Solving discrete systems of nonlinear equations
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 ndimensional 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 ndimensional 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 CournotNash equilibrium in a Cournot oligopoly model. We further obtain a discrete analogue of the wellknown BorsukUlam theorem and a theorem for the existence of a solution for the discrete nonlinear complementarity problem.
If you experience problems downloading a file, check if you have the proper application to view it first. In case of further problems read the IDEAS help page. Note that these files are not on the IDEAS site. Please be patient as the files may be large.
As the access to this document is restricted, you may want to look for a different version under "Related research" (further below) or search for a different version of it.
Volume (Year): 214 (2011)
Issue (Month): 3 (November)
Pages: 493500
Handle:  RePEc:eee:ejores:v:214:y:2011:i:3:p:493500 
Contact details of provider:  Web page: http://www.elsevier.com/locate/eor

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.:
 van der Laan, G. & Talman, A.J.J. & Yang, Z.F., 2007.
"Computing integral solutions of complementarity problems,"
Other publications TiSEM
6f3abdc6b96144668e602, Tilburg University, School of Economics and Management.
 Gerard van der Laan & Dolf Talman & Zaifu Yang, 2005. "Computing Integral Solutions of Complementarity Problems," Tinbergen Institute Discussion Papers 05006/1, Tinbergen Institute.
 van der Laan, G. & Talman, A.J.J. & Yang, Z.F., 2005. "Computing Integral Solutions of Complementarity Problems," Discussion Paper 20055, Tilburg University, Center for Economic Research.
 Talman, A.J.J. & Yang, Z.F., 2009. "A discrete multivariate mean value theorem with applications," Other publications TiSEM d48f2a19dcc240e490855, Tilburg University, School of Economics and Management.
 Herings, P.J.J. & Koshevoy, G.A. & Talman, A.J.J. & Yang, Z.F., 2002.
"A General Existence Thorem of Zero Points,"
Discussion Paper
2002107, Tilburg University, Center for Economic Research.
 Herings P. JeanJacques & Koshevoy Gleb A. & Talman Dolf & Yang Zaifu, 2002. "A General Existence Theorem of Zero Points," Research Memorandum 055, Maastricht University, Maastricht Research School of Economics of Technology and Organization (METEOR).
 Herings, P. JeanJacques, 2017. "A general existence theorem of zero points," Center for Mathematical Economics Working Papers 343, Center for Mathematical Economics, Bielefeld University.
 Herings, P.J.J. & Koshevoy, G.A. & Talman, A.J.J. & Yang, Z.F., 2004. "A general existence theorem of zero points," Other publications TiSEM 2670fc3ab9914443a8fb3, Tilburg University, School of Economics and Management.
 Talman, A.J.J. & Yang, Z.F., 2006.
"A Discrete Multivariate Mean Value Theorem with Applications,"
Discussion Paper
2006106, Tilburg University, Center for Economic Research.
 Talman, Dolf & Yang, Zaifu, 2009. "A discrete multivariate mean value theorem with applications," European Journal of Operational Research, Elsevier, vol. 192(2), pages 374381, January.
 Iimura, Takuya, 2003. "A discrete fixed point theorem and its applications," Journal of Mathematical Economics, Elsevier, vol. 39(7), pages 725742, September.
 Iimura, Takuya & Murota, Kazuo & Tamura, Akihisa, 2005. "Discrete fixed point theorem reconsidered," Journal of Mathematical Economics, Elsevier, vol. 41(8), pages 10301036, December.
 Herbert E. Scarf, 1967. "The Approximation of Fixed Points of a Continuous Mapping," Cowles Foundation Discussion Papers 216R, Cowles Foundation for Research in Economics, Yale University.
 Talman, A.J.J. & van der Laan, G., 1979. "A restart algorithm for computing fixed points without an extra dimension," Other publications TiSEM 1f2102f8e6da4e9ca2ed9, Tilburg University, School of Economics and Management.
 van der Laan, G. & Talman, A.J.J. & Yang, Z.F., 2007. "A vector labeling method for solving discrete zero point and complementarity problems," Other publications TiSEM 070869d04e424d3485f9b, Tilburg University, School of Economics and Management.
 Herings, P.J.J. & Talman, A.J.J. & Yang, Z.F., 1999.
"Variational Inequality Problems With a Continuum of Solutions : Existence and Computation,"
Discussion Paper
199972, Tilburg University, Center for Economic Research.
 Herings, P.J.J. & Talman, A.J.J. & Yang, Z.F., 2001. "Variational inequality problems with a continuum of solutions : Existence and computation," Other publications TiSEM 50bc0af9976f4c1c94e0f, Tilburg University, School of Economics and Management.
This item is not listed on Wikipedia, on a reading list or among the top items on IDEAS.
When requesting a correction, please mention this item's handle: RePEc:eee:ejores:v:214:y:2011:i:3:p:493500. See general information about how to correct material in RePEc.
For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: (Dana Niculescu)
If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.
If references are entirely missing, you can add them using this form.
If the full references list an item that is present in RePEc, but the system did not link to it, you can help with this form.
If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your profile, as there may be some citations waiting for confirmation.
Please note that corrections may take a couple of weeks to filter through the various RePEc services.