On a parameterized system of nonlinear equations with economic applications

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

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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.
Original languageEnglish
Pages (from-to)644-671
JournalJournal of Optimization Theory and Applications
Volume154
Issue number2
Publication statusPublished - 2012

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Connected Set
System of Nonlinear Equations
Nonlinear equations
Affine Function
Economics
Face
Coincidence Theorem
Zero Point
Compact Set
Convex Sets
Existence Theorem
Fans
Fixed point theorem
Euclidean space
Intersection
Boundary conditions
Generalise

Cite this

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title = "On a parameterized system of nonlinear equations with economic applications",
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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On a parameterized system of nonlinear equations with economic applications. / Talman, A.J.J.; Yang, Z.F.

In: Journal of Optimization Theory and Applications, Vol. 154, No. 2, 2012, p. 644-671.

Research output: Contribution to journalArticleScientificpeer-review

TY - JOUR

T1 - On a parameterized system of nonlinear equations with economic applications

AU - Talman, A.J.J.

AU - Yang, Z.F.

PY - 2012

Y1 - 2012

N2 - 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.

AB - 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.

M3 - Article

VL - 154

SP - 644

EP - 671

JO - Journal of Optimization Theory and Applications

JF - Journal of Optimization Theory and Applications

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