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On single-valuedness of set-valued maps satisfying linear inclusions. (English) Zbl 1163.26353
Let $$X,Y$$ be vector spaces, $${\mathcal P}_0(Y)$$ the family of all nonempty subsets of $$Y$$ and $$F:X\to {\mathcal P}_0(Y)$$. 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.

##### MSC:
 26E25 Set-valued functions 54C60 Set-valued maps in general topology 26A51 Convexity of real functions in one variable, generalizations
##### Keywords:
Set-valued map; linear inclusion; single-valuedness
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