Abstract
We prove theorems on separation by sphere or (in a general case) by the boundary of a shifted quasiball of two closed disjoint subsets of a Banach space one of which is prox-regular or weakly convex and the other is a summand of a ball or quasiball. These separation theorems are applied for proving some theorems on the continuity of the intersection of two multifunctions, the values of one of them being prox-regular or weakly convex (nonconvex, in general), and the values of the other being convex and summands of a ball or quasiball. As a corollary, a theorem on the continuity of a multifunction with values bounded by the graphs of two functions is obtained.
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Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 21, No. 4, pp. 23–65, 2016.
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Ivanov, G.E., Lopushanski, M.S. A Separation Theorem for Nonconvex Sets and its Applications. J Math Sci 245, 125–154 (2020). https://doi.org/10.1007/s10958-020-04683-7
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DOI: https://doi.org/10.1007/s10958-020-04683-7