General Existence Theorem of Zero Points

P. Jean-Jacques Herings, G. A. Koshevoy, Adolphus Talman and Zaifu Yang
Additional contact information
G. A. Koshevoy: Central Institute of Mathematics and Economics

Journal of Optimization Theory and Applications, 2004, vol. 120, issue 2, No 7, 375-394

Abstract: Abstract Let X be a nonempty, compact, convex set in $$\mathbb{R}^n$$ and let φ be an upper semicontinuous mapping from X to the collection of nonempty, compact, convex subsets of $$\mathbb{R}^n$$ . It is well known that such a mapping has a stationary point on X; i.e., there exists a point X such that its image under φ has a nonempty intersection with the normal cone of X at the point. In the case where, for every point in X, it holds that the intersection of the image under φ with the normal cone of X at the point is either empty or contains the origin 0 n , then φ must have a zero point on X; i.e., there exists a point in X such that 0 n lies in the image of the point. Another well-known condition for the existence of a zero point follows from the Ky Fan coincidence theorem, which says that, if for every point the intersection of the image with the tangent cone of X at the point is nonempty, 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 previous two results are obtained as special cases. We discuss also what kind of solutions may exist when no further conditions are stated on the mapping φ. Finally, we show how our results can be used to establish several new intersection results on a compact, convex set.

Keywords: Stationary points; zero points; fixed points; normal cones; tangent cones; intersection points (search for similar items in EconPapers)
Date: 2004
References: View references in EconPapers View complete reference list from CitEc
Citations: View citations in EconPapers (7)

Downloads: (external link)
http://link.springer.com/10.1023/B:JOTA.0000015689.71020.f0 Abstract (text/html)
Access to the full text of the articles in this series is restricted.

Related works:
Working Paper: A general existence theorem of zero points (2017)
Working Paper: A general existence theorem of zero points (2004)
Working Paper: A General Existence Thorem of Zero Points (2002)
Working Paper: A General Existence Thorem of Zero Points (2002)
Working Paper: A general existence theorem of zero points (2002)
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:joptap:v:120:y:2004:i:2:d:10.1023_b:jota.0000015689.71020.f0

Ordering information: This journal article can be ordered from
http://www.springer. ... cs/journal/10957/PS2

DOI: 10.1023/B:JOTA.0000015689.71020.f0

Access Statistics for this article

Journal of Optimization Theory and Applications is currently edited by Franco Giannessi and David G. Hull

More articles in Journal of Optimization Theory and Applications from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().