January  2018, 14(1): 65-79. doi: 10.3934/jimo.2017037

Gap functions and Hausdorff continuity of solution mappings to parametric strong vector quasiequilibrium problems

1. 

Department of Mathematics, Teacher College, Can Tho University, Can Tho, Viet Nam

2. 

Center of Research and Development Duy Tan University K7/25, Quang Trung, Danang, VietNam

3. 

Department of Mathematics, Dong Thap University, Dong Thap, Viet Nam

* Corresponding author

Received  November 2015 Revised  February 2017 Published  April 2017

Fund Project: This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 101.01-2017.18

In this paper, we consider parametric strong vector quasiequilibrium problems in Hausdorff topological vector spaces. Firstly, we introduce parametric gap functions for these problems, and study the continuity property of these functions. Next, we present two key hypotheses related to the gap functions for the considered problems and also study characterizations of these hypotheses. Then, afterwards, we prove that these hypotheses are not only sufficient but also necessary for the Hausdorff lower semicontinuity and Hausdorff continuity of solution mappings to these problems. Finally, as applications, we derive several results on Hausdorff (lower) continuity properties of the solution mappings in the special cases of variational inequalities of the Minty type and the Stampacchia type.

Citation: 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
References:
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M. Ait Mansour and L. Scrimali, Hölder continuity of solutions to elastic traffic networkmodels, J Global Optim., 40 (2008), 175-184. doi: 10.1007/s10898-007-9190-9. Google Scholar

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L. Q. Anh and P. Q. Khanh, Semicontinuity of the solution sets of parametric multivalued vector quasiequilibrium problems, J. Math. Anal. Appl., 294 (2004), 699-711. doi: 10.1016/j.jmaa.2004.03.014. Google Scholar

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L. Q. Anh and P. Q. Khanh, On the Hölder continuity of solutions to parametric multivalued vector equilibrium problems, J. Math. Anal. Appl., 321 (2006), 308-315. doi: 10.1016/j.jmaa.2005.08.018. Google Scholar

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L. Q. Anh and P. Q. Khanh, Uniqueness and Hölder continuity of the solution to multivalued equilibrium problems in metric spaces, J. Global Optim., 37 (2007), 449-465. doi: 10.1007/s10898-006-9062-8. Google Scholar

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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. Google Scholar

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L. Q. Anh and P. Q. Khanh, Sensitivity analysis for multivalued quasiequilibrium problems in metric spaces: Hölder continuity of solutions, J. Global Optim., 42 (2008), 515-531. doi: 10.1007/s10898-007-9268-4. Google Scholar

[8]

L. Q. Anh and P. Q. Khanh, Hölder continuity of the unique solution to quasiequilibrium problems in metric spaces, J. Optim. Theory Appl., 141 (2009), 37-54. doi: 10.1007/s10957-008-9508-x. Google Scholar

[9]

L. Q. Anh and P. Q. Khanh, Continuity of solution maps of parametric quasiequilibrium problems, J. Global Optim., 46 (2010), 247-259. doi: 10.1007/s10898-009-9422-2. Google Scholar

[10]

L. Q. AnhP. Q. Khanh and T. N. Tam, On Hölder continuity of approximate solutions to parametric equilibrium problems, Nonlinear Anal., 75 (2012), 2293-2303. doi: 10.1016/j.na.2011.10.029. Google Scholar

[11]

L. Q. AnhA. Y. Kruger and N. H. Thao, On Hölder calmness of solution mappings in parametric equilibrium problems, TOP, 22 (2014), 331-342. doi: 10.1007/s11750-012-0259-3. Google Scholar

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L. Q. Anh and T. N. Tam, Hausdorff continuity of approximate solution maps to parametric primal and dual equilibrium problems, TOP, 24 (2016), 242-258. doi: 10.1007/s11750-015-0390-z. Google Scholar

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M. Bianchi and R. Pini, Sensitivity for parametric vector equilibria, Optimization, 55 (2006), 221-230. doi: 10.1080/02331930600662732. Google Scholar

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

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

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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. Google Scholar

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

[22]

