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Zbl 1163.26353
Nikodem, Kazimierz; Popa, Dorian
On single-valuedness of set-valued maps satisfying linear inclusions.
(English)
[J] Banach J. Math. Anal. 3, No. 1, 44-51, electronic only (2009). ISSN 1735-8787/e

Let $X,Y$ be vector spaces, ${\cal P}_0(Y)$ the family of all nonempty subsets of $Y$ and $F:X\to {\cal P}_0(Y)$. The main result of the paper says that if $F$ satisfies $\alpha F(x)+\beta F(y)\subset F(\gamma x+\delta y$), $x,y\in X$, where $\alpha ,\beta ,\gamma ,\delta$ are non-zero reals, and $F(x_0)$ is a singleton for some $x_0\in X$, then $F$ is single-valued of the form $F(x)=a(x)+c$, where $a:X\to Y$ is additive and $c\in Y$ is a constant. The authors also give two results on the single-valuedness of convex processes and $(\alpha ,\beta )$-convex processes. The presented theorems generalize many earlier results.
[Andrzej Nowak (Katowice)]
MSC 2000:
*26E25 Set-valued real functions
54C60 Set-valued maps
26A51 Convexity, generalizations (one real variable)

Keywords: Set-valued map; linear inclusion; single-valuedness

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