Fixed points, equilibria and maximal elements in linear topological spaces. (English) Zbl 0632.47041

Let \(T: K\to 2^ K\) be a multifunction where K is a compact, convex subset of a Hausdorff linear topological space. Browder (1968) proved:
Theorem - If we have
a) for each \(x_ 0\in K\), T(x) is a non-empty convex subset of K,
b) for each \(x_ 0\in K\), \(T^{-1}(x)=\{y\in K|\) \(x\in T(y)\}\) is open in K,
then there is a point \(x_ 0\in K\) such that \(x_ 0\in T(x_ 0).\)
This paper studies two generalizations of this theorem. More precisely he looks at cases where K is not necessarily compact. This also generalizes more theorems of Browder’s type. At the end he makes two applications. One is to find a maximal element of a multifunction and the second is to find an equilibrium for a qualitative game.
Reviewer: D.Goncalves


47H10 Fixed-point theorems
47H05 Monotone operators and generalizations
91B50 General equilibrium theory
55M20 Fixed points and coincidences in algebraic topology
54H25 Fixed-point and coincidence theorems (topological aspects)
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