J. W. Chen and Z. P. Wan, Semicontinuity for parametric Minty vector quasivariational inequalities in Hausdorff topological vector spaces, Comput. Appl. Math., 33 (2014), 111-129. doi: 10.1007/s40314-013-0047-1. Google Scholar

[23]

Y. P. Fang and R. Hu, Parametric well-posedness for variational inequalities defined bifunctions, Comput. Math. Appl., 53 (2007), 1306-1316. doi: 10.1016/j.camwa.2006.09.009. Google Scholar

[24]

J. Y. Fu, Generalized vector quasiequilibrium problems, Math. Meth. Oper. Res., 52 (2000), 57-64. doi: 10.1007/s001860000058. Google Scholar

[25]

F. Giannessi, On Minty variational principle, New Trends in Mathematical Programming, Kluwer Academic Publishers, Dordrecht, Boston, London, 1 (1998), 93–99. doi: 10.1007/978-1-4757-2878-1_8. Google Scholar

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Chr. (Tammer) Gerstewitz, Nichtkonvexe dualitat in der vektaroptimierung, Wissenschaftliche Zeitschrift der Technischen Hochschule Leuna-Mersebung, 25 (1983), 357-364. Google Scholar

[27]

N. X. Hai and P. Q. Khanh, Existence of solutions to general quasiequilibrium problems and applications, J. Optim. Theory Appl., 133 (2007), 317-327. doi: 10.1007/s10957-007-9170-8. Google Scholar

[28]

N. J. HuangJ. Li and H. B. Thompson, Stability for parametric implicit vector equilibrium problems, Math. Comput. Modelling, 43 (2006), 1267-1274. doi: 10.1016/j.mcm.2005.06.010. Google Scholar

[29]

A. N. Iusem and W. Sosa, New existence results for equilibrium problems, Nonlinear Anal., 52 (2003), 621-635. doi: 10.1016/S0362-546X(02)00154-2. Google Scholar

[30]

A. N. IusemG. Kassay and W. Sosa, On certain conditions for the existence of solutions of equilibrium problems, Math. Program., 116 (2009), 259-273. doi: 10.1007/s10107-007-0125-5. Google Scholar

[31]

G. KassayJ. Kolumbán and Z. Páles, Factorization of Minty and Stampacchia variational inequality systems, European J. Oper. Res., 143 (2002), 377-389. doi: 10.1016/S0377-2217(02)00290-4. Google Scholar

[32]

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

[33]

W. K. KimS. Kum and K. H. Lee, Semicontinuity of the solution multifunctions of the parametric generalized operator equilibrium problems, Nonlinear Anal., 71 (2009), 2182-2187. doi: 10.1016/j.na.2009.04.036. Google Scholar

[34]

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

[35]

S. Komlósi, On the Stampacchia and Minty Variational Inequalities, in: G. Giorgi, F. A. Rossi (Eds. ), Generalized Convexity and Optimization for Economic and Financial Decisions, Pitagora Editrice, Bologna, 1999. Google Scholar

[36]

I. Konnov and S. Schaible, Duality for equilibrium problems under generalized monotonicity, J. Optim. Theory Appl., 104 (2000), 395-408. doi: 10.1023/A:1004665830923. Google Scholar

[37]

C. S. Lalitha and G. Bhatia, Stability of parametric quasivariational inequality of the Minty type, J. Optim. Theory. Appl., 148 (2011), 281-300. doi: 10.1007/s10957-010-9755-5. Google Scholar

[38]

S. J. LiX. B. LiL. N. Wang and K. L. Teo, The Hölder continuity of solutions to generalized vector equilibrium problems, European J. Oper. Res., 199 (2009), 334-338. doi: 10.1016/j.ejor.2008.12.024. Google Scholar

[39]

S. J. Li and X. B. Li, Hölder continuity of solutions to parametric weak generalized Ky Fan inequality, J. Optim. Theory Appl., 149 (2011), 540-553. doi: 10.1007/s10957-011-9803-9. Google Scholar

[40]

S. J. LiK. L. TeoX. Q. Yang and S. Y. Wu, Gap functions and existence of solutions to generalized vector quasiequilibrium problems, J. Global Optim., 34 (2006), 427-440. doi: 10.1007/s10898-005-2193-5. Google Scholar

