# American Institute of Mathematical Sciences

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

## Semicontinuity of approximate solution mappings to generalized vector equilibrium problems

 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.
Citation: Qilin Wang, Shengji Li. Semicontinuity of approximate solution mappings to generalized vector equilibrium problems. Journal of Industrial and 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. [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. [3] J. P. Aubin and I. Ekeland, Applied Nonlinear Analysis, Wiley, New York, 1984. [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. [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. [6] C. R. Chen and S. J. Li, On the solution continuity of parametric generalized systems, Pac. J. Optim., 6 (2010), 141-151. [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. [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. [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. [10] J. F. Fu, Generalized Vector quasi-equilibrium problems, Math.Methods Oper.Res., 52 (2000), 57-64. doi: 10.1007/s001860000058. [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. [12] F. Giannessi, 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 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. [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. [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. [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. [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. [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. [20] K. Kimura and J. C. Yao, Sensitivity analysis of vector equilibrium problems, Taiwanese J. Math., 12 (2008), 649-669. [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. [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. [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. [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. [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. [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. [27] T. Tanino, Stability and sensitivity analysis in convex vector optimization, SIAM J. Control. Optim., 26 (1988), 521-536. doi: 10.1137/0326031. [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. [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. [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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##### 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, 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. [3] J. P. Aubin and I. Ekeland, Applied Nonlinear Analysis, Wiley, New York, 1984. [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. [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. [6] C. R. Chen and S. J. Li, On the solution continuity of parametric generalized systems, Pac. J. Optim., 6 (2010), 141-151. [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. [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. [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. [10] J. F. Fu, Generalized Vector quasi-equilibrium problems, Math.Methods Oper.Res., 52 (2000), 57-64. doi: 10.1007/s001860000058. [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. [12] F. Giannessi, 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 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. [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. [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. [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. [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. [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. [20] K. Kimura and J. C. Yao, Sensitivity analysis of vector equilibrium problems, Taiwanese J. Math., 12 (2008), 649-669. [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. [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. [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. [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. [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. [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. [27] T. Tanino, Stability and sensitivity analysis in convex vector optimization, SIAM J. Control. Optim., 26 (1988), 521-536. doi: 10.1137/0326031. [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. [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. [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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