Properties of fixed point set of a multivalued map (Q2498184)

This paper studies the set of fixed points of multivalued maps defined in a Banach space. The results obtained are in slightly different directions. But in general a crucial hypothesis is that the multivalued function is \(*\)-nonexpansive with values in \(CC(C)\) (the set of nonempty closed convex subsets of \(C\)) or in \(K(C)\) (the set of nonempty compact subsets of \(C\)). The notion of \(*\)-nonexpansiveness (which does not imply continuity) is a generalization of the notion of nonexpansive maps. Let me illustrate two types of results. In one result, which is Theorem 3.9, let \(X\) be a uniformly convex Banach space and \(C\) a nonempty closed convex subset of \(X\). If \(T:C \to CC(C)\) is a \(*\)-nonexpansive map which is asymptotically contractive, then the set \(F(T)\) (the set of fixed points of \(T\)) is nonempty, closed convex and bounded. For the other result, let \(C\) be a nonempty separable closed bounded convex subset of a Banach space \(X\) and \(T:\Omega \times C\to K(C)\) a \(*\)-nonexpansive random operator. Then the author shows that under certain hypotheses the fixed point set function of \(F\) given by \(F(\omega)=\{ x\in C:x\in T(\omega,x)\}\) is measurable. Let me finish by saying that a version of the Ky-Fan best approximation theorem is also obtained.