凸距離空間におけるBergeの最大値定理の逆問題

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  • トツキョリ クウカン ニ オケル Berge ノ サイダイチ テイリ ノ ギャクモンダイ
  • <Article>Inverse of the Berge Maximum Theorem in Convex Metric Spaces
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The Berge maximum theorem is a fundamental and important theorem in the general equilibrium theory of mathematical economics. Komiya studied an inverse problem of this theorem and obtained interesting results in finite dimensional spaces. Recently, Komiya's re- sult was extended to some infinite dimensional spaces. In this paper, we study an inverse of the Berge maximum theorem in some convex metric spaces, that is, we deal with the following problem : Let X be a metric space and let Y be a convex metric space. Let Γ : X-ο Y be a nonempty compact convex-valued upper semicontinuous multi-valued mapping. Then dose there exist a continuous function f : X × Y ⟶ R such that (i) Γ(x)={y ⋳ Y : f(x, y) = max_<z⋳y> f(x, z)} for any x⋳X ; (ii) f (x,・) is quasi-concave for any x⋳X? Our main result gives an affirmative answer to this problem.

source:Economic journal of Chiba University

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