New versions of the Fan-Browder fixed point theorem and existence of economic equilibria

Park, Sehie

doi:10.1155/s1687182004308089

Cited by 6 publications

(16 citation statements)

References 10 publications

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“…They also then apply some of their results to disjunctive logic programs (with nonmonotone negation). Close in spirit, using mainly Banach spaces, topological spaces, and metric spaces in place of complete lattices, are works of the mathematical community such as [2,10,19,23,49,40,34,35,43,50,51]. We point out that these works do not cover our results.…”

Section: Conclusion and Related Workmentioning

confidence: 92%

On Fixed-Points of Multivalued Functions on Complete Lattices and Their Application to Generalized Logic Programs

Straccia¹,

Ojeda‐Aciego²,

Damásio³

2009

SIAM J. Comput.

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Abstract. Unlike monotone single-valued functions, multivalued mappings may have zero, one, or (possibly infinitely) many minimal fixed-points. The contribution of this work is twofold. First, we overview and investigate the existence and computation of minimal fixed-points of multivalued mappings, whose domain is a complete lattice and whose range is its power set. Second, we show how these results are applied to a general form of logic programs, where the truth space is a complete lattice. We show that a multivalued operator can be defined whose fixed-points are in one-to-one correspondence with the models of the logic program. The topic of this work is the overview and investigation of the fixed-points of multivalued functions f : L → 2 L (multivalued functions are also called correspondences, or set-valued functions, in the literature). Such functions naturally arise, e.g., in the specification of the semantics of nondeterministic programming languages [7,8,11,18,31,36,37,44], in game theory [6,33,45,53], and in disjunctive logic programming [22,27,32,42,52]; those of the latter case motivated our work. Informally, (i) in the first case, the meaning of a nondeterministic 1 program P may be seen as a function p : S → 2 S , where S is the set of states a program may assume. The image of p is a finite nonempty set, as at a given step of a program execution, due to a nondeterministic statement, more than one successive state is possible. The semantics of a program is then related to the fixed-points of such functions (s ∈ p(s)); (ii) in the second case, a game is represented as a function g : S → 2 S , where S is the strategy space of the involved players, and fixed-points (s ∈ g(s)) are related to the so-called Nash equilibria of the game. The image of g is a nonempty (usually finite) set, as at each step of the game, more than one incomparable strategic choice is possible; and

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Section: Conclusion and Related Workmentioning

confidence: 92%

On Fixed-Points of Multivalued Functions on Complete Lattices and Their Application to Generalized Logic Programs

Straccia¹,

Ojeda‐Aciego²,

Damásio³

2009

SIAM J. Comput.

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“…In economics, however, an algebraic structure is indispensable so long as we consider production and growth/contraction. The Hausdorff property is also removable as shown in Park (), and yet is assumed here to employ a familiar proof method due to Browder (, Theorem , p. 285) based on the partition of unity.…”

Section: Fixed Point Theorems For Maps With Their Range Contained In mentioning

confidence: 99%

“…The most important element in the method of Urai (, Corollary ) is the use of an auxiliary map, which is linked to a given map only through the subset of the domain which consists of non‐fixed points . An improvement on some of Urai's results has been made by Park (), using a generalization of the KKM (Knaster–Kuratowski–Mazurkiewicz) theorem. Later, without knowing these works and under more restrictive conditions, Herings et al .…”

Section: Introductionmentioning

confidence: 99%

Fixed Point Theorems for Discontinuous Maps on a Non‐convex Domain

Fujimoto

2013

Metroeconomica

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This paper introduces economists to some fixed point theorems for discontinuous mappings with non‐convex images on a non‐convex domain. These theorems have recently been developed based on a new approach by mathematical economists and mathematicians. The new method of proof is first transformed into a sort of metatheorem, which is then used to obtain a set of necessary and sufficient conditions for a map to have a fixed point. Some fixed point theorems for discontinuous maps are then explained in more concrete cases. The formulations are intended for easier applications towards economic models involving discontinuity as well as non‐convexity.

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“…where K is a given set and h : K × K → R is a given function such that h(x, x) ≥ 0 for all x ∈ K , which is a unified model of several problems; see for example [8][9][10][11][12][13][14][15][16], and references therein.…”

Section: Introductionmentioning

confidence: 99%

On a generalized vector equilibrium problem with bounds

Mitrović

Merkle

2010

Applied Mathematics Letters

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New versions of the Fan-Browder fixed point theorem and existence of economic equilibria

Abstract: We introduce a generalized form of the Fan-Browder fixed point theorem and apply to game-theoretic and economic equilibrium existence problem under the more generous restrictions. Consequently, we state some of recent results of Urai (2000) in more general and efficient forms

Cited by 6 publications

References 10 publications

On Fixed-Points of Multivalued Functions on Complete Lattices and Their Application to Generalized Logic Programs

On Fixed-Points of Multivalued Functions on Complete Lattices and Their Application to Generalized Logic Programs

Fixed Point Theorems for Discontinuous Maps on a Non‐convex Domain

On a generalized vector equilibrium problem with bounds

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