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Measurability of fixed point sets of multivalued random operators. (English) Zbl 0913.47057

Authors’ summary: Let \((M,d)\) be a complete separable metric space, \((\Omega,\Sigma)\) a measurable space with \(\Sigma\) a \(\sigma\)-algebra of subsets of \(\Omega\), and \(T: \Omega\times M\to{\mathcal C}{\mathcal B}(M)\) a multivalued random operator. The measurability of the function \(F\) of fixed point sets of \(T\) defined by \(F(\omega):= \{x\in M: x\in T(\omega, x)\}\) is studied. In particular, it is proved that \(F\) is measurable provided \(T\) is a random contraction, or \(M\) is a weakly compact convex separable subset of a Banach space and \(T\) is a random multivalued nonexpansive mapping such that \(I- T(\omega,\cdot)\) is demiclosed at \(0\) for every \(\omega\in\Omega\). The same result is also verified for a single-valued random nonexpansive mapping in a uniformly smooth Banach space.

MSC:

47H10 Fixed-point theorems
47H40 Random nonlinear operators
47H04 Set-valued operators
60H25 Random operators and equations (aspects of stochastic analysis)
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