### Abstract

Original language | English |
---|---|

Pages (from-to) | 644-671 |

Journal | Journal of Optimization Theory and Applications |

Volume | 154 |

Issue number | 2 |

Publication status | Published - 2012 |

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### Cite this

*Journal of Optimization Theory and Applications*,

*154*(2), 644-671.

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*Journal of Optimization Theory and Applications*, vol. 154, no. 2, pp. 644-671.

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

Research output: Contribution to journal › Article › Scientific › peer-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

SN - 0022-3239

IS - 2

ER -