DOI QR코드

DOI QR Code

A SIMPLE PROOF OF THE SION MINIMAX THEOREM

  • Park, Se-Hie (THE NATIONAL ACADEMY OF SCIENCES, DEPARTMENT OF MATHEMATICAL SCIENCES SEOUL NATIONAL UNIVERSITY)
  • Received : 2009.04.09
  • Published : 2010.09.30

Abstract

For convex subsets X of a topological vector space E, we show that a KKM principle implies a Fan-Browder type fixed point theorem and that this theorem implies generalized forms of the Sion minimax theorem.

References

  1. F. E. Browder, The fixed point theory of multi-valued mappings in topological vector spaces, Math. Ann. 177 (1968), 283-301. https://doi.org/10.1007/BF01350721
  2. Y. J. Cho, J. K. Kim, and B. Y. Lee, Remarks on KKM maps and applications, J. Adv. Res. Appl. Math. 1 (2009), no. 1, 1-8.
  3. K. Fan, A generalization of Tychonoff’s fixed point theorem, Math. Ann. 142 (1960), 305-310. https://doi.org/10.1007/BF01353421
  4. J. Kindler, A simple proof of Sion’s minimax theorem, Amer. Math. Monthly 112 (2005), no. 4, 356-358. https://doi.org/10.2307/30037472
  5. B. Knaster, K. Kuratowski, and S. Mazurkiewicz, Ein Beweis des Fixpunktsatzes fur n-Dimensionale Simplexe, Fund. Math. 14 (1929), 132-137. https://doi.org/10.4064/fm-14-1-132-137
  6. S. Park, Ninety years of the Brouwer fixed point theorem, Vietnam J. Math. 27 (1999), no. 3, 187-222.
  7. M. Sion, On general minimax theorems, Pacific J. Math. 8 (1958), 171-176. https://doi.org/10.2140/pjm.1958.8.171
  8. J. von Neumann, Zur Theorie der Gesellschaftsspiele, Math. Ann. 100 (1928), no. 1, 295-320. https://doi.org/10.1007/BF01448847

Cited by

  1. On the von Neumann–Sion minimax theorem in KKM spaces vol.23, pp.10, 2010, https://doi.org/10.1016/j.aml.2010.06.011
  2. Minimax Problems for Set-Valued Mappings vol.33, pp.2, 2012, https://doi.org/10.1080/01630563.2011.610915