等价于Fan-Knaster-Kuratowski-Mazurkiewicz定理的不动点定理

AFIXED POINT THEOREM EQUIVALENT TO THE FAN-KNASTER-KURATOWSKI-MAZURKIEWICZ THEOREM

最近Tarafdar(J.Math.Anal.Appl.128(1987),475—479)获得了一个不动点定理并证明了这个定理与Fan 的定理(Math.Ann.266(1984),519—537)等价,后一个定理是著名的Knaster-Kuratowski-Mazurkiewicz 定理(Fund.Math.14(1929),132—137)的推广,本文主要目的是推广Tarafdar 的不动点定理并证明所得的一个不动点定理和Fan 的定理等价.

In this paper,the main result is the following.Theorem Let Y be a convex subset of a Hausdorff topological vector space E and X((?)Y)≠φ be a compactconvex subset of E.Let f:X→2~x be a set valued mapping such that(i).for each x∈X,f(x)is a nonempty convex subset;(ii)for each y∈Y,F^(-1)(y)={x∈X:y∈f(x)}contains a relatively open subest O,of Y(O,may beempty for some y)and(iii)U_((?)∈x)O_(?)=XThen there exists a point x∈ X such that x∈f(x).