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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 (Y) the family of all nonempty subsets of Y and F:X𝒫 0 (Y). The main result of the paper says that if F satisfies αF(x)+βF(y)F(γx+δy), x,yX, where α,β,γ,δ are non-zero reals, and F(x 0 ) is a singleton for some x 0 X, then F is single-valued of the form F(x)=a(x)+c, where a:XY is additive and cY is a constant. The authors also give two results on the single-valuedness of convex processes and (α,β)-convex processes. The presented theorems generalize many earlier results.
26E25Set-valued real functions
54C60Set-valued maps (general topology)
26A51Convexity, generalizations (one real variable)