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

Mehta, Ghanshyam

Commentat. Math. Univ. Carol. 28, 377-385 (1987).

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

MSC:

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)

Keywords:

multifunction; maximal element of a multifunction; equilibrium for a qualitative game