Generalized Variation of Mappings with Applications to Composition Operators and Multifunctions |
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Authors: | Chistyakov Vyacheslav V |
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Institution: | (1) Department of Mathematics, University of Nizhny Novgorod, 23 Gagarin Avenue, Nizhny Novgorod, 603600, Russia |
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Abstract: | We study (set-valued) mappings of bounded -variation defined on the compact interval I and taking values in metric or normed linear spaces X. We prove a new structural theorem for these mappings and extend Medvedev's criterion from real valued functions onto mappings with values in a reflexive Banach space, which permits us to establish an explicit integral formula for the -variation of a metric space valued mapping. We show that the linear span GV
(I;X) of the set of all mappings of bounded -variation is automatically a Banach algebra provided X is a Banach algebra. If h:I× X Y is a given mapping and the composition operator is defined by (f)(t)=h(t,f(t)), where tI and f:I X, we show that :GV
(I;X) GV
(I;Y) is Lipschitzian if and only if h(t,x)=h0(t)+h1(t)x, tI, xX. This result is further extended to multivalued composition operators with values compact convex sets. We prove that any (not necessarily convex valued) multifunction of bounded -variation with respect to the Hausdorff metric, whose graph is compact, admits regular selections of bounded -variation. |
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Keywords: | metric space valued mappings bounded -variation" target="_blank">gif" alt="PHgr" align="BASELINE" BORDER="0">-variation set-valued mappings regular selections Lipschitzian Nemytskii composition operators |
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