Fixed set theory for closed correspondences with applications to self-similarity and games

Ahmet Ok

    Research output: Contribution to journalArticle

    Abstract

    This paper sketches a highly elementary order-theoretic approach which allows one to derive general existence theorems concerning the largest and minimal fixed and almost-fixed sets of closed (and condensing) families of commuting set-valued self-maps defined on a compact topological (complete and bounded metric, resp.) space. Certain converses of these results are also established, thereby providing new characterizations of compact (complete, resp.) spaces. We also study the asymptotic behavior of fixed sets sequences associated with a uniformly convergent sequence of closed correspondences, and give two applications. The first application provides two theorems on the existence of a compact self-similar set with respect to infinite families of continuous functions. The second application uses the present fixed set theory to derive a result on the existence of rationalizable outcomes in normal-form games with possibly discontinuous payoff functions.

    Original languageEnglish (US)
    Pages (from-to)309-330
    Number of pages22
    JournalNonlinear Analysis, Theory, Methods and Applications
    Volume56
    Issue number3
    DOIs
    StatePublished - Feb 2004

    Fingerprint

    Self-similarity
    Set Theory
    Set theory
    Correspondence
    Game
    Closed
    Self-similar Set
    Discontinuous Functions
    Convergent Sequence
    Compact Set
    Converse
    Existence Theorem
    Normal Form
    Continuous Function
    Asymptotic Behavior
    Metric
    Theorem
    Family

    Keywords

    • Closed correspondences
    • Condensing correspondences
    • Fixed points
    • Fixed sets
    • Nash equilibrium
    • Rationalizability in games
    • Self-similar sets

    ASJC Scopus subject areas

    • Analysis
    • Applied Mathematics

    Cite this

    Fixed set theory for closed correspondences with applications to self-similarity and games. / Ok, Ahmet.

    In: Nonlinear Analysis, Theory, Methods and Applications, Vol. 56, No. 3, 02.2004, p. 309-330.

    Research output: Contribution to journalArticle

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