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January  2013, 9(1): 1-12. doi: 10.3934/jimo.2013.9.1

Generalized minimax theorems for two set-valued mappings

Qingbang Zhang 1, , Caozong Cheng 2, and  Xuanxuan Li 2,

1. 

College of Economic Mathematics, Southwestern University of Finance and Economics, Chengdu, Sichuan 611130, China
2. 

College of Applied Sciences, Beijing University of Technology, Beijing 100124, China, China

Received  July 2011 Revised  April 2012 Published  December 2012

The generalized minimax theorems of two real set-valued mappings, which generalize and improve the corresponding conclusions of D. Kuroiwa (Appl. Math. Lett., 1996, 9: 97-101.) and Li et.al.(J. Optim. Theory Appl., 2000, 106: 183-200; J. Math. Anal. Appl., 2003, 281: 707-723.), and the generalized minimax theorems of two set-valued mappings valued in Fréchet spaces are proved. Some examples are given also to show the results.
Keywords: naturally $S$-quasiconcave, set-valued mapping, Generalized minimax theorem, Hausdorff topological vector spaces..
Mathematics Subject Classification: Primary: 49J53; Secondary: 49K3.
Citation: Qingbang Zhang, Caozong Cheng, Xuanxuan Li. Generalized minimax theorems for two set-valued mappings. Journal of Industrial & Management Optimization, 2013, 9 (1) : 1-12. doi: 10.3934/jimo.2013.9.1
References:
[1]

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

[2]

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

[3]

K. Fan, Sur un théorème minimax, C. R. Acad. Sci. Paris, 259 (1964), 3925-3928.  Google Scholar

[4]

C. W. Ha, Minimax and fixed point theorems, Mathematische Annalen, 248 (1980), 73-77.  Google Scholar

[5]

D. Kuroiwa, Convexity for set-valued maps, Appl. Math. Lett., 9 (1996), 97-101. doi: 10.1016/0893-9659(96)00020-1.  Google Scholar

[6]

S. J. Li, G. Y. Chen and G. M. Lee, Minimax theorems for set-valued mappings, J. Optim. Theory Appl., 106 (2000), 183-199. doi: 10.1023/A:1004667309814.  Google Scholar

[7]

S. J. Li, G. Y. Chen, K. L. Teo and X. Q. Yang, Generalized minimax inequalities for set-valued mappings, J. Math. Anal. Appl., 281 (2003), 707-723. doi: 10.1016/S0022-247X(03)00197-5.  Google Scholar

[8]

D. T. Luc, "Theory of Vector Optimization," in "Lecture Notes in Economics and Mathematical Sesytems" 319, 1988.  Google Scholar

[9]

J. W. Nieuwenhuis, Some minimax theorems in vector-valued functions, J. Optim. Theory Appl., 40 (1983), 463-475. doi: 10.1007/BF00933511.  Google Scholar

[10]

W. Rudin, "Functional Analysis," McGraw-Hill, Inc., 1973.  Google Scholar

[11]

K. K. Tan, J. Yu and X. Z. Yuan, Existence theorems for saddle points of vector-valued maps, J. Optim. Theory Appl., 89 (1996), 731-747. doi: 10.1007/BF02275357.  Google Scholar

[12]

T. Tanaka, Generalied quasiconvexities, cone saddle points and minimax theorems for vector-valued functions, J. Optim. Theory Appl., 81 (1994), 355-377. doi: 10.1007/BF02191669.  Google Scholar

[13]

Q. B. Zhang and C. Z. Cheng, Some fixed-point theorems and minimax inequalities in FC-space, J. Math. Anal. Appl., 328 (2007), 1369-1377. doi: 10.1016/j.jmaa.2006.06.027.  Google Scholar

[14]

Q. B. Zhang, M. J. Liu and C. Z. Cheng, Generalized saddle points theorems for set-valued mappings in locally generalized convex spaces, Nonlinear Anal., 71 (2009), 212-218. doi: 10.1016/j.na.2008.10.040.  Google Scholar

show all references

References:
[1]

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

[2]

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

[3]

K. Fan, Sur un théorème minimax, C. R. Acad. Sci. Paris, 259 (1964), 3925-3928.  Google Scholar

[4]

C. W. Ha, Minimax and fixed point theorems, Mathematische Annalen, 248 (1980), 73-77.  Google Scholar

[5]

D. Kuroiwa, Convexity for set-valued maps, Appl. Math. Lett., 9 (1996), 97-101. doi: 10.1016/0893-9659(96)00020-1.  Google Scholar

[6]

S. J. Li, G. Y. Chen and G. M. Lee, Minimax theorems for set-valued mappings, J. Optim. Theory Appl., 106 (2000), 183-199. doi: 10.1023/A:1004667309814.  Google Scholar

[7]

S. J. Li, G. Y. Chen, K. L. Teo and X. Q. Yang, Generalized minimax inequalities for set-valued mappings, J. Math. Anal. Appl., 281 (2003), 707-723. doi: 10.1016/S0022-247X(03)00197-5.  Google Scholar

[8]

D. T. Luc, "Theory of Vector Optimization," in "Lecture Notes in Economics and Mathematical Sesytems" 319, 1988.  Google Scholar

[9]

J. W. Nieuwenhuis, Some minimax theorems in vector-valued functions, J. Optim. Theory Appl., 40 (1983), 463-475. doi: 10.1007/BF00933511.  Google Scholar

[10]

W. Rudin, "Functional Analysis," McGraw-Hill, Inc., 1973.  Google Scholar

[11]

K. K. Tan, J. Yu and X. Z. Yuan, Existence theorems for saddle points of vector-valued maps, J. Optim. Theory Appl., 89 (1996), 731-747. doi: 10.1007/BF02275357.  Google Scholar

[12]

T. Tanaka, Generalied quasiconvexities, cone saddle points and minimax theorems for vector-valued functions, J. Optim. Theory Appl., 81 (1994), 355-377. doi: 10.1007/BF02191669.  Google Scholar

[13]

Q. B. Zhang and C. Z. Cheng, Some fixed-point theorems and minimax inequalities in FC-space, J. Math. Anal. Appl., 328 (2007), 1369-1377. doi: 10.1016/j.jmaa.2006.06.027.  Google Scholar

[14]

Q. B. Zhang, M. J. Liu and C. Z. Cheng, Generalized saddle points theorems for set-valued mappings in locally generalized convex spaces, Nonlinear Anal., 71 (2009), 212-218. doi: 10.1016/j.na.2008.10.040.  Google Scholar

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