[41]

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

[42]

X. B. Li and S. J. Li, Continuity of approximate solution mappings for parametric equilibrium problems, J. Global Optim., 51 (2011), 541-548. doi: 10.1007/s10898-010-9641-6. Google Scholar

[43]

X. J. LongN. J. Huang and K. L. Teo, Existence and stability of solutions for generalized strong vector quasi-equilibrium problem, Math. Comput. Modelling, 47 (2008), 445-451. doi: 10.1016/j.mcm.2007.04.013. Google Scholar

[44]

D. T. Luc, Theory of Vector Optimization, Lecture Notes in Economic and Mathematical Systems 319, Springer-verlag, Berlin, 1989. doi: 10.1007/978-3-642-50280-4. Google Scholar

[45]

B. S. Mordukhovich, Variational Analysis and Generalized Differentiation, Ⅰ: Basic Theory. Grundlehren Series (Fundamental Principles of Mathematical Sciences), 330. Springer, Berlin, 2006. Google Scholar

[46]

B. S. Mordukhovich, Variational Analysis and Generalized Differentiation, Ⅱ: Applications. Grundlehren Series (Fundamental Principles of Mathematical Sciences), 331, Springer, Berlin, 2006. Google Scholar

[47]

Z. Y. PengX. M. Yang and J. W. Peng, On the Lower semicontinuity of the solution mappings to parametric weak generalized Ky Fan inequality, J. Optim. Theory Appl., 152 (2012), 256-264. doi: 10.1007/s10957-011-9883-6. Google Scholar

[48]

Z. Y. PengY. Zhao and X. M. Yang, Semicontinuity of approximate solution mappings to parametric set-valued weak vector equilibrium problems, Numer. Funct. Anal. Optim., 36 (2015), 481-500. doi: 10.1080/01630563.2015.1013551. Google Scholar

[49]

I. Sadeqi and C. Alizadeh, Existence of solutions of generalized vector equilibrium problems in reflexive Banach spaces, Nonlinear Anal., 74 (2011), 2226-2234. doi: 10.1016/j.na.2010.11.027. Google Scholar

[50]

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

[51]

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

[52]

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

show all references

References:
[1]

M. Ait Mansour and H. Riahi, Sensitivity analysis for abstract equilibrium problems, J. Math. Anal. Appl., 306 (2005), 684-691. doi: 10.1016/j.jmaa.2004.10.011. Google Scholar

[2]

M. Ait Mansour and L. Scrimali, Hölder continuity of solutions to elastic traffic networkmodels, J Global Optim., 40 (2008), 175-184. doi: 10.1007/s10898-007-9190-9. Google Scholar

[3]

L. Q. Anh and P. Q. Khanh, Semicontinuity of the solution sets of parametric multivalued vector quasiequilibrium problems, J. Math. Anal. Appl., 294 (2004), 699-711. doi: 10.1016/j.jmaa.2004.03.014. Google Scholar

[4]

L. Q. Anh and P. Q. Khanh, On the Hölder continuity of solutions to parametric multivalued vector equilibrium problems, J. Math. Anal. Appl., 321 (2006), 308-315. doi: 10.1016/j.jmaa.2005.08.018. Google Scholar

[5]

L. Q. Anh and P. Q. Khanh, Uniqueness and Hölder continuity of the solution to multivalued equilibrium problems in metric spaces, J. Global Optim., 37 (2007), 449-465. doi: 10.1007/s10898-006-9062-8. Google Scholar

[6]

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. Google Scholar

[7]

L. Q. Anh and P. Q. Khanh, Sensitivity analysis for multivalued quasiequilibrium problems in metric spaces: Hölder continuity of solutions, J. Global Optim., 42 (2008), 515-531. doi: 10.1007/s10898-007-9268-4. Google Scholar

[8]

L. Q. Anh and P. Q. Khanh, Hölder continuity of the unique solution to quasiequilibrium problems in metric spaces, J. Optim. Theory Appl., 141 (2009), 37-54. doi: 10.1007/s10957-008-9508-x. Google Scholar

[9]

