Mehta定理在H-空間中的延伸與應用

Abstract

在最佳化理論中，證明固定點、最大元素和抽象經濟平衡點的存在性時，Mehta的基本定理常常扮演一個很重要的角色。在本篇論文裡，我們將推廣Himmelberg的測度到更一般的l.c.-空間，並且發展積H-空間中一些關於投影以及H-凸性的性質。而其中主要的結果推廣了Mehta在巴拿赫空間中以及Kim在局部凸的拓樸向量空間中所得到的結論。另外，在引進凝聚映射的概念下，我們利用這個結果推廣了Tarafdar的固定點定理而不需要緊緻的條件。文中也討論了這些結果在抽象經濟問題中的應用。

A remarkable fundamental theorem established by Mehta plays an important role in proving existence of fixed points, maximal elements, and equilibria in abstract economies. In this paper, we extend Himmelberg's measure of precompactness to the general setting of l.c.-spaces and develop related propositions about the projections and H-convexity in a product $H$-space. The key result generalizes Mehta's theorem in Banach spaces and Kim's theorem in locally convex topological vector spaces. Involving a kind of condensing mappings, we prove some rather general fixed point theorems without any compact condition. Other applications about maximal elements and abstract economy are discussed.