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Remarks on the stability of functional equations. (English) Zbl 0549.39006

Let (G,+) be an abelian group and let X be a Banach space. If f:GX is a function such that f(x+y)+f(x-y)-2f(x)-2f(y)δ for every x,yG and some δ>0, then there exists a unique function g: GX satisfying the equation g(x+y)+g(x-y)=2g(x)+2g(y) for every x,yG such that f(x)-g(x)δ/2 for every xG·

In the second part there is a short proof of a stability theorem of D. H. Hyers and S. M. Ulam [Proc. Am. Math. Soc. 3, 821-828 (1952; Zbl 0047.295)] for the inequality f(tx+(1-t)y)=tf(x)+(1-t)f(y)· Finally, the author gives a counterexample for Jensen-convex functions.

Reviewer: A.Smajdor

39B72Systems of functional equations and inequalities
39B52Functional equations for functions with more general domains and/or ranges
[1]Hyers, D. H.,On the stability of the linear functional equation. Proc. Nat. Acad. Sci. U.S.A.27 (1941), 411–416.
[2]Hyers, D. H.,Transformations with bounded m-th differences. Pacific J. Math.11 (1961), 591–602.
[3]Hyers, D. H. andUlam, M.,Approximately convex functions. Proc. Amer. Math. Soc.3 (1952), 821–828. · doi:10.1090/S0002-9939-1952-0049962-5
[4]Rockafellar, R. T.,Convex analysis. Princeton University Press, Princeton, New Jersey, 1970.
[5]Ulam, S. M.,A collection of mathematical problems. Interscience Publishers, Inc., New York, 1960.