Discrete fixed point analysis and its applications

Yang

2009

J Optim Theory Appl

Tucker's well-known combinatorial lemma states that, for any given symmetric triangulation of the n-dimensional unit cube and for any integer labeling that assigns to each vertex of the triangulation a label from the set {±1, ±2, . . . , ±n} with the property that antipodal vertices on the boundary of the cube are assigned opposite labels, the triangulation admits a 1-dimensional simplex whose two vertices have opposite labels. In this paper, we are concerned with an arbitrary finite set D of integral vectors in the n-dimensional Euclidean space and an integer labeling that assigns to each element of D a label from the set {±1, ±2, . . . , ±n}. Using a constructive approach, we prove two combinatorial theorems of Tucker type. The theorems state that, under some mild conditions, there exists two integral vectors in D having opposite labels and being cell-connected in the sense that both belong to the set {0, 1} n + q for some integral vector q. These theorems are used to show in a constructive way the existence of an integral solution to a system of nonlinear equations under certain natural conditions. An economic application is provided.

Section: Solving Discrete Systems Of Nonlinear Equationsmentioning

confidence: 99%

Combinatorial Integer Labeling Theorems on Finite Sets with Applications

Laan

Yang

2009

J Optim Theory Appl

“…However, these points are not vertices of any simplex of the K-triangulation of R 2 . In Yang (2004), the more general but more complex class of interior zero point excludable functions is introduced. A function f : Z n !…”

Section: Basic Conceptsmentioning

confidence: 99%

“…In addition to these existence results, Yang (accepted for publication) also studies discrete nonlinear complementarity problems and presents several sufficient conditions for the existence of solution for this class of problems. In Danilov and Koshevoy (accepted for publication) and Yang (2004) a class of more general but more complex functions is introduced for the existence of a discrete fixed point. All the results mentioned above are proved using the machinery of topology such as the Brouwer fixed point theorem or Borsuk-Ulam theorem.…”

Section: Introductionmentioning

confidence: 99%

A discrete multivariate mean value theorem with applications

European Journal of Operational Research

Yang

2009

Self Cite

We establish a discrete multivariate mean value theorem for the class of positive maximum component sign preserving functions. A constructive and combinatorial proof is given based upon a simplicial algorithm and vector labeling. Moreover, we apply this theorem to a discrete nonlinear complementarity problem and an economic equilibrium problem with indivisibilities and show the existence of solutions in both problems under certain mild conditions.

“…More precisely, the existence of a fixed point is guaranteed when f satisfies the so-called locally gross direction preserving property. A discrete version of this property was originally used in [17] to prove the existence of a fixed point in case the domain is a discrete set. For x ∈ R n and > 0, let B(x, ) denote the n-dimensional ball in R n with center x and radius .…”

Section: An Existence Theoremmentioning

confidence: 99%

A fixed point theorem for discontinuous functions

Herings

Laan

Operations Research Letters

et al. 2008

Self Cite

A fixed point theorem for discontinuous functionsHerings, P.J.J.; van der Laan, G.; Talman, Dolf; Yang, Z.F. General rightsCopyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.-Users may download and print one copy of any publication from the public portal for the purpose of private study or research -You may not further distribute the material or use it for any profit-making activity or commercial gain -You may freely distribute the URL identifying the publication in the public portal Take down policyIf you believe that this document breaches copyright, please contact us providing details, and we will remove access to the work immediately and investigate your claim. AbstractAny function from a non-empty polytope into itself that is locally gross direction preserving is shown to have the fixed point property. Brouwer's fixed point theorem for continuous functions is a special case. We discuss the application of the result in the area of non-cooperative game theory.