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October  2014, 10(4): 1225-1234. doi: 10.3934/jimo.2014.10.1225

Lower semicontinuity of the solution mapping to a parametric generalized vector equilibrium problem

1. 

College of Sciences, Chongqing Jiaotong University, Chongqing, 400074

2. 

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

Received  March 2013 Revised  September 2013 Published  February 2014

This paper deals with the lower semicontinuity of the solution mapping to a parametric generalized vector equilibrium problem. Under new assumptions, which do not contain any information about solution mappings, we establish the lower semicontinuity of the solution mapping to a parametric generalized vector equilibrium problem by using a scalarization method. These results improve the corresponding ones in recent literature. Some examples are given to illustrate our results.
Citation: Qilin Wang, Shengji Li. Lower semicontinuity of the solution mapping to a parametric generalized vector equilibrium problem. Journal of Industrial & Management Optimization, 2014, 10 (4) : 1225-1234. doi: 10.3934/jimo.2014.10.1225
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.  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.  doi: 10.1007/s10957-007-9250-9.  Google Scholar

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J. P. Aubin and I. Ekeland, Applied Nonlinear Analysis,, Wiley, (1984).   Google Scholar

[4]

Berge, Topological Spaces,, Oliver and Boyd, (1963).   Google Scholar

[5]

M. Bianchi, N. Hadjisavvas and S. Schaible, Vector equilibrium problems with generalized monotone bifunctions,, J. Optim. Theory Appl., 92 (1997), 527.  doi: 10.1023/A:1022603406244.  Google Scholar

[6]

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.  doi: 10.3934/jimo.2007.3.519.  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.  doi: 10.1007/s10898-008-9376-9.  Google Scholar

[8]

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

[9]

Y. H. Cheng and D. L. Zhu, Global stability results for the weak vector variational inequality,, J. Glob. Optim., 32 (2005), 543.  doi: 10.1007/s10898-004-2692-9.  Google Scholar

[10]

F. Ferro, A minimax theorem for vector-valued functions,, J. Optim. Theory Appl., 60 (1989), 19.  doi: 10.1007/BF00938796.  Google Scholar

[11]

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

[12]

F. Giannessi, Vector Variational Inequalities and Vector Equilibria: Mathematical Theories,, Kluwer Academic Publishers, (2000).  doi: 10.1007/978-1-4613-0299-5.  Google Scholar

[13]

X. H. Gong, Connectedness of the solution sets and scalarization for vector equilibrium problems,, J. Optim. Theory Appl., 133 (2007), 151.  doi: 10.1007/s10957-007-9196-y.  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.  doi: 10.1007/s10957-008-9379-1.  Google Scholar

[15]

X. H. Gong, Continuity of the solution set to parametric weak vector equilibrium problems,, J. Optim. Theory Appl., 139 (2008), 35.  doi: 10.1007/s10957-008-9429-8.  Google Scholar

[16]

X. H. Gong and J. C. Yao, Connectedness of the set of efficient solutions for generalized systems,, J. Optim. Theory Appl., 138 (2008), 189.  doi: 10.1007/s10957-008-9378-2.  Google Scholar

[17]

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

[18]

B. T. Kien, On the lower semicontinuity of optimal solution sets,, Optimization, 54 (2005), 123.  doi: 10.1080/02331930412331330379.  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.  doi: 10.1007/s10898-007-9210-9.  Google Scholar

[20]

K. Kimura and J. C.Yao, Semicontinuity of solution mappings of parametric generalized strong vector equilibrium problems,, J. Ind. Manag. Optim., 4 (2008), 167.  doi: 10.3934/jimo.2008.4.167.  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.   Google Scholar

[22]

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

[23]

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.  doi: 10.1007/s10957-007-9190-4.  Google Scholar

[24]

Z. F. Li and G. Y. Chen, Lagrangian Multipliers, saddle points and duality in vector optimization of set-valued maps,, J. Math. Anal. Appl., 215 (1997), 297.  doi: 10.1006/jmaa.1997.5568.  Google Scholar

[25]

Z. F. Li and S. Y. Wang, Lagrange Multipliers and saddle points in multiobjective programming,, J. Optim. Theory Appl., 83 (1994), 63.  doi: 10.1007/BF02191762.  Google Scholar

[26]

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), 297.  doi: 10.1023/A:1014830925232.  Google Scholar

[27]

S. J. Li and Z. M. Fang, On the stability of a dual weak vector variational inequality problem,, J. Ind. Manag. Optim., 4 (2008), 155.  doi: 10.3934/jimo.2008.4.155.  Google Scholar

[28]

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

[29]

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.  doi: 10.1007/s10957-010-9736-8.  Google Scholar

[30]

S. J. Li, H. M. Liu and C. R. Chen, Lower semicomtinuity of parametric generalized weak vector equilibrium problems,, Bull. Aust. Math. Soc., 81 (2010), 85.  doi: 10.1017/S0004972709000628.  Google Scholar

[31]

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.  doi: 10.1007/s10898-012-9985-1.  Google Scholar

[32]

