Browse > Article
http://dx.doi.org/10.4134/BKMS.2013.50.5.1639

MINIMAX PROBLEMS OF UNIFORMLY SAME-ORDER SET-VALUED MAPPINGS  

Zhang, Yu (College of Mathematics and Statistics Chongqing University, College of Statistics and Mathematics Yunnan University of Finance and Economics)
Li, Shengjie (College of Mathematics and Statistics Chongqing University)
Publication Information
Bulletin of the Korean Mathematical Society / v.50, no.5, 2013 , pp. 1639-1650 More about this Journal
Abstract
In this paper, a class of set-valued mappings is introduced, which is called uniformly same-order. For this sort of mappings, some minimax problems, in which the minimization and the maximization of set-valued mappings are taken in the sense of vector optimization, are investigated without any hypotheses of convexity.
Keywords
minimax theorem; cone loose saddle point; uniformly same-order mapping; vector optimization;
Citations & Related Records
연도 인용수 순위
  • Reference
1 S. Park, The Fan minimax inequality implies the Nash equilibrium theorem, Appl. Math. Lett. 24 (2011), no. 12, 2206-2210.   DOI   ScienceOn
2 D. S. Shi and C. Ling, Minimax theorems and cone saddle points of uniformly same-order vector-valued functions, J. Optim. Theory Appl. 84 (1995), no. 3, 575-587.   DOI
3 K. K. Tan, J. Yu, and X. Z. Yuan, Existence theorems for saddle points of vector-valued maps, J. Optim. Theory Appl. 89 (1996), no. 3, 731-747.   DOI
4 T. Tanaka, Some minimax problems of vector-valued functions, J. Optim. Theory Appl. 59 (1988), no. 3, 505-524.   DOI
5 M. G. Yang, J. P. Xu, N. J. Huang, and S. J. Yu, Minimax theorems for vector-valued mappings in abstract convex spaces, Taiwanese J. Math. 14 (2010), no. 2, 719-732.   DOI
6 Z. Yang and Y. J. Pu, Generalized Browder-type fixed point theorem with strongly geodesic convexity on Hadamard manifolds with applications, Indian J. Pure Appl. Math. 43 (2012), no. 2, 129-144.   DOI
7 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), no. 1-2, 212-218.   DOI   ScienceOn
8 Y. Zhang and S. J. Li, Ky Fan minimax inequalities for set-valued mappings, Fixed Point Theory Appl. 64 (2012), 12 pp.
9 Y. Zhang, S. J. Li, and S. K. Zhu, Minimax problems for set-valued mappings, Numer. Funct. Anal. Optim. 33 (2012), no. 2, 239-253.   DOI
10 K. Fan, A minimax inequality and applications, In: Inequalities, III (Proc. Third Sympos., Univ. California, Los Angeles, Calif., 1969; dedicated to the memory of Theodore S. Motzkin), pp. 103-13. Academic Press, New York, 1972.
11 F. Ferro, A minimax theorem for vector-valued functions, J. Optim. Theory Appl. 60 (1989), no. 1, 19-31.   DOI
12 X. H. Gong, The strong minimax theorem and strong saddle points of vector-valued functions, Nonlinear Anal. 68 (2008), no. 8, 2228-2241.   DOI   ScienceOn
13 J. Jahn, Vector Optimization, Theory, Applications, and Extensions, Springer, Berlin, Germany, 2004.
14 I. S. Kim and Y. T. Kim, Loose saddle points of set-valued maps in topological vector spaces, Appl. Math. Lett. 12 (1999), no. 8, 21-26.
15 W. K. Kim and S. Kum, On a non-compact generalization of Fan's minimax theorem, Taiwanese J. Math. 14 (2010), no. 2, 347-358.   DOI
16 G. Y. Li, A note on nonconvex minimax theorem with separable homogeneous polynomials, J. Optim. Theory Appl. 150 (2011), no. 1, 194-203.   DOI
17 S. J. Li, G. Y. Chen, and G. M. Lee, Minimax theorems for set-valued mappings, J. Optim. Theory Appl. 106 (2000), no. 1, 183-200.   DOI   ScienceOn
18 X. B. Li, S. J. Li, and Z. M. Fang, A minimax theorem for vector valued functions in lexicographic order, Nonlinear Anal. 73 (2010), no. 4, 1101-1108.   DOI   ScienceOn
19 L. J. Lin and Y. L. Tsai, On vector quasi-saddle points of set-valued maps, In: Eberkard, A., Hadjisavvas, N., Luc, D. T., (eds.) Generalized Convexity, Generalized Monotonicity and Applications, pp. 311-319. Kluwer Academic Publishers, Dordrecht, The Nether-lands, 2005.
20 D. T. Luc and C. Vargas, A saddlepoint theorem for set-valued maps, Nonlinear Anal. 18 (1992), no. 1, 1-7.   DOI   ScienceOn
21 S. S. Chang, G. M. Lee, and B. S. Lee, Minimax inequalities for vector-valued mappings on W-spaces, J. Math. Anal. Appl. 198 (1996), no. 2, 371-380.   DOI   ScienceOn
22 J. W. Nieuwenhuis, Some minimax theorems in vector-valued functions, J. Optim. Theory Appl. 40 (1983), no. 3, 463-475.   DOI
23 J. P. Aubin and I. Ekeland, Applied Nonlinear Analysis, John Wiley and Sons, New York, 1984.
24 J. P. Aubin and H. Frankowska, Set-Valued Analysis, Birkhauser, Boston, 1990.
25 G. Y. Chen, A generalized section theorem and a minimax inequality for a vector-valued mapping, Optimization 22 (1991), no. 5, 745-754.   DOI
26 G. Y. Chen and X. X. Huang, Ekeland's $\epsilon$-variational principle for set-valued mappings, Math. Methods Oper. Res. 48 (1998), no. 2, 181-186.   DOI
27 G. Y. Chen, X. X. Huang, and X. Q. Yang, Vector Optimization, Set-Valued and Variational Analysis. Springer, Berlin, Heidelberg, 2005.
28 Y. J. Cho, S. S. Chang, J. S. Jung, S. M. Kang, and X. Wu, Minimax theorems in probabilistic metric spaces, Bull. Austral. Math. Soc. 51 (1995), no. 1, 103-119.   DOI
29 Y. J. Cho, M. R. Delavar, S. A. Mohammadzadeh, and M. Roohi, Coincidence theorems and minimax inequalities in abstract convex spaces, J. Inequal. Appl. 126 (2011), 14 pp.