A General Existence Theorem of Zero Points

METEOR Working Paper No. RM 02/049

16 Pages Posted: 27 Mar 2003

See all articles by P. Jean-Jacques Herings

P. Jean-Jacques Herings

Tilburg University

Gleb A. Koshevoy

Russian Academy of Sciences

Dolf Talman

Tilburg University - Department of Econometrics & Operations Research

Zaifu Yang

Yokohama National University - Faculty of Business Administration; Tilburg University - Department of Econometrics & Operations Research

Date Written: November 27, 2002

Abstract

Let X be a non-empty, compact, convex set in R and o an upper semi-continuous mapping from X to the collection of non-empty, compact, convex subsets of R;. 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 o 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 o with the normal cone of X at the point is either empty or contains the origin O; then o must have a zero point on X, i.e. there exists a point in X satisfying that O 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 o. Finally, we show how our results can be used to establish several new intersection results on a compact, convex set.

Keywords: stationary point, zero point, fixed point, normal cone, tangent cone, intersection point

JEL Classification: C62

Suggested Citation

Herings, P. Jean-Jacques and Koshevoy, Gleb A. and Talman, Dolf J. J. and Yang, Zaifu, A General Existence Theorem of Zero Points (November 27, 2002). METEOR Working Paper No. RM 02/049, Available at SSRN: https://ssrn.com/abstract=372000 or http://dx.doi.org/10.2139/ssrn.372000

P. Jean-Jacques Herings (Contact Author)

Tilburg University ( email )

Department of Econometrics and Operations Research
P.O. Box 90153
Tilburg, 5000 LE
Netherlands
+31 13 4668797 (Phone)
5000 LE (Fax)

HOME PAGE: http://https://sites.google.com/view/jean-jacques-herings/home

Gleb A. Koshevoy

Russian Academy of Sciences ( email )

Central Institute of Mathematics and Economics
117418 Moscow
Russia

Dolf J. J. Talman

Tilburg University - Department of Econometrics & Operations Research ( email )

Tilburg, 5000 LE
Netherlands
+31 13 466 2346 (Phone)

Zaifu Yang

Yokohama National University - Faculty of Business Administration ( email )

79-4 Tokiwa-dai Hodogaya-ku
Yokohama, Kanagawa, 2408501
Japan

Tilburg University - Department of Econometrics & Operations Research ( email )

P.O. Box 90153
5000 LE Tilburg
Netherlands

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