# Solving discrete systems of nonlinear equations

## Abstract

We study the existence problem of a zero point of a function defined on a finite set of elements of the integer lattice of the n-dimensional Euclidean space . It is assumed that the set is integrally convex, which implies that the convex hull of the set can be subdivided in simplices such that every vertex is an element of and each simplex of the triangulation lies in an n-dimensional cube of size one. With respect to this triangulation we assume that the function satisfies some property that replaces continuity. Under this property and some boundary condition the function has a zero point. To prove this we use a simplicial algorithm that terminates with a zero point within a finite number of iterations. The standard technique of applying a fixed point theorem to a piecewise linear approximation cannot be applied, because the 'continuity property' is too weak to assure that a zero point of the piecewise linear approximation induces a zero point of the function itself. We apply the main existence result to prove the existence of a pure Cournot-Nash equilibrium in a Cournot oligopoly model. We further obtain a discrete analogue of the well-known Borsuk-Ulam theorem and a theorem for the existence of a solution for the discrete nonlinear complementarity problem.## Download Info

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## Bibliographic Info

Article provided by Elsevier in its journal European Journal of Operational Research.
**Volume (Year):** 214 (2011)

**Issue (Month):** 3 (November)

**Pages:** 493-500

**Handle:**RePEc:eee:ejores:v:214:y:2011:i:3:p:493-500

**Contact details of provider:**

Web page: http://www.elsevier.com/locate/eor

## Related research

**Keywords:**Discrete system of equations Triangulation Simplicial algorithm Fixed point Zero point;

**Other versions of this item:**

- Gerard van der Laan & Dolf Talman & Zaifu Yang, 2009.
"
**Solving Discrete Systems of Nonlinear Equations**," Tinbergen Institute Discussion Papers 09-062/1, Tinbergen Institute. - Laan, G. van der & Talman, A.J.J. & Yang, Z.F., 2008.
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**Solving Discrete Systems of Nonlinear Equations**," Discussion Paper 2008-105, Tilburg University, Center for Economic Research. - Laan, G. van der & Talman, A.J.J. & Yang, Z.F., 2011.
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**Solving discrete systems of nonlinear equations**," Open Access publications from Tilburg University urn:nbn:nl:ui:12-4839550, Tilburg University.

**Find related papers by JEL classification:**

**C61**- Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Optimization Techniques; Programming Models; Dynamic Analysis**C62**- Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Existence and Stability Conditions of Equilibrium**C68**- Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Computable General Equilibrium Models**C72**- Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Noncooperative Games**C58**- Mathematical and Quantitative Methods - - Econometric Modeling - - - Financial Econometrics

## References

**References listed on IDEAS**

Please report citation or reference errors to , or , if you are the registered author of the cited work, log in to your RePEc Author Service profile, click on "citations" and make appropriate adjustments.:

- Gerard van der Laan & Dolf Talman & Zaifu Yang, 2004.
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**Solving Discrete Zero Point Problems**," Tinbergen Institute Discussion Papers 04-112/1, Tinbergen Institute.

- Laan, G. van der & Talman, A.J.J. & Yang, Z.F., 2004.
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**Solving discrete zero point problems**," Discussion Paper 2004-113, Tilburg University, Center for Economic Research.

- Laan, G. van der & Talman, A.J.J. & Yang, Z.F., 2004.
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- Talman, Dolf & Yang, Zaifu, 2009.
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**A discrete multivariate mean value theorem with applications**," European Journal of Operational Research, Elsevier, vol. 192(2), pages 374-381, January.

- Talman, A.J.J. & Yang, Z.F., 2006.
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**A Discrete Multivariate Mean Value Theorem with Applications**," Discussion Paper 2006-106, Tilburg University, Center for Economic Research. - Talman, A.J.J. & Yang, Z.F., 2009.
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**A discrete multivariate mean value theorem with applications**," Open Access publications from Tilburg University urn:nbn:nl:ui:12-3129885, Tilburg University.

- Talman, A.J.J. & Yang, Z.F., 2006.
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- Iimura, Takuya & Murota, Kazuo & Tamura, Akihisa, 2005.
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**Discrete fixed point theorem reconsidered**," Journal of Mathematical Economics, Elsevier, vol. 41(8), pages 1030-1036, December. - Laan, G. van der & Talman, A.J.J. & Yang, Z.F., 2007.
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**A vector labeling method for solving discrete zero point and complementarity problems**," Open Access publications from Tilburg University urn:nbn:nl:ui:12-284192, Tilburg University. - Laan, G. van der & Talman, A.J.J. & Yang, Z.F., 2005.
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**Computing Integral Solutions of Complementarity Problems**," Discussion Paper 2005-5, Tilburg University, Center for Economic Research.

- Laan, G. van der & Talman, A.J.J. & Yang, Z.F., 2007.
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**Computing integral solutions of complementarity problems**," Open Access publications from Tilburg University urn:nbn:nl:ui:12-327117, Tilburg University. - Gerard van der Laan & Dolf Talman & Zaifu Yang, 2005.
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**Computing Integral Solutions of Complementarity Problems**," Tinbergen Institute Discussion Papers 05-006/1, Tinbergen Institute.

- Laan, G. van der & Talman, A.J.J. & Yang, Z.F., 2007.
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- Herings, P.J.J. & Talman, A.J.J. & Yang, Z.F., 2001.
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**Variational inequality problems with a continuum of solutions: Existence and computation**," Open Access publications from Tilburg University urn:nbn:nl:ui:12-86829, Tilburg University.

- Herings, P.J.J. & Talman, A.J.J. & Yang, Z.F., 1999.
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**Variational Inequality Problems With a Continuum of Solutions: Existence and Computation**," Discussion Paper 1999-72, Tilburg University, Center for Economic Research.

- Herings, P.J.J. & Talman, A.J.J. & Yang, Z.F., 1999.
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- Herbert E. Scarf, 1967.
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**The Approximation of Fixed Points of a Continuous Mapping**," Cowles Foundation Discussion Papers 216R, Cowles Foundation for Research in Economics, Yale University. - Iimura, Takuya, 2003.
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**A discrete fixed point theorem and its applications**," Journal of Mathematical Economics, Elsevier, vol. 39(7), pages 725-742, September. - Talman, A.J.J. & Laan, G. van der, 1979.
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**A restart algorithm for computing fixed points without an extra dimension**," Open Access publications from Tilburg University urn:nbn:nl:ui:12-153012, Tilburg University.

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