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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 & 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.  Google Scholar

[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.  Google Scholar

[3]

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

[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.  Google Scholar

[5]

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

[6]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

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

[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.  Google Scholar

[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.  Google Scholar

[22]

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

[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.  Google Scholar

[24]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[3]

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

[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.  Google Scholar

[5]

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

[6]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

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

[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.  Google Scholar

[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.  Google Scholar

[22]

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

[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.  Google Scholar

[24]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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