On a Parameterized System of Nonlinear Equations with Economic Applications

Adolphus Talman and Zaifu Yang

Journal of Optimization Theory and Applications, 2012, vol. 154, issue 2, No 16, 644-671

Abstract: 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.

Keywords: Parameterized system of nonlinear equations; Fixed point; Variational inequality; Algorithm; Equilibrium (search for similar items in EconPapers)
Date: 2012
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DOI: 10.1007/s10957-012-0037-2

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