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October 2017, 13(4): 1685-1699. doi: 10.3934/jimo.2017013

Continuity of approximate solution maps to vector equilibrium problems

1. 

Department of Mathematics, Teacher College, Can Tho University, Can Tho, 900000, Viet Nam

2. 

Department of Mathematics, Vo Truong Toan University, Hau Giang, Viet Nam

3. 

Department of Mathematics, Nam Can Tho University, Can Tho, 900000, Viet Nam

* Corresponding author

Received  October 2015 Revised  October 2016 Published  December 2016

Fund Project: This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 101.01-2014.44

This paper considers the parametric primal and dual vector equilibrium problems in locally convex Hausdorff topological vector spaces. Based on linear scalarization technique, we establish sufficient conditions for the continuity of approximate solution maps to these problems. As applications, some new results for vector optimization problem and vector variational inequality are derived. Our results are new and improve the existing ones in the literature.

Citation: 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
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.

[2]

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

[3]

L. Q. Anh and P. Q. Khanh, Semicontinuity of the approximate solution sets of multivalued quasiequilibrium problems, Numer. Funct. Anal. Optim., 29 (2008), 24-42. doi: 10.1080/01630560701873068.

[4]

L. Q. Anh and P. Q. Khanh, Various kinds of semicontinuity and solution sets of parametric multivalued symmetric vector quasiequilibrium problems, J. Glob. Optim., 41 (2008), 539-558. doi: 10.1007/s10898-007-9264-8.

[5]

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

[6]

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.

[7]

L. Q. AnhP. Q. Khanh 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.

[8]

Q. H. AnsariI. 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.

[9]

J. P. Aubin and H. Frankowska, Set-Valued Analysis, Birkhäuser, Boston, 1990.

[10]

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

[11]

M. Bianchi and S. Schaible, Generalized monotone bifunctions and equilibrium problems, J. Optim. Theory Appl., 90 (1996), 31-43. doi: 10.1007/BF02192244.

[12]

M. BianchiN. Hadjisavas and S. Schaible, Equilibrium problems with generalized monotone bifunctions, J. Optim. Theory Appl., 92 (1997), 527-542. doi: 10.1023/A:1022603406244.

[13]

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

[14]

C. R. ChenS. L. Li and K. L. Teo, Solution semicontinuity of parametric generalized vector equilibrium problems, J. Glob. Optim., 45 (2009), 309-318. doi: 10.1007/s10898-008-9376-9.

[15]

Y. H. Cheng and D. L. Zhu, Global stability results for the weak vector variational inequality, J. Glob. Optim., 32 (2005), 543-550. doi: 10.1007/s10898-004-2692-9.

[16]

K. Fan, A minimax inequality and applications, In: Shisha O (ed) Inequality III Academic Press, New York, (1972), 103–113.

[17] A. V. Fiacco, Introduction to Sensitivity and Stability Analysis in Nonlinear Programming, Academic Press, London, 1983.
[18]

X. H. Gong, Continuity of the solution set to parametric vector equilibrium problem, J. Optim. Theory Appl., 139 (2008), 35-46. doi: 10.1007/s10957-008-9429-8.

[19]

N. X. HaiP. Q. Khanh and N. H. Quan, On the existence of solutions to quasivariational inclusion problems, J. Glob. Optim., 45 (2009), 565-581. doi: 10.1007/s10898-008-9390-y.

[20]

J. Jahn, Mathematical Vector Optimization in Partially Ordered Linear Spaces, Peter Lang, Frankfurt, 1986.

[21]

P. Q. Khanh and V. S. T. Long, Invariant-point theorems and existence of solutions to optimization-related problems, J. Global. Optim., 58 (2014), 545-564. doi: 10.1007/s10898-013-0065-y.

[22]

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.

[23]

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.

[24]

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

[25]

S. J. LiH. M. Liu and C. R. Chen, Lower semicontinuity of parametric generalized weak vector equilibrium problems, Bull. Aust. Math. Soc., 81 (2010), 85-95. doi: 10.1017/S0004972709000628.

