A discrete fixed point theorem and its applications

Iimura, Takuya

doi:10.1016/s0304-4068(03)00007-7

“…After the completion of the manuscript, the authors learned that a Russian group of V. Danilov a nd G. Koshevoy a l s o noticed the incorrectness of Iimura's proof in Iimura (2003).…”

Section: Concluding Remarkmentioning

confidence: 99%

“…

AbstractThe aim of this note is to indicate an example that demonstrates the incorrectness of Iimura's discrete xed point t heorem (Iimura 2003) and to present a corrected statement using the concept of integrally convex sets.JEL classication: C650, C620

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mentioning

confidence: 99%

Discrete fixed point theorem reconsidered

Iimura

¹

,

Murota

²

,

Tamura

³

2005

Journal of Mathematical Economics

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“…In recent years several papers have appeared in which the existence of a discrete zero point was established for functions satisfying some property replacing continuity. Direction preserving functions are considered in Iimura (2003), Danilov and Koshevoy (2004), Iimura, Murota and Tamura (2005), and van der Laan, Yang (2006, 2007b). A function f : Z n → IR n is direction preserving if for any two cell-connected points x and y in Z n , i.e., points within a same cube of size one, it holds that f j (x)f j (y) ≥ 0 for every component j = 1, .…”

Section: Introductionmentioning

confidence: 99%

Solving Discrete Systems of Nonlinear Equations

Laan

¹

,

Talman

²

,

Yang

³

2009

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In this paper we study the existence problem of a zero point of a function defined on a finite set of elements of the integer lattice Z n of the n-dimensional Euclidean space IR n . 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 Z n 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.

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“…Direction preserving functions are considered in Iimura (2003), Danilov and Koshevoy (2004), Iimura, Murota and Tamura (2005), and van der Laan, Yang (2006, 2007b). A function f : Z n → IR n is direction preserving if for any two cell-connected points x and y in Z n , i.e., points within a same cube of size one, it holds that f j (x)f j (y) ≥ 0 for every component j = 1, .…”

Section: Introductionmentioning

confidence: 99%

Solving Discrete Systems of Nonlinear Equations

Laan

¹

,

Talman

²

,

Yang

³

2008

View full text Add to dashboard Cite

In this paper we study the existence problem of a zero point of a function defined on a finite set of elements of the integer lattice Z n of the n-dimensional Euclidean space IR n . 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 Z n 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.

show abstract

A discrete fixed point theorem and its applications

Cited by 40 publications

References 18 publications

Discrete fixed point theorem reconsidered

Discrete fixed point theorem reconsidered

Solving Discrete Systems of Nonlinear Equations

Solving Discrete Systems of Nonlinear Equations

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