\(L_ p\)-continuous extreme selectors of multifunctions with decomposable values: Existence theorems
Let \(T\) be a metrizable compact space with a measure, \(X\) a separable Banach space and \(M\) a \(\sigma\)-compact metric space. Main results: (1) Let \(\Gamma : M \multimap L_p (T,X)\), \(1 \leq p \leq \infty\) be a multifunction with decomposable, convex, weakly compact values. If \(\Gamma\) is continuous in the Hausdorff metric and bounded then there exists a continuous selector \(u : M \to L_p (T,X)\) of \(\Gamma\) such that \(u (\xi) \in \text{ext} \Gamma (\xi)\), \(\xi \in M\) where ext denotes the set of extreme points. (2) Let \(F : M \multimap L_p (T,X)\) be a multifunction with decomposable, closed, bounded values. If the multifunction \(\Gamma (\xi) = \overline {\text{co}} F(\xi)\) satisfies the assumptions of the above theorem, then there exists a continuous selector \(u\) of \(F\) such that \(u(\xi) \in \text{ext} \overline {\text{co}} F(\xi)\), \(\xi \in M\). Some applications to multimaps appearing in differential inclusions are presented.
scientific article; zbMATH DE number 894254
scientific article