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On single-valuedness of set-valued maps satisfying linear inclusions. (English) Zbl 1163.26353
Let $X,Y$ be vector spaces, ${𝒫}_{0}\left(Y\right)$ the family of all nonempty subsets of $Y$ and $F:X\to {𝒫}_{0}\left(Y\right)$. The main result of the paper says that if $F$ satisfies $\alpha F\left(x\right)+\beta F\left(y\right)\subset F\left(\gamma x+\delta y$), $x,y\in X$, where $\alpha ,\beta ,\gamma ,\delta$ are non-zero reals, and $F\left({x}_{0}\right)$ is a singleton for some ${x}_{0}\in X$, then $F$ is single-valued of the form $F\left(x\right)=a\left(x\right)+c$, 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 $\left(\alpha ,\beta \right)$-convex processes. The presented theorems generalize many earlier results.
##### MSC:
 26E25 Set-valued real functions 54C60 Set-valued maps (general topology) 26A51 Convexity, generalizations (one real variable)
##### Keywords:
Set-valued map; linear inclusion; single-valuedness