### 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 language | English (US) |
---|---|

Pages (from-to) | 309-330 |

Number of pages | 22 |

Journal | Nonlinear Analysis, Theory, Methods and Applications |

Volume | 56 |

Issue number | 3 |

DOIs | |

State | Published - Feb 2004 |

### Fingerprint

### 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.

Research output: Contribution to journal › Article

*Nonlinear Analysis, Theory, Methods and Applications*, vol. 56, no. 3, pp. 309-330. https://doi.org/10.1016/j.na.2003.08.001

}

TY - JOUR

T1 - Fixed set theory for closed correspondences with applications to self-similarity and games

AU - Ok, Ahmet

PY - 2004/2

Y1 - 2004/2

N2 - 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.

AB - 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.

KW - Closed correspondences

KW - Condensing correspondences

KW - Fixed points

KW - Fixed sets

KW - Nash equilibrium

KW - Rationalizability in games

KW - Self-similar sets

UR - http://www.scopus.com/inward/record.url?scp=0346970679&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0346970679&partnerID=8YFLogxK

U2 - 10.1016/j.na.2003.08.001

DO - 10.1016/j.na.2003.08.001

M3 - Article

VL - 56

SP - 309

EP - 330

JO - Nonlinear Analysis, Theory, Methods and Applications

JF - Nonlinear Analysis, Theory, Methods and Applications

SN - 0362-546X

IS - 3

ER -