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A general existence theorem of zero points

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  • Herings, P. Jean-Jacques

    (Center for Mathematical Economics, Bielefeld University)

  • Koshevoy, Gleb A.

    (Center for Mathematical Economics, Bielefeld University)

  • Talman, Dolf

    (Center for Mathematical Economics, Bielefeld University)

  • Yang, Zaifu

    (Center for Mathematical Economics, Bielefeld University)

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.
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Suggested Citation

  • Herings, P. Jean-Jacques & Koshevoy, Gleb A. & Talman, Dolf & Yang, Zaifu, 2017. "A general existence theorem of zero points," Center for Mathematical Economics Working Papers 343, Center for Mathematical Economics, Bielefeld University.
    • Handle: RePEc:bie:wpaper:343
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    References listed on IDEAS

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    1. P. J. J. Herings & A. J. J. Talman, 1998. "Intersection Theorems with a Continuum of Intersection Points," Journal of Optimization Theory and Applications, Springer, vol. 96(2), pages 311-335, February.
    2. van der Laan, G. & Talman, A.J.J. & Yang, Z., 1994. "Intersection theorems on polytopes," Other publications TiSEM b1440966-d5e0-44ab-9b1c-f, Tilburg University, School of Economics and Management.
    3. 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.
    4. Herings, P.J.J. & van der Laan, G. & Talman, A.J.J., 2001. "Quantity constrained equilibria," Research Memorandum 023, Maastricht University, Maastricht Research School of Economics of Technology and Organization (METEOR).
    5. Robert M. Freund, 1986. "Combinatorial Theorems on the Simplotope that Generalize Results on the Simplex and Cube," Mathematics of Operations Research, INFORMS, vol. 11(1), pages 169-179, February.
    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. Z. Yang, 2001. "An Intersection Theorem on an Unbounded Set and Its Application to the Fair Allocation Problem," Journal of Optimization Theory and Applications, Springer, vol. 110(2), pages 429-443, August.
    8. 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.
    9. Herings, P.J.J. & van der Laan, G. & Talman, A.J.J., 2001. "Quantity Constrained Equilibria," Discussion Paper 2001-93, Tilburg University, Center for Economic Research.
    10. Jean-Jacques Herings & Dolf Talman & Zaifu Yang, 1996. "The Computation of a Continuum of Constrained Equilibria," Mathematics of Operations Research, INFORMS, vol. 21(3), pages 675-696, August.
    11. Ichiishi, Tatsuro & Idzik, Adam, 1991. "Closed Covers of Compact Convex Polyhedra," International Journal of Game Theory, Springer;Game Theory Society, vol. 20(2), pages 161-169.
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    Cited by:

    1. van der Laan, Gerard & Talman, Dolf & Yang, Zaifu, 2011. "Solving discrete systems of nonlinear equations," European Journal of Operational Research, Elsevier, vol. 214(3), pages 493-500, November.
    2. G. Laan & A. J. J. Talman & Z. Yang, 2010. "Combinatorial Integer Labeling Theorems on Finite Sets with Applications," Journal of Optimization Theory and Applications, Springer, vol. 144(2), pages 391-407, February.
    3. Dolf Talman & Zaifu Yang, 2012. "On a Parameterized System of Nonlinear Equations with Economic Applications," Journal of Optimization Theory and Applications, Springer, vol. 154(2), pages 644-671, August.
    4. Gerard van der Laan & Dolf Talman & Zaifu Yang, 2005. "Solving Discrete Zero Point Problems with Vector Labeling," Tinbergen Institute Discussion Papers 05-106/1, Tinbergen Institute.
    5. Talman, Dolf & Yang, Zaifu, 2009. "A discrete multivariate mean value theorem with applications," European Journal of Operational Research, Elsevier, vol. 192(2), pages 374-381, January.
    6. van der Laan, G. & Talman, A.J.J. & Yang, Z.F., 2007. "Combinatorial Integer Labeling Thorems on Finite Sets with an Application to Discrete Systems of Nonlinear Equations," Other publications TiSEM 264c28a5-10b6-44e1-9694-4, Tilburg University, School of Economics and Management.
    7. van der Laan, G. & Talman, A.J.J. & Yang, Z.F., 2007. "A vector labeling method for solving discrete zero point and complementarity problems," Other publications TiSEM 070869d0-4e42-4d34-85f9-b, Tilburg University, School of Economics and Management.
    8. Gerard van der Laan & Dolf Talman & Zaifu Yang, 2004. "Solving Discrete Zero Point Problems," Tinbergen Institute Discussion Papers 04-112/1, Tinbergen Institute.

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