### Abstract

Original language | English |
---|---|

Place of Publication | Tilburg |

Publisher | Operations research |

Number of pages | 17 |

Volume | 2002-107 |

Publication status | Published - 2002 |

### Publication series

Name | CentER Discussion Paper |
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Volume | 2002-107 |

### Fingerprint

### Keywords

- stationary point
- zero point

### Cite this

*A General Existence Thorem of Zero Points*. (CentER Discussion Paper; Vol. 2002-107). Tilburg: Operations research.

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**A General Existence Thorem of Zero Points.** / Herings, P.J.J.; Koshevoy, G.A.; Talman, A.J.J.; Yang, Z.F.

Research output: Working paper › Discussion paper › Other research output

TY - UNPB

T1 - A General Existence Thorem of Zero Points

AU - Herings, P.J.J.

AU - Koshevoy, G.A.

AU - Talman, A.J.J.

AU - Yang, Z.F.

N1 - Pagination: 17

PY - 2002

Y1 - 2002

N2 - Let X be a non-empty, compact, convex set in Rn and ° an upper semi-continuous mapping from X to the collection of non-empty, compact, convex subsets of Rn.It is well known that such a mapping has a stationary point on X, i.e. there exists a point in X satisfying that its image under ° has a non-empty intersection with the normal cone of X at the point.In case for every point in X it holds that the intersection of the image under ° with the normal cone of X at the point is either empty orcontains the origin 0n , then ° must have a zero point on X, i.e. there exists a point in X satisfying that 0n lies in the image of the point.Another well-known condition for the existence of a zero point follows from Ky Fan's coincidence theorem, which says that if for every point the intersection of the image with the tangent cone of X at the point is non-empty, the mapping must have a zero point.In this paper we extend all these existence results by giving a general zero point existence theorem, of which the two results are obtained as special cases.We also discuss what kind of solutions may exist when no further conditions are stated on the mapping °.Finally, we show how our results can be used to establish several new intersection results on a compact, convex set.

AB - Let X be a non-empty, compact, convex set in Rn and ° an upper semi-continuous mapping from X to the collection of non-empty, compact, convex subsets of Rn.It is well known that such a mapping has a stationary point on X, i.e. there exists a point in X satisfying that its image under ° has a non-empty intersection with the normal cone of X at the point.In case for every point in X it holds that the intersection of the image under ° with the normal cone of X at the point is either empty orcontains the origin 0n , then ° must have a zero point on X, i.e. there exists a point in X satisfying that 0n lies in the image of the point.Another well-known condition for the existence of a zero point follows from Ky Fan's coincidence theorem, which says that if for every point the intersection of the image with the tangent cone of X at the point is non-empty, the mapping must have a zero point.In this paper we extend all these existence results by giving a general zero point existence theorem, of which the two results are obtained as special cases.We also discuss what kind of solutions may exist when no further conditions are stated on the mapping °.Finally, we show how our results can be used to establish several new intersection results on a compact, convex set.

KW - stationary point

KW - zero point

M3 - Discussion paper

VL - 2002-107

T3 - CentER Discussion Paper

BT - A General Existence Thorem of Zero Points

PB - Operations research

CY - Tilburg

ER -