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On continuous approximations for multifunctions. (English) Zbl 0595.47037
Problems concerning the approximation of convex valued multifunctions by continuous ones are considered. Approximation results of the type obtaind by Gel’man, Cellina, and Hukuhara for Pompeiu-Hausdorff upper semicontinuous multifunctions are shown to hold for some larger classes of multifunctions. Moreover, it is proved that Pompeiu-Hausdorff semicontinuous multifunctions, with convex bounded values, are continuous almost everywhere (in the sense of the Baire category). As an application, an alternative proof is given of Kenderov’s theorem stating that a maximal monotone operator is almost everywhere single-valued.

47H05Monotone operators (with respect to duality) and generalizations
58C06Set-valued and function-space valued mappings on manifolds
54C60Set-valued maps (general topology)
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