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A Full Characterization on Fixed-Point Theorem, Minimax Inequality, Saddle Point, and KKM Theorem

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  • Tian, Guoqiang

Abstract

This paper provides necessary and sufficient conditions for fixed-point theorems, minimax inequalities and some related theorems defined on arbitrary topological spaces that may be discrete, continuum, non-compact or non-convex. We establish a single condition, γ-recursive transfer lower semicontinuity, which fully characterizes the existence of equilibrium of minimax inequality without imposing any kind of convexity nor any restriction on topological space. The result then is employed to fully characterize fixed point theory, saddle point theory, and the FKKM theory.

Suggested Citation

  • Tian, Guoqiang, 2012. "A Full Characterization on Fixed-Point Theorem, Minimax Inequality, Saddle Point, and KKM Theorem," MPRA Paper 57929, University Library of Munich, Germany, revised Jul 2014.
    • Handle: RePEc:pra:mprapa:57929
    as

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    File URL: https://mpra.ub.uni-muenchen.de/57929/1/MPRA_paper_57929.pdf
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    References listed on IDEAS

    as
    1. Tian, Guoqiang, 1994. "Generalized KKM theorem, minimax inequalities and their applications," MPRA Paper 41217, University Library of Munich, Germany.
    2. Rabia Nessah & Guoqiang Tian, 2013. "Existence of Solution of Minimax Inequalities, Equilibria in Games and Fixed Points Without Convexity and Compactness Assumptions," Journal of Optimization Theory and Applications, Springer, vol. 157(1), pages 75-95, April.
    3. Tian, Guoqiang & Zhou, Jianxin, 1995. "Transfer continuities, generalizations of the Weierstrass and maximum theorems: A full characterization," Journal of Mathematical Economics, Elsevier, vol. 24(3), pages 281-303.
    4. Shafer, Wayne & Sonnenschein, Hugo, 1975. "Equilibrium in abstract economies without ordered preferences," Journal of Mathematical Economics, Elsevier, vol. 2(3), pages 345-348, December.
    5. Lin, Y. Joseph & Tian, Guoqiang, 1993. "Minimax inequality equivalent to the Fan-Knaster-Kuratowski-Mazurkiewicz Theorem," MPRA Paper 41220, University Library of Munich, Germany.
    6. M. B. Lignola, 1997. "Ky Fan Inequalities and Nash Equilibrium Points without Semicontinuity and Compactness," Journal of Optimization Theory and Applications, Springer, vol. 94(1), pages 137-145, July.
    7. Guoqiang Tian, 1993. "Necessary and Sufficient Conditions for Maximization of a Class of Preference Relations," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 60(4), pages 949-958.
    8. Q. H. Ansari & Y. C. Lin & J. C. Yao, 2000. "General KKM Theorem with Applications to Minimax and Variational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 104(1), pages 17-57, January.
    9. Tian, Guoqiang & Zhou, Jianxin, 1992. "Transfer Method for Characterizing the Existence of Maximal Elements of Binary Relations on Compact or Noncompact Sets," MPRA Paper 41227, University Library of Munich, Germany.
    Full references (including those not matched with items on IDEAS)

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    More about this item

    Keywords

    Fixed-point theorems; minimax inequalities; saddle points; FKKM theorems; recursive transfer continuity;
    All these keywords.

    JEL classification:

    • D00 - Microeconomics - - General - - - General

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