[26]

S. J. LiH. M. LiuY. Zhang and Z. M. Fang, Continuity of solution mappings to parametric generalized strong vector equilibrium problems, J. Global Optim., 55 (2013), 597-610. doi: 10.1007/s10898-012-9985-1.

[27]

H. Nikaido and K. Isoda, Note on non-copperative convex games, Pacific J. Math., 5 (1955), 807-815. doi: 10.2140/pjm.1955.5.807.

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.

[2]

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

[3]

L. Q. Anh and P. Q. Khanh, Semicontinuity of the approximate solution sets of multivalued quasiequilibrium problems, Numer. Funct. Anal. Optim., 29 (2008), 24-42. doi: 10.1080/01630560701873068.

[4]

L. Q. Anh and P. Q. Khanh, Various kinds of semicontinuity and solution sets of parametric multivalued symmetric vector quasiequilibrium problems, J. Glob. Optim., 41 (2008), 539-558. doi: 10.1007/s10898-007-9264-8.

[5]

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

[6]

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.

[7]

L. Q. AnhP. Q. Khanh 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.

[8]

Q. H. AnsariI. 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.

[9]

J. P. Aubin and H. Frankowska, Set-Valued Analysis, Birkhäuser, Boston, 1990.

[10]

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

[11]

M. Bianchi and S. Schaible, Generalized monotone bifunctions and equilibrium problems, J. Optim. Theory Appl., 90 (1996), 31-43. doi: 10.1007/BF02192244.

[12]

M. BianchiN. Hadjisavas and S. Schaible, Equilibrium problems with generalized monotone bifunctions, J. Optim. Theory Appl., 92 (1997), 527-542. doi: 10.1023/A:1022603406244.

[13]

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

[14]

C. R. ChenS. L. Li and K. L. Teo, Solution semicontinuity of parametric generalized vector equilibrium problems, J. Glob. Optim., 45 (2009), 309-318. doi: 10.1007/s10898-008-9376-9.

[15]

Y. H. Cheng and D. L. Zhu, Global stability results for the weak vector variational inequality, J. Glob. Optim., 32 (2005), 543-550. doi: 10.1007/s10898-004-2692-9.

[16]

K. Fan, A minimax inequality and applications, In: Shisha O (ed) Inequality III Academic Press, New York, (1972), 103–113.

[17] A. V. Fiacco, Introduction to Sensitivity and Stability Analysis in Nonlinear Programming, Academic Press, London, 1983.
[18]

X. H. Gong, Continuity of the solution set to parametric vector equilibrium problem, J. Optim. Theory Appl., 139 (2008), 35-46. doi: 10.1007/s10957-008-9429-8.

[19]

N. X. HaiP. Q. Khanh and N. H. Quan, On the existence of solutions to quasivariational inclusion problems, J. Glob. Optim., 45 (2009), 565-581. doi: 10.1007/s10898-008-9390-y.

[20]

J. Jahn, Mathematical Vector Optimization in Partially Ordered Linear Spaces, Peter Lang, Frankfurt, 1986.

[21]

P. Q. Khanh and V. S. T. Long, Invariant-point theorems and existence of solutions to optimization-related problems, J. Global. Optim., 58 (2014), 545-564. doi: 10.1007/s10898-013-0065-y.

[22]

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.

[23]

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.

[24]

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

[25]

S. J. LiH. M. Liu and C. R. Chen, Lower semicontinuity of parametric generalized weak vector equilibrium problems, Bull. Aust. Math. Soc., 81 (2010), 85-95. doi: 10.1017/S0004972709000628.

[26]

S. J. LiH. M. LiuY. Zhang and Z. M. Fang, Continuity of solution mappings to parametric generalized strong vector equilibrium problems, J. Global Optim., 55 (2013), 597-610. doi: 10.1007/s10898-012-9985-1.

[27]

H. Nikaido and K. Isoda, Note on non-copperative convex games, Pacific J. Math., 5 (1955), 807-815. doi: 10.2140/pjm.1955.5.807.

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