### Abstract

This paper aims to characterize the direction preserving condition that guarantees the existence of a fixed point of discrete mappings defined on an integer rectangle X into itself. We deal with a discrete fixed point theorem based on Brouwer's fixed point theorem, which depends on the simplicial decomposition of the convex hull of X. We first review an arbitrary simplicial decomposition in ℝ^{2} and the Preudenthal decomposition in ℝ^{n}. Next we characterize the direction preserving condition for an arbitrary consistent simplicial decomposition in ℝ^{n}, which implies a sufficient condition for the strategic game to have a pure-strategy equilibrium.

Original language | English |
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Pages (from-to) | 1535-1545 |

Number of pages | 11 |

Journal | Journal of Nonlinear and Convex Analysis |

Volume | 18 |

Issue number | 8 |

Publication status | Published - 2017 |

Externally published | Yes |

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### All Science Journal Classification (ASJC) codes

- Analysis
- Geometry and Topology
- Control and Optimization
- Applied Mathematics

### Cite this

*Journal of Nonlinear and Convex Analysis*,

*18*(8), 1535-1545.

**A discrete fixed point theorem utilizing the direction preserving condition.** / Kawasaki, Hidefumi; Hashiyama, Shuhei.

Research output: Contribution to journal › Article

*Journal of Nonlinear and Convex Analysis*, vol. 18, no. 8, pp. 1535-1545.

}

TY - JOUR

T1 - A discrete fixed point theorem utilizing the direction preserving condition

AU - Kawasaki, Hidefumi

AU - Hashiyama, Shuhei

PY - 2017

Y1 - 2017

N2 - This paper aims to characterize the direction preserving condition that guarantees the existence of a fixed point of discrete mappings defined on an integer rectangle X into itself. We deal with a discrete fixed point theorem based on Brouwer's fixed point theorem, which depends on the simplicial decomposition of the convex hull of X. We first review an arbitrary simplicial decomposition in ℝ2 and the Preudenthal decomposition in ℝn. Next we characterize the direction preserving condition for an arbitrary consistent simplicial decomposition in ℝn, which implies a sufficient condition for the strategic game to have a pure-strategy equilibrium.

AB - This paper aims to characterize the direction preserving condition that guarantees the existence of a fixed point of discrete mappings defined on an integer rectangle X into itself. We deal with a discrete fixed point theorem based on Brouwer's fixed point theorem, which depends on the simplicial decomposition of the convex hull of X. We first review an arbitrary simplicial decomposition in ℝ2 and the Preudenthal decomposition in ℝn. Next we characterize the direction preserving condition for an arbitrary consistent simplicial decomposition in ℝn, which implies a sufficient condition for the strategic game to have a pure-strategy equilibrium.

UR - http://www.scopus.com/inward/record.url?scp=85030759333&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85030759333&partnerID=8YFLogxK

M3 - Article

VL - 18

SP - 1535

EP - 1545

JO - Journal of Nonlinear and Convex Analysis

JF - Journal of Nonlinear and Convex Analysis

SN - 1345-4773

IS - 8

ER -