Y. D. Xu and S. J. Li, On the lower semicontinuity of the solution mappings to a para- metric generalized strong vector equilibrium problem,, Positivity, 17 (2013), 341.  doi: 10.1007/s11117-012-0170-z.  Google Scholar

[33]

X. M. Yang, D. Li and S. Y. Wang, Near-subconvexlikeness in vector optimization with set-valued functions,, J. Optim. Theory Appl., 110 (2001), 413.  doi: 10.1023/A:1017535631418.  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.  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.  doi: 10.1007/s10957-007-9250-9.  Google Scholar

[3]

J. P. Aubin and I. Ekeland, Applied Nonlinear Analysis,, Wiley, (1984).   Google Scholar

[4]

Berge, Topological Spaces,, Oliver and Boyd, (1963).   Google Scholar

[5]

M. Bianchi, N. Hadjisavvas and S. Schaible, Vector equilibrium problems with generalized monotone bifunctions,, J. Optim. Theory Appl., 92 (1997), 527.  doi: 10.1023/A:1022603406244.  Google Scholar

[6]

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.  doi: 10.3934/jimo.2007.3.519.  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.  doi: 10.1007/s10898-008-9376-9.  Google Scholar

[8]

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

[9]

Y. H. Cheng and D. L. Zhu, Global stability results for the weak vector variational inequality,, J. Glob. Optim., 32 (2005), 543.  doi: 10.1007/s10898-004-2692-9.  Google Scholar

[10]

F. Ferro, A minimax theorem for vector-valued functions,, J. Optim. Theory Appl., 60 (1989), 19.  doi: 10.1007/BF00938796.  Google Scholar

[11]

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

[12]

F. Giannessi, Vector Variational Inequalities and Vector Equilibria: Mathematical Theories,, Kluwer Academic Publishers, (2000).  doi: 10.1007/978-1-4613-0299-5.  Google Scholar

[13]

X. H. Gong, Connectedness of the solution sets and scalarization for vector equilibrium problems,, J. Optim. Theory Appl., 133 (2007), 151.  doi: 10.1007/s10957-007-9196-y.  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.  doi: 10.1007/s10957-008-9379-1.  Google Scholar

[15]

X. H. Gong, Continuity of the solution set to parametric weak vector equilibrium problems,, J. Optim. Theory Appl., 139 (2008), 35.  doi: 10.1007/s10957-008-9429-8.  Google Scholar

[16]

X. H. Gong and J. C. Yao, Connectedness of the set of efficient solutions for generalized systems,, J. Optim. Theory Appl., 138 (2008), 189.  doi: 10.1007/s10957-008-9378-2.  Google Scholar

[17]

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

[18]

B. T. Kien, On the lower semicontinuity of optimal solution sets,, Optimization, 54 (2005), 123.  doi: 10.1080/02331930412331330379.  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.  doi: 10.1007/s10898-007-9210-9.  Google Scholar

[20]

K. Kimura and J. C.Yao, Semicontinuity of solution mappings of parametric generalized strong vector equilibrium problems,, J. Ind. Manag. Optim., 4 (2008), 167.  doi: 10.3934/jimo.2008.4.167.  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.   Google Scholar

[22]

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

[23]

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.  doi: 10.1007/s10957-007-9190-4.  Google Scholar

[24]

Z. F. Li and G. Y. Chen, Lagrangian Multipliers, saddle points and duality in vector optimization of set-valued maps,, J. Math. Anal. Appl., 215 (1997), 297.  doi: 10.1006/jmaa.1997.5568.  Google Scholar

[25]

Z. F. Li and S. Y. Wang, Lagrange Multipliers and saddle points in multiobjective programming,, J. Optim. Theory Appl., 83 (1994), 63.  doi: 10.1007/BF02191762.  Google Scholar

[26]

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), 297.  doi: 10.1023/A:1014830925232.  Google Scholar

[27]

S. J. Li and Z. M. Fang, On the stability of a dual weak vector variational inequality problem,, J. Ind. Manag. Optim., 4 (2008), 155.  doi: 10.3934/jimo.2008.4.155.  Google Scholar

[28]

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

[29]

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.  doi: 10.1007/s10957-010-9736-8.  Google Scholar

[30]

S. J. Li, H. M. Liu and C. R. Chen, Lower semicomtinuity of parametric generalized weak vector equilibrium problems,, Bull. Aust. Math. Soc., 81 (2010), 85.  doi: 10.1017/S0004972709000628.  Google Scholar

[31]

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.  doi: 10.1007/s10898-012-9985-1.  Google Scholar

[32]

Y. D. Xu and S. J. Li, On the lower semicontinuity of the solution mappings to a para- metric generalized strong vector equilibrium problem,, Positivity, 17 (2013), 341.  doi: 10.1007/s11117-012-0170-z.  Google Scholar

[33]

X. M. Yang, D. Li and S. Y. Wang, Near-subconvexlikeness in vector optimization with set-valued functions,, J. Optim. Theory Appl., 110 (2001), 413.  doi: 10.1023/A:1017535631418.  Google Scholar

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