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April  2017, 13(2): 967-975. doi: 10.3934/jimo.2016056

Continuity of the solution mappings to parametric generalized non-weak vector Ky Fan inequalities

Yangdong Xu a,, and  Shengjie Li b,

a. 

College of Sciences, Chongqing University of Posts and Telecommunications, Chongqing 400065, China
b. 

College of Mathematics and Statistics, Chongqing University, Chongqing 401331, China

1Corresponding author

Received  December 2014 Revised  June 2016 Published  August 2016

In this paper, by using the nonlinear scalarization method and under some new assumptions, which do not involve any information on the solution set, we establish the continuity of solution mappings of parametric generalized non-weak vector Ky Fan inequality with moving cones. The results are new and improve corresponding ones in the literature. Some examples are given to illustrate our results.

Keywords: Upper semicontinuity, lower semicontinuity, Ky Fan inequality, nonlinear scalarization method.
Mathematics Subject Classification: 90C26, 90C46.
Citation: Yangdong Xu, Shengjie Li. Continuity of the solution mappings to parametric generalized non-weak vector Ky Fan inequalities. Journal of Industrial & Management Optimization, 2017, 13 (2) : 967-975. doi: 10.3934/jimo.2016056
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), 699-711.  doi: 10.1016/j.jmaa.2004.03.014.  Google Scholar

[2]

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

[3]

J. -P. Aubin and I. Ekeland, Applied Nonlinear Analysis, Wiley, New York, 2006.  Google Scholar

[4]

C. Berge, Topological Spaces, Oliver and Boy1, London, 1963, Dover Publications, Inc., Mineola, NY, 1997.  Google Scholar

[5]

B. Chen and N. J. Huang, Continuity of the solution mapping to parametric generalized vector equilibrium problems, J. Global Optim., 56 (2013), 1515-1528.  doi: 10.1007/s10898-012-9904-5.  Google Scholar

[6]

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

[7]

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

[8]

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

[9]

K. Fan, Extension of two fixed point theorems of F.E., Browder, Math. Z., 112 (1969), 234-240.  doi: 10.1007/BF01110225.  Google Scholar

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F. Ferro, A minimax theorem for vector-valued functions, J. Optim. Theory Appl., 60 (1989), 19-31.  doi: 10.1007/BF00938796.  Google Scholar

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F. Giannessi, Vector Variational Inequalities and Vector Equilibria: Mathematical Theories, Kluwer Academic Publishers, Dordrecht, 2000. doi: 10.1007/978-1-4613-0299-5.  Google Scholar

[12]

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

[13]

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

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Y. Han and X. H. Gong, Lower semicontinuity of solution mapping to parametric generalized strong vector equilibrium problems, Appl. Math. Lett., 28 (2014), 38-41.  doi: 10.1016/j.aml.2013.09.006.  Google Scholar

[15]

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

[16]

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

[17]

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), 507-515.  doi: 10.1007/s10957-010-9736-8.  Google Scholar

[18]

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

[19]

Z. Y. Peng and S. S. Chang, On the lower semicontinuity of the set of efficient solutions to parametric generalized systems, Optim. Lett., 8 (2014), 159-169.  doi: 10.1007/s11590-012-0544-y.  Google Scholar

[20]

Z. Y. Peng, X. 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

[21]

P. H. Sach, New nonlinear scalarization functions and applications, Nonlinear Anal., 75 (2012), 2281-2292.  doi: 10.1016/j.na.2011.10.028.  Google Scholar

[22]

P. H. Sach and N. B. Minh, Continuity of solution mappings in some parametric non-weak vector Ky Fan inequalities, J. Global Optim., 57 (2013), 1401-1418.  doi: 10.1007/s10898-012-0015-0.  Google Scholar

[23]

P. H. Sach and L. A. Tuan, New scalarizing approach to the stability analysis in parametric generalized Ky Fan inequality problems, J. Optim. Theory Appl., 157 (2013), 347-364.  doi: 10.1007/s10957-012-0105-7.  Google Scholar

[24]

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), 1225-1234.  doi: 10.3934/jimo.2014.10.1225.  Google Scholar

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), 699-711.  doi: 10.1016/j.jmaa.2004.03.014.  Google Scholar

[2]

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

[3]

J. -P. Aubin and I. Ekeland, Applied Nonlinear Analysis, Wiley, New York, 2006.  Google Scholar

[4]

C. Berge, Topological Spaces, Oliver and Boy1, London, 1963, Dover Publications, Inc., Mineola, NY, 1997.  Google Scholar

[5]

B. Chen and N. J. Huang, Continuity of the solution mapping to parametric generalized vector equilibrium problems, J. Global Optim., 56 (2013), 1515-1528.  doi: 10.1007/s10898-012-9904-5.  Google Scholar

[6]

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

[7]

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

[8]

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

[9]

K. Fan, Extension of two fixed point theorems of F.E., Browder, Math. Z., 112 (1969), 234-240.  doi: 10.1007/BF01110225.  Google Scholar

[10]

F. Ferro, A minimax theorem for vector-valued functions, J. Optim. Theory Appl., 60 (1989), 19-31.  doi: 10.1007/BF00938796.  Google Scholar

[11]

F. Giannessi, Vector Variational Inequalities and Vector Equilibria: Mathematical Theories, Kluwer Academic Publishers, Dordrecht, 2000. doi: 10.1007/978-1-4613-0299-5.  Google Scholar

[12]

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

[13]

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

[14]

Y. Han and X. H. Gong, Lower semicontinuity of solution mapping to parametric generalized strong vector equilibrium problems, Appl. Math. Lett., 28 (2014), 38-41.  doi: 10.1016/j.aml.2013.09.006.  Google Scholar

[15]

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

[16]

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

[17]

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), 507-515.  doi: 10.1007/s10957-010-9736-8.  Google Scholar

[18]

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

[19]

Z. Y. Peng and S. S. Chang, On the lower semicontinuity of the set of efficient solutions to parametric generalized systems, Optim. Lett., 8 (2014), 159-169.  doi: 10.1007/s11590-012-0544-y.  Google Scholar

[20]

Z. Y. Peng, X. 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

[21]

P. H. Sach, New nonlinear scalarization functions and applications, Nonlinear Anal., 75 (2012), 2281-2292.  doi: 10.1016/j.na.2011.10.028.  Google Scholar

[22]

P. H. Sach and N. B. Minh, Continuity of solution mappings in some parametric non-weak vector Ky Fan inequalities, J. Global Optim., 57 (2013), 1401-1418.  doi: 10.1007/s10898-012-0015-0.  Google Scholar

[23]

P. H. Sach and L. A. Tuan, New scalarizing approach to the stability analysis in parametric generalized Ky Fan inequality problems, J. Optim. Theory Appl., 157 (2013), 347-364.  doi: 10.1007/s10957-012-0105-7.  Google Scholar

[24]

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), 1225-1234.  doi: 10.3934/jimo.2014.10.1225.  Google Scholar

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