An extension of an additive selection theorem of Z. Gajda and R. Ger to vector relator spaces (Q2874373)

\textit{Z. Gajda} and \textit{R. Ger} [in: General inequalities 5, 5th Int. Conf., Oberwolfach/FRG 1986, ISNM 80, 281--291 (1987; Zbl 0639.39014)] consider the following problem which is closely related to the classical Hyers-Ulam stability problem: Given a subadditive multifunction from an abelian semigroup into the family of all nonempty closed convex subsets of a Banach space, does there exist an additive selection of it? It was shown that the answer is affirmative if the values of the multifunction are uniformly bounded. The following extension of this result is proved in the present paper: If \(F\) is a closed-valued, 2-sublinear relation of a commutative semigroup \(U\) to a separated, sequentially complete vector relator space \(X(\mathcal R)\) such that the sequence \((2^{-n}F(2^nx))^\infty_{n=0}\) is infinitesimal for all \(u\in U\), then \(F\) has an additive selection \(f\).