L. Q. Anh and P. Q. Khanh, Continuity of solution maps of parametric quasiequilibrium problems, J. Global Optim., 46 (2010), 247-259. doi: 10.1007/s10898-009-9422-2. Google Scholar

[10]

L. Q. AnhP. Q. Khanh and T. N. Tam, On Hölder continuity of approximate solutions to parametric equilibrium problems, Nonlinear Anal., 75 (2012), 2293-2303. doi: 10.1016/j.na.2011.10.029. Google Scholar

[11]

L. Q. AnhA. Y. Kruger and N. H. Thao, On Hölder calmness of solution mappings in parametric equilibrium problems, TOP, 22 (2014), 331-342. doi: 10.1007/s11750-012-0259-3. Google Scholar

[12]

L. Q. Anh and T. N. Tam, Hausdorff continuity of approximate solution maps to parametric primal and dual equilibrium problems, TOP, 24 (2016), 242-258. doi: 10.1007/s11750-015-0390-z. Google Scholar

[13]

J. P. Aubin and I. Ekeland, Applied Nonlinear Analysis, John Wiley and Sons, New York, 1984. Google Scholar

[15]

E. Bednarczuk, Strong pseudomonotonicity, sharp efficiency and stability for parametric vector equilibria, ESAIM Proc., 17 (2007), 9-18. doi: 10.1051/proc:071702. Google Scholar

[16]

M. Bianchi and R. Pini, A note on stability for parametric equilibrium problems, Oper. Res. Lett., 31 (2003), 445-450. doi: 10.1016/S0167-6377(03)00051-8. Google Scholar

[17]

M. Bianchi and R. Pini, Sensitivity for parametric vector equilibria, Optimization, 55 (2006), 221-230. doi: 10.1080/02331930600662732. Google Scholar

[18]

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

[19]

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

[20]

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. Google Scholar

[21]

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

[22]

J. W. Chen and Z. P. Wan, Semicontinuity for parametric Minty vector quasivariational inequalities in Hausdorff topological vector spaces, Comput. Appl. Math., 33 (2014), 111-129. doi: 10.1007/s40314-013-0047-1. Google Scholar

[23]

Y. P. Fang and R. Hu, Parametric well-posedness for variational inequalities defined bifunctions, Comput. Math. Appl., 53 (2007), 1306-1316. doi: 10.1016/j.camwa.2006.09.009. Google Scholar

[24]

J. Y. Fu, Generalized vector quasiequilibrium problems, Math. Meth. Oper. Res., 52 (2000), 57-64. doi: 10.1007/s001860000058. Google Scholar

[25]

F. Giannessi, On Minty variational principle, New Trends in Mathematical Programming, Kluwer Academic Publishers, Dordrecht, Boston, London, 1 (1998), 93–99. doi: 10.1007/978-1-4757-2878-1_8. Google Scholar

[26]

Chr. (Tammer) Gerstewitz, Nichtkonvexe dualitat in der vektaroptimierung, Wissenschaftliche Zeitschrift der Technischen Hochschule Leuna-Mersebung, 25 (1983), 357-364. Google Scholar

[27]

N. X. Hai and P. Q. Khanh, Existence of solutions to general quasiequilibrium problems and applications, J. Optim. Theory Appl., 133 (2007), 317-327. doi: 10.1007/s10957-007-9170-8. Google Scholar

[28]

N. J. HuangJ. Li and H. B. Thompson, Stability for parametric implicit vector equilibrium problems, Math. Comput. Modelling, 43 (2006), 1267-1274. doi: 10.1016/j.mcm.2005.06.010. Google Scholar

[29]

A. N. Iusem and W. Sosa, New existence results for equilibrium problems, Nonlinear Anal., 52 (2003), 621-635. doi: 10.1016/S0362-546X(02)00154-2. Google Scholar

[30]

A. N. IusemG. Kassay and W. Sosa, On certain conditions for the existence of solutions of equilibrium problems, Math. Program., 116 (2009), 259-273. doi: 10.1007/s10107-007-0125-5. Google Scholar

[31]

