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.
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Communicated by Francesco Zirilli.
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Talman, D., Yang, Z. On a Parameterized System of Nonlinear Equations with Economic Applications. J Optim Theory Appl 154, 644–671 (2012). https://doi.org/10.1007/s10957-012-0037-2
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DOI: https://doi.org/10.1007/s10957-012-0037-2