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 and 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), 699-711. 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), 271-284. doi: 10.1007/s10957-007-9250-9.

[3]

J. P. Aubin and I. Ekeland, "Applied Nonlinear Analysis," Wiley, New York, 1984.

[4]

C. R. Chen and S. J. Li, On the solution continuity of parametric generalized systems, Pac. J. Optim., 6 (2010), 141-151.

[5]

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

[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), 519-528. doi: 10.3934/jimo.2007.3.519.

[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), 2417-2425. doi: 10.1016/j.camwa.2010.08.036.

[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.

[9]

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

[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), 197-205. doi: 10.1007/s10957-008-9379-1.

[11]

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.

[12]

X. H. Gong, Connectedness of the solution sets and scalarization for vector equilibrium problems, J. Optim. Theory Appl., 133 (2007), 151-161. doi: 10.1007/s10957-007-9196-y.

[13]

X. H. Gong and J. C. Yao, Connectedness of the set of efficient solutions for generalized systems, J. Optim. Theory Appl., 138 (2008), 189-196. doi: 10.1007/s10957-008-9378-2.

[14]

P. Q. Khanh and L. M. Luu, Upper semicontinuity of the solution set to parametric vector quasivariational inequalities, J. Global Optim., 32 (2005), 569-580. doi: 10.1007/s10898-004-2694-7.

[15]

K. Kimura and J. C. Yao, Sensitivity analysis of solution mappings of parametric vector quasi-equilibrium problems, J. Global Optim., 41 (2008), 187-202. doi: 10.1007/s10898-007-9210-9.

[16]

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

[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), 155-165. doi: 10.3934/jimo.2008.4.155.

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

[19]

M. M. Wong, Lower semicontinuity of the solution map to a parametric vector variational inequality, J. Global Optim., 46 (2010), 435-446. doi: 10.1007/s10898-009-9447-6.

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.

[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), 271-284. doi: 10.1007/s10957-007-9250-9.

[3]

J. P. Aubin and I. Ekeland, "Applied Nonlinear Analysis," Wiley, New York, 1984.

[4]

C. R. Chen and S. J. Li, On the solution continuity of parametric generalized systems, Pac. J. Optim., 6 (2010), 141-151.

[5]

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

[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), 519-528. doi: 10.3934/jimo.2007.3.519.

[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), 2417-2425. doi: 10.1016/j.camwa.2010.08.036.

[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.

[9]

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

[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), 197-205. doi: 10.1007/s10957-008-9379-1.

[11]

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.

[12]

X. H. Gong, Connectedness of the solution sets and scalarization for vector equilibrium problems, J. Optim. Theory Appl., 133 (2007), 151-161. doi: 10.1007/s10957-007-9196-y.

[13]

X. H. Gong and J. C. Yao, Connectedness of the set of efficient solutions for generalized systems, J. Optim. Theory Appl., 138 (2008), 189-196. doi: 10.1007/s10957-008-9378-2.

[14]

P. Q. Khanh and L. M. Luu, Upper semicontinuity of the solution set to parametric vector quasivariational inequalities, J. Global Optim., 32 (2005), 569-580. doi: 10.1007/s10898-004-2694-7.

[15]

K. Kimura and J. C. Yao, Sensitivity analysis of solution mappings of parametric vector quasi-equilibrium problems, J. Global Optim., 41 (2008), 187-202. doi: 10.1007/s10898-007-9210-9.

[16]

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

[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), 155-165. doi: 10.3934/jimo.2008.4.155.

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

[19]

M. M. Wong, Lower semicontinuity of the solution map to a parametric vector variational inequality, J. Global Optim., 46 (2010), 435-446. doi: 10.1007/s10898-009-9447-6.

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