# MINIMAX PROBLEMS OF UNIFORMLY SAME-ORDER SET-VALUED MAPPINGS

• Zhang, Yu ;
• Li, Shengjie
• Received : 2012.10.15
• Published : 2013.09.30
• 28 4

#### 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

#### References

1. J. P. Aubin and I. Ekeland, Applied Nonlinear Analysis, John Wiley and Sons, New York, 1984.
2. J. P. Aubin and H. Frankowska, Set-Valued Analysis, Birkhauser, Boston, 1990.
3. 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. https://doi.org/10.1006/jmaa.1996.0088
4. G. Y. Chen, A generalized section theorem and a minimax inequality for a vector-valued mapping, Optimization 22 (1991), no. 5, 745-754. https://doi.org/10.1080/02331939108843716
5. 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. https://doi.org/10.1007/s001860050020
6. G. Y. Chen, X. X. Huang, and X. Q. Yang, Vector Optimization, Set-Valued and Variational Analysis. Springer, Berlin, Heidelberg, 2005.
7. 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. https://doi.org/10.1017/S0004972700013939
8. 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.
9. 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.
10. F. Ferro, A minimax theorem for vector-valued functions, J. Optim. Theory Appl. 60 (1989), no. 1, 19-31. https://doi.org/10.1007/BF00938796
11. X. H. Gong, The strong minimax theorem and strong saddle points of vector-valued functions, Nonlinear Anal. 68 (2008), no. 8, 2228-2241. https://doi.org/10.1016/j.na.2007.01.056
12. J. Jahn, Vector Optimization, Theory, Applications, and Extensions, Springer, Berlin, Germany, 2004.
13. 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.
14. 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. https://doi.org/10.11650/twjm/1500405793
15. G. Y. Li, A note on nonconvex minimax theorem with separable homogeneous polynomials, J. Optim. Theory Appl. 150 (2011), no. 1, 194-203. https://doi.org/10.1007/s10957-011-9827-1
16. 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. https://doi.org/10.1023/A:1004667309814
17. 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. https://doi.org/10.1016/j.na.2010.04.047
18. 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.
19. D. T. Luc and C. Vargas, A saddlepoint theorem for set-valued maps, Nonlinear Anal. 18 (1992), no. 1, 1-7. https://doi.org/10.1016/0362-546X(92)90044-F
20. J. W. Nieuwenhuis, Some minimax theorems in vector-valued functions, J. Optim. Theory Appl. 40 (1983), no. 3, 463-475. https://doi.org/10.1007/BF00933511
21. S. Park, The Fan minimax inequality implies the Nash equilibrium theorem, Appl. Math. Lett. 24 (2011), no. 12, 2206-2210. https://doi.org/10.1016/j.aml.2011.06.027
22. 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. https://doi.org/10.1007/BF02191986
23. 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. https://doi.org/10.1007/BF02275357
24. T. Tanaka, Some minimax problems of vector-valued functions, J. Optim. Theory Appl. 59 (1988), no. 3, 505-524. https://doi.org/10.1007/BF00940312
25. 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. https://doi.org/10.11650/twjm/1500405816
26. 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. https://doi.org/10.1007/s13226-012-0008-1
27. 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. https://doi.org/10.1016/j.na.2008.10.040
28. Y. Zhang and S. J. Li, Ky Fan minimax inequalities for set-valued mappings, Fixed Point Theory Appl. 64 (2012), 12 pp.
29. 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. https://doi.org/10.1080/01630563.2011.610915

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