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April  2015, 11(2): 661-671. doi: 10.3934/jimo.2015.11.661

Stability of solution mapping for parametric symmetric vector equilibrium problems

Xiao-Bing Li 1, , Xian-Jun Long 2, and  Zhi Lin 1,

1. 

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

College of Mathematics and Statistics, Chongqing Technology and Business University, Chongqing 400067

Received  October 2013 Revised  May 2014 Published  September 2014

This paper is concerned with the stability for a parametric symmetric vector equilibrium problem. A parametric gap function for the parametric symmetric vector equilibrium problem is introduced and investigated. By virtue of this function, we establish the sufficient and necessary conditions for the Hausdorff lower semicontinuity of solution mapping to a parametric symmetric vector equilibrium problem. The results presented in this paper generalize and improve the corresponding results in the recent literature.
Keywords: parametric gap function., solution mapping, semicontinuity, Symmetric vector equilibrium problem, continuity.
Mathematics Subject Classification: 34D10, 46N10, 49K40, 91B5.
Citation: Xiao-Bing Li, Xian-Jun Long, Zhi Lin. Stability of solution mapping for parametric symmetric vector equilibrium problems. Journal of Industrial and Management Optimization, 2015, 11 (2) : 661-671. doi: 10.3934/jimo.2015.11.661
References:
[1]

Q. H. Ansari, I. V. Konnov and J. C. Yao, Existence of a solution and variational principles for vector equilibrium problems, J. Optim. Theory Appl., 110 (2001), 481-492. doi: 10.1023/A:1017581009670.

[2]

Q. H. Ansari, I. V. Konnov and J. C. Yao, Characterizations for vector equilibrium problems, J. Optim. Theory Appl., 113 (2002), 435-447. doi: 10.1023/A:1015366419163.

[3]

J. P. Aubin and H. Frankowska, Set-Valued Analysis, Systems & Control: Foundations & Applications, 2, Birkhäuser Boston, Inc., Boston, 1990.

[4]

B. Bank, J. Guddat, D. Klattle, B. Kummer and K. Tammar, Non-Linear Parametric Optimization, Akademie-Verlag, Berlin, 1982. doi: 10.1007/978-3-0348-6328-5.

[5]

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

[6]

E. Blum and W. Oettli, From optimization and variational inequalities to equilibrium problems, Math. Stud., 63 (1994), 123-145.

[7]

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

[8]

C. R. Chen and S. J. Li, Semicontinuity of the solution 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 Z. M. Fang, On the solution semicontinuity to a parametric generalize vector quasivariational inequality, Comput. Math. Appl., 60 (2010), 2417-2425. doi: 10.1016/j.camwa.2010.08.036.

[10]

G. Y. Chen, X. X. Huang and X. Q. Yang, Vector Optimization: Set-valued and Variational Anyasis, in: Lecture Notes in Econonics and Mathematical Systems, Vol.541. Springer, Berlin, 2005.

[11]

G. Y. Chen, X. Q. Yang and H. Yu, A nonlinear scalarization function and generalized quai-vector equilibrium problem, J. Global Optim., 32 (2005), 451-466. doi: 10.1007/s10898-003-2683-2.

[12]

J. C. Chen and X. H. Gong, The stability of set of solutions for symmetric quasi-equilibrium problems, J. Optim. Theory Appl., 136 (2008), 359-374. doi: 10.1007/s10957-007-9309-7.

[13]

A. P. Farajzadeh, On the symmetric vector quasi-equilibrium problems, J. Math. Anal. Appl., 322 (2006), 1099-1110. doi: 10.1016/j.jmaa.2005.09.079.

[14]

J. Y. Fu, Symmetric vector quasi-equilibrium problems, J. Math. Anal. Appl., 285 (2003), 708-713. doi: 10.1016/S0022-247X(03)00479-7.

[15]

F. Giannessi, Theorem of the alternative, quadratic programs, and comlementarity problems, Variational Inequalities and Complementarity, Wiley, New York, (1980), 151-186.

[16]

X. H. Gong and J. C. Yao, Lower semicontinuity of the set of efficient solutions for generalized sytems, J. Optim. Theory Appl., 138 (2008), 197-205. doi: 10.1007/s10957-008-9379-1.

[17]

N. J. Huang, J. Li and H. B.Thompson, Implicit vector equilibrium problems with applications, Math. Comput. Modelling., 37 (2003), 1343-1356. doi: 10.1016/S0895-7177(03)90045-8.

[18]

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

[19]

B. T. Kien, On the lower semicontinuity of optimal solution sets, Optimization, 54 (2005), 123-130. doi: 10.1080/02331930412331330379.

[20]

K. Kimura and J. C. Yao, Semicontinuity of solutiong mappings of parametric generalized vector equilibrium problems, J. Optim. Theory Appl., 138 (2008), 429-443. doi: 10.1007/s10957-008-9386-2.

[21]

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.

[22]

M. A. Noor and W. Oettli, On general nonlinear complementary problems and quasi-equilibria, Le Matematiche, 49 (1994), 313-331.

[23]

W. Y. Zhang, Well-posedness for convex symmetric vector quasi-equilibrium problems, J. Math. Anal. Appl., 387 (2012), 909-915. doi: 10.1016/j.jmaa.2011.09.052.

[24]

J. Zhao, The lower semicontinuity of optimal solution sets, J. Math. Anal. Appl., 207 (1997), 240-254. doi: 10.1006/jmaa.1997.5288.

