On single-valuedness of set-valued maps satisfying linear inclusions. (English) Zbl 1163.26353

Nikodem, Kazimierz; Popa, Dorian

Banach J. Math. Anal. 3, No. 1, 44-51 (2009).

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.

Reviewer: Andrzej Nowak (Katowice)

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