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The paper presents recent results concerning the problem of the existence of those selections, which preserve the properties of a given set-valued mapping of one real variable taking on compact values from a metric space. The properties considered are the boundedness of Jordan, essential or generalized variation, Lipschitz or absolute continuity. Selection theorems are obtained by virtue of a single compactness argument, which is the exact generalization of the Helly selection principle. For set-valued mappings with the above properties we obtain a Castaing-type representation and prove the existence of multivalued selections and selections which pass through the boundaries of the images of the set-valued mapping and which are nearest in variation to a given mapping. Multivalued Lipschitzian superposition operators acting on mappings of bounded generalized variation are characterized, and solutions of bounded generalized variation to functional inclusions and embeddings, including variable set-valued operators in the right hand side, are obtained. Bibliography contains 113 items.
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Tom
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1--82
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Bibliogr. 113 poz.
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autor
- Department of Mathematics State University Higher School of Economics Bol`shaya Pecherskaya St. 25 Nizhny Novgorod 603600 Russia, czeslaw@mail.ru
Bibliografia
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