[25]

R. Y. Zhong and N. J. Huang, On the stability of solution mapping for parametric generalized vector quasiequilibrium problems, Comput. Math. Appl., 63 (2012), 807-815. doi: 10.1016/j.camwa.2011.11.046.

[26]

R. Y. Zhong and N. J. Huang, Lower semicontinuity for parametric weak vector variational inequalities in Reflexive Banach Spaces, J. Optim. Theory Appl., 150 (2011), 317-326. doi: 10.1007/s10957-011-9843-1.

[27]

R. Y. Zhong, N. J. Huang and M. M. Wong, Connectedness and path-connecedness of solution sets to symmtric vector equilibrium problems, Taiwanese J. Math., 13 (2009), 821-836.

show all references

References:
[1]

Q. H. Ansari, I. V. Konnov and J. C. Yao, Existence of a solution and variational principles for vector equilibrium problems, J. Optim. Theory Appl., 110 (2001), 481-492. doi: 10.1023/A:1017581009670.

[2]

Q. H. Ansari, I. V. Konnov and J. C. Yao, Characterizations for vector equilibrium problems, J. Optim. Theory Appl., 113 (2002), 435-447. doi: 10.1023/A:1015366419163.

[3]

J. P. Aubin and H. Frankowska, Set-Valued Analysis, Systems & Control: Foundations & Applications, 2, Birkhäuser Boston, Inc., Boston, 1990.

[4]

B. Bank, J. Guddat, D. Klattle, B. Kummer and K. Tammar, Non-Linear Parametric Optimization, Akademie-Verlag, Berlin, 1982. doi: 10.1007/978-3-0348-6328-5.

[5]

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

[6]

E. Blum and W. Oettli, From optimization and variational inequalities to equilibrium problems, Math. Stud., 63 (1994), 123-145.

[7]

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

[8]

C. R. Chen and S. J. Li, Semicontinuity of the solution 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 Z. M. Fang, On the solution semicontinuity to a parametric generalize vector quasivariational inequality, Comput. Math. Appl., 60 (2010), 2417-2425. doi: 10.1016/j.camwa.2010.08.036.

[10]

G. Y. Chen, X. X. Huang and X. Q. Yang, Vector Optimization: Set-valued and Variational Anyasis, in: Lecture Notes in Econonics and Mathematical Systems, Vol.541. Springer, Berlin, 2005.

[11]

G. Y. Chen, X. Q. Yang and H. Yu, A nonlinear scalarization function and generalized quai-vector equilibrium problem, J. Global Optim., 32 (2005), 451-466. doi: 10.1007/s10898-003-2683-2.

[12]

J. C. Chen and X. H. Gong, The stability of set of solutions for symmetric quasi-equilibrium problems, J. Optim. Theory Appl., 136 (2008), 359-374. doi: 10.1007/s10957-007-9309-7.

[13]

A. P. Farajzadeh, On the symmetric vector quasi-equilibrium problems, J. Math. Anal. Appl., 322 (2006), 1099-1110. doi: 10.1016/j.jmaa.2005.09.079.

[14]

J. Y. Fu, Symmetric vector quasi-equilibrium problems, J. Math. Anal. Appl., 285 (2003), 708-713. doi: 10.1016/S0022-247X(03)00479-7.

[15]

F. Giannessi, Theorem of the alternative, quadratic programs, and comlementarity problems, Variational Inequalities and Complementarity, Wiley, New York, (1980), 151-186.

[16]

X. H. Gong and J. C. Yao, Lower semicontinuity of the set of efficient solutions for generalized sytems, J. Optim. Theory Appl., 138 (2008), 197-205. doi: 10.1007/s10957-008-9379-1.

[17]

N. J. Huang, J. Li and H. B.Thompson, Implicit vector equilibrium problems with applications, Math. Comput. Modelling., 37 (2003), 1343-1356. doi: 10.1016/S0895-7177(03)90045-8.

[18]

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

[19]

B. T. Kien, On the lower semicontinuity of optimal solution sets, Optimization, 54 (2005), 123-130. doi: 10.1080/02331930412331330379.

[20]

K. Kimura and J. C. Yao, Semicontinuity of solutiong mappings of parametric generalized vector equilibrium problems, J. Optim. Theory Appl., 138 (2008), 429-443. doi: 10.1007/s10957-008-9386-2.

[21]

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.

[22]

M. A. Noor and W. Oettli, On general nonlinear complementary problems and quasi-equilibria, Le Matematiche, 49 (1994), 313-331.

[23]

W. Y. Zhang, Well-posedness for convex symmetric vector quasi-equilibrium problems, J. Math. Anal. Appl., 387 (2012), 909-915. doi: 10.1016/j.jmaa.2011.09.052.

[24]

J. Zhao, The lower semicontinuity of optimal solution sets, J. Math. Anal. Appl., 207 (1997), 240-254. doi: 10.1006/jmaa.1997.5288.

[25]

R. Y. Zhong and N. J. Huang, On the stability of solution mapping for parametric generalized vector quasiequilibrium problems, Comput. Math. Appl., 63 (2012), 807-815. doi: 10.1016/j.camwa.2011.11.046.

[26]

R. Y. Zhong and N. J. Huang, Lower semicontinuity for parametric weak vector variational inequalities in Reflexive Banach Spaces, J. Optim. Theory Appl., 150 (2011), 317-326. doi: 10.1007/s10957-011-9843-1.

[27]

R. Y. Zhong, N. J. Huang and M. M. Wong, Connectedness and path-connecedness of solution sets to symmtric vector equilibrium problems, Taiwanese J. Math., 13 (2009), 821-836.

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