Almost continuous multi-maps and \(M\)-retracts (Q1047486)

Let \(X\) be a metric space and \(CB(X)\) the family of all nonempty, closed and bounded subsets of \(X\), endowed with the Hausdorff metric. The author gives some results on fixed points of multifunctions \(f:X\to CB(X)\) almost continuous in the sense of Stallings. He proves that each connectivity or almost continuous mapping \(f:[0,1]\to CB([0,1])\) has a fixed point. Another result says that if \(X\) has the multivalued fixed point property and \(f:X\to CB(X)\) is almost continuous, then \(f\) has a fixed point. Let \(Y\) be a compact subset of the Hilbert cube \(Q\). The final theorem gives sufficient conditions for a continuous multifunction \(r:Y\to CB(Y)\), with \(y\in r(y)\) for \(y\in Y\), to have an almost continuous extension \(r:Q\to CB(Y)\). It allows to characterize compact almost continuous \(M\)-retracts of \(Q\).