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 & 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. 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. doi: 10.1007/s10957-007-9250-9.

[3]

J. P. Aubin and I. Ekeland, Applied Nonlinear Analysis,, Wiley, (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. 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. 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.

[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.

[8]

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.

[9]

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

[10]

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

[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.

[12]

F. Giannessi, Vector Variational Inequalities and Vector Equilibria: Mathematical Theories,, Kluwer Academic Publishers, (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. 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. 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. 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. 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. 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. 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. 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.

[21]

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

[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. doi: 10.1023/A:1014830925232.

[23]

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.

[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. 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. 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. doi: 10.1023/A:1023656507786.

[27]

T. Tanino, Stability and sensitivity analysis in convex vector optimization,, SIAM J. Control. Optim., 26 (1988), 521. 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. 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. 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. doi: 10.1016/j.aml.2012.11.010.

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.

[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.

[3]

J. P. Aubin and I. Ekeland, Applied Nonlinear Analysis,, Wiley, (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. 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. 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.

[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.

[8]

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.

[9]

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

[10]

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

[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.

[12]

F. Giannessi, Vector Variational Inequalities and Vector Equilibria: Mathematical Theories,, Kluwer Academic Publishers, (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. 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. 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. 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. 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. 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. 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. 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.

[21]

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

[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. doi: 10.1023/A:1014830925232.

[23]

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.

[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. 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. 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. doi: 10.1023/A:1023656507786.

[27]

T. Tanino, Stability and sensitivity analysis in convex vector optimization,, SIAM J. Control. Optim., 26 (1988), 521. 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. 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. 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. doi: 10.1016/j.aml.2012.11.010.

[1]

Kenji Kimura, Jen-Chih Yao. Semicontinuity of solution mappings of parametric generalized strong vector equilibrium problems. Journal of Industrial & Management Optimization, 2008, 4 (1) : 167-181. doi: 10.3934/jimo.2008.4.167

[2]

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

[3]

Nguyen Ba Minh, Pham Huu Sach. Strong vector equilibrium problems with LSC approximate solution mappings. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-19. doi: 10.3934/jimo.2018165

[4]

Xin Zuo, Chun-Rong Chen, Hong-Zhi Wei. Solution continuity of parametric generalized vector equilibrium problems with strictly pseudomonotone mappings. Journal of Industrial & Management Optimization, 2017, 13 (1) : 477-488. doi: 10.3934/jimo.2016027

[5]

Qiusheng Qiu, Xinmin Yang. Scalarization of approximate solution for vector equilibrium problems. Journal of Industrial & Management Optimization, 2013, 9 (1) : 143-151. doi: 10.3934/jimo.2013.9.143

[6]

Lam Quoc Anh, Pham Thanh Duoc, Tran Ngoc Tam. Continuity of approximate solution maps to vector equilibrium problems. Journal of Industrial & Management Optimization, 2017, 13 (4) : 1685-1699. doi: 10.3934/jimo.2017013

[7]

Yu Han, Nan-Jing Huang. Some characterizations of the approximate solutions to generalized vector equilibrium problems. Journal of Industrial & Management Optimization, 2016, 12 (3) : 1135-1151. doi: 10.3934/jimo.2016.12.1135

[8]

Lam Quoc Anh, Nguyen Van Hung. Gap functions and Hausdorff continuity of solution mappings to parametric strong vector quasiequilibrium problems. Journal of Industrial & Management Optimization, 2018, 14 (1) : 65-79. doi: 10.3934/jimo.2017037

[9]

Yonghai Wang. On the upper semicontinuity of pullback attractors with applications to plate equations. Communications on Pure & Applied Analysis, 2010, 9 (6) : 1653-1673. doi: 10.3934/cpaa.2010.9.1653

[10]

Micol Amar, Virginia De Cicco. Lower semicontinuity for polyconvex integrals without coercivity assumptions. Evolution Equations & Control Theory, 2014, 3 (3) : 363-372. doi: 10.3934/eect.2014.3.363

[11]

C. R. Chen, S. J. Li. Semicontinuity of the solution set map to a set-valued weak vector variational inequality. Journal of Industrial & Management Optimization, 2007, 3 (3) : 519-528. doi: 10.3934/jimo.2007.3.519

[12]

Chunrong Chen, Zhimiao Fang. A note on semicontinuity to a parametric generalized Ky Fan inequality. Numerical Algebra, Control & Optimization, 2012, 2 (4) : 779-784. doi: 10.3934/naco.2012.2.779

[13]

Yonghai Wang, Chengkui Zhong. Upper semicontinuity of pullback attractors for nonautonomous Kirchhoff wave models. Discrete & Continuous Dynamical Systems - A, 2013, 33 (7) : 3189-3209. doi: 10.3934/dcds.2013.33.3189

[14]

Ahmed Y. Abdallah. Upper semicontinuity of the attractor for a second order lattice dynamical system. Discrete & Continuous Dynamical Systems - B, 2005, 5 (4) : 899-916. doi: 10.3934/dcdsb.2005.5.899

[15]

María Astudillo, Marcelo M. Cavalcanti. On the upper semicontinuity of the global attractor for a porous medium type problem with large diffusion. Evolution Equations & Control Theory, 2017, 6 (1) : 1-13. doi: 10.3934/eect.2017001

[16]

Zhijian Yang, Yanan Li. Upper semicontinuity of pullback attractors for non-autonomous Kirchhoff wave equations. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 1-14. doi: 10.3934/dcdsb.2019036

[17]

Vítor Araújo. Semicontinuity of entropy, existence of equilibrium states and continuity of physical measures. Discrete & Continuous Dynamical Systems - A, 2007, 17 (2) : 371-386. doi: 10.3934/dcds.2007.17.371

[18]

Adela Capătă. Optimality conditions for strong vector equilibrium problems under a weak constraint qualification. Journal of Industrial & Management Optimization, 2015, 11 (2) : 563-574. doi: 10.3934/jimo.2015.11.563

[19]

Zaiyun Peng, Xinmin Yang, Kok Lay Teo. On the Hölder continuity of approximate solution mappings to parametric weak generalized Ky Fan Inequality. Journal of Industrial & Management Optimization, 2015, 11 (2) : 549-562. doi: 10.3934/jimo.2015.11.549

[20]

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

2018 Impact Factor: 1.025

Metrics

  • PDF downloads (13)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]