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Continuous selections for a family of nonconvex-valued mappings with noncompact domain. (English. Russian original) Zbl 0861.54018

Sib. Math. J. 35, No. 3, 479-494 (1994); translation from Sib. Mat. Zh. 35, No. 3, 537-553 (1994).
The existence of a continuous selection for multifunctions of the form \[ H(x) = F(x) \cap \bigcap^N_{i=1} \bigl(G_i(x) + f_i(x) \cdot B_i \bigr) \] is exhibited. \(H\) acts between a separable metric space \(S\) and the space \(L_1(T,X)\) where \(T\) is a topological space endowed with positive Radon measure adn \(X\) a separable Banach space. \(F\) and \(G_i: S\to L_1(T,X)\) are lsc with closed, decomposable values, \(f_i\) are single-valued lsc functions and \(B_i: =\{x\in L_1(T,X) : P_i(x) <1\}\) where the \(P_i\) constitute a family of continuous seminorms with some special properties. The paper contains applications of such selection and numerous proofs that many particular seminorms are suitable.

MSC:

54C65 Selections in general topology
28B20 Set-valued set functions and measures; integration of set-valued functions; measurable selections
49J24 Optimal control problems with differential inclusions (existence) (MSC2000)
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