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On a Parameterized System of Nonlinear Equations with Economic Applications

Author

Listed:
  • Dolf Talman

    (Tilburg University)

  • Zaifu Yang

    (University of York)

Abstract

We study a parameterized system of nonlinear equations. Given a nonempty, compact, and convex set, an affine function, and a point-to-set mapping from the set to the Euclidean space containing the set, we constructively prove that, under certain (boundary) conditions on the mapping, there exists a connected set of zero points of the mapping, i.e., the origin is an element of the image for every point in the connected set, such that the connected set has a nonempty intersection with both the face at which the affine function is minimized and the face at which that function is maximized. This result generalizes and unifies several well-known existence theorems including Browder’s fixed point theorem and Ky Fan’s coincidence theorem. An economic application with constrained equilibria is also discussed.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:joptap:v:154:y:2012:i:2:d:10.1007_s10957-012-0037-2
    DOI: 10.1007/s10957-012-0037-2
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    1. Herings, P. Jean-Jacques & van der Laan, Gerard & Talman, Dolf, 2009. "Equilibria with coordination failures," Journal of Mathematical Economics, Elsevier, vol. 45(1-2), pages 23-37, January.
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