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Space of continuous set-valued mappings with closed unbounded values

Авторы: Tolstonogov A.A.

Журнал: Trudy Instituta Matematiki i Mekhaniki URO RAN

Том: 24

Номер: 1

Год: 2018

Отчётный год: 2018

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Эволюционные уравнения и управляемые системы: теория, численный анализ и приложения

DOI: 10.21538/0134-4889-2018-24-1-200-208

Аннотация: We consider a space of continuous multivalued mappings defined on a locally compact space T with countable base. Values of these mappings are closed not necessarily bounded sets from a metric space (X, d (.)) in which closed balls are compact. The space (X, d (.)) is locally compact and separable. Let Y be a dense countable set from X. The distance rho(A, B) between sets A and B from the family CL (X) of all nonempty closed subsets of X is defined as rho(A, B) = Sigma(infinity)(X-1) 1/2(i) |d(y(i), A) - d(y(i), B)| / 1 + |d(y(i), A) - d(y(i), B)|, where d (y(i), A) is the distance from a point y(i) is an element of Y to the set A. This distance is independent of the choice of the set Y, and the function rho(A, B) is a metric on the space CL(X). The convergence of a sequence of sets A(n), n >= 1, from the metric space (CL(X), rho(.)) is equivalent to the Kuratowski convergence of this sequence. We prove the completeness and separability of the space (CL(X), rho(.)) and give necessary and sufficient conditions for the compactness of sets in this space. The space C(T, CL(X)) of all continuous mappings from T to (CL(X), rho(.)) is endowed with the topology of uniform convergence on compact sets from T. We prove the completeness and separability of the space C(T, CL (X)) and give necessary and sufficient conditions for the compactness of sets in this space. These results are reformulated for the space C(T, CCL(X)), where T - [0, 1], X is a finite-dimensional Euclidean space, and CCL(X) is the space of all nonempty closed convex sets from X with the metric rho(.). This space plays a crucial role in the study of sweeping processes. A counterexample showing the significance of the assumption of the compactness of closed balls from X is given.

Индексируется WOS: Q5

Индексируется Scopus: Нет

Индексируется УБС: Нет

Индексируется РИНЦ: Нет

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