On a parameterized system of nonlinear equations with economic applications

A.J.J. Talman, Z.F. Yang

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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.
Original languageEnglish
Pages (from-to)644-671
JournalJournal of Optimization Theory and Applications
Issue number2
Publication statusPublished - 2012


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