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Variational inequality problems with a continuum of solutions : existence and computation

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Author Info
Herings, P.J.J.
Talman, D.
Yang, Z. (Tilburg University, Center for Economic Research)

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Abstract

In this paper three sufficient conditions are provided under each of which an upper semi-continuous point-to-set mapping defined on an arbitrary polytope has a connected set of zero points that connect two distinct faces of the polytope. Furthermore, we obtain an existence theorem of a connected set of solutions to a nonlinear variational inequality problem over arbitrary polytopes. These results follow in a constructive way by designing a new simplicial algorithm. The algorithm operates on a triangulation of the polytope and generates a piecewise linear path of points connecting two distinct faces of the polytope. Each point on the path is an approximate zero point. As the mesh size of the triangulation goes to zero, the path converges to a connected set of zero points linking the two distinct faces. As a consequence, our results generalize Browder's fixed point theorem (1960) and an earlier result by the authors (1996) on the n-dimensional unit cube. An application in economics is also discussed.

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Paper provided by Tilburg University, Center for Economic Research in its series Discussion Paper with number 72.

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Date of creation: 1999
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Handle: RePEc:dgr:kubcen:199972

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Please report citation or reference errors to , or , if you are the registered author of the cited work, log in to your RePEc Author Service profile, click on "citations" and make appropriate adjustments.:
  1. Demarzo, Peter M. & Eaves, B. Curtis, 1996. "Computing equilibria of GEI by relocalization on a Grassmann manifold," Journal of Mathematical Economics, Elsevier, vol. 26(4), pages 479-497. [Downloadable!] (restricted)
  2. Herbert E. Scarf, 1967. "The Approximation of Fixed Points of a Continuous Mapping," Cowles Foundation Discussion Papers 216R, Cowles Foundation, Yale University. [Downloadable!]
  3. Herings, P.J.J. & Talman, D., 1994. "Intersection Theorems with a Continuum of Intersection Points," Discussion Paper 79, Tilburg University, Center for Economic Research. [Downloadable!]
  4. Yamamoto, Yoshitsugu, 1993. "A Path-Following Procedure to Find a Proper Equilibrium of Finite Games," International Journal of Game Theory, Springer, vol. 22(3), pages 249-59.
  5. Herings, P. Jean-Jacques, 1998. "On the existence of a continuum of constrained equilibria1," Journal of Mathematical Economics, Elsevier, vol. 30(3), pages 257-273, October. [Downloadable!] (restricted)
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(explanations, Please report citation or reference errors to , or , if you are the registered author of the cited work, log in to your RePEc Author Service profile, click on "citations" and make appropriate adjustments.)

  1. Herings, P.J.J. & Koshevoy, G.A. & Talman, D. & Yang, Z., 2002. "A general existence theorem of zero points," Discussion Paper 107, Tilburg University, Center for Economic Research. [Downloadable!]
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  2. Talman, A.J.J. & Yamamoto,, 2001. "Continuum of zero points of a mapping on a compact, convex set," Discussion Paper 56, Tilburg University, Center for Economic Research. [Downloadable!]
  3. Talman, A.J.J. & Yang, Z., 2004. "The computation of a coincidence of two mappings," Discussion Paper 100, Tilburg University, Center for Economic Research. [Downloadable!]
  4. Jean-Jacques Herings & Gerard van der Laan & Dolf Talman & Zaifu Yang, 2004. "A Fixed Point Theorem for Discontinuous Functions," Tinbergen Institute Discussion Papers 05-004/1, Tinbergen Institute. [Downloadable!]
  5. Talman, D. & Yang, Z., 2003. "On the connectedness of coincidences and zero points of mappings," Discussion Paper 73, Tilburg University, Center for Economic Research. [Downloadable!]
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