Fixed points of compact multifunctions

Himmelberg, C. J.

doi:10.1016/0022-247x(72)90128-x

Cited by 188 publications

(67 citation statements)

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“…Our theorem is one of the most general results and substantially extends a large number of known theorems, including Kakutani's theorem [26] for Euclidean spaces and Himmelberg's theorem [17] for locally convex topological vector spaces. Because these theorems were so useful in various problems in mathematical sciences including economic and game theories, it seems to be quite natural to generalize known results on applications of theorems of Kakutani and Himmelberg in view of our theorem.…”

Section: Introductionsupporting

confidence: 55%

Fixed points, intersection theorems, variational inequalities, and equilibrium theorems

Park

2000

International Journal of Mathematics and Mathematical Sciences

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Abstract. From a fixed point theorem for compact acyclic maps defined on admissible convex sets in the sense of Klee, we first deduce collectively fixed point theorems, intersection theorems for sets with convex sections, and quasi-equilibrium theorems. These quasi-equilibrium theorems are applied to give simple and unified proofs of the known variational inequalities of the Hartman-Stampacchia-Browder type. Moreover, from our new fixed point theorem, we deduce new variational inequalities which can be used to obtain fixed point results for convex-valued maps. Finally, various general economic equilibrium theorems are deduced in the forms of the Nash type, the Tarafdar type, and the Yannelis-Prabhakar type. Our results are stated for not-necessarily locally convex topological vector spaces and for abstract economies with arbitrary number of commodities and agents. Our new results extend a lot of known works with much simpler proofs.

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Section: Introductionsupporting

confidence: 55%

Fixed points, intersection theorems, variational inequalities, and equilibrium theorems

Park

2000

International Journal of Mathematics and Mathematical Sciences

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“…Thus G is u.s.c. It now follows by a theorem of Himmelberg [2] that there is an x E S with x E G(x). This implies that x E Q(y) for some y E Fx with dp(Fx, S) = dP(y, S).…”

mentioning

confidence: 99%

A generalization to multifunctions of Fan’s best approximation theorem

Sehgal¹,

Singh²

1988

Proc. Amer. Math. Soc.

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ABSTRACT. We prove a theorem for set valued mappings in an approximatively compact, convex subset of a locally convex space, and then derive results due to Ky Fan and S. Reich as corollaries.Let E be a locally convex Hausdorff topological vector sapee, S a nonempty subset of E and p a continuous seminorm on E. It is a well-known result (see the proof in Sehgal [8] or Ky Fan [1]) that if S is compact and convex and f:S->E is a continuous map, then there exists an a; G S satisfying (1) p(fx-x) =dp(fx,S) =min{p(fx,-y)\ yES}.Since then a number of authors have provided either an extension of the above theorem to set valued mappings or have weakened the compactness condition therein. Some of these results are (d) SEHGAL AND SINGH (1985). Let S Ç E with int(S) ^ 0 and cl(S) convex and let /: S -► 2E be a continuous condensing multifunction with convex, compact values and with a bounded range. Then for each w E int(S), there exists a continuous seminorm p = p(w) satisfying (1) [6].Our aim in this presentation is to prove (a) for multifonctions and derive some results as easy corollaries.For definitions and terminologies we refer to Reich [5] (see also [3]). DEFINITION. A subset S of E is approximatively p-compact iff for each y E E and a net {xa} in S satisfying p(xa -y)-+ dp(y, S) there is a subnet {xp} and an x E S such that Xß -► x.Clearly a compact set in E is approximatively compact. The converse, however, may fail. For example, the closed unit ball of an infinite dimensional uniformly convex Banach space is approximatively norm compact but not compact.

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“…and has nonempty closed values, then F has a closed graph. Himmelberg [1] defined that a subset A of a t.v.s. E is said to be almost convex if, for any neighborhood V of the origin 0 in E and for any finite set {w 1 ,...,w n } of points of A, there exist z 1 ,...,z n ∈ A such that z i − w i ∈ V for all i, and co{z 1 ,...,z n } ⊂ A.…”

Section: Preliminariesmentioning

confidence: 99%

Remarks on Extensions of the Himmelberg Fixed Point Theorem

Komiya

Park

2007

Fixed Point Theory Appl

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Recommended by Anthony To-Ming LauRecently, Jafari and Sehgal obtained an extension of the Himmelberg fixed point theorem based on the Kakutani fixed-point theorem. We give generalizations of the extension to almost convex sets instead of convex sets. We also give generalizations for a large class B of better admissible multimaps instead of the Kakutani maps. Our arguments are based on the KKM principle and some of previous results due to the second author.

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Fixed points of compact multifunctions

Cited by 188 publications

References 1 publication

Fixed points, intersection theorems, variational inequalities, and equilibrium theorems

Fixed points, intersection theorems, variational inequalities, and equilibrium theorems

A generalization to multifunctions of Fan’s best approximation theorem

Remarks on Extensions of the Himmelberg Fixed Point Theorem

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