NDLI: General Existence Theorem of Zero Points

Content Provider	Springer Nature Link
Author	Herings, P. J. J. Koshevoy, G. A. Talman, A. J. J. Yang, Z.
Copyright Year	2004
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.
Starting Page	375
Ending Page	394
Page Count	20
File Format	PDF
ISSN	00223239
Journal	Journal of Optimization Theory and Applications
Volume Number	120
Issue Number	2
e-ISSN	15732878
Language	English
Publisher	Kluwer Academic Publishers-Plenum Publishers
Publisher Date	2004-01-01
Publisher Place	New York
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Theory of Computation Applications of Mathematics Optimization Calculus of Variations and Optimal Control Engineering Operation Research/Decision Theory
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics Control and Optimization Management Science and Operations Research

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1	Ministry of Education (GoI), Department of Higher Education	Sanctioning Authority	https://www.education.gov.in/ict-initiatives
2	Indian Institute of Technology Kharagpur	Host Institute of the Project: The host institute of the project is responsible for providing infrastructure support and hosting the project	https://www.iitkgp.ac.in
3	National Digital Library of India Office, Indian Institute of Technology Kharagpur	The administrative and infrastructural headquarters of the project	Dr. B. Sutradhar bsutra@ndl.gov.in
4	Project PI / Joint PI	Principal Investigator and Joint Principal Investigators of the project	Dr. B. Sutradhar bsutra@ndl.gov.in Prof. Saswat Chakrabarti will be added soon
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