Generalized minimax theorems for two set-valued mappings

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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

Abstract
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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 and 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.
[2]	F. Ferro, A minimax theorem for vector-valued functions, J. Optim. Theory Appl., 60 (1989), 19-31. doi: 10.1007/BF00938796.
[3]	K. Fan, Sur un théorème minimax, C. R. Acad. Sci. Paris, 259 (1964), 3925-3928.
[4]	C. W. Ha, Minimax and fixed point theorems, Mathematische Annalen, 248 (1980), 73-77.
[5]	D. Kuroiwa, Convexity for set-valued maps, Appl. Math. Lett., 9 (1996), 97-101. doi: 10.1016/0893-9659(96)00020-1.
[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.
[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.
[8]	D. T. Luc, "Theory of Vector Optimization," in "Lecture Notes in Economics and Mathematical Sesytems" 319, 1988.
[9]	J. W. Nieuwenhuis, Some minimax theorems in vector-valued functions, J. Optim. Theory Appl., 40 (1983), 463-475. doi: 10.1007/BF00933511.
[10]	W. Rudin, "Functional Analysis," McGraw-Hill, Inc., 1973.
[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.
[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.
[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.
[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.

show all references

References:

[1]	J. P. Aubin and I. Ekeland, "Applied Nonlinear Analysis," John Wiley and Sons, New York, 1984.
[2]	F. Ferro, A minimax theorem for vector-valued functions, J. Optim. Theory Appl., 60 (1989), 19-31. doi: 10.1007/BF00938796.
[3]	K. Fan, Sur un théorème minimax, C. R. Acad. Sci. Paris, 259 (1964), 3925-3928.
[4]	C. W. Ha, Minimax and fixed point theorems, Mathematische Annalen, 248 (1980), 73-77.
[5]	D. Kuroiwa, Convexity for set-valued maps, Appl. Math. Lett., 9 (1996), 97-101. doi: 10.1016/0893-9659(96)00020-1.
[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.
[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.
[8]	D. T. Luc, "Theory of Vector Optimization," in "Lecture Notes in Economics and Mathematical Sesytems" 319, 1988.
[9]	J. W. Nieuwenhuis, Some minimax theorems in vector-valued functions, J. Optim. Theory Appl., 40 (1983), 463-475. doi: 10.1007/BF00933511.
[10]	W. Rudin, "Functional Analysis," McGraw-Hill, Inc., 1973.
[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.
[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.
[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.
[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.

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