G. KassayJ. Kolumbán and Z. Páles, Factorization of Minty and Stampacchia variational inequality systems, European J. Oper. Res., 143 (2002), 377-389. doi: 10.1016/S0377-2217(02)00290-4. Google Scholar

[32]

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

[33]

W. K. KimS. Kum and K. H. Lee, Semicontinuity of the solution multifunctions of the parametric generalized operator equilibrium problems, Nonlinear Anal., 71 (2009), 2182-2187. doi: 10.1016/j.na.2009.04.036. Google Scholar

[34]

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

[35]

S. Komlósi, On the Stampacchia and Minty Variational Inequalities, in: G. Giorgi, F. A. Rossi (Eds. ), Generalized Convexity and Optimization for Economic and Financial Decisions, Pitagora Editrice, Bologna, 1999. Google Scholar

[36]

I. Konnov and S. Schaible, Duality for equilibrium problems under generalized monotonicity, J. Optim. Theory Appl., 104 (2000), 395-408. doi: 10.1023/A:1004665830923. Google Scholar

[37]

C. S. Lalitha and G. Bhatia, Stability of parametric quasivariational inequality of the Minty type, J. Optim. Theory. Appl., 148 (2011), 281-300. doi: 10.1007/s10957-010-9755-5. Google Scholar

[38]

S. J. LiX. B. LiL. N. Wang and K. L. Teo, The Hölder continuity of solutions to generalized vector equilibrium problems, European J. Oper. Res., 199 (2009), 334-338. doi: 10.1016/j.ejor.2008.12.024. Google Scholar

[39]

S. J. Li and X. B. Li, Hölder continuity of solutions to parametric weak generalized Ky Fan inequality, J. Optim. Theory Appl., 149 (2011), 540-553. doi: 10.1007/s10957-011-9803-9. Google Scholar

[40]

S. J. LiK. L. TeoX. Q. Yang and S. Y. Wu, Gap functions and existence of solutions to generalized vector quasiequilibrium problems, J. Global Optim., 34 (2006), 427-440. doi: 10.1007/s10898-005-2193-5. Google Scholar

[41]

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

[42]

X. B. Li and S. J. Li, Continuity of approximate solution mappings for parametric equilibrium problems, J. Global Optim., 51 (2011), 541-548. doi: 10.1007/s10898-010-9641-6. Google Scholar

[43]

X. J. LongN. J. Huang and K. L. Teo, Existence and stability of solutions for generalized strong vector quasi-equilibrium problem, Math. Comput. Modelling, 47 (2008), 445-451. doi: 10.1016/j.mcm.2007.04.013. Google Scholar

[44]

D. T. Luc, Theory of Vector Optimization, Lecture Notes in Economic and Mathematical Systems 319, Springer-verlag, Berlin, 1989. doi: 10.1007/978-3-642-50280-4. Google Scholar

[45]

B. S. Mordukhovich, Variational Analysis and Generalized Differentiation, Ⅰ: Basic Theory. Grundlehren Series (Fundamental Principles of Mathematical Sciences), 330. Springer, Berlin, 2006. Google Scholar

[46]

B. S. Mordukhovich, Variational Analysis and Generalized Differentiation, Ⅱ: Applications. Grundlehren Series (Fundamental Principles of Mathematical Sciences), 331, Springer, Berlin, 2006. Google Scholar

[47]

Z. Y. PengX. M. Yang and J. W. Peng, On the Lower semicontinuity of the solution mappings to parametric weak generalized Ky Fan inequality, J. Optim. Theory Appl., 152 (2012), 256-264. doi: 10.1007/s10957-011-9883-6. Google Scholar

[48]

Z. Y. PengY. Zhao and X. M. Yang, Semicontinuity of approximate solution mappings to parametric set-valued weak vector equilibrium problems, Numer. Funct. Anal. Optim., 36 (2015), 481-500. doi: 10.1080/01630563.2015.1013551. Google Scholar

[49]

I. Sadeqi and C. Alizadeh, Existence of solutions of generalized vector equilibrium problems in reflexive Banach spaces, Nonlinear Anal., 74 (2011), 2226-2234. doi: 10.1016/j.na.2010.11.027. Google Scholar

[50]

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

[51]

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

[52]

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

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