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October  2016, 12(4): 1303-1309. doi: 10.3934/jimo.2016.12.1303

Semicontinuity of approximate solution mappings to generalized vector equilibrium problems

Qilin Wang 1, and  Shengji Li 2,

1. 

College of Sciences, Chongqing Jiaotong University, Chongqing, 400074
2. 

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

Received  July 2014 Revised  October 2015 Published  January 2016

In this paper, the lower semicontinuity of the approximate solution mapping to generalized strong vector equilibrium problems is established by using a new proof method which is different from the ones used in the literature. Simultaneously, we also obtain the upper semicontinuity of the approximate solution mapping without the assumptions about monotonicity and approximate solution mappings. Some examples are given to illustrate our results.
Keywords: approximate solution mappings., upper semicontinuity, Generalized strong vector equilibrium problems, lower semicontinuity.
Mathematics Subject Classification: 49K40, 90C31, 91B5.
Citation: Qilin Wang, Shengji Li. Semicontinuity of approximate solution mappings to generalized vector equilibrium problems. Journal of Industrial & Management Optimization, 2016, 12 (4) : 1303-1309. doi: 10.3934/jimo.2016.12.1303
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, On the stability of the solution sets of general multivalued vector quasiequilibrium problems, J. Optim. Theory Appl., 135 (2007), 271-284. doi: 10.1007/s10957-007-9250-9.  Google Scholar

[3]

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

[4]

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

[5]

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

[6]

C. R. Chen and S. J. Li, On the solution continuity of parametric generalized systems, Pac. J. Optim., 6 (2010), 141-151.  Google Scholar

[7]

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

[8]

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

[9]

C. Chiang, O. Chadli and J. C. Yao, Genralized Vector equilibrium problems with trifunctions, J. Glob. Optim., 30 (2004), 135-154. doi: 10.1007/s10898-004-8273-0.  Google Scholar

[10]

J. F. Fu, Generalized Vector quasi-equilibrium problems, Math.Methods Oper.Res., 52 (2000), 57-64. doi: 10.1007/s001860000058.  Google Scholar

[11]

J. F. Fu, Vector equilibrium problems, existence theorems and convexity of solution set, J. Glob. Optim., 31 (2005), 109-119. doi: 10.1007/s10898-004-4274-2.  Google Scholar

[12]

F. Giannessi, 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 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

[15]

Y. Han and X. H. Gong, Lower semicontinuity of solution mapping to parametric generalized strong vector equilibrium problems, Appl. Math. Lett., 28 (2014), 38-41. doi: 10.1016/j.aml.2013.09.006.  Google Scholar

[16]

N. J. Huang, J. Li and H. B. Thompson, Stability for parametric implicit vector equilibrium problems, Math. Comput. Model., 43 (2006), 1267-1274. doi: 10.1016/j.mcm.2005.06.010.  Google Scholar

[17]

P. Q. Khanh and L. M. Luu, Lower and upper semicontinuity of the solution sets and approximate solution sets to parametric multivalued quasivariational inequalities, J. Optim. Theory Appl., 133 (2007), 329-339. doi: 10.1007/s10957-007-9190-4.  Google Scholar

[18]

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

[19]

K. Kimura and J. C. Yao, Sensitivity analysis of solution mappings of parametric vector quasi-equilibrium problems, J. Glob. Optim., 41 (2008), 187-202. doi: 10.1007/s10898-007-9210-9.  Google Scholar

[20]

K. Kimura and J. C. Yao, Sensitivity analysis of vector equilibrium problems, Taiwanese J. Math., 12 (2008), 649-669.  Google Scholar

[21]

K. Kimura and J. C. Yao, Sensitivity analysis of solution mappings of parametric generalized quasivector equilibrium problems, Taiwanese J. Math., 12 (2008), 2233-2268.  Google Scholar

[22]

S. J. Li, G. Y. Chen and K. L. Teo, On the stability of generalized vector quasivariational inequality problems, J. Optim. Theory Appl., 113 (2002), 283-295. doi: 10.1023/A:1014830925232.  Google Scholar

[23]

S. J. Li and C. R. Chen, Stability of weak vector variational inequality, Nonlinear Anal., 70 (2009), 1528-1535. doi: 10.1016/j.na.2008.02.032.  Google Scholar

[24]

S. J. Li and Z. M. Fang, Lower semicontinuity of the solution mappings to a parametric generalized Ky Fan inequality, J. Optim. Theory Appl., 147 (2010), 507-515. doi: 10.1007/s10957-010-9736-8.  Google Scholar

[25]

