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A General Existence Thorem of Zero Points

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

    (Tilburg University, Center for Economic Research)

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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Bibliographic Info

Paper provided by Tilburg University, Center for Economic Research in its series Discussion Paper with number 2002-107.

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Date of creation: 2002
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Handle: RePEc:dgr:kubcen:2002107

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Keywords: stationary point; zero point;

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  1. Laan, G. van der & Talman, A.J.J. & Yang, Z.F., 1999. "Intersection theorems on polytypes," Open Access publications from Tilburg University urn:nbn:nl:ui:12-78480, Tilburg University.
  2. Ichiishi, Tatsuro & Idzik, Adam, 1991. "Closed Covers of Compact Convex Polyhedra," International Journal of Game Theory, Springer, vol. 20(2), pages 161-69.
  3. Herings, P.J.J. & Talman, A.J.J. & Zang, Z., 1994. "The computation of a continuum of constrained equilibria," Discussion Paper 1994-38, Tilburg University, Center for Economic Research.
  4. Herbert E. Scarf, 1967. "The Approximation of Fixed Points of a Continuous Mapping," Cowles Foundation Discussion Papers 216R, Cowles Foundation for Research in Economics, Yale University.
  5. P. Jean-Jacques Herings & Gerard van der Laan & Dolf Talman, 2001. "Quantity Constrained Equilibria," Tinbergen Institute Discussion Papers 01-116/1, Tinbergen Institute.
  6. Herings, P.J.J. & Talman, A.J.J. & Yang, Z.F., 1999. "Variational Inequality Problems With a Continuum of Solutions: Existence and Computation," Discussion Paper 1999-72, Tilburg University, Center for Economic Research.
  7. Herings, P.J.J. & Laan, G. van der & Talman, A.J.J., 2001. "Quantity Constrained Equilibria," Discussion Paper 2001-93, Tilburg University, Center for Economic Research.
  8. Herings, P.J.J. & Talman, A.J.J., 1994. "Intersection theorems with a continuum of intersection points," Discussion Paper 1994-79, Tilburg University, Center for Economic Research.
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