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Subadditive multifunctions and Hyers-Ulam stability. (English) Zbl 0639.39014
General inequalities 5, 5th Int. Conf., Oberwolfach/FRG 1986, ISNM 80, 281-291 (1987).
[For the entire collection see Zbl 0621.00004.]
Let F be a set-valued function from a semigroup \((S,+)\) into a Banach space \((X,\| \cdot \|)\) with nonempty closed convex values. If F is subadditive (i.e. \(F(x+y)\subset F(x)+F(y)\) for all x,y\(\in S)\) and sup\(\{\) diam F(x),x\(\in S\}<\infty\), then there exists exactly one additive selection of F. This result allows to derive the classical Hyers’ stability theorem of D. H. Hyer [Proc. Nat. Acad. Sci. U.S.A. 27, 222-224 (1941; Zbl 0061.264)]. An abstract analogue of this theorem is obtained, too.
Reviewer: A.Smajdor

39B72 Systems of functional equations and inequalities
54C60 Set-valued maps in general topology
54C65 Selections in general topology