2012, 2(4): 779-784. doi: 10.3934/naco.2012.2.779

A note on semicontinuity to a parametric generalized Ky Fan inequality

1. 

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

2. 

Chongqing Police College, Chongqing 401331, China

Received  September 2011 Revised  May 2012 Published  November 2012

In this note, the continuity results of weak vector solutions and global vector solutions to a parametric generalized Ky Fan inequality are established by using a new scalarization method. Our results improve the corresponding ones of Li and Fang (J. Optim. Theory Appl. 147: 507-515, 2010).
Citation: 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
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.  Google Scholar

[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.  Google Scholar

[3]

J. P. Aubin and I. Ekeland, "Applied Nonlinear Analysis,", Wiley, (1984).   Google Scholar

[4]

C. R. Chen and S. J. Li, On the solution continuity of parametric generalized systems,, Pac. J. Optim., 6 (2010), 141.   Google Scholar

[5]

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

[6]

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.  Google Scholar

[7]

C. R. Chen, S. J. Li and Z. M. Fang, On the solution semicontinuity to a parametric generalized vector quasivariational inequality,, Comput. Math. Appl., 60 (2010), 2417.  doi: 10.1016/j.camwa.2010.08.036.  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.  doi: 10.1007/s10898-004-2692-9.  Google Scholar

[9]

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

[10]

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

[11]

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.  Google Scholar

[12]

X. H. Gong, Connectedness of the solution sets and scalarization for vector equilibrium problems,, J. Optim. Theory Appl., 133 (2007), 151.  doi: 10.1007/s10957-007-9196-y.  Google Scholar

[13]

X. H. Gong and J. C. Yao, Connectedness of the set of efficient solutions for generalized systems,, J. Optim. Theory Appl., 138 (2008), 189.  doi: 10.1007/s10957-008-9378-2.  Google Scholar

[14]

P. Q. Khanh and L. M. Luu, Upper semicontinuity of the solution set to parametric vector quasivariational inequalities,, J. Global Optim., 32 (2005), 569.  doi: 10.1007/s10898-004-2694-7.  Google Scholar

[15]

K. Kimura and J. C. Yao, Sensitivity analysis of solution mappings of parametric vector quasi-equilibrium problems,, J. Global Optim., 41 (2008), 187.  doi: 10.1007/s10898-007-9210-9.  Google Scholar

[16]

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

[17]

S. J. Li and Z. M. Fang, On the stability of a dual weak vector variational inequality problem,, J. Ind. Manag. Optim., 4 (2008), 155.  doi: 10.3934/jimo.2008.4.155.  Google Scholar

[18]

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.  Google Scholar

[19]

M. M. Wong, Lower semicontinuity of the solution map to a parametric vector variational inequality,, J. Global Optim., 46 (2010), 435.  doi: 10.1007/s10898-009-9447-6.  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.  doi: 10.1016/j.jmaa.2004.03.014.  Google Scholar

[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.  Google Scholar

[3]

J. P. Aubin and I. Ekeland, "Applied Nonlinear Analysis,", Wiley, (1984).   Google Scholar

[4]

C. R. Chen and S. J. Li, On the solution continuity of parametric generalized systems,, Pac. J. Optim., 6 (2010), 141.   Google Scholar

[5]

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

[6]

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.  Google Scholar

[7]

C. R. Chen, S. J. Li and Z. M. Fang, On the solution semicontinuity to a parametric generalized vector quasivariational inequality,, Comput. Math. Appl., 60 (2010), 2417.  doi: 10.1016/j.camwa.2010.08.036.  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.  doi: 10.1007/s10898-004-2692-9.  Google Scholar

[9]

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

[10]

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

[11]

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.  Google Scholar

[12]

X. H. Gong, Connectedness of the solution sets and scalarization for vector equilibrium problems,, J. Optim. Theory Appl., 133 (2007), 151.  doi: 10.1007/s10957-007-9196-y.  Google Scholar

[13]

X. H. Gong and J. C. Yao, Connectedness of the set of efficient solutions for generalized systems,, J. Optim. Theory Appl., 138 (2008), 189.  doi: 10.1007/s10957-008-9378-2.  Google Scholar

[14]

P. Q. Khanh and L. M. Luu, Upper semicontinuity of the solution set to parametric vector quasivariational inequalities,, J. Global Optim., 32 (2005), 569.  doi: 10.1007/s10898-004-2694-7.  Google Scholar

[15]

K. Kimura and J. C. Yao, Sensitivity analysis of solution mappings of parametric vector quasi-equilibrium problems,, J. Global Optim., 41 (2008), 187.  doi: 10.1007/s10898-007-9210-9.  Google Scholar

[16]

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

[17]

S. J. Li and Z. M. Fang, On the stability of a dual weak vector variational inequality problem,, J. Ind. Manag. Optim., 4 (2008), 155.  doi: 10.3934/jimo.2008.4.155.  Google Scholar

[18]

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.  Google Scholar

[19]

M. M. Wong, Lower semicontinuity of the solution map to a parametric vector variational inequality,, J. Global Optim., 46 (2010), 435.  doi: 10.1007/s10898-009-9447-6.  Google Scholar

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