Chen, Hong-Zhi Wei. Solution continuity of parametric generalized vector equilibrium problems with strictly pseudomonotone mappings. Journal of Industrial & Management Optimization, 2017, 13 (1) : 477-488. doi: 10.3934/jimo.2016027
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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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C. Berge, Topological Spaces, Oliver and Boyd, London, 1963.  Google Scholar

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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.  Google Scholar

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G. Y. Chen, X. X. Huang and X. Q. Yang, Vector Optimization: Set-Valued and Variational Analysis, Springer, Berlin, 2005.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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X. H. Gong, K. Kimura and J. C. Yao, Sensitivity analysis of strong vector equilibrium problems, J. Nonlinear Convex Anal., 9 (2008), 83-94.   Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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[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.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

J. P. Aubin and I. Ekeland, Applied Nonlinear Analysis, John Wiley and Sons, New York, 1984.  Google Scholar

[5]

C. Berge, Topological Spaces, Oliver and Boyd, London, 1963.  Google Scholar

[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.  Google Scholar

[7]

G. Y. Chen, X. X. Huang and X. Q. Yang, Vector Optimization: Set-Valued and Variational Analysis, Springer, Berlin, 2005.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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