A General Existence Theorem of Zero Points

P. Jean-Jacques Herings (), Gleb A. Koshevoy, Dolf Talman () and Zaifu Yang
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Zaifu Yang: METEOR

No 55, Research Memoranda from Maastricht : METEOR, Maastricht Research School of Economics of Technology and Organization

Abstract: Let X be a non-empty, compact, convex set in R and
an upper semi-continuous mapping from X to the collection of non-empty, compact, convex subsets in R. Its is well knwon that such a mapping has a stationary point in 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 or contains the origin 0, then
must have a zero point on X, i.e. there exists a point in X satisfying that 0 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 in 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.

Keywords: Economics (search for similar items in EconPapers)
Date: 2002

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Working Paper: A general existence theorem of zero points (2002)
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