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On the question of the existence of continuous branches of multivalued mappings with nonconvex images in spaces of summable functions
A. I. Bulgakov
Abstract:
Let
$B$ be a Banach space with norm
$\|\cdot\|$, and let
$(E,\mathfrak M)$ be a compact topological space with
$\sigma$-algebra of measurable sets
$\mathfrak M$ on which a nonnegative regular Borel measure
$\mu$ is given. Further, let
$L_1(E,B)$ be the Banach space of Bochner-integrable functions
$u\colon E\to B$, with the norm
$\|u\|_{L_1(E,B)}=\int_E\|u(t)\|\,d\mu$, and let
$\Phi\colon K\to2^{L_1(E,B)}$ be a multivalued mapping and
$P\colon K\to L_1(E,B)$ a single-valued mapping, where
$K$ is a compact topological space. Under certain assumptions it is proved that for any
$\varepsilon>0$ there exists a continuous mapping
$g\colon K\to L_1(E,B)$ such that the following conditions hold for any
$x\in K$:
$g(x)\in\Phi(x)$, and $\|P(x)-g(x)\|_{L_1(E,B)}<\rho_{L_1(E,B)}[P(x),\Phi(x)]+\varepsilon$, where
$\rho_{L_1(E,B)}[\,\cdot\,{,}\,\cdot\,]$ is the distance in
$L_1(E,B)$ from a point to a set.
Bibliography: 11 titles.
UDC:
517.965
MSC: Primary
54C65; Secondary
46E30 Received: 13.01.1987
Fulltext:
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