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Minimax inequalities for set-valued mappings

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Abstract

In this paper, by using the finite intersection property, we first obtain two types of minimax inequalities for set-valued mappings, which improve and generalize the corresponding results in Ferro (J Optim Theory Appl 60:19–31, 1989) and Li et al. (J Math Anal Appl 281:707–723, 2003). Then, by using the Ky Fan lemma and the Kakutani–Fan–Glicksberg fixed point theorem, we also investigate some Ky Fan minimx inequalities for set-valued mappings.

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References

  1. Aubin J.P., Ekeland I.: Applied Nonlinear Analysis. Wiley, New York (1984)

    MATH  Google Scholar 

  2. Chang S.S., Lee G.M., Lee B.S.: Minimax inequalities for vector-valued mappings on W-spaces. J. Math. Anal. Appl. 198, 371–380 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  3. Chen G.Y.: A generalized section theorem and a minimax inequality for a vector-valued mapping. Optimization 22, 745–754 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  4. Chen G.Y., Huang X.X., Hou S.H.: General Ekeland’s variational principle for set-valued mappings. J. Optim. Theory Appl. 106, 151–164 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  5. Fan K.: A generalization of Tychondff’s fixed point theorem. Math. Ann. 142, 305–310 (1961)

    Article  MathSciNet  MATH  Google Scholar 

  6. Fan K.: A Minimax Inequality and Applications. Inequalities III, pp. 103–111. Academic Press, New York (1972)

    Google Scholar 

  7. Ferro F.: A minimax theorem for vector-valued functions. J. Optim. Theory Appl. 60, 19–31 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  8. Ferro F.: A minimax theorem for vector-valued functions, part 2. J. Optim. Theory Appl. 68, 35–48 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  9. Gong X.H.: The strong minimax theorem and strong saddle points of vector-valued functions. Nonlinear Anal. 68, 2228–2241 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  10. Ha C.W.: Minimax and fixed point theorems. Math. Ann. 248, 73–77 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  11. Ha C.W.: On a minimax inequality of Ky Fan. Proc. Am. Math. Soc. 99, 680–682 (1987)

    Article  MATH  Google Scholar 

  12. Holmes R.B.: Geometric Functional and It’s Applications. Springer, New York (1975)

    Book  Google Scholar 

  13. Jahn J.: Vector Optimization: Theory, Applications, and Extensions. Springer, Berlin (2004)

    MATH  Google Scholar 

  14. Li S.J., Chen G.Y., Lee G.M.: Minimax theorems for set-valued mappings. J. Optim. Theory Appl. 106, 183–200 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  15. Li S.J., Chen G.Y., Teo K.L., Yang X.Q.: Generalized minimax inequalities for set-valued mappings. J. Math. Anal. Appl. 281, 707–723 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  16. Li X.B., Li S.J., Fang Z.M.: A minimax theorem for vector valued functions in lexicographic order. Nonlinear Anal. 73, 1101–1108 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  17. Li Z.F., Wang S.Y.: A type of minimax inequality for vector-valued mappings. J. Math. Anal. Appl. 227, 68–80 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  18. Luo, X.Q.: On some generalized Ky Fan minimax inequalities. Fixed Point Theory Appl. Article ID 194671 (2009)

  19. Nieuwenhuis J.W.: Some minimax theorems in vector-valued functions. J. Optim. Theory Appl. 40, 463–475 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  20. Shi D.S., Ling C.: Minimax theorems and cone saddle points of uniformly same-order vector-valued functions. J. Optim. Theory Appl. 84, 575–587 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  21. Tanaka T.: Some minimax problems of vector-valued functions. J. Optim. Theory Appl. 59, 505–524 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  22. Tanaka T.: Two types of minimax theorems for vector-valued functions. J. Optim. Theory Appl. 68, 321–334 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  23. Tanaka T.: Generalized quasiconvexities, cone saddle points, and minimax theorem for vector-valued functions. J. Optim. Theory Appl. 81, 355–377 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  24. Zeng J., Li S.J.: An Ekeland’s variational principle for set-valued mappings with application. J. Comput. Appl. Math. 230, 477–484 (2009)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. College of Mathematics and Statistics, Chongqing University, Chongqing, 401331, China

    Y. Zhang, S. J. Li & M. H. Li

Authors
  1. Y. Zhang
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  2. S. J. Li
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  3. M. H. Li
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Corresponding author

Correspondence to S. J. Li.

Additional information

This research was partially supported by the National Natural Science Foundation of China (Grant number: 11171362) and the Fundamental Research Funds for the Central Universities (Grant number: CDJXS11100016).

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Zhang, Y., Li, S.J. & Li, M.H. Minimax inequalities for set-valued mappings. Positivity 16, 751–770 (2012). https://doi.org/10.1007/s11117-011-0144-6

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  • DOI: https://doi.org/10.1007/s11117-011-0144-6

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