S. J. Li, H. M. Liu, Y. Zhang and Z. M. Fang, Continuity of 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

[26]

L. J. Lin, Q. H. Ansari and J. Y. Wu, Geometric properties and coincidence theorems with applications to generalized vector equilibrium problems, J. Optim. Theory Appl., 117 (2003), 121-137. doi: 10.1023/A:1023656507786.  Google Scholar

[27]

T. Tanino, Stability and sensitivity analysis in convex vector optimization, SIAM J. Control. Optim., 26 (1988), 521-536. doi: 10.1137/0326031.  Google Scholar

[28]

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

[29]

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

[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, On the stability of the solution sets of general multivalued vector quasiequilibrium problems, J. Optim. Theory Appl., 135 (2007), 271-284. doi: 10.1007/s10957-007-9250-9.  Google Scholar

[3]

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

[4]

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

[5]

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

[6]

C. R. Chen and S. J. Li, On the solution continuity of parametric generalized systems, Pac. J. Optim., 6 (2010), 141-151.  Google Scholar

[7]

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

[8]

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

[9]

C. Chiang, O. Chadli and J. C. Yao, Genralized Vector equilibrium problems with trifunctions, J. Glob. Optim., 30 (2004), 135-154. doi: 10.1007/s10898-004-8273-0.  Google Scholar

[10]

J. F. Fu, Generalized Vector quasi-equilibrium problems, Math.Methods Oper.Res., 52 (2000), 57-64. doi: 10.1007/s001860000058.  Google Scholar

[11]

J. F. Fu, Vector equilibrium problems, existence theorems and convexity of solution set, J. Glob. Optim., 31 (2005), 109-119. doi: 10.1007/s10898-004-4274-2.  Google Scholar

[12]

F. Giannessi, 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 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

[15]

Y. Han and X. H. Gong, Lower semicontinuity of solution mapping to parametric generalized strong vector equilibrium problems, Appl. Math. Lett., 28 (2014), 38-41. doi: 10.1016/j.aml.2013.09.006.  Google Scholar

[16]

N. J. Huang, J. Li and H. B. Thompson, Stability for parametric implicit vector equilibrium problems, Math. Comput. Model., 43 (2006), 1267-1274. doi: 10.1016/j.mcm.2005.06.010.  Google Scholar

[17]

P. Q. Khanh and L. M. Luu, Lower and upper semicontinuity of the solution sets and approximate solution sets to parametric multivalued quasivariational inequalities, J. Optim. Theory Appl., 133 (2007), 329-339. doi: 10.1007/s10957-007-9190-4.  Google Scholar

[18]

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

[19]

K. Kimura and J. C. Yao, Sensitivity analysis of solution mappings of parametric vector quasi-equilibrium problems, J. Glob. Optim., 41 (2008), 187-202. doi: 10.1007/s10898-007-9210-9.  Google Scholar

[20]

K. Kimura and J. C. Yao, Sensitivity analysis of vector equilibrium problems, Taiwanese J. Math., 12 (2008), 649-669.  Google Scholar

[21]

K. Kimura and J. C. Yao, Sensitivity analysis of solution mappings of parametric generalized quasivector equilibrium problems, Taiwanese J. Math., 12 (2008), 2233-2268.  Google Scholar

[22]

S. J. Li, G. Y. Chen and K. L. Teo, On the stability of generalized vector quasivariational inequality problems, J. Optim. Theory Appl., 113 (2002), 283-295. doi: 10.1023/A:1014830925232.  Google Scholar

[23]

S. J. Li and C. R. Chen, Stability of weak vector variational inequality, Nonlinear Anal., 70 (2009), 1528-1535. doi: 10.1016/j.na.2008.02.032.  Google Scholar

[24]

S. J. Li and Z. M. Fang, Lower semicontinuity of the solution mappings to a parametric generalized Ky Fan inequality, J. Optim. Theory Appl., 147 (2010), 507-515. doi: 10.1007/s10957-010-9736-8.  Google Scholar

[25]

S. J. Li, H. M. Liu, Y. Zhang and Z. M. Fang, Continuity of 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

[26]

L. J. Lin, Q. H. Ansari and J. Y. Wu, Geometric properties and coincidence theorems with applications to generalized vector equilibrium problems, J. Optim. Theory Appl., 117 (2003), 121-137. doi: 10.1023/A:1023656507786.  Google Scholar

[27]

T. Tanino, Stability and sensitivity analysis in convex vector optimization, SIAM J. Control. Optim., 26 (1988), 521-536. doi: 10.1137/0326031.  Google Scholar

[28]

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

[29]

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

[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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