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CM-Selectors for pairs of oppositely semicontinuous multivalued maps with $𝕃_{p}$-decomposable values

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Nguyêñ H., Juniewicz M., Ziemińska J.

Studia Mathematica

2001

tom 144

nr 2

135-152

We present a new continuous selection theorem, which unifies in some sense two well known selection theorems; namely we prove that if F is an H-upper semicontinuous multivalued map on a separable metric space X, G is a lower semicontinuous multivalued map on X, both F and G take nonconvex $L_{p}(T,E)$-decomposable closed values, the measure space T with a σ-finite measure μ is nonatomic, 1 ≤ p < ∞, $L_{p}(T,E)$ is the Bochner-Lebesgue space of functions defined on T with values in a Banach space E, F(x) ∩ G(x) ≠ ∅ for all x ∈ X, then there exists a CM-selector for the pair (F,G), i.e. a continuous selector for G (as in the theorem of H. Antosiewicz and A. Cellina (1975), A. Bressan (1980), S. Łojasiewicz, Jr. (1982), generalized by A. Fryszkowski (1983), A. Bressan and G. Colombo (1988)) which is simultaneously an ε-approximate continuous selector for F (as in the theorem of A. Cellina, G. Colombo and A. Fonda (1986), A. Bressan and G. Colombo (1988)).

Ograniczanie wyników

1 Studia Mathematica

1 Juniewicz M.

1 Nguyêñ H.

1 Ziemińska J.

1 2001

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CM-Selectors for pairs of oppositely semicontinuous multivalued maps with $𝕃_{p}$-decomposable values