On Maps Of Bounded p-Variation With p>1 |
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Authors: | Chistyakov V. V. Galkin O. E. |
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Affiliation: | (1) Department of Mathematics, University of Nizhny Novgorod, 23 Gagarin Avenue, Nizhny Novgorod, 603600, Russia |
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Abstract: | This paper addresses properties of maps of bounded p-variation (p>1) in the sense of N. Wiener, which are defined on a subset of the real line and take values in metric or normed spaces. We prove the structural theorem for these maps and study their continuity properties. We obtain the existence of a Hölder continuous path of minimal p-variation between two points and establish the compactness theorem relative to the p-variation, which is an analog of the well-known Helly selection principle in the theory of functions of bounded variation. We prove that the space of maps of bounded p-variation with values in a Banach space is also a Banach space. We give an example of a Hölder continuous of exponent 0<<1 set-valued map with no continuous selection. In the case p=1 we show that a compact absolutely continuous set-valued map from the compact interval into subsets of a Banach space admits an absolutely continuous selection. |
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Keywords: | maps of bounded p-variation maps with values in metric spaces Hö lder continuous maps minimal paths Helly's selection principle set-valued maps selections |
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