A General Existence Thorem of Zero Points

P.J.J. Herings, G.A. Koshevoy, A.J.J. Talman, Z.F. Yang

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Abstract

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.
Original languageEnglish
Place of PublicationTilburg
PublisherOperations research
Number of pages17
Volume2002-107
Publication statusPublished - 2002

Publication series

NameCentER Discussion Paper
Volume2002-107

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Zero Point
Intersection
Normal Cone
Compact Convex Set
Coincidence Theorem
Tangent Cone
Upper Semicontinuous
Stationary point
Existence Theorem
Existence Results
Subset

Keywords

  • stationary point
  • zero point

Cite this

Herings, P. J. J., Koshevoy, G. A., Talman, A. J. J., & Yang, Z. F. (2002). A General Existence Thorem of Zero Points. (CentER Discussion Paper; Vol. 2002-107). Tilburg: Operations research.
Herings, P.J.J. ; Koshevoy, G.A. ; Talman, A.J.J. ; Yang, Z.F. / A General Existence Thorem of Zero Points. Tilburg : Operations research, 2002. (CentER Discussion Paper).
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abstract = "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.",
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Herings, PJJ, Koshevoy, GA, Talman, AJJ & Yang, ZF 2002 'A General Existence Thorem of Zero Points' CentER Discussion Paper, vol. 2002-107, Operations research, Tilburg.

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

Tilburg : Operations research, 2002. (CentER Discussion Paper; Vol. 2002-107).

Research output: Working paperDiscussion paperOther research output

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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.

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Herings PJJ, Koshevoy GA, Talman AJJ, Yang ZF. A General Existence Thorem of Zero Points. Tilburg: Operations research. 2002. (CentER Discussion Paper).